{"collection":"philosophy-ethics","full_text_available":true,"graph_url":"https://unchartedknowledge.5gfusion.net/v2/api/page/gutenberg_74864/graph","html_url":"https://unchartedknowledge.5gfusion.net/v2/page/gutenberg_74864","id":"gutenberg_74864","license":"Public Domain (US)","rights_basis":"gutenberg_metadata_copyright_false","source":"gutenberg","source_url":"https://www.gutenberg.org/ebooks/74864","summary":null,"tags":["Category: Philosophy & Ethics","Category: Research Methods/Statistics/Information Sys","Category: Science - Physics","Logic","Science -- Methodology"],"text":"<b>Transcriber\u2019s notes</b>:\n\nThe text of this book has been preserved as closely as practicable\nto its original form. However, the author used some unusual symbols,\nand I have taken the liberty of using Unicode characters with similar\nappearance (\ua58c \u1515) as substitutes, disregarding their official meaning\nand aware that they might not display on all devices. An archaic symbol\nused by the author to indicate the mathematical \u2018factorial\u2019 function\nhas been replaced by the modern equivalent, viz. ! Unusual placements\nof various sub- and superscripted symbols remain as in the original text.\n\nIn this plain-text version, italic text is denoted by *asterisks*;\nsuperscripted and subscripted characters are enclosed within {curly}\nbrackets preceded by a caret (^) or underline (_) respectively.\n\nInconsistencies of punctuation have been corrected silently, but\ninconsistent spellings such as *Roemer, R\u00f6mer, R\u0153mer* have not been\naltered. A list of corrected spellings is appended at the end of the\nbook.\n\nFootnotes have been renumbered consecutively and positioned below the\nrelevant paragraphs. A missing footnote marker has been inserted on\np.751 after tracking down the original document. A missing negative\nsymbol has been added to an exponent in a formula on p.327.\n\nThere is a misleading calculation on p.194 and the table that follows,\nregarding progressive powers of two: ((2^{2})^{2})^{2} is equivalent to\n(16)^{2} which equals 256 not 65,356 as stated, but 2^{16} does equal\n65,356.\n\n[sic] has been inserted on p.179 alongside a statement that the\nalphabet contains 24 letters; however, the statement may well be\ncorrect given that it was written in 1704 by a Flemish author and the\nlanguage is not specified.\n\nA large arithmetic triangle on p.184 cannot be displayed optimally in\nthis plain-text version for lack of space\n\n\n\n\nTHE PRINCIPLES OF SCIENCE.\n\n[Illustration]\n\n[Illustration: THE LOGICAL MACHINE.]\n\n\n\n                      THE PRINCIPLES OF SCIENCE:\n\n                         *A TREATISE ON LOGIC*\n\n                                 *AND*\n\n                         *SCIENTIFIC METHOD.*\n\n\n                                  BY\n\n                          W. STANLEY JEVONS,\n\n                 LL.D. (EDINB.), M.A. (LOND.), F.R.S.\n\n\n                                London:\n\n                           MACMILLAN AND CO.\n                                 1883.\n\n\n       *The Right of Translation and Reproduction is Reserved.*\n\n\n\n  LONDON:\n  R. Clay, Sons, & Taylor, Printers,\n  BREAD STREET HILL.\n\n\n  Stereotyped Edition.\n\n\n\n\nPREFACE\n\n*TO THE FIRST EDITION*.\n\n\nIt may be truly asserted that the rapid progress of the physical\nsciences during the last three centuries has not been accompanied by\na corresponding advance in the theory of reasoning. Physicists speak\nfamiliarly of Scientific Method, but they could not readily describe\nwhat they mean by that expression. Profoundly engaged in the study\nof particular classes of natural phenomena, they are usually too\nmuch engrossed in the immense and ever-accumulating details of their\nspecial sciences to generalise upon the methods of reasoning which they\nunconsciously employ. Yet few will deny that these methods of reasoning\nought to be studied, especially by those who endeavour to introduce\nscientific order into less successful and methodical branches of\nknowledge.\n\nThe application of Scientific Method cannot be restricted to the sphere\nof lifeless objects. We must sooner or later have strict sciences of\nthose mental and social phenomena, which, if comparison be possible,\nare of more interest to us than purely material phenomena. But it\nis the proper course of reasoning to proceed from the known to the\nunknown--from the evident to the obscure--from the material and\npalpable to the subtle and refined. The physical sciences may therefore\nbe properly made the practice-ground of the reasoning powers,\nbecause they furnish us with a great body-of precise and successful\ninvestigations. In these sciences we meet with happy instances of\nunquestionable deductive reasoning, of extensive generalisation, of\nhappy prediction, of satisfactory verification, of nice calculation of\nprobabilities. We can note how the slightest analogical clue has been\nfollowed up to a glorious discovery, how a rash generalisation has at\nlength been exposed, or a conclusive *experimentum crucis* has decided\nthe long-continued strife between two rival theories.\n\nIn following out my design of detecting the general methods of\ninductive investigation, I have found that the more elaborate and\ninteresting processes of quantitative induction have their necessary\nfoundation in the simpler science of Formal Logic. The earlier, and\nprobably by far the least attractive part of this work, consists,\ntherefore, in a statement of the so-called Fundamental Laws of\nThought, and of the all-important Principle of Substitution, of which,\nas I think, all reasoning is a development. The whole procedure of\ninductive inquiry, in its most complex cases, is foreshadowed in\nthe combinational view of Logic, which arises directly from these\nfundamental principles. Incidentally I have described the mechanical\narrangements by which the use of the important form called the Logical\nAlphabet, and the whole working of the combinational system of Formal\nLogic, may be rendered evident to the eye, and easy to the mind and\nhand.\n\nThe study both of Formal Logic and of the Theory of Probabilities has\nled me to adopt the opinion that there is no such thing as a distinct\nmethod of induction as contrasted with deduction, but that induction\nis simply an inverse employment of deduction. Within the last century\na reaction has been setting in against the purely empirical procedure\nof Francis Bacon, and physicists have learnt to advocate the use of\nhypotheses. I take the extreme view of holding that Francis Bacon,\nalthough he correctly insisted upon constant reference to experience,\nhad no correct notions as to the logical method by which from\nparticular facts we educe laws of nature. I endeavour to show that\nhypothetical anticipation of nature is an essential part of inductive\ninquiry, and that it is the Newtonian method of deductive reasoning\ncombined with elaborate experimental verification, which has led to all\nthe great triumphs of scientific research.\n\nIn attempting to give an explanation of this view of Scientific Method,\nI have first to show that the sciences of number and quantity repose\nupon and spring from the simpler and more general science of Logic.\nThe Theory of Probability, which enables us to estimate and calculate\nquantities of knowledge, is then described, and especial attention\nis drawn to the Inverse Method of Probabilities, which involves, as\nI conceive, the true principle of inductive procedure. No inductive\nconclusions are more than probable, and I adopt the opinion that the\ntheory of probability is an essential part of logical method, so\nthat the logical value of every inductive result must be determined\nconsciously or unconsciously, according to the principles of the\ninverse method of probability.\n\nThe phenomena of nature are commonly manifested in quantities of\ntime, space, force, energy, &c., and the observation, measurement,\nand analysis of the various quantitative conditions or results\ninvolved, even in a simple experiment, demand much employment of\nsystematic procedure. I devote a book, therefore, to a simple and\ngeneral description of the devices by which exact measurement is\neffected, errors eliminated, a probable mean result attained, and\nthe probable error of that mean ascertained. I then proceed to the\nprincipal, and probably the most interesting, subject of the book,\nillustrating successively the conditions and precautions requisite\nfor accurate observation, for successful experiment, and for the sure\ndetection of the quantitative laws of nature. As it is impossible to\ncomprehend aright the value of quantitative laws without constantly\nbearing in mind the degree of quantitative approximation to the truth\nprobably attained, I have devoted a special chapter to the Theory of\nApproximation, and however imperfectly I may have treated this subject,\nI must look upon it as a very essential part of a work on Scientific\nMethod.\n\nIt then remains to illustrate the sound use of hypothesis, to\ndistinguish between the portions of knowledge which we owe to empirical\nobservation, to accidental discovery, or to scientific prediction.\nInteresting questions arise concerning the accordance of quantitative\ntheories and experiments, and I point out how the successive\nverification of an hypothesis by distinct methods of experiment yields\nconclusions approximating to but never attaining certainty. Additional\nillustrations of the general procedure of inductive investigations are\ngiven in a chapter on the Character of the Experimentalist, in which\nI endeavour to show, moreover, that the inverse use of deduction was\nreally the logical method of such great masters of experimental inquiry\nas Newton, Huyghens, and Faraday.\n\nIn treating Generalisation and Analogy, I consider the precautions\nrequisite in inferring from one case to another, or from one part of\nthe universe to another part; the validity of all such inferences\nresting ultimately upon the inverse method of probabilities. The\ntreatment of Exceptional Phenomena appeared to afford an interesting\nsubject for a further chapter illustrating the various modes in which\nan outstanding fact may eventually be explained. The formal part of\nthe book closes with the subject of Classification, which is, however,\nvery inadequately treated. I have, in fact, almost restricted myself to\nshowing that all classification is fundamentally carried out upon the\nprinciples of Formal Logic and the Logical Alphabet described at the\noutset.\n\nIn certain concluding remarks I have expressed the conviction which\nthe study of Logic has by degrees forced upon my mind, that serious\nmisconceptions are entertained by some scientific men as to the\nlogical value of our knowledge of nature. We have heard much of\nwhat has been aptly called the Reign of Law, and the necessity and\nuniformity of natural forces has been not uncommonly interpreted as\ninvolving the non-existence of an intelligent and benevolent Power,\ncapable of interfering with the course of natural events. Fears have\nbeen expressed that the progress of Scientific Method must therefore\nresult in dissipating the fondest beliefs of the human heart. Even\nthe \u2018Utility of Religion\u2019 is seriously proposed as a subject of\ndiscussion. It seemed to be not out of place in a work on Scientific\nMethod to allude to the ultimate results and limits of that method. I\nfear that I have very imperfectly succeeded in expressing my strong\nconviction that before a rigorous logical scrutiny the Reign of Law\nwill prove to be an unverified hypothesis, the Uniformity of Nature an\nambiguous expression, the certainty of our scientific inferences to a\ngreat extent a delusion. The value of science is of course very high,\nwhile the conclusions are kept well within the limits of the data on\nwhich they are founded, but it is pointed out that our experience is\nof the most limited character compared with what there is to learn,\nwhile our mental powers seem to fall infinitely short of the task of\ncomprehending and explaining fully the nature of any one object. I\ndraw the conclusion that we must interpret the results of Scientific\nMethod in an affirmative sense only. Ours must be a truly positive\nphilosophy, not that false negative philosophy which, building on a few\nmaterial facts, presumes to assert that it has compassed the bounds\nof existence, while it nevertheless ignores the most unquestionable\nphenomena of the human mind and feelings.\n\nIt is approximately certain that in freely employing illustrations\ndrawn from many different sciences, I have frequently fallen into\nerrors of detail. In this respect I must throw myself upon the\nindulgence of the reader, who will bear in mind, as I hope, that the\nscientific facts are generally mentioned purely for the purpose of\nillustration, so that inaccuracies of detail will not in the majority\nof cases affect the truth of the general principles illustrated.\n\n  *December 15, 1873.*\n\n\n\n\nPREFACE\n\n*TO THE SECOND EDITION*.\n\n\nFew alterations of importance have been made in preparing this second\nedition. Nevertheless, advantage has been taken of the opportunity to\nrevise very carefully both the language and the matter of the book.\nCorrespondents and critics having pointed out inaccuracies of more\nor less importance in the first edition, suitable corrections and\nemendations have been made. I am under obligations to Mr. C.\u00a0J. Monro,\nM.A., of Barnet, and to Mr. W.\u00a0H. Brewer, M.A., one of Her Majesty\u2019s\nInspectors of Schools, for numerous corrections.\n\nAmong several additions which have been made to the text, I may mention\nthe abstract (p.\u00a0143) of Professor Clifford\u2019s remarkable investigation\ninto the number of types of compound statement involving four classes\nof objects. This inquiry carries forward the inverse logical problem\ndescribed in the preceding sections. Again, the need of some better\nlogical method than the old Barbara Celarent, &c., is strikingly shown\nby Mr. Venn\u2019s logical problem, described at p.\u00a090. A great number\nof candidates in logic and philosophy were tested by Mr. Venn with\nthis problem, which, though simple in reality, was solved by very few\nof those who were ignorant of Boole\u2019s Logic. Other evidence could\nbe adduced by Mr. Venn of the need for some better means of logical\ntraining. To enable the logical student to test his skill in the\nsolution of inductive logical problems, I have given (p.\u00a0127) a series\nof ten problems graduated in difficulty.\n\nTo prevent misapprehension, it should be mentioned that, throughout\nthis edition, I have substituted the name *Logical Alphabet* for\n*Logical Abecedarium*, the name applied in the first edition to the\nexhaustive series of logical combinations represented in terms of\n*A*, *B*, *C*, *D* (p.\u00a094). It was objected by some readers that\n*Abecedarium* is a long and unfamiliar name.\n\nTo the chapter on Units and Standards of Measurement, I have added two\nsections, one (p.\u00a0325) containing a brief statement of the Theory of\nDimensions, and the other (p.\u00a0319) discussing Professor Clerk Maxwell\u2019s\nvery original suggestion of a Natural System of Standards for the\nmeasurement of space and time, depending upon the length and rapidity\nof waves of light.\n\nIn my description of the Logical Machine in the *Philosophical\nTransactions* (vol.\u00a0160, p.\u00a0498), I said--\u201cIt is rarely indeed that\nany invention is made without some anticipation being sooner or later\ndiscovered; but up to the present time I am totally unaware of even\na single previous attempt to devise or construct a machine which\nshould perform the operations of logical inference; and it is only,\nI believe, in the satirical writings of Swift that an allusion to an\nactual reasoning machine is to be found.\u201d Before the paper was printed,\nhowever, I was able to refer (p.\u00a0518) to the ingenious designs of the\nlate Mr. Alfred Smee as attempts to represent thought mechanically. Mr.\nSmee\u2019s machines indeed were never constructed, and, if constructed,\nwould not have performed actual logical inference. It has now just\ncome to light, however, that the celebrated Lord Stanhope actually did\nconstruct a mechanical device, capable of representing syllogistic\ninferences in a concrete form. It appears that logic was one of the\nfavourite studies of this truly original and ingenious nobleman. There\nremain fragments of a logical work, printed by the Earl at his own\npress, which show that he had arrived, before the year 1800, at the\nprinciple of the quantified predicate. He puts forward this principle\nin the most explicit manner, and proposes to employ it throughout his\nsyllogistic system. Moreover, he converts negative propositions into\naffirmative ones, and represents these by means of the copula \u201cis\nidentic with.\u201d Thus he anticipated, probably by the force of his own\nunaided insight, the main points of the logical method originated in\nthe works of George Bentham and George Boole, and developed in this\nwork. Stanhope, indeed, has no claim to priority of discovery, because\nhe seems never to have published his logical writings, although they\nwere put into print. There is no trace of them in the British Museum\nLibrary, nor in any other library or logical work, so far as I am\naware. Both the papers and the logical contrivance have been placed\nby the present Earl Stanhope in the hands of the Rev. Robert Harley,\nF.R.S., who will, I hope, soon publish a description of them.[1]\n\n  [1] Since the above was written Mr. Harley has read an account of\n  Stanhope\u2019s logical remains at the Dublin Meeting (1878) of the\n  British Association. The paper will be printed in *Mind*. (Note added\n  November, 1878.)\n\nBy the kindness of Mr. Harley, I have been able to examine Stanhope\u2019s\nlogical contrivance, called by him the Demonstrator. It consists of a\nsquare piece of bay-wood with a square depression in the centre, across\nwhich two slides can be pushed, one being a piece of red glass, and\nthe other consisting of wood coloured gray. The extent to which each\nof these slides is pushed in is indicated by scales and figures along\nthe edges of the aperture, and the simple rule of inference adopted\nby Stanhope is: \u201cTo the gray add the red and subtract the *holon*,\u201d\nmeaning by holon (\u1f45\u03bb\u03bf\u03bd) the whole width of the aperture. This rule\nof inference is a curious anticipation of De Morgan\u2019s numerically\ndefinite syllogism (see below, p.\u00a0168), and of inferences founded on\nwhat Hamilton called \u201cUltra-total distribution.\u201d Another curious point\nabout Stanhope\u2019s device is, that one slide can be drawn out and pushed\nin again at right angles to the other, and the overlapping part of the\nslides then represents the probability of a conclusion, derived from\ntwo premises of which the probabilities are respectively represented\nby the projecting parts of the slides. Thus it appears that Stanhope\nhad studied the logic of probability as well as that of certainty, here\nagain anticipating, however obscurely, the recent progress of logical\nscience. It will be seen, however, that between Stanhope\u2019s Demonstrator\nand my Logical Machine there is no resemblance beyond the fact that\nthey both perform logical inference.\n\nIn the first edition I inserted a section (vol.\u00a0i. p.\u00a025), on\n\u201cAnticipations of the Principle of Substitution,\u201d and I have reprinted\nthat section unchanged in this edition (p.\u00a021). I remark therein that,\n\u201cIn such a subject as logic it is hardly possible to put forth any\nopinions which have not been in some degree previously entertained. The\ngerm at least of every doctrine will be found in earlier writings, and\nnovelty must arise chiefly in the mode of harmonising and developing\nideas.\u201d I point out, as Professor T.\u00a0M. Lindsay had previously done,\nthat Beneke had employed the name and principle of substitution, and\nthat doctrines closely approximating to substitution were stated by the\nPort Royal Logicians more than 200 years ago.\n\nI have not been at all surprised to learn, however, that other\nlogicians have more or less distinctly stated this principle of\nsubstitution during the last two centuries. As my friend and successor\nat Owens College, Professor Adamson, has discovered, this principle can\nbe traced back to no less a philosopher than Leibnitz.\n\nThe remarkable tract of Leibnitz,[2] entitled \u201cNon inelegans Specimen\nDemonstrandi in Abstractis,\u201d commences at once with a definition\ncorresponding to the principle:--\n\n\u201cEadem sunt quorum unum potest substitui alteri salva veritate. Si sint\n*A* et *B*, et *A* ingrediatur aliquam propositionem veram, et ibi in\naliquo loco ipsius *A* pro ipso substituendo *B* fiat nova propositio\n\u00e6que itidem vera, idque semper succedat in quacunque tali propositione,\n*A* et *B* dicuntur esse eadem; et contra, si eadem sint *A* et *B*,\nprocedet substitutio quam dixi.\u201d\n\n  [2] Leibnitii *Opera Philosophica qu\u00e6 extant*. Erdmann, Pars I.\n  Berolini, 1840, p.\u00a094.\n\nLeibnitz, then, explicitly adopts the principle of substitution, but\nhe puts it in the form of a definition, saying that those things are\nthe same which can be substituted one for the other, without affecting\nthe truth of the proposition. It is only after having thus tested\nthe sameness of things that we can turn round and say that *A* and\n*B*, being the same, may be substituted one for the other. It would\nseem as if we were here in a vicious circle; for we are not allowed\nto substitute *A* for *B*, unless we have ascertained by trial that\nthe result is a true proposition. The difficulty does not seem to be\nremoved by Leibnitz\u2019 proviso, \u201cidque semper succedat in quacunque\ntali propositione.\u201d How can we learn that because *A* and *B* may\nbe mutually substituted in some propositions, they may therefore\nbe substituted in others; and what is the criterion of likeness of\npropositions expressed in the word \u201ctali\u201d? Whether the principle\nof substitution is to be regarded as a postulate, an axiom, or a\ndefinition, is just one of those fundamental questions which it seems\nimpossible to settle in the present position of philosophy, but this\nuncertainty will not prevent our making a considerable step in logical\nscience.\n\nLeibnitz proceeds to establish in the form of a theorem what is\nusually taken as an axiom, thus (*Opera*, p.\u00a095): \u201cTheorema I. Qu\u00e6\nsunt eadem uni tertio, eadem sunt inter se. Si *A* \u221d *B* et *B* \u221d *C*,\nerit *A* \u221d *C*. Nam si in propositione *A* \u221d *B* (vera ea hypothesi)\nsubstituitur *C* in locum *B* (quod facere licet per Def. I. quia *B* \u221d\n*C* ex hypothesi) fiet *A* \u221d *C*. Q.\u00a0E. Dem.\u201d Thus Leibnitz precisely\nanticipates the mode of treating inference with two simple identities\ndescribed at p.\u00a051 of this work.\n\nEven the mathematical axiom that \u2018equals added to equals make equals,\u2019\nis deduced from the principle of substitution. At p.\u00a095 of Erdmann\u2019s\nedition, we find: \u201cSi eidem addantur coincidentia fiunt coincidentia.\nSi *A* \u221d *B*, erit *A* + *C* \u221d *B* + *C*. Nam si in propositione *A*\n+ *C* \u221d *A* + *C* (qu\u00e6 est vera per se) pro *A* semel substituas *B*\n(quod facere licet per Def. I. quia *A* \u221d *B*) fiet *A* + *C* \u221d *B* +\n*C* Q.\u00a0E. Dem.\u201d This is unquestionably the mode of deducing the several\naxioms of mathematical reasoning from the higher axiom of substitution,\nwhich is explained in the section on mathematical inference (p.\u00a0162) in\nthis work, and which had been previously stated in my *Substitution of\nSimilars*, p.\u00a016.\n\nThere are one or two other brief tracts in which Leibnitz anticipates\nthe modern views of logic. Thus in the eighteenth tract in Erdmann\u2019s\nedition (p.\u00a092), called \u201cFundamenta Calculi Ratiocinatoris\u201d, he says:\n\u201cInter ea quorum unum alteri substitui potest, salvis calculi legibus,\ndicetur esse \u00e6quipollentiam.\u201d There is evidence, also, that he had\narrived at the quantification of the predicate, and that he fully\nunderstood the reduction of the universal affirmative proposition to\nthe form of an equation, which is the key to an improved view of logic.\nThus, in the tract entitled \u201cDifficultates Qu\u00e6dam Logic\u00e6,\u201d[3] he says:\n\u201cOmne *A* est *B*; id est \u00e6quivalent *AB* et *A*, seu *A* non *B* est\nnon-ens.\u201d\n\n  [3] Erdmann, p.\u00a0102.\n\nIt is curious to find, too, that Leibnitz was fully acquainted with the\nLaws of Commutativeness and \u201cSimplicity\u201d (as I have called the second\nlaw) attaching to logical symbols. In the \u201cAddenda ad Specimen Calculi\nUniversalis\u201d we read as follows.[4] \u201cTranspositio literarum in eodem\ntermino nihil mutat, ut *ab* coincidet cum *ba*, seu animal rationale\net rationale animal.\u201d\n\n\u201cRepetitio ejusdem liter\u00e6 in eodem termino est inutilis, ut *b* est\n*aa*; vel *bb* est *a*; homo est animal animal, vel homo homo est\nanimal. Sufficit enim dici *a* est *b*, seu homo est animal.\u201d\n\n  [4] Ibid. p.\u00a098.\n\nComparing this with what is stated in Boole\u2019s *Mathematical Analysis of\nLogic*, pp.\u00a017\u201318, in his *Laws of Thought*, p.\u00a029, or in this work,\npp.\u00a032\u201335, we find that Leibnitz had arrived two centuries ago at a\nclear perception of the bases of logical notation. When Boole pointed\nout that, in logic, *xx* = *x*, this seemed to mathematicians to be a\nparadox, or in any case a wholly new discovery; but here we have it\nplainly stated by Leibnitz.\n\nThe reader must not assume, however, that because Leibnitz correctly\napprehended the fundamental principles of logic, he left nothing for\nmodern logicians to do. On the contrary, Leibnitz obtained no useful\nresults from his definition of substitution. When he proceeds to\nexplain the syllogism, as in the paper on \u201cDefinitiones Logic\u00e6,\u201d[5]\nhe gives up substitution altogether, and falls back upon the notion\nof inclusion of class in class, saying, \u201cIncludens includentis est\nincludens inclusi, seu si *A* includit *B* et *B* includit *C*, etiam\n*A* includet *C*.\u201d He proceeds to make out certain rules of the\nsyllogism involving the distinction of subject and predicate, and\nin no important respect better than the old rules of the syllogism.\nLeibnitz\u2019 logical tracts are, in fact, little more than brief memoranda\nof investigations which seem never to have been followed out. They\nremain as evidence of his wonderful sagacity, but it would be difficult\nto show that they have had any influence on the progress of logical\nscience in recent times.\n\n  [5] Erdmann, p.\u00a0100.\n\nI should like to explain how it happened that these logical writings\nof Leibnitz were unknown to me, until within the last twelve months. I\nam so slow a reader of Latin books, indeed, that my overlooking a few\npages of Leibnitz\u2019 works would not have been in any case surprising.\nBut the fact is that the copy of Leibnitz\u2019 works of which I made\noccasional use, was one of the edition of Dutens, contained in Owens\nCollege Library. The logical tracts in question were not printed in\nthat edition, and with one exception, they remained in manuscript in\nthe Royal Library at Hanover, until edited by Erdmann, in 1839\u201340.\nThe tract \u201cDifficultates Qu\u00e6dam Logic\u00e6,\u201d though not known to Dutens,\nwas published by Raspe in 1765, in his collection called *\u0152uvres\nPhilosophiques de feu M^{r.} Leibnitz*; but this work had not come to\nmy notice, nor does the tract in question seem to contain any explicit\nstatement of the principle of substitution.\n\nIt is, I presume, the comparatively recent publication of Leibnitz\u2019\nmost remarkable logical tracts which explains the apparent ignorance of\nlogicians as regards their contents and importance. The most learned\nlogicians, such as Hamilton and Ueberweg, ignore Leibnitz\u2019 principle\nof substitution. In the Appendix to the fourth volume of Hamilton\u2019s\n*Lectures on Metaphysics and Logic*, is given an elaborate compendium\nof the views of logical writers concerning the ultimate basis of\ndeductive reasoning. Leibnitz is briefly noticed on p.\u00a0319, but without\nany hint of substitution. He is here quoted as saying, \u201cWhat are the\nsame with the same third, are the same with each other; that is, if *A*\nbe the same with *B*, and *C* be the same with *B*, it is necessary\nthat *A* and *C* should also be the same with one another. For this\nprinciple flows immediately from the principle of contradiction, and is\nthe ground and basis of all logic; if that fail, there is no longer any\nway of reasoning with certainty.\u201d This view of the matter seems to be\ninconsistent with that which he adopted in his posthumous tract.\n\nDr. Thomson, indeed, was acquainted with Leibnitz\u2019 tracts, and refers\nto them in his *Outline of the Necessary Laws of Thought*. He calls\nthem valuable; nevertheless, he seems to have missed the really\nvaluable point; for in making two brief quotations,[6] he omits all\nmention of the principle of substitution.\n\n  [6] Fifth Edition, 1860, p.\u00a0158.\n\nUeberweg is probably considered the best authority concerning the\nhistory of logic, and in his well-known *System of Logic and History\nof Logical Doctrines*,[7] he gives some account of the principle of\nsubstitution, especially as it is implicitly stated in the *Port Royal\nLogic*. But he omits all reference to Leibnitz in this connection,\nnor does he elsewhere, so far as I can find, supply the omission.\nHis English editor, Professor T.\u00a0M. Lindsay, in referring to my\n*Substitution of Similars*, points out how I was anticipated by Beneke;\nbut he also ignores Leibnitz. It is thus apparent that the most learned\nlogicians, even when writing especially on the history of logic,\ndisplayed ignorance of Leibnitz\u2019 most valuable logical writings.\n\n  [7] Section 120.\n\nIt has been recently pointed out to me, however, that the Rev. Robert\nHarley did draw attention, at the Nottingham Meeting of the British\nAssociation, in 1866, to Leibnitz\u2019 anticipations of Boole\u2019s laws of\nlogical notation,[8] and I am informed that Boole, about a year after\nthe publication of his *Laws of Thought*, was made acquainted with\nthese anticipations by R. Leslie Ellis.\n\n  [8] See his \u201cRemarks on Boole\u2019s Mathematical Analysis of Logic.\u201d\n  *Report of the 36th Meeting of the British Association, Transactions\n  of the Sections*, pp.\u00a03\u20136.\n\nThere seems to have been at least one other German logician who\ndiscovered, or adopted, the principle of substitution. Reusch, in his\n*Systema Logicum*, published in 1734, laboured to give a broader basis\nto the *Dictum de Omni et Nullo*. He argues, that \u201cthe whole business\nof ordinary reasoning is accomplished by the substitution of ideas in\nplace of the subject or predicate of the fundamental proposition. This\nsome call the *equation of thoughts*.\u201d But, in the hands of Reusch,\nsubstitution does not seem to lead to simplicity, since it has to be\ncarried on according to the rules of Equipollence, Reciprocation,\nSubordination, and Co-ordination.[9] Reusch is elsewhere spoken of[10]\nas the \u201ccelebrated Reusch\u201d; nevertheless, I have not been able to find\na copy of his book in London, even in the British Museum Library; it is\nnot mentioned in the printed catalogue of the Bodleian Library; Messrs.\nAsher have failed to obtain it for me by advertisement in Germany; and\nProfessor Adamson has been equally unsuccessful. From the way in which\nthe principle of substitution is mentioned by Reusch, it would seem\nlikely that other logicians of the early part of the eighteenth century\nwere acquainted with it; but, if so, it is still more curious that\nrecent historians of logical science have overlooked the doctrine.\n\n  [9] Hamilton\u2019s Lectures, vol.\u00a0iv. p.\u00a0319.\n\n  [10] Ibid. p.\u00a0326.\n\nIt is a strange and discouraging fact, that true views of logic should\nhave been discovered and discussed from one to two centuries ago, and\nyet should have remained, like George Bentham\u2019s work in this century,\nwithout influence on the subsequent progress of the science. It may be\nregarded as certain that none of the discoverers of the quantification\nof the predicate, Bentham, Hamilton, Thomson, De Morgan, and Boole,\nwere in any way assisted by the hints of the principle contained in\nprevious writers. As to my own views of logic, they were originally\nmoulded by a careful study of Boole\u2019s works, as fully stated in my\nfirst logical essay.[11] As to the process of substitution, it was\nnot learnt from any work on logic, but is simply the process of\nsubstitution perfectly familiar to mathematicians, and with which I\nnecessarily became familiar in the course of my long-continued study of\nmathematics under the late Professor De Morgan.\n\n  [11] *Pure Logic, or the Logic of Quality apart from Quantity;\n  with Remarks on Boole\u2019s System, and on the Relation of Logic and\n  Mathematics.* London, 1864, p.\u00a03.\n\nI find that the Theory of Number, which I explained in the eighth\nchapter of this work, is also partially anticipated in a single\nscholium of Leibnitz. He first gives as an axiom the now well-known law\nof Boole, as follows:--\n\n\u201cAxioma I. Si idem secum ipso sumatur, nihil constituitur novum, seu\n*A* + *A* \u221d *A*.\u201d Then follows this remarkable scholium: \u201cEquidem in\nnumeris 4\u00a0+\u00a04 facit 8, seu bini nummi binis additi faciunt quatuor\nnummos, sed tunc bini additi sunt alii a prioribus; si iidem essent\nnihil novi prodiret et perinde esset ac si joco ex tribus ovis facere\nvellemus sex numerando, primum 3 ova, deinde uno sublato residua 2, ac\ndenique uno rursus sublato residuum.\u201d\n\nTranslated this would read as follows:--\n\n\u201cAxiom I. If the same thing is taken together with itself, nothing new\narises, or *A* + *A* = *A*.\n\n\u201cScholium. In numbers, indeed, 4\u00a0+\u00a04 makes 8, or two coins added to two\ncoins make four coins, but then the two added are different from the\nformer ones; if they were the same nothing new would be produced, and\nit would be just as if we tried in joke to make six eggs out of three,\nby counting firstly the three eggs, then, one being removed, counting\nthe remaining two, and lastly, one being again removed, counting the\nremaining egg.\u201d\n\nCompare the above with pp.\u00a0156 to 162 of the present work.\n\nM. Littr\u00e9 has quite recently pointed out[12] what he thinks is an\nanalogy between the system of formal logic, stated in the following\npages, and the logical devices of the celebrated Raymond Lully. Lully\u2019s\nmethod of invention was described in a great number of medi\u00e6val books,\nbut is best stated in his *Ars Compendiosa Inveniendi Veritatem, seu\nArs Magna et Major*. This method consisted in placing various names\nof things in the sectors of concentric circles, so that when the\ncircles were turned, every possible combination of the things was\neasily produced by mechanical means. It might, perhaps, be possible to\ndiscover in this method a vague and rude anticipation of combinational\nlogic; but it is well known that the results of Lully\u2019s method were\nusually of a fanciful, if not absurd character.\n\n  [12] *La Philosophie Positive*, Mai-Juin, 1877, tom. xviii. p.\u00a0456.\n\nA much closer analogue of the Logical Alphabet is probably to be found\nin the Logical Square, invented by John Christian Lange, and described\nin a rare and unnoticed work by him which I have recently found in the\nBritish Museum.[13] This square involved the principle of bifurcate\nclassification, and was an improved form of the Ramean and Porphyrian\ntree (see below, p.\u00a0702). Lange seems, indeed, to have worked out his\nLogical Square into a mechanical form, and he suggests that it might\nbe employed somewhat in the manner of Napier\u2019s Bones (p.\u00a065). There\nis much analogy between his Square and my Abacus, but Lange had not\narrived at a logical system enabling him to use his invention for\nlogical inference in the manner of the Logical Abacus. Another work\nof Lange is said to contain the first publication of the well known\nEulerian diagrams of proposition and syllogism.[14]\n\n  [13] *Inventum Novum Quadrati Logici*, &c., Giss\u00e6 Hassorum, 1714, 8vo.\n\n  [14] See *Ueberweg\u2019s System of Logic*, &c., translated by Lindsay,\n  p.\u00a0302.\n\nSince the first edition was published, an important work by Mr. George\nLewes has appeared, namely, his *Problems of Life and Mind*, which\nto a great extent treats of scientific method, and formulates the\nrules of philosophising. I should have liked to discuss the bearing\nof Mr. Lewes\u2019s views upon those here propounded, but I have felt it\nto be impossible in a book already filling nearly 800 pages, to enter\nupon the discussion of a yet more extensive book. For the same reason\nI have not been able to compare my own treatment of the subject of\nprobability with the views expressed by Mr. Venn in his *Logic of\nChance*. With Mr. J.\u00a0J. Murphy\u2019s profound and remarkable works on\n*Habit and Intelligence*, and on *The Scientific Basis of Faith*, I\nwas unfortunately unacquainted when I wrote the following pages. They\ncannot safely be overlooked by any one who wishes to comprehend the\ntendency of philosophy and scientific method in the present day.\n\nIt seems desirable that I should endeavour to answer some of the\ncritics who have pointed out what they consider defects in the\ndoctrines of this book, especially in the first part, which treats\nof deduction. Some of the notices of the work were indeed rather\nstatements of its contents than critiques. Thus, I am much indebted\nto M. Louis Liard, Professor of Philosophy at Bordeaux, for the very\ncareful exposition[15] of the substitutional view of logic which he\ngave in the excellent *Revue Philosophique*, edited by M. Ribot.\n(Mars, 1877, tom. iii. p.\u00a0277.) An equally careful account of the\nsystem was given by M. Riehl, Professor of Philosophy at Graz, in\nhis article on \u201cDie Englische Logik der Gegenwart,\u201d published in the\n*Vierteljahrsschrift f\u00fcr wissenschaftliche Philosophie*. (1 Heft,\nLeipzig, 1876.) I should like to acknowledge also the careful and able\nmanner in which my book was reviewed by the *New York Daily Tribune*\nand the *New York Times*.\n\n  [15] Since the above was written M. Liard has republished this\n  exposition as one chapter of an interesting and admirably lucid\n  account of the progress of logical science in England. After a brief\n  but clear introduction, treating of the views of Herschel, Mill, and\n  others concerning Inductive Logic, M. Liard describes in succession\n  the logical systems of George Bentham, Hamilton, De Morgan, Boole,\n  and that contained in the present work. The title of the book is as\n  follows:--*Les Logiciens Anglais Contemporains*. Par Louis Liard,\n  Professeur de Philosophie \u00e0 la Facult\u00e9 des Lettres de Bordeaux.\n  Paris: Librairie Germer Bailli\u00e8re. 1878. (Note added November, 1878.)\n\nThe most serious objections which have been brought against my\ntreatment of logic have regard to my failure to enter into an analysis\nof the ultimate nature and origin of the Laws of Thought. The\n*Spectator*,[16] for instance, in the course of a careful review, says\nof the principle of substitution, \u201cSurely it is a great omission not\nto discuss whence we get this great principle itself; whether it is a\npure law of the mind, or only an approximate lesson of experience; and\nif a pure product of the mind, whether there are any other products\nof the same kind, furnished by our knowing faculty itself.\u201d Professor\nRobertson, in his very acute review,[17] likewise objects to the want\nof psychological and philosophical analysis. \u201cIf the book really\ncorresponded to its title, Mr. Jevons could hardly have passed so\nlightly over the question, which he does not omit to raise, concerning\nthose undoubted principles of knowledge commonly called the Laws of\nThought.... Everywhere, indeed, he appears least at ease when he\ntouches on questions properly philosophical; nor is he satisfactory\nin his psychological references, as on pp.\u00a04, 5, where he cannot\ncommit himself to a statement without an accompaniment of \u2018probably,\u2019\n\u2018almost,\u2019 or \u2018hardly.\u2019 Reservations are often very much in place, but\nthere are fundamental questions on which it is proper to make up one\u2019s\nmind.\u201d\n\n  [16] *Spectator*, September 19, 1874, p.\u00a01178. A second portion of\n  the review appeared in the same journal for September 26, 1874,\n  p.\u00a01204.\n\n  [17] *Mind*: a Quarterly Review of Psychology and Philosophy. No. II.\n  April 1876. Vol.\u00a0I. p.\u00a0206.\n\nThese remarks appear to me to be well founded, and I must state why\nit is that I have ventured to publish an extensive work on logic,\nwithout properly making up my mind as to the fundamental nature of the\nreasoning process. The fault after all is one of omission rather than\nof commission. It is open to me on a future occasion to supply the\ndeficiency if I should ever feel able to undertake the task. But I do\nnot conceive it to be an essential part of any treatise to enter into\nan ultimate analysis of its subject matter. Analyses must always end\nsomewhere. There were good treatises on light which described the laws\nof the phenomenon correctly before it was known whether light consisted\nof undulations or of projected particles. Now we have treatises on the\nUndulatory Theory which are very valuable and satisfactory, although\nthey leave us in almost complete doubt as to what the vibrating medium\nreally is. So I think that, in the present day, we need a correct and\nscientific exhibition of the formal laws of thought, and of the forms\nof reasoning based on them, although we may not be able to enter into\nany complete analysis of the nature of those laws. What would the\nscience of geometry be like now if the Greek geometers had decided that\nit was improper to publish any propositions before they had decided\non the nature of an axiom? Where would the science of arithmetic be\nnow if an analysis of the nature of number itself were a necessary\npreliminary to a development of the results of its laws? In recent\ntimes there have been enormous additions to the mathematical sciences,\nbut very few attempts at psychological analysis. In the Alexandrian and\nearly medi\u00e6val schools of philosophy, much attention was given to the\nnature of unity and plurality chiefly called forth by the question of\nthe Trinity. In the last two centuries whole sciences have been created\nout of the notion of plurality, and yet speculation on the nature of\nplurality has dwindled away. This present treatise contains, in the\neighth chapter, one of the few recent attempts to analyse the notion of\nnumber itself.\n\nIf further illustration is needed, I may refer to the differential\ncalculus. Nobody calls in question the formal truth of the results of\nthat calculus. All the more exact and successful parts of physical\nscience depend upon its use, and yet the mathematicians who have\ncreated so great a body of exact truths have never decided upon the\nbasis of the calculus. What is the nature of a limit or the nature of\nan infinitesimal? Start the question among a knot of mathematicians,\nand it will be found that hardly two agree, unless it is in regarding\nthe question itself as a trifling one. Some hold that there are no such\nthings as infinitesimals, and that it is all a question of limits.\nOthers would argue that the infinitesimal is the necessary outcome of\nthe limit, but various shades of intermediate opinion spring up.\n\nNow it is just the same with logic. If the forms of deductive and\ninductive reasoning given in the earlier part of this treatise are\ncorrect, they constitute a definite addition to logical science, and\nit would have been absurd to decline to publish such results because I\ncould not at the same time decide in my own mind about the psychology\nand philosophy of the subject. It comes in short to this, that my book\nis a book on Formal Logic and Scientific Method, and not a book on\npsychology and philosophy.\n\nIt may be objected, indeed, as the *Spectator* objects, that Mill\u2019s\nSystem of Logic is particularly strong in the discussion of the\npsychological foundations of reasoning, so that Mill would appear to\nhave successfully treated that which I feel myself to be incapable of\nattempting at present. If Mill\u2019s analysis of knowledge is correct, then\nI have nothing to say in excuse for my own deficiencies. But it is\nwell to do one thing at a time, and therefore I have not occupied any\nconsiderable part of this book with controversy and refutation. What I\nhave to say of Mill\u2019s logic will be said in a separate work, in which\nhis analysis of knowledge will be somewhat minutely analysed. It will\nthen be shown, I believe, that Mill\u2019s psychological and philosophical\ntreatment of logic has not yielded such satisfactory results as some\nwriters seem to believe.[18]\n\n  [18] Portions of this work have already been published in my\n  articles, entitled \u201cJohn Stuart Mill\u2019s Philosophy Tested,\u201d printed\n  in the *Contemporary Review* for December, 1877, vol.\u00a0xxxi. p.\u00a0167,\n  and for January and April, 1878, vol.\u00a0xxxi. p.\u00a0256, and vol.\u00a0xxxii.\n  p.\u00a088. (Note added in November, 1878.)\n\nVarious minor but still important criticisms were made by Professor\nRobertson, a few of which have been noticed in the text (pp.\u00a027, 101).\nIn other cases his objections hardly admit of any other answer than\nsuch as consists in asking the reader to judge between the work and the\ncriticism. Thus Mr. Robertson asserts[19] that the most complex logical\nproblems solved in this book (up to p.\u00a0102 of this edition) might be\nmore easily and shortly dealt with upon the principles and with the\nrecognised methods of the traditional logic. The burden of proof here\nlies upon Mr. Robertson, and his only proof consists in a single case,\nwhere he is able, as it seems to me accidentally, to get a special\nconclusion by the old form of dilemma. It would be a long labour to\ntest the old logic upon every result obtained by my notation, and I\nmust leave such readers as are well acquainted with the syllogistic\nlogic to pronounce upon the comparative simplicity and power of the\nnew and old systems. For other acute objections brought forward by Mr.\nRobertson, I must refer the reader to the article in question.\n\n  [19] *Mind*, vol.\u00a0i. p.\u00a0222.\n\nOne point in my last chapter, that on the Results and Limits of\nScientific Method, has been criticised by Professor W.\u00a0K. Clifford in\nhis lecture[20] on \u201cThe First and the Last Catastrophe.\u201d In vol.\u00a0ii.\np.\u00a0438 of the first edition (p.\u00a0744 of this edition) I referred to\ncertain inferences drawn by eminent physicists as to a limit to\nthe antiquity of the present order of things. \u201cAccording to Sir W.\nThomson\u2019s deductions from Fourier\u2019s *theory of heat*, we can trace down\nthe dissipation of heat by conduction and radiation to an infinitely\ndistant time when all things will be uniformly cold. But we cannot\nsimilarly trace the Heat-history of the Universe to an infinite\ndistance in the past. For a certain negative value of the time, the\nformul\u00e6 give impossible values, indicating that there was some initial\ndistribution of heat which could not have resulted, according to known\nlaws of nature, from any previous distribution.\u201d\n\n  [20] *Fortnightly Review*, New Series, April 1875, p.\u00a0480. Lecture\n  reprinted by the Sunday Lecture Society, p.\u00a024.\n\nNow according to Professor Clifford I have here misstated Thomson\u2019s\nresults. \u201cIt is not according to the known laws of nature, it is\naccording to the known laws of conduction of heat, that Sir William\nThomson is speaking.... All these physical writers, knowing what they\nwere writing about, simply drew such conclusions from the facts which\nwere before them as could be reasonably drawn. They say, here is a\nstate of things which could not have been produced by the circumstances\nwe are at present investigating. Then your speculator comes, he reads\na sentence and says, \u2018Here is an opportunity for me to have my fling.\u2019\nAnd he has his fling, and makes a purely baseless theory about the\nnecessary origin of the present order of nature at some definite point\nof time, which might be calculated.\u201d\n\nProfessor Clifford proceeds to explain that Thomson\u2019s formul\u00e6 only give\na limit to the heat history of, say, the earth\u2019s crust in the solid\nstate. We are led back to the time when it became solidified from the\nfluid condition. There is discontinuity in the history of the solid\nmatter, but still discontinuity which is within our comprehension.\nStill further back we should come to discontinuity again, when the\nliquid was formed by the condensation of heated gaseous matter. Beyond\nthat event, however, there is no need to suppose further discontinuity\nof law, for the gaseous matter might consist of molecules which had\nbeen falling together from different parts of space through infinite\npast time. As Professor Clifford says (p.\u00a0481) of the bodies of the\nuniverse, \u201cWhat they have actually done is to fall together and get\nsolid. If we should reverse the process we should see them separating\nand getting cool, and as a limit to that, we should find that all these\nbodies would be resolved into molecules, and all these would be flying\naway from each other. There would be no limit to that process, and we\ncould trace it as far back as ever we liked to trace it.\u201d\n\nAssuming that I have erred, I should like to point out that I have\nerred in the best company, or more strictly, being a speculator, I\nhave been led into error by the best physical writers. Professor Tait,\nin his *Sketch of Thermodynamics*, speaking of the laws discovered by\nFourier for the motion of heat in a solid, says, \u201cTheir mathematical\nexpressions point also to the fact that a uniform distribution of heat,\nor a distribution tending to become uniform, must have arisen from some\nprimitive distribution of heat of a kind not capable of being produced\nby known laws from any previous distribution.\u201d In the latter words it\nwill be seen that there is no limitation to the laws of conduction,\nand, although I had carefully referred to Sir W. Thomson\u2019s original\npaper, it is not unnatural that I should take Professor Tait\u2019s\ninterpretation of its meaning.[21]\n\n  [21] Sir W. Thomson\u2019s words are as follows (*Cambridge Mathematical\n  Journal*, Nov. 1842, vol.\u00a0iii. p.\u00a0174). \u201cWhen *x* is negative, the\n  state represented cannot be the result of any *possible* distribution\n  of temperature which has previously existed.\u201d There is no limitation\n  in the sentence to the laws of conduction, but, as the whole paper\n  treats of the results of conduction in a solid, it may no doubt be\n  understood that there is a *tacit* limitation. See also a second\n  paper on the subject in the same journal for February, 1844, vol.\u00a0iv.\n  p.\u00a067, where again there is no expressed limitation.\n\nIn his new work *On some Recent Advances in Physical Science*,\nProfessor Tait has recurred to the subject as follows:[22] \u201cA profound\nlesson may be learned from one of the earliest little papers of Sir W.\nThomson, published while he was an undergraduate at Cambridge, where\nhe shows that Fourier\u2019s magnificent treatment of the conduction of\nheat [in a solid body] leads to formul\u00e6 for its distribution which\nare intelligible (and of course capable of being fully verified by\nexperiment) for all time future, but which, except in particular\ncases, when extended to time past, remain intelligible for a finite\nperiod only, and *then* indicate a state of things which could\nnot have resulted under known laws from any conceivable previous\ndistribution [of heat in the body]. So far as heat is concerned, modern\ninvestigations have shown that a previous distribution of the *matter*\ninvolved may, by its potential energy, be capable of producing such\na state of things at the moment of its aggregation; but the example\nis now adduced not for its bearing on heat alone, but as a simple\nillustration of the fact that all portions of our Science, especially\nthat beautiful one, the Dissipation of Energy, point unanimously to a\nbeginning, to a state of things incapable of being derived by present\nlaws [of tangible matter and its energy] from any conceivable previous\narrangement.\u201d As this was published nearly a year after Professor\nClifford\u2019s lecture, it may be inferred that Professor Tait adheres to\nhis original opinion that the theory of heat does give evidence of \u201ca\nbeginning.\u201d\n\n  [22] Pp. 25\u201326. The parentheses are in the original, and show\n  Professor Tait\u2019s corrections in the verbatim reports of his lectures.\n  The subject is treated again on pp.\u00a0168\u20139.\n\nI may add that Professor Clerk Maxwell\u2019s words seem to countenance the\nsame view, for he says,[23] \u201cThis is only one of the cases in which a\nconsideration of the dissipation of energy leads to the determination\nof a superior limit to the antiquity of the observed order of things.\u201d\nThe expression \u201cobserved order of things\u201d is open to much ambiguity,\nbut in the absence of qualification I should take it to include the\naggregate of the laws of nature known to us. I should interpret\nProfessor Maxwell as meaning that the theory of heat indicates the\noccurrence of some event of which our science cannot give any further\nexplanation. The physical writers thus seem not to be so clear about\nthe matter as Professor Clifford assumes.\n\n  [23] *Theory of Heat* 1871, p.\u00a0245.\n\nSo far as I may venture to form an independent opinion on the subject,\nit is to the effect that Professor Clifford is right, and that the\nknown laws of nature do not enable us to assign a \u201cbeginning.\u201d Science\nleads us backwards into infinite past duration. But that Professor\nClifford is right on this point, is no reason why we should suppose\nhim to be right in his other opinions, some of which I am sure are\nwrong. Nor is it a reason why other parts of my last chapter should be\nwrong. The question only affects the single paragraph on pp.\u00a0744\u20135 of\nthis book, which might, I believe, be struck out without necessitating\nany alteration in the rest of the text. It is always to be remembered\nthat the failure of an argument in favour of a proposition does not,\ngenerally speaking, add much, if any, probability to the contradictory\nproposition. I cannot conclude without expressing my acknowledgments\nto Professor Clifford for his kind expressions regarding my work as a\nwhole.\n\n  2, The Chestnuts,\n  West Heath,\n  Hampstead, N.\u00a0W.\n\n  *August 15, 1877.*\n\n\n\n\nCONTENTS.\n\n\n  BOOK I.\n\n  FORMAL LOGIC, DEDUCTIVE AND INDUCTIVE.\n\n\n  CHAPTER I.\n\n  INTRODUCTION.\n\n  SECTION      PAGE\n\n  1. Introduction                                                 1\n\n  2. The Powers of Mind concerned in the Creation of Science      4\n\n  3. Laws of Identity and Difference                              5\n\n  4. The Nature of the Laws of Identity and Difference            6\n\n  5. The Process of Inference                                     9\n\n  6. Deduction and Induction                                     11\n\n  7. Symbolic Expression of Logical Inference                    13\n\n  8. Expression of Identity and Difference                       14\n\n  9. General Formula of Logical Inference                        17\n\n  10. The Propagating Power of Similarity                        20\n\n  11. Anticipations of the Principle of Substitution             21\n\n  12. The Logic of Relatives                                     22\n\n\n  CHAPTER II.\n\n  TERMS.\n\n  1. Terms                                                       24\n\n  2. Twofold meaning of General Names                            25\n\n  3. Abstract Terms                                              27\n\n  4. Substantial Terms                                           28\n\n  5. Collective Terms                                            29\n\n  6. Synthesis of Terms                                          30\n\n  7. Symbolic Expression of the Law of Contradiction             31\n\n  8. Certain Special Conditions of Logical Symbols               32\n\n\n  CHAPTER III.\n\n  PROPOSITIONS.\n\n  1. Propositions                                                36\n\n  2. Simple Identities                                           37\n\n  3. Partial Identities                                          40\n\n  4. Limited Identities                                          42\n\n  5. Negative Propositions                                       43\n\n  6. Conversion of Propositions                                  46\n\n  7. Twofold Interpretation of Propositions                      47\n\n\n  CHAPTER IV.\n\n  DEDUCTIVE REASONING.\n\n  1. Deductive Reasoning                                         49\n\n  2. Immediate Inference                                         50\n\n  3. Inference with Two Simple Identities                        51\n\n  4. Inference with a Simple and a Partial Identity              53\n\n  5. Inference of a Partial from Two Partial Identities          55\n\n  6. On the Ellipsis of Terms in Partial Identities              57\n\n  7. Inference of a Simple from Two Partial Identities           58\n\n  8. Inference of a Limited from Two Partial Identities          59\n\n  9. Miscellaneous Forms of Deductive Inference                  60\n\n  10. Fallacies                                                  62\n\n\n  CHAPTER V.\n\n  DISJUNCTIVE PROPOSITIONS.\n\n  1. Disjunctive Propositions                                    66\n\n  2. Expression of the Alternative Relation                      67\n\n  3. Nature of the Alternative Relation                          68\n\n  4. Laws of the Disjunctive Relation                            71\n\n  5. Symbolic Expression of the Law of Duality                   73\n\n  6. Various Forms of the Disjunctive Proposition                74\n\n  7. Inference by Disjunctive Propositions                       76\n\n\n  CHAPTER VI.\n\n  THE INDIRECT METHOD OF INFERENCE.\n\n  1. The Indirect Method of Inference                            81\n\n  2. Simple Illustrations                                        83\n\n  3. Employment of the Contrapositive Proposition                84\n\n  4. Contrapositive of a Simple Identity                         86\n\n  5. Miscellaneous Examples of the Method                        88\n\n  6. Mr. Venn\u2019s Problem                                          90\n\n  7. Abbreviation of the Process                                 91\n\n  8. The Logical Alphabet                                        94\n\n  9. The Logical Slate                                           95\n\n  10. Abstraction of Indifferent Circumstances                   97\n\n  11. Illustrations of the Indirect Method                       98\n\n  12. Second Example                                             99\n\n  13. Third Example                                             100\n\n  14. Fourth Example                                            101\n\n  15. Fifth Example                                             101\n\n  16. Fallacies Analysed by the Indirect Method                 102\n\n  17. The Logical Abacus                                        104\n\n  18. The Logical Machine                                       107\n\n  19. The Order of Premises                                     114\n\n  20. The Equivalence of Propositions                           115\n\n  21. The Nature of Inference                                   118\n\n\n  CHAPTER VII.\n\n  INDUCTION.\n\n  1. Induction                                                  121\n\n  2. Induction an Inverse Operation                             122\n\n  3. Inductive Problems for Solution by the Reader              126\n\n  4. Induction of Simple Identities                             127\n\n  5. Induction of Partial Identities                            130\n\n  6. Solution of the Inverse or Inductive Problem, involving\n       Two Classes                                              134\n\n  7. The Inverse Logical Problem, involving Three Classes       137\n\n  8. Professor Clifford on the Types of Compound Statement\n       involving Four Classes                                   143\n\n  9. Distinction between Perfect and Imperfect Induction        146\n\n  10. Transition from Perfect to Imperfect Induction            149\n\n\n  BOOK II.\n\n  NUMBER, VARIETY, AND PROBABILITY.\n\n\n  CHAPTER VIII.\n\n  PRINCIPLES OF NUMBER.\n\n  1. Principles of Number                                       153\n\n  2. The Nature of Numbe                                        156\n\n  3. Of Numerical Abstraction                                   158\n\n  4. Concrete and Abstract Number                               159\n\n  5. Analogy of Logical and Numerical Terms                     160\n\n  6. Principle of Mathematical Inference                        162\n\n  7. Reasoning by Inequalities                                  165\n\n  8. Arithmetical Reasoning                                     167\n\n  9. Numerically Definite Reasoning                             168\n\n  10. Numerical meaning of Logical Conditions                   171\n\n\n  CHAPTER IX.\n\n  THE VARIETY OF NATURE, OR THE DOCTRINE OF COMBINATIONS\n  AND PERMUTATIONS.\n\n  1. The Variety of Nature                                      173\n\n  2. Distinction of Combinations and Permutations               177\n\n  3. Calculation of Number of Combinations                      180\n\n  4. The Arithmetical Triangle                                  182\n\n  5. Connexion between the Arithmetical Triangle and the\n       Logical Alphabet                                         189\n\n  6. Possible Variety of Nature and Art                         190\n\n  7. Higher Orders of Variety                                   192\n\n\n  CHAPTER X.\n\n  THEORY OF PROBABILITY.\n\n  1. Theory of Probability                                      197\n\n  2. Fundamental Principles of the Theory                       200\n\n  3. Rules for the Calculation of Probabilities                 203\n\n  4. The Logical Alphabet in questions of Probability           205\n\n  5. Comparison of the Theory with Experience                   206\n\n  6. Probable Deductive Arguments                               209\n\n  7. Difficulties of the Theory                                 213\n\n\n  CHAPTER XI.\n\n  PHILOSOPHY OF INDUCTIVE INFERENCE.\n\n  1. Philosophy of Inductive Inference                          218\n\n  2. Various Classes of Inductive Truths                        219\n\n  3. The Relation of Cause and Effect                           220\n\n  4. Fallacious Use of the Term Cause                           221\n\n  5. Confusion of Two Questions                                 222\n\n  6. Definition of the Term Cause                               224\n\n  7. Distinction of Inductive and Deductive Results             226\n\n  8. The Grounds of Inductive Inference                         228\n\n  9. Illustrations of the Inductive Process                     229\n\n  10. Geometrical Reasoning                                     233\n\n  11. Discrimination of Certainty and Probability               235\n\n\n  CHAPTER XII.\n\n  THE INDUCTIVE OR INVERSE APPLICATION OF THE THEORY\n  OF PROBABILITY.\n\n  1. The Inductive or Inverse Application of the Theory         240\n\n  2. Principle of the Inverse Method                            242\n\n  3. Simple Applications of the Inverse Method                  244\n\n  4. The Theory of Probability in Astronomy                     247\n\n  5. The General Inverse Problem                                250\n\n  6. Simple Illustration of the Inverse Problem                 253\n\n  7. General Solution of the Inverse Problem                    255\n\n  8. Rules of the Inverse Method                                257\n\n  9. Fortuitous Coincidences                                    261\n\n  10. Summary of the Theory of Inductive Inference              265\n\n\n  BOOK III.\n\n  METHODS OF MEASUREMENT.\n\n\n  CHAPTER XIII.\n\n  THE EXACT MEASUREMENT OF PHENOMENA.\n\n  1. The Exact Measurement of Phenomena                         270\n\n  2. Division of the Subject                                    274\n\n  3. Continuous quantity                                        274\n\n  4. The Fallacious Indications of the Senses                   276\n\n  5. Complexity of Quantitative Questions                       278\n\n  6. The Methods of Accurate Measurement                        282\n\n  7. Conditions of Accurate Measurement                         282\n\n  8. Measuring Instruments                                      284\n\n  9. The Method of Repetition                                   288\n\n  10. Measurements by Natural Coincidence                       292\n\n  11. Modes of Indirect Measurement                             296\n\n  12. Comparative Use of Measuring Instruments                  299\n\n  13. Systematic Performance of Measurements                    300\n\n  14. The Pendulum                                              302\n\n  15. Attainable Accuracy of Measurement                        303\n\n\n  CHAPTER XIV.\n\n  UNITS AND STANDARDS OF MEASUREMENT.\n\n  1. Units and Standards of Measurement                         305\n\n  2. Standard Unit of Time                                      307\n\n  3. The Unit of Space and the Bar Standard                     312\n\n  4. The Terrestrial Standard                                   314\n\n  5. The Pendulum Standard                                      315\n\n  6. Unit of Density                                            316\n\n  7. Unit of Mass                                               317\n\n  8. Natural System of Standards                                319\n\n  9. Subsidiary Units                                           320\n\n  10. Derived Units                                             321\n\n  11. Provisional Units                                         323\n\n  12. Theory of Dimensions                                      325\n\n  13. Natural Constants                                         328\n\n  14. Mathematical Constants                                    330\n\n  15. Physical Constants                                        331\n\n  16. Astronomical Constants                                    332\n\n  17. Terrestrial Numbers                                       333\n\n  18. Organic Numbers                                           333\n\n  19. Social Numbers                                            334\n\n\n  CHAPTER XV.\n\n  ANALYSIS OF QUANTITATIVE PHENOMENA.\n\n  1. Analysis of Quantitative Phenomena                         335\n\n  2. Illustrations of the Complication of Effects               336\n\n  3. Methods of Eliminating Error                               339\n\n  4. Method of Avoidance of Error                               340\n\n  5. Differential Method                                        344\n\n  6. Method of Correction                                       346\n\n  7. Method of Compensation                                     350\n\n  8. Method of Reversal                                         354\n\n\n  CHAPTER XVI.\n\n  THE METHOD OF MEANS.\n\n  1. The Method of Means                                        357\n\n  2. Several Uses of the Mean Result                            359\n\n  3. The Mean and the Average                                   360\n\n  4. On the Average or Fictitious Mean                          363\n\n  5. The Precise Mean Result                                    365\n\n  6. Determination of the Zero Point                            368\n\n  7. Determination of Maximum Points                            371\n\n\n  CHAPTER XVII.\n\n  THE LAW OF ERROR.\n\n  1. The Law of Error                                           374\n\n  2. Establishment of the Law of Error                          375\n\n  3. Herschel\u2019s Geometrical Proof                               377\n\n  4. Laplace\u2019s and Quetelet\u2019s Proof of the Law                  378\n\n  5. Logical Origin of the Law of Error                         383\n\n  6. Verification of the Law of Error                           383\n\n  7. The Probable Mean Result                                   385\n\n  8. The Probable Error of Results                              386\n\n  9. Rejection of the Mean Result                               389\n\n  10. Method of Least Squares                                   393\n\n  11. Works upon the Theory of Probability                      394\n\n  12. Detection of Constant Errors                              396\n\n\n  BOOK IV.\n\n  INDUCTIVE INVESTIGATION.\n\n\n  CHAPTER XVIII.\n\n  OBSERVATION.\n\n  1. Observation                                                399\n\n  2. Distinction of Observation and Experiment                  400\n\n  3. Mental Conditions of Correct Observation                   402\n\n  4. Instrumental and Sensual Conditions of Correct Observation 404\n\n  5. External Conditions of Correct Observation                 407\n\n  6. Apparent Sequence of Events                                409\n\n  7. Negative Arguments from Non-Observation                    411\n\n\n  CHAPTER XIX.\n\n  EXPERIMENT.\n\n  1. Experiment                                                 416\n\n  2. Exclusion of Indifferent Circumstances                     419\n\n  3. Simplification of Experiments                              422\n\n  4. Failure in the Simplification of Experiments               424\n\n  5. Removal of Usual Conditions                                426\n\n  6. Interference of Unsuspected Conditions                     428\n\n  7. Blind or Test Experiments                                  433\n\n  8. Negative Results of Experiment                             434\n\n  9. Limits of Experiment                                       437\n\n\n  CHAPTER XX.\n\n  METHOD OF VARIATIONS.\n\n  1. Method of Variations                                       439\n\n  2. The Variable and the Variant                               440\n\n  3. Measurement of the Variable                                441\n\n  4. Maintenance of Similar Conditions                          443\n\n  5. Collective Experiments                                     445\n\n  6. Periodic Variations                                        447\n\n  7. Combined Periodic Changes                                  450\n\n  8. Principle of Forced Vibrations                             451\n\n  9. Integrated Variations                                      452\n\n\n  CHAPTER XXI.\n\n  THEORY OF APPROXIMATION.\n\n  1. Theory of Approximation                                    456\n\n  2. Substitution of Simple Hypotheses                          458\n\n  3. Approximation to Exact Laws                                462\n\n  4. Successive Approximations to Natural Conditions            465\n\n  5. Discovery of Hypothetically Simple Laws                    470\n\n  6. Mathematical Principles of Approximation                   471\n\n  7. Approximate Independence of Small Effects                  475\n\n  8. Four Meanings of Equality                                  479\n\n  9. Arithmetic of Approximate Quantities                       481\n\n\n  CHAPTER XXII.\n\n  QUANTITATIVE INDUCTION.\n\n  1. Quantitative Induction                                     483\n\n  2. Probable Connexion of Varying Quantities                   484\n\n  3. Empirical Mathematical Laws                                487\n\n  4. Discovery of Rational Formul\u00e6                              489\n\n  5. The Graphical Method                                       492\n\n  6. Interpolation and Extrapolation                            495\n\n  7. Illustrations of Empirical Quantitative Laws               499\n\n  8. Simple Proportional Variation                              501\n\n\n  CHAPTER XXIII.\n\n  THE USE OF HYPOTHESIS.\n\n  1. The Use of Hypothesis                                      504\n\n  2. Requisites of a good Hypothesis                            510\n\n  3. Possibility of Deductive Reasoning                         511\n\n  4. Consistency with the Laws of Nature                        514\n\n  5. Conformity with Facts                                      516\n\n  6. Experimentum Crucis                                        518\n\n  7. Descriptive Hypotheses                                     522\n\n\n  CHAPTER XXIV.\n\n  EMPIRICAL KNOWLEDGE, EXPLANATION AND PREDICTION.\n\n  1. Empirical Knowledge, Explanation and Prediction            525\n\n  2. Empirical Knowledge                                        526\n\n  3. Accidental Discovery                                       529\n\n  4. Empirical Observations subsequently Explained              532\n\n  5. Overlooked Results of Theory                               534\n\n  6. Predicted Discoveries                                      536\n\n  7. Predictions in the Science of Light                        538\n\n  8. Predictions from the Theory of Undulations                 540\n\n  9. Prediction in other Sciences                               542\n\n  10. Prediction by Inversion of Cause and Effect               545\n\n  11. Facts known only by Theory                                547\n\n\n  CHAPTER XXV.\n\n  ACCORDANCE OF QUANTITATIVE THEORIES.\n\n  1. Accordance of Quantitative Theories                        551\n\n  2. Empirical Measurements                                     552\n\n  3. Quantities indicated by Theory, but Empirically Measured   553\n\n  4. Explained Results of Measurement                           554\n\n  5. Quantities determined by Theory and verified by\n       Measurement                                              555\n\n  6. Quantities determined by Theory and not verified           556\n\n  7. Discordance of Theory and Experiment                       558\n\n  8. Accordance of Measurements of Astronomical Distances       560\n\n  9. Selection of the best Mode of Measurement                  563\n\n  10. Agreement of Distinct Modes of Measurement                564\n\n  11. Residual Phenomena                                        569\n\n\n  CHAPTER XXVI.\n\n  CHARACTER OF THE EXPERIMENTALIST.\n\n  1. Character of the Experimentalist                           574\n\n  2. Error of the Baconian Method                               576\n\n  3. Freedom of Theorising                                      577\n\n  4. The Newtonian Method, the True Organum                     581\n\n  5. Candour and Courage of the Philosophic Mind                586\n\n  6. The Philosophic Character of Faraday                       587\n\n  7. Reservation of Judgment                                    592\n\n\n  BOOK V.\n\n  GENERALISATION, ANALOGY, AND CLASSIFICATION.\n\n\n  CHAPTER XXVII.\n\n  GENERALISATION.\n\n  1. Generalisation                                             594\n\n  2. Distinction of Generalisation and Analogy                  596\n\n  3. Two Meanings of Generalisation                             597\n\n  4. Value of Generalisation                                    599\n\n  5. Comparative Generality of Properties                       600\n\n  6. Uniform Properties of all Matter                           603\n\n  7. Variable Properties of Matter                              606\n\n  8. Extreme Instances of Properties                            607\n\n  9. The Detection of Continuity                                610\n\n  10. The Law of Continuity                                     615\n\n  11. Failure of the Law of Continuity                          619\n\n  12. Negative Arguments on the Principle of Continuity         621\n\n  13. Tendency to Hasty Generalisation                          623\n\n\n  CHAPTER XXVIII.\n\n  ANALOGY.\n\n  1. Analogy                                                    627\n\n  2. Analogy as a Guide in Discovery                            629\n\n  3. Analogy in the Mathematical Sciences                       631\n\n  4. Analogy in the Theory of Undulations                       635\n\n  5. Analogy in Astronomy                                       638\n\n  6. Failures of Analogy                                        641\n\n\n  CHAPTER XXIX.\n\n  EXCEPTIONAL PHENOMENA.\n\n  1. Exceptional Phenomena                                      644\n\n  2. Imaginary or False Exceptions                              647\n\n  3. Apparent but Congruent Exceptions                          649\n\n  4. Singular Exceptions                                        652\n\n  5. Divergent Exceptions                                       655\n\n  6. Accidental Exceptions                                      658\n\n  7. Novel and Unexplained Exceptions                           661\n\n  8. Limiting Exceptions                                        663\n\n  9. Real Exceptions to Supposed Laws                           666\n\n  10. Unclassed Exceptions                                      668\n\n\n  CHAPTER XXX.\n\n  CLASSIFICATION.\n\n  1. Classification                                             673\n\n  2. Classification involving Induction                         675\n\n  3. Multiplicity of Modes of Classification                    677\n\n  4. Natural and Artificial Systems of Classification           679\n\n  5. Correlation of Properties                                  681\n\n  6. Classification in Crystallography                          685\n\n  7. Classification an Inverse and Tentative Operation          689\n\n  8. Symbolic Statement of the Theory of Classification         692\n\n  9. Bifurcate Classification                                   694\n\n  10. The Five Predicates                                       698\n\n  11. Summum Genus and Infima Species                           701\n\n  12. The Tree of Porphyry                                      702\n\n  13. Does Abstraction imply Generalisation?                    704\n\n  14. Discovery of Marks or Characteristics                     708\n\n  15. Diagnostic Systems of Classification                      710\n\n  16. Index Classifications                                     714\n\n  17. Classification in the Biological Sciences                 718\n\n  18. Classification by Types                                   722\n\n  19. Natural Genera and Species                                724\n\n  20. Unique or Exceptional Objects                             728\n\n  21. Limits of Classification                                  730\n\n\n  BOOK VI.\n\n  CHAPTER XXXI.\n\n  REFLECTIONS ON THE RESULTS AND LIMITS OF SCIENTIFIC METHOD.\n\n  1. Reflections on the Results and Limits of Scientific Method 735\n\n  2. The Meaning of Natural Law                                 737\n\n  3. Infiniteness of the Universe                               738\n\n  4. The Indeterminate Problem of Creation                      740\n\n  5. Hierarchy of Natural Laws                                  742\n\n  6. The Ambiguous Expression--\u201cUniformity of Nature\u201d           745\n\n  7. Possible States of the Universe                            749\n\n  8. Speculations on the Reconcentration of Energy              751\n\n  9. The Divergent Scope for New Discovery                      752\n\n  10. Infinite Incompleteness of the Mathematical Sciences      754\n\n  11. The Reign of Law in Mental and Social Phenomena           759\n\n  12. The Theory of Evolution                                   761\n\n  13. Possibility of Divine Interference                        765\n\n  14. Conclusion                                                766\n\n\n  INDEX                                                         773\n\n\n\n\nTHE PRINCIPLES OF SCIENCE.\n\n\n\n\nCHAPTER I.\n\nINTRODUCTION.\n\n\nScience arises from the discovery of Identity amidst Diversity. The\nprocess may be described in different words, but our language must\nalways imply the presence of one common and necessary element. In\nevery act of inference or scientific method we are engaged about a\ncertain identity, sameness, similarity, likeness, resemblance, analogy,\nequivalence or equality apparent between two objects. It is doubtful\nwhether an entirely isolated phenomenon could present itself to our\nnotice, since there must always be some points of similarity between\nobject and object. But in any case an isolated phenomenon could be\nstudied to no useful purpose. The whole value of science consists\nin the power which it confers upon us of applying to one object the\nknowledge acquired from like objects; and it is only so far, therefore,\nas we can discover and register resemblances that we can turn our\nobservations to account.\n\nNature is a spectacle continually exhibited to our senses, in which\nphenomena are mingled in combinations of endless variety and novelty.\nWonder fixes the mind\u2019s attention; memory stores up a record of each\ndistinct impression; the powers of association bring forth the record\nwhen the like is felt again. By the higher faculties of judgment and\nreasoning the mind compares the new with the old, recognises essential\nidentity, even when disguised by diverse circumstances, and expects to\nfind again what was before experienced. It must be the ground of all\nreasoning and inference that *what is true of one thing will be true\nof its equivalent*, and that under carefully ascertained conditions\n*Nature repeats herself*.\n\nWere this indeed a Chaotic Universe, the powers of mind employed in\nscience would be useless to us. Did Chance wholly take the place of\norder, and did all phenomena come out of an *Infinite Lottery*, to use\nCondorcet\u2019s expression, there could be no reason to expect the like\nresult in like circumstances. It is possible to conceive a world in\nwhich no two things should be associated more often, in the long run,\nthan any other two things. The frequent conjunction of any two events\nwould then be purely fortuitous, and if we expected conjunctions to\nrecur continually, we should be disappointed. In such a world we might\nrecognise the same kind of phenomenon as it appeared from time to time,\njust as we might recognise a marked ball as it was occasionally drawn\nand re-drawn from a ballot-box; but the approach of any phenomenon\nwould be in no way indicated by what had gone before, nor would it be a\nsign of what was to come after. In such a world knowledge would be no\nmore than the memory of past coincidences, and the reasoning powers, if\nthey existed at all, would give no clue to the nature of the present,\nand no presage of the future.\n\nHappily the Universe in which we dwell is not the result of chance,\nand where chance seems to work it is our own deficient faculties which\nprevent us from recognising the operation of Law and of Design. In\nthe material framework of this world, substances and forces present\nthemselves in definite and stable combinations. Things are not in\nperpetual flux, as ancient philosophers held. Element remains element;\niron changes not into gold. With suitable precautions we can calculate\nupon finding the same thing again endowed with the same properties.\nThe constituents of the globe, indeed, appear in almost endless\ncombinations; but each combination bears its fixed character, and\nwhen resolved is found to be the compound of definite substances.\nMisapprehensions must continually occur, owing to the limited extent\nof our experience. We can never have examined and registered possible\nexistences so thoroughly as to be sure that no new ones will occur and\nfrustrate our calculations. The same outward appearances may cover\nany amount of hidden differences which we have not yet suspected. To\nthe variety of substances and powers diffused through nature at its\ncreation, we should not suppose that our brief experience can assign\na limit, and the necessary imperfection of our knowledge must be ever\nborne in mind.\n\nYet there is much to give us confidence in Science. The wider our\nexperience, the more minute our examination of the globe, the\ngreater the accumulation of well-reasoned knowledge,--the fewer in\nall probability will be the failures of inference compared with the\nsuccesses. Exceptions to the prevalence of Law are gradually reduced\nto Law themselves. Certain deep similarities have been detected among\nthe objects around us, and have never yet been found wanting. As the\nmeans of examining distant parts of the universe have been acquired,\nthose similarities have been traced there as here. Other worlds and\nstellar systems may be almost incomprehensively different from ours in\nmagnitude, condition and disposition of parts, and yet we detect there\nthe same elements of which our own limbs are composed. The same natural\nlaws can be detected in operation in every part of the universe within\nthe scope of our instruments; and doubtless these laws are obeyed\nirrespective of distance, time, and circumstance.\n\nIt is the prerogative of Intellect to discover what is uniform and\nunchanging in the phenomena around us. So far as object is different\nfrom object, knowledge is useless and inference impossible. But so\nfar as object resembles object, we can pass from one to the other. In\nproportion as resemblance is deeper and more general, the commanding\npowers of knowledge become more wonderful. Identity in one or other\nof its phases is thus always the bridge by which we pass in inference\nfrom case to case; and it is my purpose in this treatise to trace out\nthe various forms in which the one same process of reasoning presents\nitself in the ever-growing achievements of Scientific Method.\n\n\n*The Powers of Mind concerned in the Creation of Science.*\n\nIt is no part of the purpose of this work to investigate the nature\nof mind. People not uncommonly suppose that logic is a branch of\npsychology, because reasoning is a mental operation. On the same\nground, however, we might argue that all the sciences are branches\nof psychology. As will be further explained, I adopt the opinion of\nMr. Herbert Spencer, that logic is really an objective science, like\nmathematics or mechanics. Only in an incidental manner, then, need\nI point out that the mental powers employed in the acquisition of\nknowledge are probably three in number. They are substantially as\nProfessor Bain has stated them[24]:--\n\n  [24] *The Senses and the Intellect*, Second Ed., pp.\u00a05, 325, &c.\n\n  1. The Power of Discrimination.\n  2. The Power of Detecting Identity.\n  3. The Power of Retention.\n\n\nWe exert the first power in every act of perception. Hardly can we have\na sensation or feeling unless we discriminate it from something else\nwhich preceded. Consciousness would almost seem to consist in the break\nbetween one state of mind and the next, just as an induced current of\nelectricity arises from the beginning or the ending of the primary\ncurrent. We are always engaged in discrimination; and the rudiment of\nthought which exists in the lower animals probably consists in their\npower of feeling difference and being agitated by it.\n\nYet had we the power of discrimination only, Science could not be\ncreated. To know that one feeling differs from another gives purely\nnegative information. It cannot teach us what will happen. In such a\nstate of intellect each sensation would stand out distinct from every\nother; there would be no tie, no bridge of affinity between them. We\nwant a unifying power by which the present and the future may be linked\nto the past; and this seems to be accomplished by a different power of\nmind. Lord Bacon has pointed out that different men possess in very\ndifferent degrees the powers of discrimination and identification. It\nmay be said indeed that discrimination necessarily implies the action\nof the opposite process of identification; and so it doubtless does in\nnegative points. But there is a rare property of mind which consists\nin penetrating the disguise of variety and seizing the common elements\nof sameness; and it is this property which furnishes the true measure\nof intellect. The name of \u201cintellect\u201d expresses the interlacing of the\ngeneral and the single, which is the peculiar province of mind.[25] To\n*cogitate* is the Latin *coagitare*, resting on a like metaphor. Logic,\nalso, is but another name for the same process, the peculiar work of\nreason; for \u03bb\u03bf\u03b3\u03bf\u03c2 is derived from \u03bb\u03b5\u03b3\u03b5\u03b9\u03bd, which like the Latin *legere*\nmeant originally to gather. Plato said of this unifying power, that if\nhe met the man who could detect *the one in the many*, he would follow\nhim as a god.\n\n  [25] Max M\u00fcller, *Lectures on the Science of Language*, Second\n  Series, vol. ii. p.\u00a063; or Sixth Edition, vol.\u00a0ii. p.\u00a067. The view\n  of the etymological meaning of \u201cintellect\u201d is given above on the\n  authority of Professor Max M\u00fcller. It seems to be opposed to the\n  ordinary opinion, according to which the Latin *intelligere* means to\n  choose between, to see a difference between, to discriminate, instead\n  of to unite.\n\n\n*Laws of Identity and Difference.*\n\nAt the base of all thought and science must lie the laws which express\nthe very nature and conditions of the discriminating and identifying\npowers of mind. These are the so-called Fundamental Laws of Thought,\nusually stated as follows:--\n\n  1. The Law of Identity. *Whatever is, is.*\n\n  2. The Law of Contradiction. *A thing cannot both be and not be.*\n\n  3. The Law of Duality. *A thing must either be or not be.*\n\nThe first of these statements may perhaps be regarded as a description\nof identity itself, if so fundamental a notion can admit of\ndescription. A thing at any moment is perfectly identical with itself,\nand, if any person were unaware of the meaning of the word \u201cidentity,\u201d\nwe could not better describe it than by such an example.\n\nThe second law points out that contradictory attributes can never be\njoined together. The same object may vary in its different parts;\nhere it may be black, and there white; at one time it may be hard and\nat another time soft; but at the same time and place an attribute\ncannot be both present and absent. Aristotle truly described this law\nas the first of all axioms--one of which we need not seek for any\ndemonstration. All truths cannot be proved, otherwise there would be an\nendless chain of demonstration; and it is in self-evident truths like\nthis that we find the simplest foundations.\n\nThe third of these laws completes the other two. It asserts that at\nevery step there are two possible alternatives--presence or absence,\naffirmation or negation. Hence I propose to name this law the Law of\nDuality, for it gives to all the formul\u00e6 of reasoning a dual character.\nIt asserts also that between presence and absence, existence and\nnon-existence, affirmation and negation, there is no third alternative.\nAs Aristotle said, there can be no mean between opposite assertions: we\nmust either affirm or deny. Hence the inconvenient name by which it has\nbeen known--The Law of Excluded Middle.\n\nIt may be allowed that these laws are not three independent and\ndistinct laws; they rather express three different aspects of the\nsame truth, and each law doubtless presupposes and implies the other\ntwo. But it has not hitherto been found possible to state these\ncharacters of identity and difference in less than the threefold\nformula. The reader may perhaps desire some information as to the\nmode in which these laws have been stated, or the way in which they\nhave been regarded, by philosophers in different ages of the world.\nAbundant information on this and many other points of logical history\nwill be found in Ueberweg\u2019s *System of Logic*, of which an excellent\ntranslation has been published by Professor T.\u00a0M. Lindsay (see\npp.\u00a0228\u2013281).\n\n\n*The Nature of the Laws of Identity and Difference.*\n\nI must at least allude to the profoundly difficult question concerning\nthe nature and authority of these Laws of Identity and Difference.\nAre they Laws of Thought or Laws of Things? Do they belong to mind or\nto material nature? On the one hand it may be said that science is a\npurely mental existence, and must therefore conform to the laws of\nthat which formed it. Science is in the mind and not in the things,\nand the properties of mind are therefore all important. It is true\nthat these laws are verified in the observation of the exterior world;\nand it would seem that they might have been gathered and proved by\ngeneralisation, had they not already been in our possession. But\non the other hand, it may well be urged that we cannot prove these\nlaws by any process of reasoning or observation, because the laws\nthemselves are presupposed, as Leibnitz acutely remarked, in the very\nnotion of a proof. They are the prior conditions of all thought and\nall knowledge, and even to question their truth is to allow them true.\nHartley ingeniously refined upon this argument, remarking that if the\nfundamental laws of logic be not certain, there must exist a logic of\na second order whereby we may determine the degree of uncertainty: if\nthe second logic be not certain, there must be a third; and so on *ad\ninfinitum*. Thus we must suppose either that absolutely certain laws of\nthought exist, or that there is no such thing as certainty whatever.[26]\n\n  [26] Hartley on Man, vol.\u00a0i. p.\u00a0359.\n\nLogicians, indeed, appear to me to have paid insufficient attention to\nthe fact that mistakes in reasoning are always possible, and of not\nunfrequent occurrence. The Laws of Thought are often called necessary\nlaws, that is, laws which cannot but be obeyed. Yet as a matter of\nfact, who is there that does not often fail to obey them? They are\nthe laws which the mind ought to obey rather than what it always does\nobey. Our thoughts cannot be the criterion of truth, for we often\nhave to acknowledge mistakes in arguments of moderate complexity,\nand we sometimes only discover our mistakes by collision between our\nexpectations and the events of objective nature.\n\nMr. Herbert Spencer holds that the laws of logic are objective\nlaws,[27] and he regards the mind as being in a state of constant\neducation, each act of false reasoning or miscalculation leading to\nresults which are likely to prevent similar mistakes from being again\ncommitted. I am quite inclined to accept such ingenious views; but at\nthe same time it is necessary to distinguish between the accumulation\nof knowledge, and the constitution of the mind which allows of the\nacquisition of knowledge. Before the mind can perceive or reason at\nall it must have the conditions of thought impressed upon it. Before\na mistake can be committed, the mind must clearly distinguish the\nmistaken conclusion from all other assertions. Are not the Laws of\nIdentity and Difference the prior conditions of all consciousness and\nall existence? Must they not hold true, alike of things material and\nimmaterial? and if so, can we say that they are only subjectively true\nor objectively true? I am inclined, in short, to regard them as true\nboth \u201cin the nature of thought and things,\u201d as I expressed it in my\nfirst logical essay;[28] and I hold that they belong to the common\nbasis of all existence. But this is one of the most difficult questions\nof psychology and metaphysics which can be raised, and it is hardly\none for the logician to decide. As the mathematician does not inquire\ninto the nature of unity and plurality, but develops the formal laws of\nplurality, so the logician, as I conceive, must assume the truth of the\nLaws of Identity and Difference, and occupy himself in developing the\nvariety of forms of reasoning in which their truth may be manifested.\n\n  [27] *Principles of Psychology*, Second Ed., vol.\u00a0ii. p.\u00a086.\n\n  [28] *Pure Logic, or the Logic of Quality apart from Quantity*, 1864,\n  pp. 10, 16, 22, 29, 36, &c.\n\nAgain, I need hardly dwell upon the question whether logic treats of\nlanguage, notions, or things. As reasonably might we debate whether a\nmathematician treats of symbols, quantities, or things. A mathematician\ncertainly does treat of symbols, but only as the instruments whereby\nto facilitate his reasoning concerning quantities; and as the axioms\nand rules of mathematical science must be verified in concrete objects\nin order that the calculations founded upon them may have any validity\nor utility, it follows that the ultimate objects of mathematical\nscience are the things themselves. In like manner I conceive that\nthe logician treats of language so far as it is essential for the\nembodiment and exhibition of thought. Even if reasoning can take place\nin the inner consciousness of man without the use of any signs, which\nis doubtful, at any rate it cannot become the subject of discussion\nuntil by some system of material signs it is manifested to other\npersons. The logician then uses words and symbols as instruments of\nreasoning, and leaves the nature and peculiarities of language to the\ngrammarian. But signs again must correspond to the thoughts and things\nexpressed, in order that they shall serve their intended purpose. We\nmay therefore say that logic treats ultimately of thoughts and things,\nand immediately of the signs which stand for them. Signs, thoughts, and\nexterior objects may be regarded as parallel and analogous series of\nphenomena, and to treat any one of the three series is equivalent to\ntreating either of the other series.\n\n\n*The Process of Inference.*\n\nThe fundamental action of our reasoning faculties consists in inferring\nor carrying to a new instance of a phenomenon whatever we have\npreviously known of its like, analogue, equivalent or equal. Sameness\nor identity presents itself in all degrees, and is known under various\nnames; but the great rule of inference embraces all degrees, and\naffirms that *so far as there exists sameness, identity or likeness,\nwhat is true of one thing will be true of the other*. The great\ndifficulty doubtless consists in ascertaining that there does exist\na sufficient degree of likeness or sameness to warrant an intended\ninference; and it will be our main task to investigate the conditions\nunder which reasoning is valid. In this place I wish to point out that\nthere is something common to all acts of inference, however different\ntheir apparent forms. The one same rule lends itself to the most\ndiverse applications.\n\nThe simplest possible case of inference, perhaps, occurs in the use of\na *pattern*, *example*, or, as it is commonly called, a *sample*. To\nprove the exact similarity of two portions of commodity, we need not\nbring one portion beside the other. It is sufficient that we take a\nsample which exactly represents the texture, appearance, and general\nnature of one portion, and according as this sample agrees or not with\nthe other, so will the two portions of commodity agree or differ.\nWhatever is true as regards the colour, texture, density, material of\nthe sample will be true of the goods themselves. In such cases likeness\nof quality is the condition of inference.\n\nExactly the same mode of reasoning holds true of magnitude and figure.\nTo compare the sizes of two objects, we need not lay them beside each\nother. A staff, string, or other kind of measure may be employed to\nrepresent the length of one object, and according as it agrees or not\nwith the other, so must the two objects agree or differ. In this case\nthe proxy or sample represents length; but the fact that lengths can\nbe added and multiplied renders it unnecessary that the proxy should\nalways be as large as the object. Any standard of convenient size, such\nas a common foot-rule, may be made the medium of comparison. The height\nof a church in one town may be carried to that in another, and objects\nexisting immovably at opposite sides of the earth may be vicariously\nmeasured against each other. We obviously employ the axiom that\nwhatever is true of a thing as regards its length, is true of its equal.\n\nTo every other simple phenomenon in nature the same principle of\nsubstitution is applicable. We may compare weights, densities, degrees\nof hardness, and degrees of all other qualities, in like manner. To\nascertain whether two sounds are in unison we need not compare them\ndirectly, but a third sound may be the go-between. If a tuning-fork is\nin unison with the middle C of York Minster organ, and we afterwards\nfind it to be in unison with the same note of the organ in Westminster\nAbbey, then it follows that the two organs are tuned in unison. The\nrule of inference now is, that what is true of the tuning-fork as\nregards the tone or pitch of its sound, is true of any sound in unison\nwith it.\n\nThe skilful employment of this substitutive process enables us to\nmake measurements beyond the powers of our senses. No one can count\nthe vibrations, for instance, of an organ-pipe. But we can construct\nan instrument called the *siren*, so that, while producing a sound of\nany pitch, it shall register the number of vibrations constituting the\nsound. Adjusting the sound of the siren in unison with an organ-pipe,\nwe measure indirectly the number of vibrations belonging to a sound\nof that pitch. To measure a sound of the same pitch is as good as to\nmeasure the sound itself.\n\nSir David Brewster, in a somewhat similar manner, succeeded in\nmeasuring the refractive indices of irregular fragments of transparent\nminerals. It was a troublesome, and sometimes impracticable work to\ngrind the minerals into prisms, so that the power of refracting light\ncould be directly observed; but he fell upon the ingenious device\nof compounding a liquid possessing the same refractive power as the\ntransparent fragment under examination. The moment when this equality\nwas attained could be known by the fragments ceasing to reflect or\nrefract light when immersed in the liquid, so that they became almost\ninvisible in it. The refractive power of the liquid being then measured\ngave that of the solid. A more beautiful instance of representative\nmeasurement, depending immediately upon the principle of inference,\ncould not be found.[29]\n\n  [29] Brewster, *Treatise on New Philosophical Instruments*, p.\u00a0273.\n  Concerning this method see also Whewell, *Philosophy of the Inductive\n  Sciences*, vol.\u00a0ii. p.\u00a0355; Tomlinson, *Philosophical Magazine*,\n  Fourth Series, vol.\u00a0xl. p.\u00a0328; Tyndall, in Youmans\u2019 *Modern\n  Culture*, p.\u00a016.\n\nThroughout the various logical processes which we are about\nto consider--Deduction, Induction, Generalisation, Analogy,\nClassification, Quantitative Reasoning--we shall find the one same\nprinciple operating in a more or less disguised form.\n\n\n*Deduction and Induction.*\n\nThe processes of inference always depend on the one same principle of\nsubstitution; but they may nevertheless be distinguished according as\nthe results are inductive or deductive. As generally stated, deduction\nconsists in passing from more general to less general truths; induction\nis the contrary process from less to more general truths. We may\nhowever describe the difference in another manner. In deduction we are\nengaged in developing the consequences of a law. We learn the meaning,\ncontents, results or inferences, which attach to any given proposition.\nInduction is the exactly inverse process. Given certain results or\nconsequences, we are required to discover the general law from which\nthey flow.\n\nIn a certain sense all knowledge is inductive. We can only learn the\nlaws and relations of things in nature by observing those things. But\nthe knowledge gained from the senses is knowledge only of particular\nfacts, and we require some process of reasoning by which we may\ncollect out of the facts the laws obeyed by them. Experience gives\nus the materials of knowledge: induction digests those materials, and\nyields us general knowledge. When we possess such knowledge, in the\nform of general propositions and natural laws, we can usefully apply\nthe reverse process of deduction to ascertain the exact information\nrequired at any moment. In its ultimate foundation, then, all knowledge\nis inductive--in the sense that it is derived by a certain inductive\nreasoning from the facts of experience.\n\nIt is nevertheless true,--and this is a point to which insufficient\nattention has been paid, that all reasoning is founded on the\nprinciples of deduction. I call in question the existence of any method\nof reasoning which can be carried on without a knowledge of deductive\nprocesses. I shall endeavour to show that *induction is really the\ninverse process of deduction*. There is no mode of ascertaining the\nlaws which are obeyed in certain phenomena, unless we have the power\nof determining what results would follow from a given law. Just as the\nprocess of division necessitates a prior knowledge of multiplication,\nor the integral calculus rests upon the observation and remembrance\nof the results of the differential calculus, so induction requires a\nprior knowledge of deduction. An inverse process is the undoing of\nthe direct process. A person who enters a maze must either trust to\nchance to lead him out again, or he must carefully notice the road by\nwhich he entered. The facts furnished to us by experience are a maze of\nparticular results; we might by chance observe in them the fulfilment\nof a law, but this is scarcely possible, unless we thoroughly learn the\neffects which would attach to any particular law.\n\nAccordingly, the importance of deductive reasoning is doubly supreme.\nEven when we gain the results of induction they would be of no use\nunless we could deductively apply them. But before we can gain them\nat all we must understand deduction, since it is the inversion of\ndeduction which constitutes induction. Our first task in this work,\nthen, must be to trace out fully the nature of identity in all its\nforms of occurrence. Having given any series of propositions we must be\nprepared to develop deductively the whole meaning embodied in them, and\nthe whole of the consequences which flow from them.\n\n\n*Symbolic Expression of Logical Inference.*\n\nIn developing the results of the Principle of Inference we require to\nuse an appropriate language of signs. It would indeed be quite possible\nto explain the processes of reasoning by the use of words found in the\ndictionary. Special examples of reasoning, too, may seem to be more\nreadily apprehended than general symbolic forms. But it has been shown\nin the mathematical sciences that the attainment of truth depends\ngreatly upon the invention of a clear, brief, and appropriate system\nof symbols. Not only is such a language convenient, but it is almost\nessential to the expression of those general truths which are the very\nsoul of science. To apprehend the truth of special cases of inference\ndoes not constitute logic; we must apprehend them as cases of more\ngeneral truths. The object of all science is the separation of what is\ncommon and general from what is accidental and different. In a system\nof logic, if anywhere, we should esteem this generality, and strive to\nexhibit clearly what is similar in very diverse cases. Hence the great\nvalue of *general symbols* by which we can represent the form of a\nreasoning process, disentangled from any consideration of the special\nsubject to which it is applied.\n\nThe signs required in logic are of a very simple kind. As sameness or\ndifference must exist between two things or notions, we need signs to\nindicate the things or notions compared, and other signs to denote the\nrelations between them. We need, then, (1) symbols for terms, (2) a\nsymbol for sameness, (3) a symbol for difference, and (4) one or two\nsymbols to take the place of conjunctions.\n\nOrdinary nouns substantive, such as *Iron*, *Metal*, *Electricity*,\n*Undulation*, might serve as terms, but, for the reasons explained\nabove, it is better to adopt blank letters, devoid of special\nsignification, such as A, B, C, &c. Each letter must be understood to\nrepresent a noun, and, so far as the conditions of the argument allow,\n*any noun*. Just as in Algebra, *x*, *y*, *z*, *p*, *q*, &c. are used\nfor *any quantities*, undetermined or unknown, except when the special\nconditions of the problem are taken into account, so will our letters\nstand for undetermined or unknown things.\n\nThese letter-terms will be used indifferently for nouns substantive\nand adjective. Between these two kinds of nouns there may perhaps\nbe differences in a metaphysical or grammatical point of view.\nBut grammatical usage sanctions the conversion of adjectives into\nsubstantives, and *vice vers\u00e2*; we may avail ourselves of this latitude\nwithout in any way prejudging the metaphysical difficulties which may\nbe involved. Here, as throughout this work, I shall devote my attention\nto truths which I can exhibit in a clear and formal manner, believing\nthat in the present condition of logical science, this course will lead\nto greater advantage than discussion upon the metaphysical questions\nwhich may underlie any part of the subject.\n\nEvery noun or term denotes an object, and usually implies the\npossession by that object of certain qualities or circumstances common\nto all the objects denoted. There are certain terms, however, which\nimply the absence of qualities or circumstances attaching to other\nobjects. It will be convenient to employ a special mode of indicating\nthese *negative terms*, as they are called. If the general name A\ndenotes an object or class of objects possessing certain defined\nqualities, then the term Not A will denote any object which does not\npossess the whole of those qualities; in short, Not A is the sign for\nanything which differs from A in regard to any one or more of the\nassigned qualities. If A denote \u201ctransparent object,\u201d Not A will denote\n\u201cnot transparent object.\u201d Brevity and facility of expression are of\nno slight importance in a system of notation, and it will therefore\nbe desirable to substitute for the negative term Not A a briefer\nsymbol. De Morgan represented negative terms by small Roman letters,\nor sometimes by small italic letters;[30] as the latter seem to be\nhighly convenient, I shall use *a*, *b*, *c*, ... *p*, *q*, &c., as the\nnegative terms corresponding to A, B, C, ... P, Q, &c. Thus if A means\n\u201cfluid,\u201d *a* will mean \u201cnot fluid.\u201d\n\n  [30] *Formal Logic*, p.\u00a038.\n\n\n*Expression of Identity and Difference.*\n\nTo denote the relation of sameness or identity I unhesitatingly adopt\nthe sign =, so long used by mathematicians to denote equality. This\nsymbol was originally appropriated by Robert Recorde in his *Whetstone\nof Wit*, to avoid the tedious repetition of the words \u201cis equal to;\u201d\nand he chose a pair of parallel lines, because no two things can be\nmore equal.[31] The meaning of the sign has however been gradually\nextended beyond that of equality of quantities; mathematicians have\nthemselves used it to indicate equivalence of operations. The force\nof analogy has been so great that writers in most other branches\nof science have employed the same sign. The philologist uses it to\nindicate the equivalence of meaning of words: chemists adopt it to\nsignify identity in kind and equality in weight of the elements which\nform two different compounds. Not a few logicians, for instance\nLambert, Drobitsch, George Bentham,[32] Boole,[33] have employed it\nas the copula of propositions. De Morgan declined to use it for this\npurpose, but still further extended its meaning so as to include the\nequivalence of a proposition with the premises from which it can be\ninferred;[34] and Herbert Spencer has applied it in a like manner.[35]\n\n  [31] Hallam\u2019s *Literature of Europe*, First Ed., vol.\u00a0ii. p.\u00a0444.\n\n  [32] *Outline of a New System of Logic*, London, 1827, pp.\u00a0133, &c.\n\n  [33] *An Investigation of the Laws of Thought*, pp.\u00a027, &c.\n\n  [34] *Formal Logic*, pp.\u00a082, 106. In his later work, *The Syllabus of\n  a New System of Logic*, he discontinued the use of the sign.\n\n  [35] *Principles of Psychology*, Second Ed., vol.\u00a0ii. pp.\u00a054, 55.\n\nMany persons may think that the choice of a symbol is a matter of\nslight importance or of mere convenience; but I hold that the common\nuse of this sign = in so many different meanings is really founded\nupon a generalisation of the widest character and of the greatest\nimportance--one indeed which it is a principal purpose of this work to\nexplain. The employment of the same sign in different cases would be\nunphilosophical unless there were some real analogy between its diverse\nmeanings. If such analogy exists, it is not only allowable, but highly\ndesirable and even imperative, to use the symbol of equivalence with a\ngenerality of meaning corresponding to the generality of the principles\ninvolved. Accordingly De Morgan\u2019s refusal to use the symbol in logical\npropositions indicated his opinion that there was a want of analogy\nbetween logical propositions and mathematical equations. I use the sign\nbecause I hold the contrary opinion.\n\nI conceive that the sign = as commonly employed, always denotes\nsome form or degree of sameness, and the particular form is usually\nindicated by the nature of the terms joined by it. Thus \u201c6,720 pounds =\n3 tons\u201d is evidently an equation of quantities. The formula -\u00a0\u00d7\u00a0-\u00a0=\u00a0+\nexpresses the equivalence of operations. \u201cExogens = Dicotyledons\u201d is a\nlogical identity expressing a profound truth concerning the character\nand origin of a most important group of plants.\n\nWe have great need in logic of a distinct sign for the copula, because\nthe little verb *is* (or *are*), hitherto used both in logic and\nordinary discourse, is thoroughly ambiguous. It sometimes denotes\nidentity, as in \u201cSt. Paul\u2019s is the *chef-d\u2019\u0153uvre* of Sir Christopher\nWren;\u201d but it more commonly indicates inclusion of class within class,\nor partial identity, as in \u201cBishops are members of the House of\nLords.\u201d This latter relation involves identity, but requires careful\ndiscrimination from simple identity, as will be shown further on.\n\nWhen with this sign of equality we join two nouns or logical terms, as\nin\n\n  Hydrogen = The least dense element,\n\nwe signify that the object or group of objects denoted by one term is\nidentical with that denoted by the other, in everything except the\nnames. The general formula\n\n  A = B\n\nmust be taken to mean that A and B are symbols for the same object\nor group of objects. This identity may sometimes arise from the mere\nimposition of names, but it may also arise from the deepest laws of the\nconstitution of nature; as when we say\n\n  Gravitating matter = Matter possessing inertia,\n  Exogenous plants = Dicotyledonous plants,\n  Plagihedral quartz crystals = Quartz crystals causing\n    the plane of polarisation of light to rotate.\n\nWe shall need carefully to distinguish between relations of terms which\ncan be modified at our own will and those which are fixed as expressing\nthe laws of nature; but at present we are considering only the mode of\nexpression which may be the same in either case.\n\nSometimes, but much less frequently, we require a symbol to indicate\ndifference or the absence of complete sameness. For this purpose we\nmay generalise in like manner the symbol ~, which was introduced by\nWallis to signify difference between quantities. The general formula\n\n  B ~ C\n\ndenotes that B and C are the names of two objects or groups which are\nnot identical with each other. Thus we may say\n\n  Acrogens ~ Flowering plants.\n  Snowdon ~ The highest mountain in Great Britain.\n\nI shall also occasionally use the sign \u1515 to signify in the most general\nmanner the existence of any relation between the two terms connected by\nit. Thus \u1515 might mean not only the relations of equality or inequality,\nsameness or difference, but any special relation of time, place, size,\ncausation, &c. in which one thing may stand to another. By A\u00a0\u1515\u00a0B I\nmean, then, any two objects of thought related to each other in any\nconceivable manner.\n\n\n*General Formula of Logical Inference.*\n\nThe one supreme rule of inference consists, as I have said, in the\ndirection to affirm of anything whatever is known of its like, equal\nor equivalent. The *Substitution of Similars* is a phrase which seems\naptly to express the capacity of mutual replacement existing in any\ntwo objects which are like or equivalent to a sufficient degree.\nIt is matter for further investigation to ascertain when and for\nwhat purposes a degree of similarity less than complete identity is\nsufficient to warrant substitution. For the present we think only of\nthe exact sameness expressed in the form\n\n  A = B.\n\nNow if we take the letter C to denote any third conceivable object, and\nuse the sign \u1515 in its stated meaning of *indefinite relation*, then the\ngeneral formula of all inference may be thus exhibited:--\n\n  From       A = B\u00a0\u1515\u00a0C\n  we may infer   A\u00a0\u1515\u00a0C\n\nor, in words--*In whatever relation a thing stands to a second thing,\nin the same relation it stands to the like or equivalent of that second\nthing.* The identity between A and B allows us indifferently to place\nA where B was, or B where A was; and there is no limit to the variety\nof special meanings which we can bestow upon the signs used in this\nformula consistently with its truth. Thus if we first specify only the\nmeaning of the sign \u1515, we may say that if *C is the weight of B*, then\n*C is also the weight of A*. Similarly\n\n  If C is the father of B, C is the father of A;\n  If C is a fragment of B, C is a fragment of A;\n  If C is a quality of B, C is a quality of A;\n  If C is a species of B, C is a species of A;\n  If C is the equal of B, C is the equal of A;\n\nand so on *ad infinitum*.\n\nWe may also endow with special meanings the letter-terms A, B, and C,\nand the process of inference will never be false. Thus let the sign\n\u1515\u00a0mean \u201cis height of,\u201d and let\n\n  A = Snowdon,\n  B = Highest mountain in England or Wales,\n  C = 3,590 feet;\n\nthen it obviously follows since \u201c3,590 feet is the height of Snowdon,\u201d\nand \u201cSnowdon = the highest mountain in England or Wales,\u201d that, \u201c3,590\nfeet is the height of the highest mountain in England or Wales.\u201d\n\nOne result of this general process of inference is that we may in any\naggregate or complex whole replace any part by its equivalent without\naltering the whole. To alter is to make a difference; but if in\nreplacing a part I make no difference, there is no alteration of the\nwhole. Many inferences which have been very imperfectly included in\nlogical formulas at once follow. I remember the late Prof. De Morgan\nremarking that all Aristotle\u2019s logic could not prove that \u201cBecause a\nhorse is an animal, the head of a horse is the head of an animal.\u201d I\nconceive that this amounts merely to replacing in the complete notion\n*head of a horse*, the term \u201chorse,\u201d by its equivalent *some animal* or\n*an animal*. Similarly, since\n\n  The Lord Chancellor = The Speaker of the House of Lords,\n\nit follows that\n\n  The death of the Lord Chancellor = The death of the Speaker of the\n  House of Lords;\n\nand any event, circumstance or thing, which stands in a certain\nrelation to the one will stand in like relation to the other. Milton\nreasons in this way when he says, in his Areopagitica, \u201cWho kills a\nman, kills a reasonable creature, God\u2019s image.\u201d If we may suppose him\nto mean\n\n  God\u2019s image = man = some reasonable creature,\n\nit follows that \u201cThe killer of a man is the killer of some reasonable\ncreature,\u201d and also \u201cThe killer of God\u2019s image.\u201d\n\nThis replacement of equivalents may be repeated over and over again to\nany extent. Thus if *person* is identical in meaning with *individual*,\nit follows that\n\n  Meeting of persons = meeting of individuals;\n\nand if *assemblage* = *meeting*, we may make a new replacement and show\nthat\n\n  Meeting of persons = assemblage of individuals.\n\nWe may in fact found upon this principle of substitution a most general\naxiom in the following terms[36]:--\n\n  [36] *Pure Logic, or the Logic of Quality*, p.\u00a014.\n\n  *Same parts samely related make same wholes.*\n\nIf, for instance, exactly similar bricks and other materials be used\nto build two houses, and they be similarly placed in each house, the\ntwo houses must be similar. There are millions of cells in a human\nbody, but if each cell of one person were represented by an exactly\nsimilar cell similarly placed in another body, the two persons would\nbe undistinguishable, and would be only *numerically* different. It\nis upon this principle, as we shall see, that all accurate processes\nof measurement depend. If for a weight in a scale of a balance we\nsubstitute another weight, and the equilibrium remains entirely\nunchanged, then the weights must be exactly equal. The general test of\nequality is substitution. Objects are equally bright when on replacing\none by the other the eye perceives no difference. Objects are equal in\ndimensions when tested by the same gauge they fit in the same manner.\nGenerally speaking, two objects are alike so far as when substituted\none for another no alteration is produced, and *vice vers\u00e2* when alike\nno alteration is produced by the substitution.\n\n\n*The Propagating Power of Similarity.*\n\nThe relation of similarity in all its degrees is reciprocal. So far\nas things are alike, either may be substituted for the other; and\nthis may perhaps be considered the very meaning of the relation. But\nit is well worth notice that there is in similarity a peculiar power\nof extending itself among all the things which are similar. To render\na number of things similar to each other we need only render them\nsimilar to one standard object. Each coin struck from a pair of dies\nnot only resembles the matrix or original pattern from which the dies\nwere struck, but resembles every other coin manufactured from the same\noriginal pattern. Among a million such coins there are not less than\n499,999,500,000 *pairs of coins* resembling each other. Similars to\nthe same are similars to all. It is one great advantage of printing\nthat all copies of a document struck from the same type are necessarily\nidentical each with each, and whatever is true of one copy will be true\nof every copy. Similarly, if fifty rows of pipes in an organ be tuned\nin perfect unison with one row, usually the Principal, they must be\nin unison with each other. Similarity can also reproduce or propagate\nitself *ad infinitum*: for if a number of tuning-forks be adjusted in\nperfect unison with one standard fork, all instruments tuned to any one\nfork will agree with any instrument tuned to any other fork. Standard\nmeasures of length, capacity, weight, or any other measurable quality,\nare propagated in the same manner. So far as copies of the original\nstandard, or copies of copies, or copies again of those copies, are\naccurately executed, they must all agree each with every other.\n\nIt is the capability of mutual substitution which gives such great\nvalue to the modern methods of mechanical construction, according\nto which all the parts of a machine are exact facsimiles of a fixed\npattern. The rifles used in the British army are constructed on the\nAmerican interchangeable system, so that any part of any rifle can be\nsubstituted for the same part of another. A bullet fitting one rifle\nwill fit all others of the same bore. Sir J. Whitworth has extended\nthe same system to the screws and screw-bolts used in connecting\ntogether the parts of machines, by establishing a series of standard\nscrews.\n\n\n*Anticipations of the Principle of Substitution.*\n\nIn such a subject as logic it is hardly possible to put forth any\nopinions which have not been in some degree previously entertained. The\ngerm at least of every doctrine will be found in earlier writers, and\nnovelty must arise chiefly in the mode of harmonising and developing\nideas. When I first employed the process and name of *substitution*\nin logic,[37] I was led to do so from analogy with the familiar\nmathematical process of substituting for a symbol its value as given in\nan equation. In writing my first logical essay I had a most imperfect\nconception of the importance and generality of the process, and I\ndescribed, as if they were of equal importance, a number of other laws\nwhich now seem to be but particular cases of the one general rule of\nsubstitution.\n\n  [37] *Pure Logic*, pp.\u00a018, 19.\n\nMy second essay, \u201cThe Substitution of Similars,\u201d was written shortly\nafter I had become aware of the great simplification which may be\neffected by a proper application of the principle of substitution. I\nwas not then acquainted with the fact that the German logician Beneke\nhad employed the principle of substitution, and had used the word\nitself in forming a theory of the syllogism. My imperfect acquaintance\nwith the German language had prevented me from acquiring a complete\nknowledge of Beneke\u2019s views; but there is no doubt that Professor\nLindsay is right in saying that he, and probably other logicians,\nwere in some degree familiar with the principle.[38] Even Aristotle\u2019s\ndictum may be regarded as an imperfect statement of the principle of\nsubstitution; and, as I have pointed out, we have only to modify that\ndictum in accordance with the quantification of the predicate in order\nto arrive at the complete process of substitution.[39] The Port-Royal\nlogicians appear to have entertained nearly equivalent views, for\nthey considered that all moods of the syllogism might be reduced\nunder one general principle.[40] Of two premises they regard one as\nthe *containing proposition* (propositio continens), and the other as\nthe *applicative proposition*. The latter proposition must always be\naffirmative, and represents that by which a substitution is made; the\nformer may or may not be negative, and is that in which a substitution\nis effected. They also show that this method will embrace certain cases\nof complex reasoning which had no place in the Aristotelian syllogism.\nTheir views probably constitute the greatest improvement in logical\ndoctrine made up to that time since the days of Aristotle. But a true\nreform in logic must consist, not in explaining the syllogism in one\nway or another, but in doing away with all the narrow restrictions of\nthe Aristotelian system, and in showing that there exists an infinite\nvariety of logical arguments immediately deducible from the principle\nof substitution of which the ancient syllogism forms but a small and\nnot even the most important part.\n\n  [38] Ueberweg\u2019s *System of Logic*, transl. by Lindsay, pp.\u00a0442\u2013446,\n  571, 572. The anticipations of the principle of substitution to be\n  found in the works of Leibnitz, Reusch, and perhaps other German\n  logicians, will be noticed in the preface to this second edition.\n\n  [39] *Substitution of Similars* (1869), p.\u00a09.\n\n  [40] *Port-Royal Logic*, transl. by Spencer Baynes, pp.\u00a0212\u2013219. Part\n  III. chap. x. and xi.\n\n\n*The Logic of Relatives.*\n\nThere is a difficult and important branch of logic which may be\ncalled the Logic of Relatives. If I argue, for instance, that because\nDaniel Bernoulli was the son of John, and John the brother of James,\ntherefore Daniel was the nephew of James, it is not possible to prove\nthis conclusion by any simple logical process. We require at any rate\nto assume that the son of a brother is a nephew. A simple logical\nrelation is that which exists between properties and circumstances of\nthe same object or class. But objects and classes of objects may also\nbe related according to all the properties of time and space. I believe\nit may be shown, indeed, that where an inference concerning such\nrelations is drawn, a process of substitution is really employed and an\nidentity must exist; but I will not undertake to prove the assertion\nin this work. The relations of time and space are logical relations\nof a complicated character demanding much abstract and difficult\ninvestigation. The subject has been treated with such great ability by\nPeirce,[41] De Morgan,[42] Ellis,[43] and Harley, that I will not in\nthe present work attempt any review of their writings, but merely refer\nthe reader to the publications in which they are to be found.\n\n  [41] *Description of a Notation for the Logic of Relatives, resulting\n  from an Amplification of the Conceptions of Boole\u2019s Calculus of\n  Logic.* By C.\u00a0S. Peirce. *Memoirs of the American Academy*, vol.\u00a0ix.\n  Cambridge, U.S., 1870.\n\n  [42] *On the Syllogism No IV., and on the Logic of Relations.* By\n  Augustus De Morgan. *Transactions of the Cambridge Philosophical\n  Society*, vol.\u00a0x. part ii., 1860.\n\n  [43] *Observations on Boole\u2019s Laws of Thought.* By the late R. Leslie\n  Ellis; communicated by the Rev. Robert Harley, F.R.S. *Report of the\n  British Association*, 1870. *Report of Sections*, p.\u00a012. Also, *On\n  Boole\u2019s Laws of Thought*. By the Rev. Robert Harley, F.R.S., *ibid.*\n  p. 14.\n\n\n\n\nCHAPTER II.\n\nTERMS.\n\n\nEvery proposition expresses the resemblance or difference of the things\ndenoted by its terms. As inference treats of the relation between two\nor more propositions, so a proposition expresses a relation between two\nor more terms. In the portion of this work which treats of deduction\nit will be convenient to follow the usual order of exposition. We will\nconsider in succession the various kinds of terms, propositions, and\narguments, and we commence in this chapter with terms.\n\nThe simplest and most palpable meaning which can belong to a term\nconsists of some single material object, such as Westminster Abbey,\nStonehenge, the Sun, Sirius, &c. It is probable that in early stages of\nintellect only concrete and palpable things are the objects of thought.\nThe youngest child knows the difference between a hot and a cold body.\nThe dog can recognise his master among a hundred other persons, and\nanimals of much lower intelligence know and discriminate their haunts.\nIn all such acts there is judgment concerning the likeness of physical\nobjects, but there is little or no power of analysing each object and\nregarding it as a group of qualities.\n\nThe dignity of intellect begins with the power of separating points of\nagreement from those of difference. Comparison of two objects may lead\nus to perceive that they are at once like and unlike. Two fragments of\nrock may differ entirely in outward form, yet they may have the same\ncolour, hardness, and texture. Flowers which agree in colour may differ\nin odour. The mind learns to regard each object as an aggregate of\nqualities, and acquires the power of dwelling at will upon one or other\nof those qualities to the exclusion of the rest. Logical abstraction,\nin short, comes into play, and the mind becomes capable of reasoning,\nnot merely about objects which are physically complete and concrete,\nbut about things which may be thought of separately in the mind though\nthey exist not separately in nature. We can think of the hardness of\na rock, or the colour of a flower, and thus produce abstract notions,\ndenoted by abstract terms, which will form a subject for further\nconsideration.\n\nAt the same time arise general notions and classes of objects. We\ncannot fail to observe that the quality *hardness* exists in many\nobjects, for instance in many fragments of rock; mentally joining these\ntogether, we create the class *hard object*, which will include, not\nonly the actual objects examined, but all others which may happen to\nagree with them, as they agree with each other. As our senses cannot\npossibly report to us all the contents of space, we cannot usually\nset any limits to the number of objects which may fall into any such\nclass. At this point we begin to perceive the power and generality of\nthought, which enables us in a single act to treat of indefinitely or\neven infinitely numerous objects. We can safely assert that whatever is\ntrue of any one object coming under a class is true of any of the other\nobjects so far as they possess the common qualities implied in their\nbelonging to the class. We must not place a thing in a class unless\nwe are prepared to believe of it all that is believed of the class in\ngeneral; but it remains a matter of important consideration to decide\nhow far and in what manner we can safely undertake thus to assign\nthe place of objects in that general system of classification which\nconstitutes the body of science.\n\n\n*Twofold Meaning of General Names.*\n\nEtymologically the *meaning* of a name is that which we are caused\nto think of when the name is used. Now every general name causes us\nto think of some one or more of the objects belonging to a class; it\nmay also cause us to think of the common qualities possessed by those\nobjects. A name is said to *denote* the object of thought to which it\nmay be applied; it *implies* at the same time the possession of certain\nqualities or circumstances. The objects denoted form the *extent*\nof meaning of the term; the qualities implied form the *intent* of\nmeaning. Crystal is the name of any substance of which the molecules\nare arranged in a regular geometrical manner. The substances or objects\nin question form the extent of meaning; the circumstance of having the\nmolecules so arranged forms the intent of meaning.\n\nWhen we compare general terms together, it may often be found that\nthe meaning of one is included in the meaning of another. Thus\nall *crystals* are included among *material substances*, and all\n*opaque crystals* are included among *crystals*; here the inclusion\nis in extension. We may also have inclusion of meaning in regard to\nintension. For, as all crystals are material substances, the qualities\nimplied by the term material substance must be among those implied by\ncrystal. Again, it is obvious that while in extension of meaning opaque\ncrystals are but a part of crystals, in intension of meaning crystal\nis but part of opaque crystal. We increase the intent of meaning of a\nterm by joining to it adjectives, or phrases equivalent to adjectives,\nand the removal of such adjectives of course decreases the intensive\nmeaning. Now, concerning such changes of meaning, the following\nall-important law holds universally true:--*When the intent of meaning\nof a term is increased the extent is decreased; and* vice vers\u00e2, *when\nthe extent is increased the intent is decreased*. In short, as one is\nincreased the other is decreased.\n\nThis law refers only to logical changes. The number of steam-engines\nin the world may be undergoing a rapid increase without the intensive\nmeaning of the name being altered. The law will only be verified,\nagain, when there is a real change in the intensive meaning, and an\nadjective may often be joined to a noun without making a change.\n*Elementary metal* is identical with *metal*; *mortal man* with *man*;\nit being a *property* of all metals to be elements, and of all men to\nbe mortals.\n\nThere is no limit to the amount of meaning which a term may have. A\nterm may denote one object, or many, or an infinite number; it may\nimply a single quality, if such there be, or a group of any number\nof qualities, and yet the law connecting the extension and intension\nwill infallibly apply. Taking the general name *planet*, we increase\nits intension and decrease its extension by prefixing the adjective\n*exterior*; and if we further add *nearest to the earth*, there remains\nbut one planet, *Mars*, to which the name can then be applied. Singular\nterms, which denote a single individual only, come under the same law\nof meaning as general names. They may be regarded as general names of\nwhich the meaning in extension is reduced to a minimum. Logicians have\nerroneously asserted, as it seems to me, that singular terms are devoid\nof meaning in intension, the fact being that they exceed all other\nterms in that kind of meaning, as I have elsewhere tried to show.[44]\n\n  [44] Jevons\u2019 *Elementary Lessons in Logic*, pp.\u00a041\u201343; *Pure Logic*,\n  p.\u00a06. See also J.\u00a0S. Mill, *System of Logic*, Book I. chap. ii.\n  section 5, and Shedden\u2019s *Elements of Logic*, London, 1864, pp.\u00a014,\n  &c. Professor Robertson objects (*Mind*, vol.\u00a0i. p.\u00a0210) that I\n  confuse *singular* and *proper* names; if so, it is because I hold\n  that the same remarks apply to proper names, which do not seem to me\n  to differ logically from singular names.\n\n\n*Abstract Terms.*\n\nComparison of objects, and analysis of the complex resemblances and\ndifferences which they present, lead us to the conception of *abstract\nqualities*. We learn to think of one object as not only different from\nanother, but as differing in some particular point, such as colour, or\nweight, or size. We may then convert points of agreement or difference\ninto separate objects of thought which we call qualities and denote by\n*abstract terms*. Thus the term *redness* means something in which a\nnumber of objects agree as to colour, and in virtue of which they are\ncalled red. Redness forms, in fact, the intensive meaning of the term\nred.\n\nAbstract terms are strongly distinguished from general terms by\npossessing only one kind of meaning; for as they denote qualities\nthere is nothing which they cannot in addition imply. The adjective\n\u201cred\u201d is the name of red objects, but it implies the possession by\nthem of the quality *redness*; but this latter term has one single\nmeaning--the quality alone. Thus it arises that abstract terms are\nincapable of plurality. Red objects are numerically distinct each\nfrom each, and there are multitudes of such objects; but redness is a\nsingle quality which runs through all those objects, and is the same in\none as it is in another. It is true that we may speak of *rednesses*,\nmeaning different kinds or tints of redness, just as we may speak of\n*colours*, meaning different kinds of colours. But in distinguishing\nkinds, degrees, or other differences, we render the terms so far\nconcrete. In that they are merely red there is but a single nature in\nred objects, and so far as things are merely coloured, colour is a\nsingle indivisible quality. Redness, so far as it is redness merely,\nis one and the same everywhere, and possesses absolute oneness. In\nvirtue of this unity we acquire the power of treating all instances of\nsuch quality as we may treat any one. We possess, in short, general\nknowledge.\n\n\n*Substantial Terms.*\n\nLogicians appear to have taken little notice of a class of terms which\npartake in certain respects of the character of abstract terms and yet\nare undoubtedly the names of concrete existing things. These terms are\nthe names of substances, such as gold, carbonate of lime, nitrogen, &c.\nWe cannot speak of two golds, twenty carbonates of lime, or a hundred\nnitrogens. There is no such distinction between the parts of a uniform\nsubstance as will allow of a discrimination of numerous individuals.\nThe qualities of colour, lustre, malleability, density, &c., by which\nwe recognise gold, extend through its substance irrespective of\nparticular size or shape. So far as a substance is gold, it is one and\nthe same everywhere; so that terms of this kind, which I propose to\ncall *substantial terms*, possess the peculiar unity of abstract terms.\nYet they are not abstract; for gold is of course a tangible visible\nbody, entirely concrete, and existing independently of other bodies.\n\nIt is only when, by actual mechanical division, we break up the uniform\nwhole which forms the meaning of a substantial term, that we introduce\nnumber. *Piece of gold* is a term capable of plurality; for there may\nbe a great many pieces discriminated either by their various shapes and\nsizes, or, in the absence of such marks, by simultaneously occupying\ndifferent parts of space. In substance they are one; as regards the\nproperties of space they are many.[45] We need not further pursue this\nquestion, which involves the distinction between unity and plurality,\nuntil we consider the principles of number in a subsequent chapter.\n\n  [45] Professor Robertson has criticised my introduction of\n  \u201cSubstantial Terms\u201d (*Mind*, vol.\u00a0i. p.\u00a0210), and objects, perhaps\n  correctly, that the distinction if valid is extra-logical. I am\n  inclined to think, however, that the doctrine of terms is, strictly\n  speaking, for the most part extra-logical.\n\n\n*Collective Terms.*\n\nWe must clearly distinguish between the *collective* and the *general\nmeanings* of terms. The same name may be used to denote the whole body\nof existing objects of a certain kind, or any one of those objects\ntaken separately. \u201cMan\u201d may mean the aggregate of existing men, which\nwe sometimes describe as *mankind*; it is also the general name\napplying to any man. The vegetable kingdom is the name of the whole\naggregate of *plants*, but \u201cplant\u201d itself is a general name applying\nto any one or other plant. Every material object may be conceived as\ndivisible into parts, and is therefore collective as regards those\nparts. The animal body is made up of cells and fibres, a crystal\nof molecules; wherever physical division, or as it has been called\n*partition*, is possible, there we deal in reality with a collective\nwhole. Thus the greater number of general terms are at the same time\ncollective as regards each individual whole which they denote.\n\nIt need hardly be pointed out that we must not infer of a collective\nwhole what we know only of the parts, nor of the parts what we know\nonly of the whole. The relation of whole and part is not one of\nidentity, and does not allow of substitution. There may nevertheless be\nqualities which are true alike of the whole and of its parts. A number\nof organ-pipes tuned in unison produce an aggregate of sound which\nis of exactly the same pitch as each separate sound. In the case of\nsubstantial terms, certain qualities may be present equally in each\nminutest part as in the whole. The chemical nature of the largest mass\nof pure carbonate of lime is the same as the nature of the smallest\nparticle. In the case of abstract terms, again, we cannot draw a\ndistinction between whole and part; what is true of redness in any case\nis always true of redness, so far as it is merely red.\n\n\n*Synthesis of Terms.*\n\nWe continually combine simple terms together so as to form new terms\nof more complex meaning. Thus, to increase the intension of meaning of\na term we write it with an adjective or a phrase of adjectival nature.\nBy joining \u201cbrittle\u201d to \u201cmetal,\u201d we obtain a combined term, \u201cbrittle\nmetal,\u201d which denotes a certain portion of the metals, namely, such as\nare selected on account of possessing the quality of *brittleness*.\nAs we have already seen, \u201cbrittle metal\u201d possesses less extension and\ngreater intension than metal. Nouns, prepositional phrases, participial\nphrases and subordinate propositions may also be added to terms so as\nto increase their intension and decrease their extension.\n\nIn our symbolic language we need some mode of indicating this junction\nof terms, and the most convenient device will be the juxtaposition of\nthe letter-terms. Thus if A mean brittle, and B mean metal, then AB\nwill mean brittle metal. Nor need there be any limit to the number of\nletters thus joined together, or the complexity of the notions which\nthey may represent.\n\nThus if we take the letters\n\n  P = metal,\n  Q = white,\n  R = monovalent,\n  S = of specific gravity 10\u00b75,\n  T = melting above 1000\u00b0\u00a0C.,\n  V = good conductor of heat and electricity,\n\nthen we can form a combined term PQRSTV, which will denote \u201ca white\nmonovalent metal, of specific gravity 10\u00b75, melting above 1000\u00b0\u00a0C., and\na good conductor of heat and electricity.\u201d\n\nThere are many grammatical usages concerning the junction of words and\nphrases to which we need pay no attention in logic. We can never say\nin ordinary language \u201cof wood table,\u201d meaning \u201ctable of wood;\u201d but we\nmay consider \u201cof wood\u201d as logically an exact equivalent of \u201cwooden\u201d; so\nthat if\n\n  X = of wood,\n  Y = table,\n\nthere is no reason why, in our symbols, XY should not be just as\ncorrect an expression for \u201ctable of wood \u201d as YX. In this case indeed\nwe might substitute for \u201cof wood \u201d the corresponding adjective\n\u201cwooden,\u201d but we should often fail to find any adjective answering\nexactly to a phrase. There is no single word by which we could express\nthe notion \u201cof specific gravity 10\u00b75:\u201d but logically we may consider\nthese words as forming an adjective; and denoting this by S and metal\nby P, we may say that SP means \u201cmetal of specific gravity 10\u00b75.\u201d It\nis one of many advantages in these blank letter-symbols that they\nenable us completely to neglect all grammatical peculiarities and to\nfix our attention solely on the purely logical relations involved.\nInvestigation will probably show that the rules of grammar are mainly\nfounded upon traditional usage and have little logical signification.\nThis indeed is sufficiently proved by the wide grammatical differences\nwhich exist between languages, though the logical foundation must be\nthe same.\n\n\n*Symbolic Expression of the Law of Contradiction.*\n\nThe synthesis of terms is subject to the all-important Law of\nThought, described in a previous section (p.\u00a05) and called the Law of\nContradiction, It is self-evident that no quality can be both present\nand absent at the same time and place. This fundamental condition\nof all thought and of all existence is expressed symbolically by a\nrule that a term and its negative shall never be allowed to come\ninto combination. Such combined terms as A*a*, B*b*, C*c*, &c., are\nself-contradictory and devoid of all intelligible meaning. If they\ncould represent anything, it would be what cannot exist, and cannot\neven be imagined in the mind. They can therefore only enter into our\nconsideration to suffer immediate exclusion. The criterion of false\nreasoning, as we shall find, is that it involves self-contradiction,\nthe affirming and denying of the same statement. We might represent\nthe object of all reasoning as the separation of the consistent and\npossible from the inconsistent and impossible; and we cannot make any\nstatement except a truism without implying that certain combinations of\nterms are contradictory and excluded from thought. To assert that \u201call\nA\u2019s are B\u2019s\u201d is equivalent to the assertion that \u201cA\u2019s which are not B\u2019s\ncannot exist.\u201d\n\nIt will be convenient to have the means of indicating the exclusion of\nthe self-contradictory, and we may use the familiar sign for *nothing*,\nthe cipher 0. Thus the second law of thought may be symbolised in the\nforms\n\n  A*a* = 0     AB*b* = 0     ABC*a* = 0\n\nWe may variously describe the meaning of 0 in logic as the\n*non-existent*, the *impossible*, the *self-inconsistent*, the\n*inconceivable*. Close analogy exists between this meaning and its\nmathematical signification.\n\n\n*Certain Special Conditions of Logical Symbols.*\n\nIn order that we may argue and infer truly we must treat our logical\nsymbols according to the fundamental laws of Identity and Difference.\nBut in thus using our symbols we shall frequently meet with\ncombinations of which the meaning will not at first sight be apparent.\nIf in one case we learn that an object is \u201cyellow and round,\u201d and in\nanother case that it is \u201cround and yellow,\u201d there arises the question\nwhether these two descriptions are identical in meaning or not. Again,\nif we proved that an object was \u201cround round,\u201d the meaning of such an\nexpression would be open to doubt. Accordingly we must take notice,\nbefore proceeding further, of certain special laws which govern the\ncombination of logical terms.\n\nIn the first place the combination of a logical term with itself is\nwithout effect, just as the repetition of a statement does not alter\nthe meaning of the statement; \u201ca round round object\u201d is simply \u201ca round\nobject.\u201d What is yellow yellow is merely yellow; metallic metals cannot\ndiffer from metals, nor circular circles from circles. In our symbolic\nlanguage we may similarly hold that AA is identical with A, or\n\n  A = AA = AAA = &c.\n\nThe late Professor Boole is the only logician in modern times who has\ndrawn attention to this remarkable property of logical terms;[46]\nbut in place of the name which he gave to the law, I have proposed\nto call it The Law of Simplicity.[47] Its high importance will only\nbecome apparent when we attempt to determine the relations of logical\nand mathematical science. Two symbols of quantity, and only two, seem\nto obey this law; we may say that 1\u00a0\u00d7\u00a01\u00a0=\u00a01, and 0\u00a0\u00d7\u00a00\u00a0=\u00a00 (taking 0\nto mean absolute zero or 1\u00a0\u2013\u00a01); there is apparently no other number\nwhich combined with itself gives an unchanged result. I shall point\nout, however, in the chapter upon Number, that in reality all numerical\nsymbols obey this logical principle.\n\n  [46] *Mathematical Analysis of Logic*, Cambridge, 1847, p.\u00a017. *An\n  Investigation of the Laws of Thought*, London, 1854, p.\u00a031.\n\n  [47] *Pure Logic*, p.\u00a015.\n\nIt is curious that this Law of Simplicity, though almost unnoticed\nin modern times, was known to Bo\u00ebthius, who makes a singular remark\nin his treatise *De Trinitate et Unitate Dei* (p.\u00a0959). He says: \u201cIf\nI should say sun, sun, sun, I should not have made three suns, but I\nshould have named one sun so many times.\u201d[48] Ancient discussions about\nthe doctrine of the Trinity drew more attention to subtle questions\nconcerning the nature of unity and plurality than has ever since been\ngiven to them.\n\n  [48] \u201cVelut si dicam, Sol, Sol, Sol, non tres soles effecerim, sed\n  uno toties pr\u00e6dicaverim.\u201d\n\nIt is a second law of logical symbols that order of combination is a\nmatter of indifference. \u201cRich and rare gems\u201d are the same as \u201crare and\nrich gems,\u201d or even as \u201cgems, rich and rare.\u201d Grammatical, rhetorical,\nor poetic usage may give considerable significance to order of\nexpression. The limited power of our minds prevents our grasping many\nideas at once, and thus the order of statement may produce some effect,\nbut not in a simply logical manner. All life proceeds in the succession\nof time, and we are obliged to write, speak, or even think of things\nand their qualities one after the other; but between the things and\ntheir qualities there need be no such relation of order in time or\nspace. The sweetness of sugar is neither before nor after its weight\nand solubility. The hardness of a metal, its colour, weight, opacity,\nmalleability, electric and chemical properties, are all coexistent\nand coextensive, pervading the metal and every part of it in perfect\ncommunity, none before nor after the others. In our words and symbols\nwe cannot observe this natural condition; we must name one quality\nfirst and another second, just as some one must be the first to sign a\npetition, or to walk foremost in a procession. In nature there is no\nsuch precedence.\n\nI find that the opinion here stated, to the effect that relations of\nspace and time do not apply to many of our ideas, is clearly adopted by\nHume in his celebrated *Treatise on Human Nature* (vol.\u00a0i. p.\u00a0410). He\nsays:[49]--\u201cAn object may be said to be no where, when its parts are\nnot so situated with respect to each other, as to form any figure or\nquantity; nor the whole with respect to other bodies so as to answer\nto our notions of contiguity or distance. Now this is evidently the\ncase with all our perceptions and objects, except those of sight and\nfeeling. A moral reflection cannot be placed on the right hand or on\nthe left hand of a passion, nor can a smell or sound be either of a\ncircular or a square figure. These objects and perceptions, so far from\nrequiring any particular place, are absolutely incompatible with it,\nand even the imagination cannot attribute it to them.\u201d\n\n  [49] Book i., Part iv., Section 5.\n\nA little reflection will show that knowledge in the highest perfection\nwould consist in the *simultaneous* possession of a multitude of\nfacts. To comprehend a science perfectly we should have every fact\npresent with every other fact. We must write a book and we must read\nit successively word by word, but how infinitely higher would be\nour powers of thought if we could grasp the whole in one collective\nact of consciousness! Compared with the brutes we do possess some\nslight approximation to such power, and it is conceivable that in the\nindefinite future mind may acquire an increase of capacity, and be less\nrestricted to the piecemeal examination of a subject. But I wish here\nto make plain that there is no logical foundation for the successive\ncharacter of thought and reasoning unavoidable under our present mental\nconditions. *We are logically weak and imperfect in respect of the\nfact that we are obliged to think of one thing after another.* We must\ndescribe metal as \u201chard and opaque,\u201d or \u201copaque and hard,\u201d but in the\nmetal itself there is no such difference of order; the properties are\nsimultaneous and coextensive in existence.\n\nSetting aside all grammatical peculiarities which render a substantive\nless moveable than an adjective, and disregarding any meaning indicated\nby emphasis or marked order of words, we may state, as a general law of\nlogic, that AB is identical with BA, or AB = BA. Similarly, ABC = ACB =\nBCA = &c.\n\nBoole first drew attention in recent years to this property of logical\nterms, and he called it the property of Commutativeness.[50] He not\nonly stated the law with the utmost clearness, but pointed out that\nit is a Law of Thought rather than a Law of Things. I shall have in\nvarious parts of this work to show how the necessary imperfection of\nour symbols expressed in this law clings to our modes of expression,\nand introduces complication into the whole body of mathematical\nformul\u00e6, which are really founded on a logical basis.\n\n  [50] *Laws of Thought*, p.\u00a029. It is pointed out in the preface to\n  this Second Edition that Leibnitz was acquainted with the Laws of\n  Simplicity and of Commutativeness.\n\nIt is of course apparent that the power of commutation belongs only\nto terms related in the simple logical mode of synthesis. No one can\nconfuse \u201ca house of bricks\u201d with \u201cbricks of a house,\u201d \u201ctwelve square\nfeet\u201d with \u201ctwelve feet square,\u201d \u201cthe water of crystallization\u201d with\n\u201cthe crystallization of water.\u201d All relations which involve differences\nof time and space are inconvertible; the higher must not be made to\nchange places with the lower, nor the first with the last. For the\nparties concerned there is all the difference in the world between A\nkilling B and B killing A. The law of commutativeness simply asserts\nthat difference of order does not attach to the connection between the\nproperties and circumstances of a thing--to what I call *simple logical\nrelation*.\n\n\n\n\nCHAPTER III.\n\nPROPOSITIONS.\n\n\nWe now proceed to consider the variety of forms of propositions in\nwhich the truths of science must be expressed. I shall endeavour to\nshow that, however diverse these forms may be, they all admit the\napplication of the one same principle of inference that what is true of\na thing is true of the like or same. This principle holds true whatever\nbe the kind or manner of the likeness, provided proper regard be had to\nits nature. Propositions may assert an identity of time, space, manner,\nquantity, degree, or any other circumstance in which things may agree\nor differ.\n\nWe find an instance of a proposition concerning time in the\nfollowing:--\u201cThe year in which Newton was born, was the year in which\nGalileo died.\u201d This proposition expresses an approximate identity of\ntime between two events; hence whatever is true of the year in which\nGalileo died is true of that in which Newton was born, and *vice\nvers\u00e2*. \u201cTower Hill is the place where Raleigh was executed\u201d expresses\nan identity of place; and whatever is true of the one spot is true\nof the spot otherwise defined, but in reality the same. In ordinary\nlanguage we have many propositions obscurely expressing identities\nof number, quantity, or degree. \u201cSo many men, so many minds,\u201d is a\nproposition concerning number, that is to say, an equation; whatever\nis true of the number of men is true of the number of minds, and\n*vice vers\u00e2*. \u201cThe density of Mars is (nearly) the same as that of\nthe Earth,\u201d \u201cThe force of gravity is directly as the product of the\nmasses, and inversely as the square of the distance,\u201d are propositions\nconcerning magnitude or degree. Logicians have not paid adequate\nattention to the great variety of propositions which can be stated by\nthe use of the little conjunction *as*, together with *so*. \u201cAs the\nhome so the people,\u201d is a proposition expressing identity of manner;\nand a great number of similar propositions all indicating some kind of\nresemblance might be quoted. Whatever be the special kind of identity,\nall such expressions are subject to the great principle of inference;\nbut as we shall in later parts of this work treat more particularly of\ninference in cases of number and magnitude, we will here confine our\nattention to logical propositions which involve only notions of quality.\n\n\n*Simple Identities.*\n\nThe most important class of propositions consists of those which fall\nunder the formula\n\n  A = B,\n\nand may be called *simple identities*. I may instance, in the first\nplace, those most elementary propositions which express the exact\nsimilarity of a quality encountered in two or more objects. I may\ncompare the colour of the Pacific Ocean with that of the Atlantic, and\ndeclare them identical. I may assert that \u201cthe smell of a rotten egg\nis like that of hydrogen sulphide;\u201d \u201cthe taste of silver hyposulphite\nis like that of cane sugar;\u201d \u201cthe sound of an earthquake resembles\nthat of distant artillery.\u201d Such are propositions stating, accurately\nor otherwise, the identity of simple physical sensations. Judgments of\nthis kind are necessarily pre-supposed in more complex judgments. If\nI declare that \u201cthis coin is made of gold,\u201d I must base the judgment\nupon the exact likeness of the substance in several qualities to\nother pieces of substance which are undoubtedly gold. I must make\njudgments of the colour, the specific gravity, the hardness, and of\nother mechanical and chemical properties; each of these judgments is\nexpressed in an elementary proposition, \u201cthe colour of this coin is the\ncolour of gold,\u201d and so on. Even when we establish the identity of a\nthing with itself under a different name or aspect, it is by distinct\njudgments concerning single circumstances. To prove that the Homeric\n\u03c7\u03b1\u03bb\u03ba\u03cc\u03c2 is copper we must show the identity of each quality recorded of\n\u03c7\u03b1\u03bb\u03ba\u03cc\u03c2 with a quality of copper. To establish Deal as the landing-place\nof C\u00e6sar all material circumstances must be shown to agree. If the\nmodern Wroxeter is the ancient Uriconium, there must be the like\nagreement of all features of the country not subject to alteration by\ntime.\n\nSuch identities must be expressed in the form A = B. We may say\n\n  Colour of Pacific Ocean = Colour of Atlantic Ocean.\n  Smell of rotten egg = Smell of hydrogen sulphide.\n\nIn these and similar propositions we assert identity of single\nqualities or causes of sensation. In the same form we may also express\nidentity of any group of qualities, as in\n\n  \u03c7\u03b1\u03bb\u03ba\u03cc\u03c2 = Copper.\n  Deal = Landing-place of C\u00e6sar.\n\nA multitude of propositions involving singular terms fall into the same\nform, as in\n\n  The Pole star = The slowest-moving star.\n  Jupiter = The greatest of the planets.\n  The ringed planet = The planet having seven satellites.\n  The Queen of England = The Empress of India.\n  The number two = The even prime number.\n  Honesty = The best policy.\n\nIn mathematical and scientific theories we often meet with simple\nidentities capable of expression in the same form. Thus in mechanical\nscience \u201cThe process for finding the resultant of forces = the process\nfor finding the resultant of simultaneous velocities.\u201d Theorems in\ngeometry often give results in this form, as\n\n  Equilateral triangles = Equiangular triangles.\n  Circle = Finite plane curve of constant curvature.\n  Circle = Curve of least perimeter.\n\nThe more profound and important laws of nature are often expressible in\nthe form of simple identities; in addition to some instances which have\nalready been given, I may suggest,\n\n  Crystals of cubical system = Crystals not possessing the power of\n  double refraction.\n\nAll definitions are necessarily of this form, whether the objects\ndefined be many, few, or singular. Thus we may say,\n\n  Common salt = Sodium chloride.\n  Chlorophyl = Green colouring matter of leaves.\n  Square = Equal-sided rectangle.\n\nIt is an extraordinary fact that propositions of this elementary form,\nall-important and very numerous as they are, had no recognised place\nin Aristotle\u2019s system of Logic. Accordingly their importance was\noverlooked until very recent times, and logic was the most deformed\nof sciences. But it is impossible that Aristotle or any other person\nshould avoid constantly using them; not a term could be defined\nwithout their use. In one place at least Aristotle actually notices a\nproposition of the kind. He observes: \u201cWe sometimes say that that white\nthing is Socrates, or that the object approaching is Callias.\u201d[51] Here\nwe certainly have simple identity of terms; but he considered such\npropositions purely accidental, and came to the unfortunate conclusion,\nthat \u201cSingulars cannot be predicated of other terms.\u201d\n\n  [51] *Prior Analytics*, i. cap. xxvii. 3.\n\nPropositions may also express the identity of extensive groups of\nobjects taken collectively or in one connected whole; as when we say,\n\n  The Queen, Lords, and Commons = The Legislature of the United Kingdom.\n\nWhen Blackstone asserts that \u201cThe only true and natural foundation of\nsociety are the wants and fears of individuals,\u201d we must interpret him\nas meaning that the whole of the wants and fears of individuals in the\naggregate form the foundation of society. But many propositions which\nmight seem to be collective are but groups of singular propositions or\nidentities. When we say \u201cPotassium and sodium are the metallic bases of\npotash and soda,\u201d we obviously mean,\n\n  Potassium = Metallic base of potash;\n  Sodium = Metallic base of soda.\n\nIt is the work of grammatical analysis to separate the various\npropositions often combined into a single sentence. Logic cannot be\nproperly required to interpret the forms and devices of language, but\nonly to treat the meaning when clearly exhibited.\n\n\n*Partial Identities.*\n\nA second highly important kind of proposition is that which I propose\nto call *a partial identity*. When we say that \u201cAll mammalia are\nvertebrata,\u201d we do not mean that mammalian animals are identical with\nvertebrate animals, but only that the mammalia form a *part of the\nclass vertebrata*. Such a proposition was regarded in the old logic as\nasserting the inclusion of one class in another, or of an object in a\nclass. It was called a universal affirmative proposition, because the\nattribute *vertebrate* was affirmed of the whole subject *mammalia*;\nbut the attribute was said to be *undistributed*, because not all\nvertebrata were of necessity involved in the proposition. Aristotle,\noverlooking the importance of simple identities, and indeed almost\ndenying their existence, unfortunately founded his system upon the\nnotion of inclusion in a class, instead of adopting the basis of\nidentity. He regarded inference as resting upon the rule that what is\ntrue of the containing class is true of the contained, in place of the\nvastly more general rule that what is true of a class or thing is true\nof the like. Thus he not only reduced logic to a fragment of its proper\nself, but destroyed the deep analogies which bind together logical and\nmathematical reasoning. Hence a crowd of defects, difficulties and\nerrors which will long disfigure the first and simplest of the sciences.\n\nIt is surely evident that the relation of inclusion rests upon the\nrelation of identity. Mammalian animals cannot be included among\nvertebrates unless they be identical with part of the vertebrates.\nCabinet Ministers are included almost always in the class Members of\nParliament, because they are identical with some who sit in Parliament.\nWe may indicate this identity with a part of the larger class in\nvarious ways; as for instance,\n\n  Mammalia = part of the vertebrata.\n  Diatomace\u00e6 = a class of plants.\n  Cabinet Ministers = some members of Parliament.\n  Iron = a metal.\n\nIn ordinary language the verbs *is* and *are* express mere inclusion\nmore often than not. *Men are mortals*, means that *men* form a part\nof the class *mortal*; but great confusion exists between this sense\nof the verb and that in which it expresses identity, as in \u201cThe sun is\nthe centre of the planetary system.\u201d The introduction of the indefinite\narticle *a* often expresses partiality; when we say \u201cIron is a metal\u201d\nwe clearly mean that iron is *one only* of several metals.\n\nCertain recent logicians have proposed to avoid the indefiniteness\nin question by what is called the Quantification of the Predicate,\nand they have generally used the little word *some* to show that only\na part of the predicate is identical with the subject. *Some* is an\n*indeterminate adjective*; it implies unknown qualities by which we\nmight select the part in question if the qualities were known, but\nit gives no hint as to their nature. I might make use of such an\nindeterminate sign to express partial identities in this work. Thus,\ntaking the special symbol V = Some, the general form of a partial\nidentity would be A = VB, and in Boole\u2019s Logic expressions of the\nkind were much used. But I believe that indeterminate symbols only\nintroduce complexity, and destroy the beauty and simple universality\nof the system which may be created without their use. A vague word\nlike *some* is only used in ordinary language by *ellipsis*, and to\navoid the trouble of attaining accuracy. We can always employ more\ndefinite expressions if we like; but when once the indefinite *some* is\nintroduced we cannot replace it by the special description. We do not\nknow whether *some* colour is red, yellow, blue, or what it is; but on\nthe other hand *red* colour is certainly *some* colour.\n\nThroughout this system of logic I shall dispense with such indefinite\nexpressions; and this can readily be done by substituting one of the\nother terms. To express the proposition \u201cAll A\u2019s are some B\u2019s\u201d I shall\nnot use the form A = VB, but\n\n  A = AB.\n\nThis formula states that the class A is identical with the class AB;\nand as the latter must be a part at least of the class B, it implies\nthe inclusion of the class A in that of B. We might represent our\nformer example thus,\n\n  Mammalia = Mammalian vertebrata.\n\nThis proposition asserts identity between a part (or it may be the\nwhole) of the vertebrata and the mammalia. If it is asked What part?\nthe proposition affords no answer, except that it is the part which is\nmammalian; but the assertion \u201cmammalia = some vertebrata\u201d tells us no\nmore.\n\nIt is quite likely that some readers will think this mode of\nrepresenting the universal affirmative proposition artificial and\ncomplicated. I will not undertake to convince them of the opposite\nat this point of my exposition. Justification for it will be found,\nnot so much in the immediate treatment of this proposition, as in the\ngeneral harmony which it will enable us to disclose between all parts\nof reasoning. I have no doubt that this is the critical difficulty in\nthe relation of logical to other forms of reasoning. Grant this mode of\ndenoting that \u201call A\u2019s are B\u2019s,\u201d and I fear no further difficulties;\nrefuse it, and we find want of analogy and endless anomaly in every\ndirection. It is on general grounds that I hope to show overwhelming\nreasons for seeking to reduce every kind of proposition to the form of\nan identity.\n\nI may add that not a few logicians have accepted this view of the\nuniversal affirmative proposition. Leibnitz, in his *Difficultates\nQu\u00e6dam Logic\u00e6*, adopts it, saying, \u201cOmne A est B; id est \u00e6quivalent AB\net A, seu A non B est nonens.\u201d Boole employed the logical equation *x*\n= *xy* concurrently with *x* = *vy*; and Spalding[52] distinctly\nsays that the proposition \u201call metals are minerals\u201d might be described\nas an assertion of *partial identity* between the two classes. Hence\nthe name which I have adopted for the proposition.\n\n  [52] *Encyclop\u00e6dia Britannica*, Eighth Ed. art. Logic, sect. 37,\n  note. 8vo. reprint, p.\u00a079.\n\n\n*Limited Identities.*\n\nAn important class of propositions have the form\n\n  AB = AC,\n\nexpressing the identity of the class AB with the class AC. In other\nwords, \u201cWithin the sphere of the class A, all the B\u2019s are all the\nC\u2019s;\u201d or again, \u201cThe B\u2019s and C\u2019s, which are A\u2019s, are identical.\u201d But\nit will be observed that nothing is asserted concerning things which\nare outside of the class A; and thus the identity is of limited extent.\nIt is the proposition B = C limited to the sphere of things called A.\nThus we may say, with some approximation to truth, that \u201cLarge plants\nare plants devoid of locomotive power.\u201d\n\nA barrister may make numbers of most general statements concerning\nthe relations of persons and things in the course of an argument, but\nit is of course to be understood that he speaks only of persons and\nthings under the English Law. Even mathematicians make statements which\nare not true with absolute generality. They say that imaginary roots\nenter into equations by pairs; but this is only true under the tacit\ncondition that the equations in question shall not have imaginary\ncoefficients.[53] The universe, in short, within which they habitually\ndiscourse is that of equations with real coefficients. These implied\nlimitations form part of that great mass of tacit knowledge which\naccompanies all special arguments.\n\n  [53] De Morgan, *On the Root of any Function*. Cambridge\n  Philosophical Transactions, 1867, vol.\u00a0xi. p.\u00a025.\n\nTo De Morgan is due the remark, that we do usually think and argue in\na limited universe or sphere of notions, even when it is not expressly\nstated.[54]\n\n  [54] *Syllabus of a proposed System of Logic*, \u00a7\u00a7 122, 123.\n\nIt is worthy of inquiry whether all identities are not really limited\nto an implied sphere of meaning. When we make such a plain statement as\n\u201cGold is malleable\u201d we obviously speak of gold only in its solid state;\nwhen we say that \u201cMercury is a liquid metal\u201d we must be understood to\nexclude the frozen condition to which it may be reduced in the Arctic\nregions. Even when we take such a fundamental law of nature as \u201cAll\nsubstances gravitate,\u201d we must mean by substance, material substance,\nnot including that basis of heat, light, and electrical undulations\nwhich occupies space and possesses many wonderful mechanical\nproperties, but not gravity. The proposition then is really of the form\n\n  Material substance = Material gravitating substance.\n\n\n*Negative Propositions.*\n\nIn every act of intellect we are engaged with a certain identity or\ndifference between things or sensations compared together. Hitherto\nI have treated only of identities; and yet it might seem that the\nrelation of difference must be infinitely more common than that of\nlikeness. One thing may resemble a great many other things, but then it\ndiffers from all remaining things in the world. Diversity may almost be\nsaid to constitute life, being to thought what motion is to a river.\nThe perception of an object involves its discrimination from all other\nobjects. But we may nevertheless be said to detect resemblance as often\nas we detect difference. We cannot, in fact, assert the existence of\na difference, without at the same time implying the existence of an\nagreement.\n\nIf I compare mercury, for instance, with other metals, and decide that\nit is *not solid*, here is a difference between mercury and solid\nthings, expressed in a negative proposition; but there must be implied,\nat the same time, an agreement between mercury and the other substances\nwhich are not solid. As it is impossible to separate the vowels of the\nalphabet from the consonants without at the same time separating the\nconsonants from the vowels, so I cannot select as the object of thought\n*solid things*, without thereby throwing together into another class\nall things which are *not solid*. The very fact of not possessing a\nquality, constitutes a new quality which may be the ground of judgment\nand classification. In this point of view, agreement and difference are\never the two sides of the same act of intellect, and it becomes equally\npossible to express the same judgment in the one or other aspect.\n\nBetween affirmation and negation there is accordingly a perfect\nequilibrium. Every affirmative proposition implies a negative one, and\n*vice vers\u00e2*. It is even a matter of indifference, in a logical point\nof view, whether a positive or negative term be used to denote a given\nquality and the class of things possessing it. If the ordinary state\nof a man\u2019s body be called *good health*, then in other circumstances\nhe is said *not to be in good health*; but we might equally describe\nhim in the latter state as *sickly*, and in his normal condition he\nwould be *not sickly*. Animal and vegetable substances are now called\n*organic*, so that the other substances, forming an immensely greater\npart of the globe, are described negatively as *inorganic*. But we\nmight, with at least equal logical correctness, have described the\npreponderating class of substances as *mineral*, and then vegetable and\nanimal substances would have been *non-mineral*.\n\nIt is plain that any positive term and its corresponding negative\ndivide between them the whole universe of thought: whatever does not\nfall into one must fall into the other, by the third fundamental Law\nof Thought, the Law of Duality. It follows at once that there are\ntwo modes of representing a difference. Supposing that the things\nrepresented by A and B are found to differ, we may indicate (see p.\u00a017)\nthe result of the judgment by the notation\n\n  A ~ B.\n\nWe may now represent the same judgment by the assertion that A agrees\nwith those things which differ from B, or that A agrees with the\nnot-B\u2019s. Using our notation for negative terms (see p.\u00a014), we obtain\n\n  A = A*b*\n\nas the expression of the ordinary negative proposition. Thus if we\ntake A to mean quicksilver, and B solid, then we have the following\nproposition:--\n\n  Quicksilver = Quicksilver not-solid.\n\nThere may also be several other classes of negative propositions, of\nwhich no notice was taken in the old logic. We may have cases where\nall A\u2019s are not-B\u2019s, and at the same time all not-B\u2019s are A\u2019s; there\nmay, in short, be a simple identity between A and not-B, which may be\nexpressed in the form\n\n  A = *b*.\n\nAn example of this form would be\n\n  Conductors of electricity = non-electrics.\n\nWe shall also frequently have to deal as results of deduction, with\nsimple, partial, or limited identities between negative terms, as in\nthe forms\n\n  *a* = *b*,   *a* = *a**b*,   *a*C = *b*C, etc.\n\nIt would be possible to represent affirmative propositions in the\nnegative form. Thus \u201cIron is solid,\u201d might be expressed as \u201cIron is not\nnot-solid,\u201d or \u201cIron is not fluid;\u201d or, taking A and *b* for the terms\n\u201ciron,\u201d and \u201cnot-solid,\u201d the form would be A ~ *b*.\n\nBut there are very strong reasons why we should employ all propositions\nin their affirmative form. All inference proceeds by the substitution\nof equivalents, and a proposition expressed in the form of an identity\nis ready to yield all its consequences in the most direct manner. As\nwill be more fully shown, we can infer *in* a negative proposition,\nbut not *by* it. Difference is incapable of becoming the ground of\ninference; it is only the implied agreement with other differing\nobjects which admits of deductive reasoning; and it will always be\nfound advantageous to employ propositions in the form which exhibits\nclearly the implied agreements.\n\n\n*Conversion of Propositions.*\n\nThe old books of logic contain many rules concerning the conversion of\npropositions, that is, the transposition of the subject and predicate\nin such a way as to obtain a new proposition which will be true when\nthe original proposition is true. The reduction of every proposition to\nthe form of an identity renders all such rules and processes needless.\nIdentity is essentially reciprocal. If the colour of the Atlantic Ocean\nis the same as that of the Pacific Ocean, that of the Pacific must\nbe the same as that of the Atlantic. Sodium chloride being identical\nwith common salt, common salt must be identical with sodium chloride.\nIf the number of windows in Salisbury Cathedral equals the number of\ndays in the year, the number of days in the year must equal the number\nof the windows. Lord Chesterfield was not wrong when he said, \u201cI will\ngive anybody their choice of these two truths, which amount to the\nsame thing; He who loves himself best is the honestest man; or, The\nhonestest man loves himself best.\u201d Scotus Erigena exactly expresses\nthis reciprocal character of identity in saying, \u201cThere are not two\nstudies, one of philosophy and the other of religion; true philosophy\nis true religion, and true religion is true philosophy.\u201d\n\nA mathematician would not think it worth while to mention that if\n*x* = *y* then also *y* = *x*. He would not consider these to be\ntwo equations at all, but one equation accidentally written in two\ndifferent manners. In written symbols one of two names must come first,\nand the other second, and a like succession must perhaps be observed\nin our thoughts: but in the relation of identity there is no need for\nsuccession in order (see p.\u00a033), each is simultaneously equal and\nidentical to the other. These remarks will hold true both of logical\nand mathematical identity; so that I shall consider the two forms\n\n  A = B and B = A\n\nto express exactly the same identity differently written. All need for\nrules of conversion disappears, and there will be no single proposition\nin the system which may not be written with either end foremost. Thus A\n= AB is the same as AB = A, *a*C = *b*C is the same as *b*C = *a*C, and\nso forth.\n\nThe same remarks are partially true of differences and inequalities,\nwhich are also reciprocal to the extent that one thing cannot differ\nfrom a second without the second differing from the first. Mars differs\nin colour from Venus, and Venus must differ from Mars. The Earth\ndiffers from Jupiter in density; therefore Jupiter must differ from the\nEarth. Speaking generally, if A ~ B we shall also have B ~ A, and these\ntwo forms may be considered expressions of the same difference. But\nthe relation of differing things is not wholly reciprocal. The density\nof Jupiter does not differ from that of the Earth in the same way that\nthat of the Earth differs from that of Jupiter. The change of sensation\nwhich we experience in passing from Venus to Mars is not the same as\nwhat we experience in passing back to Venus, but just the opposite\nin nature. The colour of the sky is lighter than that of the ocean;\ntherefore that of the ocean cannot be lighter than that of the sky, but\ndarker. In these and all similar cases we gain a notion of *direction*\nor character of change, and results of immense importance may be shown\nto rest on this notion. For the present we shall be concerned with the\nmere fact of identity existing or not existing.\n\n\n*Twofold Interpretation of Propositions.*\n\nTerms, as we have seen (p.\u00a025), may have a meaning either in extension\nor intension; and according as one or the other meaning is attributed\nto the terms of a proposition, so may a different interpretation be\nassigned to the proposition itself. When the terms are abstract we\nmust read them in intension, and a proposition connecting such terms\nmust denote the identity or non-identity of the qualities respectively\ndenoted by the terms. Thus if we say\n\n  Equality = Identity of magnitude,\n\nthe assertion means that the circumstance of being equal exactly\ncorresponds with the circumstance of being identical in magnitude.\nSimilarly in\n\n  Opacity = Incapability of transmitting light,\n\nthe quality of being incapable of transmitting light is declared to be\nthe same as the intended meaning of the word opacity.\n\nWhen general names form the terms of a proposition we may apply a\ndouble interpretation. Thus\n\n  Exogens = Dicotyledons\n\nmeans either that the qualities which belong to all exogens are the\nsame as those which belong to all dicotyledons, or else that every\nindividual falling under one name falls equally under the other.\nHence it may be said that there are two distinct fields of logical\nthought. We may argue either by the qualitative meaning of names or\nby the quantitative, that is, the extensive meaning. Every argument\ninvolving concrete plural terms might be converted into one involving\nonly abstract singular terms, and *vice vers\u00e2*. But there are reasons\nfor believing that the intensive or qualitative form of reasoning is\nthe primary and fundamental one. It is sufficient to point out that the\nextensive meaning of a name is a changeable and fleeting thing, while\nthe intensive meaning may nevertheless remain fixed. Very numerous\nadditions have been lately made to the extensive meanings both of\nplanet and element. Every iron steam-ship which is made or destroyed\nadds to or subtracts from the extensive meaning of the name steam-ship,\nwithout necessarily affecting the intensive meaning. Stage coach means\nas much as ever in one way, but in extension the class is nearly\nextinct. Chinese railway, on the other hand, is a term represented only\nby a single instance; in twenty years it may be the name of a large\nclass.\n\n\n\n\nCHAPTER IV.\n\nDEDUCTIVE REASONING.\n\n\nThe general principle of inference having been explained in the\nprevious chapters, and a suitable system of symbols provided, we have\nnow before us the comparatively easy task of tracing out the most\ncommon and important forms of deductive reasoning. The general problem\nof deduction is as follows:--*From one or more propositions called\npremises to draw such other propositions as will necessarily be true\nwhen the premises are true.* By deduction we investigate and unfold the\ninformation contained in the premises; and this we can do by one single\nrule--*For any term occurring in any proposition substitute the term\nwhich is asserted in any premise to be identical with it.* To obtain\ncertain deductions, especially those involving negative conclusions, we\nshall require to bring into use the second and third Laws of Thought,\nand the process of reasoning will then be called *Indirect Deduction*.\nIn the present chapter, however, I shall confine my attention to\nthose results which can be obtained by the process of *Direct\nDeduction*, that is, by applying to the premises themselves the rule of\nsubstitution. It will be found that we can combine into one harmonious\nsystem, not only the various moods of the ancient syllogism but a great\nnumber of equally important forms of reasoning, which had no recognised\nplace in the old logic. We can at the same time dispense entirely with\nthe elaborate apparatus of logical rules and mnemonic lines, which were\nrequisite so long as the vital principle of reasoning was not clearly\nexpressed.\n\n\n*Immediate Inference.*\n\nProbably the simplest of all forms of inference is that which has been\ncalled *Immediate Inference*, because it can be performed upon a single\nproposition. It consists in joining an adjective, or other qualifying\nclause of the same nature, to both sides of an identity, and asserting\nthe equivalence of the terms thus produced. For instance, since\n\n  Conductors of electricity = Non-electrics,\n\nit follows that\n\n  Liquid conductors of electricity = Liquid non-electrics.\n\nIf we suppose that\n\n  Plants = Bodies decomposing carbonic acid,\n\nit follows that\n\n  Microscopic plants = Microscopic bodies decomposing\n    carbonic acid.\n\nIn general terms, from the identity\n\n  A = B\n\nwe can infer the identity\n\n  AC = BC.\n\nThis is but a case of plain substitution; for by the first Law of\nThought it must be admitted that\n\n  AC = AC,\n\nand if, in the second side of this identity, we substitute for A its\nequivalent B, we obtain\n\n  AC = BC.\n\nIn like manner from the partial identity\n\n  A = AB\n\nwe may obtain\n\n  AC = ABC\n\nby an exactly similar act of substitution; and in every other case\nthe rule will be found capable of verification by the principle of\ninference. The process when performed as here described will be quite\nfree from the liability to error which I have shown[55] to exist in\n\u201cImmediate Inference by added Determinants,\u201d as described by Dr.\nThomson.[56]\n\n  [55] *Elementary Lessons in Logic*, p.\u00a086.\n\n  [56] *Outline of the Laws of Thought*, \u00a7 87.\n\n\n*Inference with Two Simple Identities.*\n\nOne of the most common forms of inference, and one to which I shall\nespecially direct attention, is practised with two simple identities.\nFrom the two statements that \u201cLondon is the capital of England\u201d and\n\u201cLondon is the most populous city in the world,\u201d we instantaneously\ndraw the conclusion that \u201cThe capital of England is the most populous\ncity in the world.\u201d Similarly, from the identities\n\n  Hydrogen = Substance of least density,\n  Hydrogen = Substance of least atomic weight,\n\nwe infer\n\n  Substance of least density = Substance of least atomic weight.\n\nThe general form of the argument is exhibited in the symbols\n\n           B = A         (1)\n           B = C         (2)\n  hence    A = C.        (3)\n\nWe may describe the result by saying that terms identical with the\nsame term are identical with each other; and it is impossible to\noverlook the analogy to the first axiom of Euclid that \u201cthings equal\nto the same thing are equal to each other.\u201d It has been very commonly\nsupposed that this is a fundamental principle of thought, incapable of\nreduction to anything simpler. But I entertain no doubt that this form\nof reasoning is only one case of the general rule of inference. We have\ntwo propositions, A = B and B = C, and we may for a moment consider\nthe second one as affirming a truth concerning B, while the former one\ninforms us that B is identical with A; hence by substitution we may\naffirm the same truth of A. It happens in this particular case that the\ntruth affirmed is identity to C, and we might, if we preferred it, have\nconsidered the substitution as made by means of the second identity in\nthe first. Having two identities we have a choice of the mode in which\nwe will make the substitution, though the result is exactly the same in\neither case.\n\nNow compare the three following formul\u00e6,\n\n  (1)   A = B = C, hence A = C\n  (2)   A = B ~ C, hence A ~ C\n  (3)   A ~ B ~ C, no inference.\n\nIn the second formula we have an identity and a difference, and we are\nable to infer a difference; in the third we have two differences and\nare unable to make any inference at all. Because A and C both differ\nfrom B, we cannot tell whether they will or will not differ from each\nother. The flowers and leaves of a plant may both differ in colour from\nthe earth in which the plant grows, and yet they may differ from each\nother; in other cases the leaves and stem may both differ from the\nsoil and yet agree with each other. Where we have difference only we\ncan make no inference; where we have identity we can infer. This fact\ngives great countenance to my assertion that inference proceeds always\nthrough identity, but may be equally well effected in propositions\nasserting difference or identity.\n\nDeferring a more complete discussion of this point, I will only mention\nnow that arguments from double identity occur very frequently, and are\nusually taken for granted, owing to their extreme simplicity. In regard\nto the equivalence of words this form of inference must be constantly\nemployed. If the ancient Greek \u03c7\u03b1\u03bb\u03ba\u03cc\u03c2 is our *copper*, then it must\nbe the French *cuivre*, the German *kupfer*, the Latin *cuprum*,\nbecause these are words, in one sense at least, equivalent to copper.\nWhenever we can give two definitions or expressions for the same term,\nthe formula applies; thus Senior defined wealth as \u201cAll those things,\nand those things only, which are transferable, are limited in supply,\nand are directly or indirectly productive of pleasure or preventive\nof pain.\u201d Wealth is also equivalent to \u201cthings which have value in\nexchange;\u201d hence obviously, \u201cthings which have value in exchange = all\nthose things, and those things only, which are transferable, &c.\u201d Two\nexpressions for the same term are often given in the same sentence,\nand their equivalence implied. Thus Thomson and Tait say,[57] \u201cThe\nnaturalist may be content to know matter as that which can be perceived\nby the senses, or as that which can be acted upon by or can exert\nforce.\u201d I take this to mean--\n\n  Matter = what can be perceived by the senses;\n  Matter = what can be acted upon by or can exert force.\n\n  [57] *Treatise on Natural Philosophy*, vol.\u00a0i. p.\u00a0161.\n\nFor the term \u201cmatter\u201d in either of these identities we may substitute\nits equivalent given in the other definition. Elsewhere they often\nemploy sentences of the form exemplified in the following:[58] \u201cThe\nintegral curvature, or whole change of direction of an arc of a plane\ncurve, is the angle through which the tangent has turned as we pass\nfrom one extremity to the other.\u201d This sentence is certainly of the\nform--\n\n  The integral curvature = the whole change of direction, &c. = the\n  angle through which the tangent has turned, &c.\n\n  [58] *Treatise on Natural Philosophy*, vol.\u00a0i. p.\u00a06.\n\nDisguised cases of the same kind of inference occur throughout all\nsciences, and a remarkable instance is found in algebraic geometry.\nMathematicians readily show that every equation of the form *y* = *mx*\n+ *c* corresponds to or represents a straight line; it is also easily\nproved that the same equation is equivalent to one of the general form\nA*x* + B*y* + C = 0, and *vice vers\u00e2*. Hence it follows that every\nequation of the form in question, that is to say, every equation of the\nfirst degree, corresponds to or represents a straight line.[59]\n\n  [59] Todhunter\u2019s *Plane Co-ordinate Geometry*, chap. ii. pp.\u00a011\u201314.\n\n\n*Inference with a Simple and a Partial Identity.*\n\nA form of reasoning somewhat different from that last considered\nconsists in inference-between a simple and a partial identity. If we\nhave two propositions of the forms\n\n  A = B,\n  B = BC,\n\nwe may then substitute for B in either proposition its equivalent in\nthe other, getting in both cases A = BC; in this we may if we like make\na second substitution for B, getting\n\n  A = AC.\n\nThus, since \u201cThe Mont Blanc is the highest mountain in Europe, and\nthe Mont Blanc is deeply covered with snow,\u201d we infer by an obvious\nsubstitution that \u201cThe highest mountain in Europe is deeply covered\nwith snow.\u201d These propositions when rigorously stated fall into the\nforms above exhibited.\n\nThis mode of inference is constantly employed when for a term we\nsubstitute its definition, or *vice vers\u00e2*. The very purpose of a\ndefinition is to allow a single noun to be employed in place of a long\ndescriptive phrase. Thus, when we say \u201cA circle is a curve of the\nsecond degree,\u201d we may substitute a definition of the circle, getting\n\u201cA curve, all points of which are at equal distances from one point, is\na curve of the second degree.\u201d The real forms of the propositions here\ngiven are exactly those shown in the symbolic statement, but in this\nand many other cases it will be sufficient to state them in ordinary\nelliptical language for sake of brevity. In scientific treatises a\nterm and its definition are often both given in the same sentence,\nas in \u201cThe weight of a body in any given locality, or the force\nwith which the earth attracts it, is proportional to its mass.\u201d The\nconjunction *or* in this statement gives the force of equivalence to\nthe parenthetic phrase, so that the propositions really are\n\n  Weight of a body = force with which the earth attracts it.\n  Weight of a body = weight, &c. proportional to its mass.\n\nA slightly different case of inference consists in substituting in a\nproposition of the form A = AB, a definition of the term B. Thus from A\n= AB and B = C we get A = AC. For instance, we may say that \u201cMetals are\nelements\u201d and \u201cElements are incapable of decomposition.\u201d\n\n  Metal = metal element.\n  Element = what is incapable of decomposition.\n\nHence\n\n  Metal = metal incapable of decomposition.\n\nIt is almost needless to point out that the form of these arguments\ndoes not suffer any real modification if some of the terms happen to be\nnegative; indeed in the last example \u201cincapable of decomposition\u201d may\nbe treated as a negative term. Taking\n\n  A = metal\n  B = element\n  C = capable of decomposition\n  *c* = incapable of decomposition;\n\nthe propositions are of the forms\n\n  A = AB\n  B = *c*\n\nwhence, by substitution,\n\n  A = A*c*.\n\n\n*Inference of a Partial from Two Partial Identities.*\n\nHowever common be the cases of inference already noticed, there is\na form occurring almost more frequently, and which deserves much\nattention, because it occupied a prominent place in the ancient\nsyllogistic system. That system strangely overlooked all the kinds of\nargument we have as yet considered, and selected, as the type of all\nreasoning, one which employs two partial identities as premises. Thus\nfrom the propositions\n\n  Sodium is a metal               (1)\n  Metals conduct electricity,     (2)\n\nwe may conclude that\n\n  Sodium conducts electricity.    (3)\n\nTaking A, B, C to represent the three terms respectively, the premises\nare of the forms\n\n  A = AB         (1)\n  B = BC.        (2)\n\nNow for B in (1) we can substitute its expression as given in (2),\nobtaining\n\n  A = ABC,       (3)\n\nor, in words, from\n\n  Sodium = sodium metal,                        (1)\n  Metal = metal conducting electricity,         (2)\n\nwe infer\n\n  Sodium = sodium metal conducting electricity, (3)\n\nwhich, in the elliptical language of common life, becomes\n\n  \u201cSodium conducts electricity.\u201d\n\nThe above is a syllogism in the mood called Barbara[60] in the truly\nbarbarous language of ancient logicians; and the first figure of the\nsyllogism contained Barbara and three other moods which were esteemed\ndistinct forms of argument. But it is worthy of notice that, without\nany real change in our form of inference, we readily include these\nthree other moods under Barbara. The negative mood Celarent will be\nrepresented by the example\n\n  [60] An explanation of this and other technical terms of the old\n  logic will be found in my *Elementary Lessons in Logic*, Sixth\n  Edition, 1876; Macmillan.\n\n           Neptune is a planet,                  (1)\n           No planet has retrograde motion;      (2)\n  Hence    Neptune has not retrograde motion.    (3)\n\nIf we put A for Neptune, B for planet, and C for \u201chaving retrograde\nmotion,\u201d then by the corresponding negative term c, we denote \u201cnot\nhaving retrograde motion.\u201d The premises now fall into the forms\n\n  A = AB            (1)\n  B = B*c*,         (2)\n\nand by substitution for B, exactly as before, we obtain\n\n  A = AB*c*.        (3)\n\nWhat is called in the old logic a particular conclusion may be deduced\nwithout any real variation in the symbols. Particular quantity is\nindicated, as before mentioned (p.\u00a041), by joining to the term an\nindefinite adjective of quantity, such as *some*, *a part of*,\n*certain*, &c., meaning that an unknown part of the term enters into\nthe proposition as subject. Considerable doubt and ambiguity arise out\nof the question whether the part may not in some cases be the whole,\nand in the syllogism at least it must be understood in this sense.[61]\nNow, if we take a letter to represent this indefinite part, we need\nmake no change in our formul\u00e6 to express the syllogisms Darii and\nFerio. Consider the example--\n\n  [61] *Elementary Lessons in Logic*, pp.\u00a067, 79.\n\n  Some metals are of less density than water,      (1)\n\n  All bodies of less density than water will float\n  upon the surface of water; hence                 (2)\n\n  Some metals will float upon the surface of\n  water.                                           (3)\n\nLet\n\n  A = some metals,\n  B = body of less density than water,\n  C = floating on the surface of water\n\nthen the propositions are evidently as before,\n\n           A = AB,        (1)\n           B = BC;        (2)\n  hence    A = ABC,       (3)\n\nThus the syllogism Darii does not really differ from Barbara. If the\nreader prefer it, we can readily employ a distinct symbol for the\nindefinite sign of quantity.\n\n  Let    P = some,\n         Q = metal,\n\nB and C having the same meanings as before. Then the premises become\n\n  PQ = PQB,       (1)\n   B = BC;        (2)\n\nhence, by substitution, as before,\n\n  PQ = PQBC.      (3)\n\nExcept that the formul\u00e6 look a little more complicated there is no\ndifference whatever.\n\nThe mood Ferio is of exactly the same character as Darii or Barbara,\nexcept that it involves the use of a negative term. Take the example,\n\n  Bodies which are equally elastic in all directions do not doubly\n  refract light;\n\n  Some crystals are bodies equally elastic in all directions;\n  therefore, some crystals do not doubly refract light.\n\nAssigning the letters as follows:--\n\n   A  = some crystals,\n   B  = bodies equally elastic in all directions,\n   C  = doubly refracting light,\n  *c* = not doubly refracting light.\n\nOur argument is of the same form as before, and may be concisely stated\nin one line,\n\n  A = AB = AB*c*.\n\nIf it is preferred to put PQ for the indefinite *some crystals*, we have\n\n  PQ = PQB = PQB*c*.\n\nThe only difference is that the negative term c takes the place of C in\nthe mood Darii.\n\n\n*Ellipsis of Terms in Partial Identities.*\n\nThe reader will probably have noticed that the conclusion which we\nobtain from premises is often more full than that drawn by the old\nAristotelian processes. Thus from \u201cSodium is a metal,\u201d and \u201cMetals\nconduct electricity,\u201d we inferred (p.\u00a055) that \u201cSodium = sodium, metal,\nconducting electricity,\u201d whereas the old logic simply concludes that\n\u201cSodium conducts electricity.\u201d Symbolically, from A = AB, and B = BC,\nwe get A = ABC, whereas the old logic gets at the most A = AC. It is\ntherefore well to show that without employing any other principles of\ninference than those already described, we may infer A = AC from A =\nABC, though we cannot infer the latter more full and accurate result\nfrom the former. We may show this most simply as follows:--\n\nBy the first Law of Thought it is evident that\n\n  AA = AA;\n\nand if we have given the proposition A = ABC, we may substitute for\nboth the A\u2019s in the second side of the above, obtaining\n\n  AA = ABC . ABC.\n\nBut from the property of logical symbols expressed in the Law of\nSimplicity (p.\u00a033) some of the repeated letters may be made to\ncoalesce, and we have\n\n  A = ABC . C.\n\nSubstituting again for ABC its equivalent A, we obtain\n\n  A = AC,\n\nthe desired result.\n\nBy a similar process of reasoning it may be shown that we can always\ndrop out any term appearing in one member of a proposition, provided\nthat we substitute for it the whole of the other member. This\nprocess was described in my first logical Essay,[62] as *Intrinsic\nElimination*, but it might perhaps be better entitled the *Ellipsis\nof Terms*. It enables us to get rid of needless terms by strict\nsubstitutive reasoning.\n\n  [62] *Pure Logic*, p.\u00a019.\n\n\n*Inference of a Simple from Two Partial Identities.*\n\nTwo terms may be connected together by two partial identities in yet\nanother manner, and a case of inference then arises which is of the\nhighest importance. In the two premises\n\n  A = AB      (1)\n  B = AB      (2)\n\nthe second member of each is the same; so that we can by obvious\nsubstitution obtain\n\n  A = B.\n\nThus, in plain geometry we readily prove that \u201cEvery equilateral\ntriangle is also an equiangular triangle,\u201d and we can with equal ease\nprove that \u201cEvery equiangular triangle is an equilateral triangle.\u201d\nThence by substitution, as explained above, we pass to the simple\nidentity,\n\n  Equilateral triangle = equiangular triangle.\n\nWe thus prove that one class of triangles is entirely identical with\nanother class; that is to say, they differ only in our way of naming\nand regarding them.\n\nThe great importance of this process of inference arises from the\nfact that the conclusion is more simple and general than either of\nthe premises, and contains as much information as both of them put\ntogether. It is on this account constantly employed in inductive\ninvestigation, as will afterwards be more fully explained, and it is\nthe natural mode by which we arrive at a conviction of the truth of\nsimple identities as existing between classes of numerous objects.\n\n\n*Inference of a Limited from Two Partial Identities.*\n\nWe have considered some arguments which are of the type treated by\nAristotle in the first figure of the syllogism. But there exist two\nother types of argument which employ a pair of partial identities. If\nour premises are as shown in these symbols,\n\n  B = AB       (1)\n  B = CB,      (2)\n\nwe may substitute for B either by (1) in (2) or by (2) in (1), and by\nboth modes we obtain the conclusion\n\n  AB = CB,      (3)\n\na proposition of the kind which we have called a limited identity\n(p.\u00a042). Thus, for example,\n\n  Potassium = potassium metal                               (1)\n  Potassium = potassium capable of floating on water;       (2)\n\nhence\n\n  Potassium metal = potassium capable of floating on water. (3)\n\nThis is really a syllogism of the mood Darapti in the third figure,\nexcept that we obtain a conclusion of a more exact character than\nthe old syllogism gives. From the premises \u201cPotassium is a metal\u201d\nand \u201cPotassium floats on water,\u201d Aristotle would have inferred that\n\u201cSome metals float on water.\u201d But if inquiry were made what the \u201csome\nmetals\u201d are, the answer would certainly be \u201cMetal which is potassium.\u201d\nHence Aristotle\u2019s conclusion simply leaves out some of the information\nafforded in the premises. It even leaves us open to interpret the\n*some metals* in a wider sense than we are warranted in doing. From\nthese distinct defects of the old syllogism the process of substitution\nis free, and the new process only incurs the possible objection of\nbeing tediously minute and accurate.\n\n\n*Miscellaneous Forms of Deductive Inference.*\n\nThe more common forms of deductive reasoning having been exhibited\nand demonstrated on the principle of substitution, there still remain\nmany, in fact an indefinite number, which may be explained with nearly\nequal ease. Such as involve the use of disjunctive propositions will\nbe described in a later chapter, and several of the syllogistic moods\nwhich include negative terms will be more conveniently treated after\nwe have introduced the symbolic use of the second and third laws of\nthought.\n\nWe sometimes meet with a chain of propositions which allow of repeated\nsubstitution, and form an argument called in the old logic a Sorites.\nTake, for instance, the premises\n\n  Iron is a metal,                                  (1)\n  Metals are good conductors of electricity,        (2)\n  Good conductors of electricity are useful for\n    telegraphic purposes.                           (3)\n\nIt obviously follows that\n\n  Iron is useful for telegraphic purposes.          (4)\n\nNow if we take our letters thus,\n\n  A = Iron, B = metal, C = good conductor of electricity, D = useful\n  for telegraphic purposes,\n\nthe premises will assume the forms\n\n  A = AB,             (1)\n  B = BC,             (2)\n  C = CD.             (3)\n\nFor B in (1) we can substitute its equivalent in (2) obtaining, as\nbefore,\n\n  A = ABC.\n\nSubstituting for C in this intermediate result its equivalent as given\nin (3), we obtain the complete conclusion\n\n  A = ABCD.           (4)\n\nThe full interpretation is that *Iron is iron, metal, good conductor of\nelectricity, useful for telegraphic purposes*, which is abridged in\ncommon language by the ellipsis of the circumstances which are not of\nimmediate importance.\n\nInstead of all the propositions being exactly of the same kind as\nin the last example, we may have a series of premises of various\ncharacter; for instance,\n\nCommon salt is sodium chloride,                               (1)\n\nSodium chloride crystallizes in a cubical form,               (2)\n\nWhat crystallizes in a cubical form does not possess the power of\ndouble refraction;                                            (3)\n\nit will follow that\n\nCommon salt does not possess the power of double refraction.  (4)\n\nTaking our letter-terms thus,\n\n  A = Common salt,\n  B = Sodium chloride,\n  C = Crystallizing in a cubical form,\n  D = Possessing the power of double refraction,\n\nwe may state the premises in the forms\n\n  A = B,              (1)\n  B = BC,             (2)\n  C = C*d*.           (3)\n\nSubstituting by (3) in (2) and then by (2) as thus altered in (1) we\nobtain\n\n  A = BC*d*,          (4)\n\nwhich is a more precise version of the common conclusion.\n\nWe often meet with a series of propositions describing the qualities or\ncircumstances of the one same thing, and we may combine them all into\none proposition by the process of substitution. This case is, in fact,\nthat which Dr. Thomson has called \u201cImmediate Inference by the sum of\nseveral predicates,\u201d and his example will serve my purpose well.[63]\nHe describes copper as \u201cA metal--of a red colour--and disagreeable\nsmell--and taste--all the preparations of which are poisonous--which is\nhighly malleable--ductile--and tenacious--with a specific gravity of\nabout 8.83.\u201d If we assign the letter A to copper, and the succeeding\nletters of the alphabet in succession to the series of predicates, we\nhave nine distinct statements, of the form A = AB (1) A = AC (2) A = AD\n(3) ... A = AK (9). We can readily combine these propositions into one\nby substituting for A in the second side of (1) its expression in (2).\nWe thus get\n\n  [63] *An Outline of the Necessary Laws of Thought*, Fifth Ed. p. 161.\n\n  A = ABC,\n\nand by repeating the process over and over again we obviously get the\nsingle proposition\n\n  A = ABCD ... JK.\n\nBut Dr. Thomson is mistaken in supposing that we can obtain in\nthis manner a *definition* of copper. Strictly speaking, the above\nproposition is only a *description* of copper, and all the ordinary descriptions\nof substances in scientific works may be summed up in this form. Thus\nwe may assert of the organic substances called Paraffins that they are\nall saturated hydrocarbons, incapable of uniting with other substances,\nproduced by heating the alcoholic iodides with zinc, and so on. It may\nbe shown that no amount of ordinary description can be equivalent to a\ndefinition of any substance.\n\n\n*Fallacies.*\n\nI have hitherto been engaged in showing that all the forms of\nreasoning of the old syllogistic logic, and an indefinite number of\nother forms in addition, may be readily and clearly explained on the\nsingle principle of substitution. It is now desirable to show that the\nsame principle will prevent us falling into fallacies. So long as we\nexactly observe the one rule of substitution of equivalents it will\nbe impossible to commit a *paralogism*, that is to break any one of\nthe elaborate rules of the ancient system. The one new rule is thus\nproved to be as powerful as the six, eight, or more rules by which the\ncorrectness of syllogistic reasoning was guarded.\n\nIt was a fundamental rule, for instance, that two negative premises\ncould give no conclusion. If we take the propositions\n\n  Granite is not a sedimentary rock,    (1)\n  Basalt is not a sedimentary rock,     (2)\n\nwe ought not to be able to draw any inference concerning the relation\nbetween granite and basalt. Taking our letter-terms thus:\n\n  A = granite,  B = sedimentary rock,  C = basalt,\n\nthe premises may be expressed in the forms\n\n  A ~ B,          (1)\n  C ~ B.          (2)\n\nWe have in this form two statements of difference; but the principle\nof inference can only work with a statement of agreement or identity\n(p.\u00a063). Thus our rule gives us no power whatever of drawing any\ninference; this is exactly in accordance with the fifth rule of the\nsyllogism.\n\nIt is to be remembered, indeed, that we claim the power of always\nturning a negative proposition into an affirmative one (p.\u00a045); and it\nmight seem that the old rule against negative premises would thus be\ncircumvented. Let us try. The premises (1) and (2) when affirmatively\nstated take the forms\n\n  A = A*b*        (1)\n  C = C*b*.       (2)\n\nThe reader will find it impossible by the rule of substitution to\ndiscover a relation between A and C. Three terms occur in the above\npremises, namely A, *b*, and C; but they are so combined that no term\noccurring in one has its exact equivalent stated in the other. No\nsubstitution can therefore be made, and the principle of the fifth rule\nof the syllogism holds true. Fallacy is impossible.\n\nIt would be a mistake, however, to suppose that the mere occurrence of\nnegative terms in both premises of a syllogism renders them incapable\nof yielding a conclusion. The old rule informed us that from two\nnegative premises no conclusion could be drawn, but it is a fact that\nthe rule in this bare form does not hold universally true; and I am not\naware that any precise explanation has been given of the conditions\nunder which it is or is not imperative. Consider the following example:\n\n  Whatever is not metallic is not capable of powerful\n    magnetic influence,                                  (1)\n  Carbon is not metallic,                                (2)\n  Therefore, carbon is not capable of powerful magnetic\n    influence.                                           (3)\n\nHere we have two distinctly negative premises (1) and (2), and yet they\nyield a perfectly valid negative conclusion (3). The syllogistic rule\nis actually falsified in its bare and general statement. In this and\nmany other cases we can convert the propositions into affirmative ones\nwhich will yield a conclusion by substitution without any difficulty.\nTo show this let\n\n  A = carbon,                B = metallic,\n  C = capable of powerful magnetic influence.\n\nThe premises readily take the forms\n\n  *b* = *bc*,      (1)\n   A  =  A*b*,     (2)\n\nand substitution for *b* in (2) by means of (1) gives the conclusion\n\n  A = A*bc*.       (3)\n\nOur principle of inference then includes the rule of negative premises\nwhenever it is true, and discriminates correctly between the cases\nwhere it does and does not hold true.\n\nThe paralogism, anciently called *the Fallacy of Undistributed Middle*,\nis also easily exhibited and infallibly avoided by our system. Let the\npremises be\n\n  Hydrogen is an element,       (1)\n  All metals are elements.      (2)\n\nAccording to the syllogistic rules the middle term \u201celement\u201d is here\nundistributed, and no conclusion can be obtained; we cannot tell then\nwhether hydrogen is or is not a metal. Represent the terms as follows\n\n  A = hydrogen,\n  B = element,\n  C = metal.\n\nThe premises then become\n\n  A = AB,          (1)\n  C = CB.          (2)\n\nThe reader will here, as in a former page (p.\u00a062), find it impossible\nto make any substitution. The only term which occurs in both premises\nis B, but it is differently combined in the two premises. For B we\nmust not substitute A, which is equivalent to AB, not to B. Nor must\nwe confuse together CB and AB, which, though they contain one common\nletter, are different aggregate terms. The rule of substitution gives\nus no right to decompose combinations; and if we adhere rigidly to the\nrule, that if two terms are stated to be equivalent we may substitute\none for the other, we cannot commit the fallacy. It is apparent that\nthe form of premises stated above is the same as that which we obtained\nby translating two negative premises into the affirmative form.\n\nThe old fallacy, technically called the *Illicit Process of the Major\nTerm*, is more easy to commit and more difficult to detect than any\nother breach of the syllogistic rules. In our system it could hardly\noccur. From the premises\n\n  All planets are subject to gravity,    (1)\n  Fixed stars are not planets,           (2)\n\nwe might inadvertently but fallaciously infer that, \u201cFixed stars are\nnot subject to gravity.\u201d To reduce the premises to symbolic form, let\n\n  A = planet\n  B = fixed star\n  C = subject to gravity;\n\nthen we have the propositions\n\n  A = AC           (1)\n  B = B*a*.        (2)\n\nThe reader will try in vain to produce from these premises by\nlegitimate substitution any relation between B and C; he could not then\ncommit the fallacy of asserting that B is not C.\n\nThere remain two other kinds of paralogism, commonly known as the\nfallacy of Four Terms and the Illicit Process of the Minor Term. They\nare so evidently impossible while we obey the rule of the substitution\nof equivalents, that it is not necessary to give any illustrations.\nWhen there are four distinct terms in two propositions as in A = B\nand C = D, there could evidently be no opening for substitution. As\nto the Illicit Process of the Minor Term it consists in a flagrant\nsubstitution for a term of another wider term which is not known to be\nequivalent to it, and which is therefore not allowed by our rule to be\nsubstituted for it.\n\n\n\n\nCHAPTER V.\n\nDISJUNCTIVE PROPOSITIONS.\n\n\nIn the previous chapter I have exhibited various cases of deductive\nreasoning by the process of substitution, avoiding the introduction of\ndisjunctive propositions; but we cannot long defer the consideration of\nthis more complex class of identities. General terms arise, as we have\nseen (p.\u00a024), from classifying or mentally uniting together all objects\nwhich agree in certain qualities, the value of this union consisting in\nthe fact that the power of knowledge is multiplied thereby. In forming\nsuch classes or general notions, we overlook or abstract the points of\ndifference which exist between the objects joined together, and fix our\nattention only on the points of agreement. But every process of thought\nmay be said to have its inverse process, which consists in undoing the\neffects of the direct process. Just as division undoes multiplication,\nand evolution undoes involution, so we must have a process which undoes\ngeneralization, or the operation of forming general notions. This\ninverse process will consist in distinguishing the separate objects or\nminor classes which are the constituent parts of any wider class. If\nwe mentally unite together certain objects visible in the sky and call\nthem planets, we shall afterwards need to distinguish the contents of\nthis general notion, which we do in the disjunctive proposition--\n\n  A planet is either Mercury or Venus or the Earth or ... or Neptune.\n\nHaving formed the very wide class \u201cvertebrate animal,\u201d we may specify\nits subordinate classes thus:--\u201cA vertebrate animal is either a\nmammal, bird, reptile, or fish.\u201d Nor is there any limit to the\nnumber of possible alternatives. \u201cAn exogenous plant is either a\nranunculus, a poppy, a crucifer, a rose, or it belongs to some one\nof the other seventy natural orders of exogens at present recognized\nby botanists.\u201d A cathedral church in England must be either that of\nLondon, Canterbury, Winchester, Salisbury, Manchester, or of one of\nabout twenty-four cities possessing such churches. And if we were to\nattempt to specify the meaning of the term \u201cstar,\u201d we should require\nto enumerate as alternatives, not only the many thousands of stars\nrecorded in catalogues, but the many millions unnamed.\n\nWhenever we thus distinguish the parts of a general notion we employ\na disjunctive proposition, in at least one side of which are several\nalternatives joined by the so-called disjunctive conjunction or,\na contracted form of *other*. There must be some relation between\nthe parts thus connected in one proposition; we may call it the\n*disjunctive* or *alternative* relation, and we must carefully inquire\ninto its nature. This relation is that of ignorance and doubt, giving\nrise to choice. Whenever we classify and abstract we must open the way\nto such uncertainty. By fixing our attention on certain attributes to\nthe exclusion of others, we necessarily leave it doubtful what those\nother attributes are. The term \u201cmolar tooth\u201d bears upon the face of\nit that it is a part of the wider term \u201ctooth.\u201d But if we meet with\nthe simple term \u201ctooth\u201d there is nothing to indicate whether it is\nan incisor, a canine, or a molar tooth. This doubt, however, may be\nresolved by further information, and we have to consider what are the\nappropriate logical processes for treating disjunctive propositions in\nconnection with other propositions disjunctive or otherwise.\n\n\n*Expression of the Alternative Relation.*\n\nIn order to represent disjunctive propositions with convenience we\nrequire a sign of the alternative relation, equivalent to one meaning\nat least of the little conjunction *or* so frequently used in common\nlanguage. I propose to use for this purpose the symbol \ua58c. In my first\nlogical essay I followed the practice of Boole and adopted the sign\n+; but this sign should not be employed unless there exists exact\nanalogy between mathematical addition and logical alternation. We shall\nfind that the analogy is imperfect, and that there is such profound\ndifference between logical and mathematical terms as should prevent\nour uniting them by the same symbol. Accordingly I have chosen a sign\n\ua58c, which seems aptly to suggest whatever degree of analogy may exist\nwithout implying more. The exact meaning of the symbol we will now\nproceed to investigate.\n\n\n*Nature of the Alternative Relation.*\n\nBefore treating disjunctive propositions it is indispensable to decide\nwhether the alternatives must be considered exclusive or unexclusive.\nBy *exclusive alternatives* we mean those which cannot contain the same\nthings. If we say \u201cArches are circular or pointed,\u201d it is certainly to\nbe understood that the same arch cannot be described as both circular\nand pointed. Many examples, on the other hand, can readily be suggested\nin which two or more alternatives may hold true of the same object. Thus\n\n  Luminous bodies are self-luminous or luminous by reflection.\n\nIt is undoubtedly possible, by the laws of optics, that the same\nsurface may at one and the same moment give off light of its own and\nreflect light from other bodies. We speak familiarly of *deaf or dumb*\npersons, knowing that the majority of those who are deaf from birth are\nalso dumb.\n\nThere can be no doubt that in a great many cases, perhaps the greater\nnumber of cases, alternatives are exclusive as a matter of fact. Any\none number is incompatible with any other; one point of time or place\nis exclusive of all others. Roger Bacon died either in 1284 or 1292; it\nis certain that he could not die in both years. Henry Fielding was born\neither in Dublin or Somersetshire; he could not be born in both places.\nThere is so much more precision and clearness in the use of exclusive\nalternatives that we ought doubtless to select them when possible. Old\nworks on logic accordingly contained a rule directing that the *Membra\ndividentia*, the parts of a division or the constituent species of a\ngenus, should be exclusive of each other.\n\nIt is no doubt owing to the great prevalence and convenience of\nexclusive divisions that the majority of logicians have held it\nnecessary to make every alternative in a disjunctive proposition\nexclusive of every other one. Aquinas considered that when this was not\nthe case the proposition was actually *false*, and Kant adopted the\nsame opinion.[64] A multitude of statements to the same effect might\nreadily be quoted, and if the question were to be determined by the\nweight of historical evidence, it would certainly go against my view.\nAmong recent logicians Hamilton, as well as Boole, took the exclusive\nside. But there are authorities to the opposite effect. Whately,\nMansel, and J.\u00a0S. Mill have all pointed out that we may often treat\nalternatives as *Compossible*, or true at the same time. Whately gives\nus an example,[65] \u201cVirtue tends to procure us either the esteem of\nmankind, or the favour of God,\u201d and he adds--\u201cHere both members are\ntrue, and consequently from one being affirmed we are not authorized to\ndeny the other. Of course we are left to conjecture in each case, from\nthe context, whether it is meant to be implied that the members are\nor are not exclusive.\u201d Mansel says,[66] \u201c*We may happen to know* that\ntwo alternatives cannot be true together, so that the affirmation of\nthe second necessitates the denial of the first; but this, as Boethius\nobserves, is a *material*, not a *formal* consequence.\u201d Mill has also\npointed out the absurdities which would arise from always interpreting\nalternatives as exclusive. \u201cIf we assert,\u201d he says,[67] \u201cthat a man\nwho has acted in some particular way must be either a knave or a fool,\nwe by no means assert, or intend to assert, that he cannot be both.\u201d\nAgain, \u201cto make an entirely unselfish use of despotic power, a man must\nbe either a saint or a philosopher.... Does the disjunctive premise\nnecessarily imply, or must it be construed as supposing, that the same\nperson cannot be both a saint and a philosopher? Such a construction\nwould be ridiculous.\u201d\n\n  [64] Mansel\u2019s *Aldrich*, p.\u00a0103, and *Prolegomena Logica*, p.\u00a0221.\n\n  [65] *Elements of Logic*, Book II. chap. iv. sect. 4.\n\n  [66] Aldrich, *Artis Logic\u00e6 Rudimenta*, p.\u00a0104.\n\n  [67] *Examination of Sir W. Hamilton\u2019s Philosophy*, pp.\u00a0452\u2013454.\n\nI discuss this subject fully because it is really the point which\nseparates my logical system from that of Boole. In his *Laws of\nThought* (p.\u00a032) he expressly says, \u201cIn strictness, the words \u2018and,\u2019\n\u2018or,\u2019 interposed between the terms descriptive of two or more classes\nof objects, imply that those classes are quite distinct, so that no\nmember of one is found in another.\u201d This I altogether dispute. In the\nordinary use of these conjunctions we do not join distinct terms only;\nand when terms so joined do prove to be logically distinct, it is by\nvirtue of a *tacit premise*, something in the meaning of the names and\nour knowledge of them, which teaches us that they are distinct. If our\nknowledge of the meanings of the words joined is defective it will\noften be impossible to decide whether terms joined by conjunctions are\nexclusive or not.\n\nIn the sentence \u201cRepentance is not a single act, but a habit or\nvirtue,\u201d it cannot be implied that a virtue is not a habit; by\nAristotle\u2019s definition it is. Milton has the expression in one of\nhis sonnets, \u201cUnstain\u2019d by gold or fee,\u201d where it is obvious that if\nthe fee is not always gold, the gold is meant to be a fee or bribe.\nTennyson has the expression \u201cwreath or anadem.\u201d Most readers would\nbe quite uncertain whether a wreath may be an anadem, or an anadem a\nwreath, or whether they are quite distinct or quite the same. From\nDarwin\u2019s *Origin of Species*, I take the expression, \u201cWhen we see any\n*part or organ* developed in a remarkable *degree or manner*.\u201d In this,\n*or* is used twice, and neither time exclusively. For if *part* and\n*organ* are not synonymous, at any rate an organ is a part. And it\nis obvious that a part may be developed at the same time both in an\nextraordinary degree and an extraordinary manner, although such cases\nmay be comparatively rare.\n\nFrom a careful examination of ordinary writings, it will thus be found\nthat the meanings of terms joined by \u201cand,\u201d \u201cor\u201d vary from absolute\nidentity up to absolute contrariety. There is no logical condition of\ndistinctness at all, and when we do choose exclusive alternatives, it\nis because our subject demands it. The matter, not the form of an\nexpression, points out whether terms are exclusive or not.[68] In\nbills, policies, and other kinds of legal documents, it is sometimes\nnecessary to express very distinctly that alternatives are not\nexclusive. The form and/or is then used, and, as Mr. J.\u00a0J. Murphy has\nremarked, this form coincides exactly in meaning with the symbol \ua58c.\n\n  [68] *Pure Logic*, pp 76, 77.\n\nIn the first edition of this work (vol.\u00a0i., p.\u00a081), I took the\ndisjunctive proposition \u201cMatter is solid, or liquid, or gaseous,\u201d and\ntreated it as an instance of exclusive alternatives, remarking that the\nsame portion of matter cannot be at once solid and liquid, properly\nspeaking, and that still less can we suppose it to be solid and\ngaseous, or solid, liquid, and gaseous all at the same time. But the\nexperiments of Professor Andrews show that, under certain conditions\nof temperature and pressure, there is no abrupt change from the liquid\nto the gaseous state. The same substance may be in such a state as to\nbe indifferently described as liquid and gaseous. In many cases, too,\nthe transition from solid to liquid is gradual, so that the properties\nof solidity are at least partially joined with those of liquidity.\nThe proposition then, instead of being an instance of exclusive\nalternatives, seems to afford an excellent instance to the opposite\neffect. When such doubts can arise, it is evidently impossible to treat\nalternatives as absolutely exclusive by the logical nature of the\nrelation. It becomes purely a question of the matter of the proposition.\n\nThe question, as we shall afterwards see more fully, is one of\nthe greatest theoretical importance, because it concerns the true\ndistinction between the sciences of Logic and Mathematics. It is the\nfoundation of number that every unit shall be distinct from every other\nunit; but Boole imported the conditions of number into the science of\nLogic, and produced a system which, though wonderful in its results,\nwas not a system of logic at all.\n\n\n*Laws of the Disjunctive Relation.*\n\nIn considering the combination or synthesis of terms (p.\u00a030), we found\nthat certain laws, those of Simplicity and Commutativeness, must be\nobserved. In uniting terms by the disjunctive symbol we shall find that\nthe same or closely similar laws hold true. The alternatives of either\nmember of a disjunctive proposition are certainly commutative. Just as\nwe cannot properly distinguish between *rich and rare gems* and *rare\nand rich gems*, so we must consider as identical the expression *rich\nor rare gems*, and *rare or rich gems*. In our symbolic language we may\nsay\n\n  A \ua58c B = B \ua58c A.\n\nThe order of statement, in short, has no effect upon the meaning of an\naggregate of alternatives, so that the Law of Commutativeness holds\ntrue of the disjunctive symbol.\n\nAs we have admitted the possibility of joining as alternatives terms\nwhich are not really different, the question arises, How shall we treat\ntwo or more alternatives when they are clearly shown to be the same?\nIf we have it asserted that P is Q or R, and it is afterwards proved\nthat Q is but another name for R, the result is that P is either R or\nR. How shall we interpret such a statement? What would be the meaning,\nfor instance, of \u201cwreath or anadem\u201d if, on referring to a dictionary,\nwe found *anadem* described as a wreath? I take it to be self-evident\nthat the meaning would then become simply \u201cwreath.\u201d Accordingly we may\naffirm the general law\n\n  A \ua58c A = A.\n\nAny number of identical alternatives may always be reduced to, and are\nlogically equivalent to, any one of those alternatives. This is a law\nwhich distinguishes mathematical terms from logical terms, because\nit obviously does not apply to the former. I propose to call it the\n*Law of Unity*, because it must really be involved in any definition\nof a mathematical unit. This law is closely analogous to the Law of\nSimplicity, AA = A; and the nature of the connection is worthy of\nattention.\n\nFew or no logicians except De Morgan have adequately noticed the close\nrelation between combined and disjunctive terms, namely, that every\ndisjunctive term is the negative of a corresponding combined term, and\n*vice vers\u00e2*. Consider the term\n\n  Malleable dense metal.\n\nHow shall we describe the class of things which are not\nmalleable-dense-metals? Whatever is included under that term must have\nall the qualities of malleability, denseness, and metallicity. Wherever\nany one or more of the qualities is wanting, the combined term will not\napply. Hence the negative of the whole term is\n\n  Not-malleable or not-dense or not-metallic.\n\nIn the above the conjunction *or* must clearly be interpreted\nas unexclusive; for there may readily be objects which are both\nnot-malleable, and not-dense, and perhaps not-metallic at the same\ntime. If in fact we were required to use *or* in a strictly exclusive\nmanner, it would be requisite to specify seven distinct alternatives\nin order to describe the negative of a combination of three terms. The\nnegatives of four or five terms would consist of fifteen or thirty-one\nalternatives. This consideration alone is sufficient to prove that the\nmeaning of *or* cannot be always exclusive in common language.\n\nExpressed symbolically, we may say that the negative of\n\n                         ABC\n  is           not-A or not-B or not-C;\n  that is,     *a* \ua58c *b* \ua58c *c*.\n\nReciprocally the negative of\n\n                    P \ua58c Q \ua58c R\n  is                  *pqr*.\n\nEvery disjunctive term, then, is the negative of a combined term, and\n*vice vers\u00e2*.\n\nApply this result to the combined term AAA, and its negative is\n\n  *a* \ua58c *a* \ua58c *a*.\n\nSince AAA is by the Law of Simplicity equivalent to A, so *a* \ua58c *a* \ua58c\n*a* must be equivalent to *a*, and the Law of Unity holds true. Each\nlaw thus necessarily presupposes the other.\n\n\n*Symbolic expression of the Law of Duality.*\n\nWe may now employ our symbol of alternation to express in a clear and\nformal manner the third Fundamental Law of Thought, which I have called\nthe Law of Duality (p.\u00a06). Taking A to represent any class or object\nor quality, and B any other class, object or quality, we may always\nassert that A either agrees with B, or does not agree. Thus we may say\n\n  A = AB \ua58c A*b*.\n\nThis is a formula which will henceforth be constantly employed, and it\nlies at the basis of reasoning.\n\nThe reader may perhaps wish to know why A is inserted in both\nalternatives of the second member of the identity, and why the law is\nnot stated in the form\n\n  A = B \ua58c *b*.\n\nBut if he will consider the contents of the last section (p.\u00a073), he\nwill see that the latter expression cannot be correct, otherwise no\nterm could have a corresponding negative term. For the negative of B\n\ua58c *b* is *b*B, or a self-contradictory term; thus if A were identical\nwith B \ua58c *b*, its negative *a* would be non-existent. To say the least,\nthis result would in most cases be an absurd one, and I see much reason\nto think that in a strictly logical point of view it would always be\nabsurd. In all probability we ought to assume as a fundamental logical\naxiom that *every term has its negative in thought*. We cannot think at\nall without separating what we think about from other things, and these\nthings necessarily form the negative notion.[69] It follows that any\nproposition of the form A = B \ua58c *b* is just as self-contradictory as\none of the form A = B*b*.\n\n  [69] *Pure Logic*, p.\u00a065. See also the criticism of this point by De\n  Morgan in the *Athen\u00e6um*, No. 1892, 30th January, 1864; p.\u00a0155.\n\nIt is convenient to recapitulate in this place the three Laws of\nThought in their symbolic form, thus\n\n  Law of Identity            A = A.\n  Law of Contradiction    A*a* = 0.\n  Law of Duality             A = AB \ua58c A*b*.\n\n\n*Various Forms of the Disjunctive Proposition.*\n\nDisjunctive propositions may occur in a great variety of forms, of\nwhich the old logicians took insufficient notice. There may be any\nnumber of alternatives, each of which may be a combination of any\nnumber of simple terms. A proposition, again, may be disjunctive in one\nor both members. The proposition\n\n  Solids or liquids or gases are electrics or conductors of electricity\n\nis an example of the doubly disjunctive form. The meaning of such a\nproposition is that whatever falls under any one or more alternatives\non one side must fall under one or more alternatives on the other side.\nFrom what has been said before, it is apparent that the proposition\n\n  A \ua58c B = C \ua58c D\n\nwill correspond to\n\n  *ab* = *cd*,\n\neach member of the latter being the negative of a member of the former\nproposition.\n\nAs an instance of a complex disjunctive proposition I may give Senior\u2019s\ndefinition of wealth, which, briefly stated, amounts to the proposition\n\u201cWealth is what is transferable, limited in supply, and either\nproductive of pleasure or preventive of pain.\u201d[70]\n\n  [70] Boole\u2019s *Laws of Thought*, p.\u00a0106. Jevons\u2019 *Pure Logic*, p.\u00a069.\n\n  Let A = wealth\n      B = transferable\n      C = limited in supply\n      D = productive of pleasure\n      E = preventive of pain.\n\nThe definition takes the form\n\n  A = BC(D \ua58c E);\n\nbut if we develop the alternatives by a method to be afterwards more\nfully considered, it becomes\n\n  A = BCDE \ua58c BCD*e* \ua58c BC*d*E.\n\nAn example of a still more complex proposition is found in De Morgan\u2019s\nwritings,[71] as follows:--\u201cHe must have been rich, and if not\nabsolutely mad was weakness itself, subjected either to bad advice or\nto most unfavourable circumstances.\u201d\n\n  [71] *On the Syllogism*, No. iii. p.\u00a012. Camb. Phil. Trans. vol.\u00a0x,\n  part i.\n\nIf we assign the letters of the alphabet in succession, thus,\n\n  A = he\n  B = rich\n  C = absolutely mad\n  D = weakness itself\n  E = subjected to bad advice\n  F = subjected to most unfavourable circumstances,\n        the proposition will take the form\n\n  A = AB{C \ua58c D (E \ua58c F)},\n\nand if we develop the alternatives, expressing some of the different\ncases which may happen, we obtain\n\n  A = ABC \ua58c AB*c*DEF \ua58c AB*c*DE*f* \ua58c AB*c*D*e*F.\n\nThe above gives the strict logical interpretation of the sentence, and\nthe first alternative ABC is capable of development into eight cases,\naccording as D, E and F are or are not present. Although from our\nknowledge of the matter, we may infer that weakness of character cannot\nbe asserted of a person absolutely mad, there is no explicit statement\nto this effect.\n\n\n*Inference by Disjunctive Propositions.*\n\nBefore we can make a free use of disjunctive propositions in the\nprocesses of inference we must consider how disjunctive terms can be\ncombined together or with simple terms. In the first place, to combine\na simple term with a disjunctive one, we must combine it with every\nalternative of the disjunctive term. A vegetable, for instance, is\neither a herb, a shrub, or a tree. Hence an exogenous vegetable is\neither an exogenous herb, or an exogenous shrub, or an exogenous tree.\nSymbolically stated, this process of combination is as follows,\n\n  A(B \ua58c C) = AB \ua58c AC.\n\nSecondly, to combine two disjunctive terms with each other, combine\neach alternative of one with each alternative of the other. Since\nflowering plants are either exogens or endogens, and are at the\nsame time either herbs, shrubs or trees, it follows that there are\naltogether six alternatives--namely, exogenous herbs, exogenous shrubs,\nexogenous trees, endogenous herbs, endogenous shrubs, endogenous trees.\nThis process of combination is shown in the general form\n\n  (A \ua58c B) (C \ua58c D \ua58c E) = AC \ua58c AD \ua58c AE \ua58c BC \ua58c BD \ua58c BE.\n\nIt is hardly necessary to point out that, however numerous the\nterms combined, or the alternatives in those terms, we may effect\nthe combination, provided each alternative is combined with each\nalternative of the other terms, as in the algebraic process of\nmultiplication.\n\nSome processes of deduction may be at once exhibited. We may\nalways, for instance, unite the same qualifying term to each side\nof an identity even though one or both members of the identity be\ndisjunctive. Thus let\n\n  A = B \ua58c C.\n\nNow it is self-evident that\n\n  AD = AD,\n\nand in one side of this identity we may for A substitute its equivalent\nB\u00a0\ua58c\u00a0C, obtaining\n\n  AD = BD \ua58c CD.\n\nSince \u201ca gaseous element is either hydrogen, or oxygen, or nitrogen,\nor chlorine, or fluorine,\u201d it follows that \u201ca free gaseous element\nis either free hydrogen, or free oxygen, or free nitrogen, or free\nchlorine, or free fluorine.\u201d\n\nThis process of combination will lead to most useful inferences when\nthe qualifying adjective combined with both sides of the proposition is\na negative of one or more alternatives. Since chlorine is a coloured\ngas, we may infer that \u201ca colourless gaseous element is either\n(colourless) hydrogen, oxygen, nitrogen, or fluorine.\u201d The alternative\nchlorine disappears because colourless chlorine does not exist. Again,\nsince \u201ca tooth is either an incisor, canine, bicuspid, or molar,\u201d\nit follows that \u201ca not-incisor tooth is either canine, bicuspid,\nor molar.\u201d The general rule is that from the denial of any of the\nalternatives the affirmation of the remainder can be inferred. Now this\nresult clearly follows from our process of substitution; for if we have\nthe proposition\n\n  A = B \ua58c C \ua58c D,\n\nand we insert this expression for A on one side of the self-evident\nidentity\n\n  A*b* = A*b*,\n\nwe obtain A*b*\u00a0=\u00a0AB*b*\u00a0\ua58c\u00a0A*b*C\u00a0\ua58c\u00a0A*b*D;\n\nand, as the first of the three alternatives is self-contradictory, we\nstrike it out according to the law of contradiction: there remains\n\n  A*b*\u00a0=\u00a0A*b*C\u00a0\ua58c\u00a0A*b*D.\n\nThus our system fully includes and explains that mood of the\nDisjunctive Syllogism technically called the *modus tollendo ponens*.\n\nBut the reader must carefully observe that the Disjunctive Syllogism of\nthe mood *ponendo tollens*, which affirms one alternative, and thence\ninfers the denial of the rest, cannot be held true in this system. If I\nsay, indeed, that\n\n  Water is either salt or fresh water,\n\nit seems evident that \u201cwater which is salt is not fresh.\u201d But this\ninference really proceeds from our knowledge that water cannot be at\nonce salt and fresh. This inconsistency of the alternatives, as I have\nfully shown, will not always hold. Thus, if I say\n\n  Gems are either rare stones or beautiful stones,       (1)\n\nit will obviously not follow that\n\n  A rare gem is not a beautiful stone,                   (2)\n\nnor that\n\n  A beautiful gem is not a rare stone.                   (3)\n\nOur symbolic method gives only true conclusions; for if we take\n\n  A = gem\n  B = rare stone\n  C = beautiful stone,\n\nthe proposition (1) is of the form\n\n             A = B \ua58c C\n  hence     AB = B \ua58c BC\n  and       AC = BC \ua58c C;\n\nbut these inferences are not equivalent to the false ones (2) and (3).\n\n\nWe can readily represent disjunctive reasoning by the *modus ponendo\ntollens*, when it is valid, by expressing the inconsistency of the\nalternatives explicitly. Thus if we resort to our instance of\n\n  Water is either salt or fresh,\n\nand take\n\n  A = Water  B = salt  C = fresh,\n\nthen the premise is apparently of the form\n\n  A = AB \ua58c AC;\n\nbut in reality there is an unexpressed condition that \u201cwhat is salt\nis not fresh,\u201d from which follows, by a process of inference to be\nafterwards described, that \u201cwhat is fresh is not salt.\u201d We have then,\nin letter-terms, the two propositions\n\n  B = B*c*\n  C = *b*C.\n\nIf we substitute these descriptions in the original proposition, we\nobtain /* A = AB*c* \ua58c A*b*C; */\n\nuniting B to each side we infer\n\n       AB = AB*c* \ua58c AB*b*C\n  or   AB = AB*c*;\n\nthat is,\n\n  Water which is salt is water salt and not fresh.\n\nI should weary the reader if I attempted to illustrate the multitude of\nforms which disjunctive reasoning may take; and as in the next chapter\nwe shall be constantly treating the subject, I must here restrict\nmyself to a single instance. A very common process of reasoning\nconsists in the determination of the name of a thing by the successive\nexclusion of alternatives, a process called by the old name *abscissio\ninfiniti*. Take the case:\n\n  Red-coloured metal is either copper or gold    (1)\n  Copper is dissolved by nitric acid             (2)\n  This specimen is red-coloured metal            (3)\n  This specimen is not dissolved by nitric acid  (4)\n  Therefore, this specimen consists of gold      (5)\n\nLet us assign the letter-symbols thus--\n\n  A = this specimen\n  B = red-coloured metal\n  C = copper\n  D = gold\n  E = dissolved by nitric acid.\n\nAssuming that the alternatives copper or gold are intended to be\nexclusive, as just explained in the case of fresh and salt water, the\npremises may be stated in the forms\n\n  B = BC*d* \ua58c B*c*D            (1)\n  C = CE                       (2)\n  A = AB                       (3)\n  A = A*e*                     (4)\n\nSubstituting for C in (1) by means of (2) we get\n\n  B = BC*d*E \ua58c B*c*D\n\nFrom (3) and (4) we may infer likewise\n\n  A = AB*e*\n\nand if in this we substitute for B its equivalent just stated, it\nfollows that\n\n  A = ABC*d*E*e* \ua58c AB*c*D*e*\n\nThe first of the alternatives being contradictory the result is\n\n  A = AB*c*D*e*\n\nwhich contains a full description of \u201cthis specimen,\u201d as furnished\nin the premises, but by ellipsis asserts that it is gold. It will be\nobserved that in the symbolic expression (1) I have explicitly stated\nwhat is certainly implied, that copper is not gold, and gold not\ncopper, without which condition the inference would not hold good.\n\n\n\n\nCHAPTER VI.\n\nTHE INDIRECT METHOD OF INFERENCE.\n\n\nThe forms of deductive reasoning as yet considered, are mostly cases\nof Direct Deduction as distinguished from those which we are now about\nto treat. The method of Indirect Deduction may be described as that\nwhich points out what a thing is, by showing that it cannot be anything\nelse. We can define a certain space upon a map, either by colouring\nthat space, or by colouring all except the space; the first mode is\npositive, the second negative. The difference, it will be readily seen,\nis exactly analogous to that between the direct and indirect modes\nof proof in geometry. Euclid often shows that two lines are equal,\nby showing that they cannot be unequal, and the proof rests upon the\nknown number of alternatives, greater, equal or less, which are alone\nconceivable. In other cases, as for instance in the seventh proposition\nof the first book, he shows that two lines must meet in a particular\npoint, by showing that they cannot meet elsewhere.\n\nIn logic we can always define with certainty the utmost number of\nalternatives which are conceivable. The Law of Duality (pp.\u00a06, 74)\nenables us always to assert that any quality or circumstance whatsoever\nis either present or absent. Whatever may be the meaning of the terms A\nand B it is certainly true that\n\n  A = AB \ua58c A*b*\n  B = AB \ua58c *a*B.\n\nThese are universal tacit premises which may be employed in the\nsolution of every problem, and which are such invariable and necessary\nconditions of all thought, that they need not be specially laid down.\nThe Law of Contradiction is a further condition of all thought and of\nall logical symbols; it enables, and in fact obliges, us to reject from\nfurther consideration all terms which imply the presence and absence\nof the same quality. Now, whenever we bring both these Laws of Thought\ninto explicit action by the method of substitution, we employ the\nIndirect Method of Inference. It will be found that we can treat not\nonly those arguments already exhibited according to the direct method,\nbut we can include an infinite multitude of other arguments which are\nincapable of solution by any other means.\n\nSome philosophers, especially those of France, have held that the\nIndirect Method of Proof has a certain inferiority to the direct\nmethod, which should prevent our using it except when obliged. But\nthere are many truths which we can prove only indirectly. We can\nprove that a number is a prime only by the purely indirect method of\nshowing that it is not any of the numbers which have divisors, and the\nremarkable process known as Eratosthenes\u2019 Sieve is the only mode by\nwhich we can select the prime numbers.[72] It bears a strong analogy to\nthe indirect method here to be described. We can prove that the side\nand diameter of a square are incommensurable, but only in the negative\nor indirect manner, by showing that the contrary supposition inevitably\nleads to contradiction.[73] Many other demonstrations in various\nbranches of the mathematical sciences proceed upon a like method.\nNow, if there is only one important truth which must be, and can only\nbe, proved indirectly, we may say that the process is a necessary and\nsufficient one, and the question of its comparative excellence or\nusefulness is not worth discussion. As a matter of fact I believe that\nnearly half our logical conclusions rest upon its employment.\n\n  [72] See Horsley, *Philosophical Transactions*, 1772; vol.\u00a0lxii.\n  p.\u00a0327. Montucla, *Histoire des Mathematiques*, vol.\u00a0i. p.\u00a0239.\n  *Penny Cyclop\u00e6dia*, article \u201cEratosthenes.\u201d\n\n  [73] Euclid, Book x. Prop. 117.\n\n\n*Simple Illustrations.*\n\nIn tracing out the powers and results of this method, we will begin\nwith the simplest possible instance. Let us take a proposition of the\ncommon form, A\u00a0=\u00a0AB, say,\n\n  *A Metal is an Element,*\n\nand let us investigate its full meaning. Any person who has had the\nleast logical training, is aware that we can draw from the above\nproposition an apparently different one, namely,\n\n  *A Not-element is a Not-metal.*\n\nWhile some logicians, as for instance De Morgan,[74] have considered\nthe relation of these two propositions to be purely self-evident, and\nneither needing nor allowing analysis, a great many more persons, as\nI have observed while teaching logic, are at first unable to perceive\nthe close connection between them. I believe that a true and complete\nsystem of logic will furnish a clear analysis of this process, which\nhas been called *Contrapositive Conversion*; the full process is as\nfollows:--\n\n  [74] *Philosophical Magazine*, December 1852; Fourth Series, vol.\u00a0iv.\n  p.\u00a0435, \u201cOn Indirect Demonstration.\u201d\n\nFirstly, by the Law of Duality we know that\n\n  *Not-element is either Metal or Not-metal.*\n\nIf it be metal, we know that it is by the premise *an element*; we\nshould thus be supposing that the same thing is an element and a\nnot-element, which is in opposition to the Law of Contradiction.\nAccording to the only other alternative, then, the not-element must be\na not-metal.\n\nTo represent this process of inference symbolically we take the premise\nin the form\n\n  A = AB.                  (1)\n\nWe observe that by the Law of Duality the term not-B is thus described\n\n  *b* = A*b* \ua58c *ab*.       (2)\n\nFor A in this proposition we substitute its description as given in\n(1), obtaining\n\n  *b* = AB*b* \ua58c *ab*.\n\nBut according to the Law of Contradiction the term AB*b* must be\nexcluded from thought, or\n\n  AB*b* = 0.\n\nHence it results that *b* is either nothing at all, or it is *ab*; and\nthe conclusion is\n\n  *b* = *ab*.\n\nAs it will often be necessary to refer to a conclusion of this kind I\nshall call it, as is usual, the *Contrapositive Proposition* of the\noriginal. The reader need hardly be cautioned to observe that from all\nA\u2019s are B\u2019s it does not follow that all not-A\u2019s are not-B\u2019s. For by the\nLaw of Duality we have\n\n  *a* = *a*B \ua58c *ab*,\n\nand it will not be found possible to make any substitution in this by\nour original premise A = AB. It still remains doubtful, therefore,\nwhether not-metal is element or not-element.\n\nThe proof of the Contrapositive Proposition given above is exactly the\nsame as that which Euclid applies in the case of geometrical notions.\nDe Morgan describes Euclid\u2019s process as follows[75]:--\u201cFrom every not-B\nis not-A he produces Every A is B, thus: If it be possible, let this A\nbe not-B, but every not-B is not-A, therefore this A is not-A, which\nis absurd: whence every A is B.\u201d Now De Morgan thinks that this proof\nis entirely needless, because common logic gives the inference without\nthe use of any geometrical reasoning. I conceive however that logic\ngives the inference only by an indirect process. De Morgan claims \u201cto\nsee identity in Every A is B and every not-B is not-A, by a process\nof thought prior to syllogism.\u201d Whether prior to syllogism or not, I\nclaim that it is not prior to the laws of thought and the process of\nsubstitutive inference, by which it may be undoubtedly demonstrated.\n\n  [75] *Philosophical Magazine*, Dec. 1852; p.\u00a0437.\n\n\n*Employment of the Contrapositive Proposition.*\n\nWe can frequently employ the contrapositive form of a proposition by\nthe method of substitution; and certain moods of the ancient syllogism,\nwhich we have hitherto passed over, may thus be satisfactorily\ncomprehended in our system. Take for instance the following syllogism\nin the mood Camestres:--\n\n  \u201cWhales are not true fish; for they do not respire water,\n     whereas true fish do respire water.\u201d\n\nLet us take\n\n  A = whale\n  B = true fish\n  C = respiring water\n\nThe premises are of the forms\n\n  A = A*c*        (1)\n  B = BC          (2)\n\nNow, by the process of contraposition we obtain from the second premise\n\n  *c* = *bc*\n\nand we can substitute this expression for *c* in (1), obtaining\n\n  A = A*bc*\n\nor \u201cWhales are not true fish, not respiring water.\u201d\n\nThe mood Cesare does not really differ from Camestres except in the\norder of the premises, and it could be exhibited in an exactly similar\nmanner.\n\nThe mood Baroko gave much trouble to the old logicians, who could\nnot *reduce* it to the first figure in the same manner as the other\nmoods, and were obliged to invent, specially for it and for Bokardo, a\nmethod of Indirect Reduction closely analogous to the indirect proof\nof Euclid. Now these moods require no exceptional treatment in this\nsystem. Let us take as an instance of Baroko, the argument\n\n  All heated solids give continuous spectra        (1)\n  Some nebul\u00e6 do not give continuous spectra       (2)\n  Therefore, some nebul\u00e6 are not heated solids     (3)\n\nTreating the little word some as an indeterminate adjective of\nselection, to which we assign a symbol like any other adjective, let\n\n  A = some\n  B = nebul\u00e6\n  C = giving continuous spectra\n  D = heated solids\n\nThe premises then become\n\n  D = DC                        (1)\n  AB = AB*c*                    (2)\n\nNow from (1) we obtain by the indirect method the contrapositive\nproposition\n\n  *c* = *cd*\n\nand if we substitute this expression for *c* in (2) we have\n\n  AB = AB*cd*\n\nthe full meaning of which is that \u201csome nebul\u00e6 do not give continuous\nspectra and are not heated solids.\u201d\n\nWe might similarly apply the contrapositive in many other instances.\nTake the argument, \u201cAll fixed stars are self-luminous; but some of the\nheavenly bodies are not self-luminous, and are therefore not fixed\nstars.\u201d Taking our terms\n\n  A = fixed stars\n  B = self-luminous\n  C = some\n  D = heavenly bodies\n\nwe have the premises\n\n   A = AB,                      (1)\n  CD = *b*CD                    (2)\n\nNow from (1) we can draw the contrapositive\n\n  *b* = *ab*\n\nand substituting this expression for *b* in (2) we obtain\n\n  CD = *ab*CD\n\nwhich expresses the conclusion of the argument that some heavenly\nbodies are not fixed stars.\n\n\n*Contrapositive of a Simple Identity.*\n\nThe reader should carefully note that when we apply the process of\nIndirect Inference to a simple identity of the form\n\n  A = B\n\nwe may obtain further results. If we wish to know what is the term\nnot-B, we have as before, by the Law of Duality,\n\n  *b* = A*b* \ua58c *ab*\n\nand substituting for A we obtain\n\n  *b* = B*b* \ua58c *ab* = *ab*.\n\nBut we may now also draw a second contrapositive; for we have\n\n  *a* = *a*B \ua58c *ab*,\n\nand substituting for B its equivalent A we have\n\n  *a* = *a*A \ua58c *ab* = *ab*.\n\nHence from the single identity A\u00a0=\u00a0B we can draw the two propositions\n\n  *a* = *ab*\n  *b* = *ab*,\n\nand observing that these propositions have a common term *ab* we can\nmake a new substitution, getting\n\n  *a* = *b*.\n\nThis result is in strict accordance with the fundamental principles of\ninference, and it may be a question whether it is not a self-evident\nresult, independent of the steps of deduction by which we have reached\nit. For where two classes are coincident like A and B, whatever is true\nof the one is true of the other; what is excluded from the one must be\nexcluded from the other similarly. Now as *a* bears to A exactly the\nsame relation that *b* bears to B, the identity of either pair follows\nfrom the identity of the other pair. In every identity, equality, or\nsimilarity, we may argue from the negative of the one side to the\nnegative of the other. Thus at ordinary temperatures\n\n  Mercury = liquid-metal,\n\nhence obviously\n\n  Not-mercury = not liquid-metal;\n\nor since\n\n  Sirius = brightest fixed star,\n\nit follows that whatever star is not the brightest is not Sirius, and\n*vice vers\u00e2*. Every correct definition is of the form A = B, and may\noften require to be applied in the equivalent negative form.\n\nLet us take as an illustration of the mode of using this result the\nargument following:\n\n  Vowels are letters which can be sounded alone,     (1)\n  The letter *w* cannot be sounded alone;            (2)\n  Therefore the letter *w* is not a vowel.           (3)\n\nHere we have a definition (1), and a comparison of a thing with that\ndefinition (2), leading to exclusion of the thing from the class\ndefined.\n\nTaking the terms\n\n  A = vowel,\n  B = letter which can be sounded alone,\n  C = letter *w*,\n\nthe premises are plainly of the forms\n\n  A = B,                          (1)\n  C = *b*C.                       (2)\n\nNow by the Indirect method we obtain from (1) the Contrapositive\n\n  *b* = *a*,\n\nand inserting in (2) the equivalent for *b* we have\n\n  C = *a*C,                       (3)\n\nor \u201cthe letter *w* is not a vowel.\u201d\n\n\n*Miscellaneous Examples of the Method.*\n\nWe can apply the Indirect Method of Inference however many may be the\nterms involved or the premises containing those terms. As the working\nof the method is best learnt from examples, I will take a case of two\npremises forming the syllogism Barbara: thus\n\n  Iron is metal                   (1)\n  Metal is element.               (2)\n\nIf we want to ascertain what inference is possible concerning the\nterm *Iron*, we develop the term by the Law of Duality. Iron must be\neither metal or not-metal; iron which is metal must be either element\nor not-element; and similarly iron which is not-metal must be either\nelement or not-element. There are then altogether four alternatives\namong which the description of iron must be contained; thus\n\n  Iron, metal, element,           (\u03b1)\n  Iron, metal, not-element,       (\u03b2)\n  Iron, not-metal, element,       (\u03b3)\n  Iron, not-metal, not-element.   (\u03b4)\n\nOur first premise informs us that iron is a metal, and if we substitute\nthis description in (\u03b3) and (\u03b4) we shall have self-contradictory\ncombinations. Our second premise likewise informs us that metal\nis element, and applying this description to (\u03b2) we again have\nself-contradiction, so that there remains only (\u03b1) as a description of\niron--our inference is\n\n  Iron = iron, metal, element.\n\nTo represent this process of reasoning in general symbols, let\n\n  A = iron\n  B = metal\n  C = element,\n\nThe premises of the problem take the forms\n\n  A = AB                   (1)\n  B = BC.                  (2)\n\nBy the Law of Duality we have\n\n  A = AB \ua58c A*b*            (3)\n  A = AC \ua58c A*c*.           (4)\n\nNow, if we insert for A in the second side of (3) its description in\n(4), we obtain what I shall call the *development of A with respect to\nB and C*, namely\n\n  A\u00a0=\u00a0ABC\u00a0\ua58c\u00a0AB*c*\u00a0\ua58c\u00a0A*b*C\u00a0\ua58c\u00a0A*bc*.      (5)\n\nWherever the letters A or B appear in the second side of (5) substitute\ntheir equivalents given in (1) and (2), and the results stated at full\nlength are\n\n  A\u00a0=\u00a0ABC\u00a0\ua58c\u00a0ABC*c*\u00a0\ua58c\u00a0AB*b*C\u00a0\ua58c\u00a0AB*b*C*c*.\n\nThe last three alternatives break the Law of Contradiction, so that\n\n  A\u00a0=\u00a0ABC\u00a0\ua58c\u00a00\u00a0\ua58c\u00a00\u00a0\ua58c\u00a00\u00a0=\u00a0ABC.\n\nThis conclusion is, indeed, no more than we could obtain by the\ndirect process of substitution, that is by substituting for B in (1),\nits description in (2) as in p.\u00a055; it is the characteristic of the\nIndirect process that it gives all possible logical conclusions, both\nthose which we have previously obtained, and an immense number of\nothers or which the ancient logic took little or no account. From the\nsame premises, for instance, we can obtain a description of the class\n*not-element* or *c*. By the Law of Duality we can develop *c* into\nfour alternatives, thus\n\n  *c*\u00a0=\u00a0AB*c*\u00a0\ua58c\u00a0A*bc*\u00a0\ua58c\u00a0*a*B*c*\u00a0\ua58c\u00a0*abc*.\n\nIf we substitute for A and B as before, we get\n\n  *c*\u00a0= ABC*c*\u00a0\ua58c\u00a0AB*bc*\u00a0\ua58c\u00a0*a*BC*c*\u00a0\ua58c\u00a0*abc*,\n\nand, striking out the terms which break the Law of Contradiction, there\nremains\n\n  *c* = *abc*,\n\nor what is not element is also not iron and not metal. This Indirect\nMethod of Inference thus furnishes a complete solution of the following\nproblem--*Given any number of logical premises or conditions, required\nthe description of any class of objects, or of any term, as governed by\nthose conditions.*\n\nThe steps of the process of inference may thus be concisely stated--\n\n1. By the Law of Duality develop the utmost number of alternatives\nwhich may exist in the description of the required class or term as\nregards the terms involved in the premises.\n\n2. For each term in these alternatives substitute its description as\ngiven in the premises.\n\n3. Strike out every alternative which is then found to break the Law of\nContradiction.\n\n4. The remaining terms may be equated to the term in question as the\ndesired description.\n\n\n*Mr. Venn\u2019s Problem.*\n\nThe need of some logical method more powerful and comprehensive than\nthe old logic of Aristotle is strikingly illustrated by Mr. Venn\nin his most interesting and able article on Boole\u2019s logic.[76] An\neasy example, originally got, as he says, by the aid of my method as\nsimply described in the *Elementary Lessons in Logic*, was proposed in\nexamination and lecture-rooms to some hundred and fifty students as a\nproblem in ordinary logic. It was answered by, at most, five or six\nof them. It was afterwards set, as an example on Boole\u2019s method, to\na small class who had attended a few lectures on the nature of these\nsymbolic methods. It was readily answered by half or more of their\nnumber.\n\n  [76] *Mind*; a Quarterly Review of Psychology and Philosophy;\n  October, 1876, vol.\u00a0i. p.\u00a0487.\n\nThe problem was as follows:--\u201cThe members of a board were all of them\neither bondholders, or shareholders, but not both; and the bondholders\nas it happened, were all on the board. What conclusion can be drawn?\u201d\nThe conclusion wanted is, \u201cNo shareholders are bondholders.\u201d Now, as\nMr. Venn says, nothing can look simpler than the following reasoning,\n*when stated*:--\u201cThere can be no bondholders who are shareholders; for\nif there were they must be either on the board, or off it. But they\nare not on it, by the first of the given statements; nor off it, by\nthe second.\u201d Yet from the want of any systematic mode of treating such\na question only five or six of some hundred and fifty students could\nsucceed in so simple a problem.\n\nBy symbolic statement the problem is instantly solved. Taking\n\n  A = member of board\n  B = bondholder\n  C = shareholder\n\nthe premises are evidently\n\n  A = AB*c* \ua58c A*b*C\n  B = AB.\n\nThe class C or shareholders may in respect of A and B be developed into\nfour alternatives,\n\n  C = ABC \ua58c A*b*C \ua58c *a*BC \ua58c *ab*C.\n\nBut substituting for A in the first and for B in the third alternative\nwe get\n\n  C\u00a0=\u00a0ABC*c*\u00a0\ua58c\u00a0AB*b*C\u00a0\ua58c\u00a0A*b*C\u00a0\ua58c\u00a0*a*ABC\u00a0\ua58c\u00a0*ab*C.\n\nThe first, second, and fourth alternatives in the above are\nself-contradictory combinations, and only these; striking them out\nthere remain\n\n  C = A*b*C \ua58c *ab*C = *b*C,\n\nthe required answer. This symbolic reasoning is, I believe, the exact\nequivalent of Mr. Venn\u2019s reasoning, and I do not believe that the\nresult can be attained in a simpler manner. Mr. Venn adds that he\ncould adduce other similar instances, that is, instances showing the\nnecessity of a better logical method.\n\n\n*Abbreviation of the Process.*\n\nBefore proceeding to further illustrations of the use of this method,\nI must point out how much its practical employment can be simplified,\nand how much more easy it is than would appear from the description.\nWhen we want to effect at all a thorough solution of a logical problem\nit is best to form, in the first place, a complete series of all the\ncombinations of terms involved in it. If there be two terms A and B,\nthe utmost variety of combinations in which they can appear are\n\n  AB          *a*B\n  A*b*        *ab*.\n\nThe term A appears in the first and second; B in the first and third;\n*a* in the third and fourth; and *b* in the second and fourth. Now if\nwe have any premise, say\n\n  A = B,\n\nwe must ascertain which of these combinations will be rendered\nself-contradictory by substitution; the second and third will have to\nbe struck out, and there will remain only\n\n   AB\n  *ba*.\n\nHence we draw the following inferences\n\n  A\u00a0=\u00a0AB,\u00a0B\u00a0=\u00a0AB,\u00a0*a*\u00a0=\u00a0*ab*,\u00a0*b*\u00a0=\u00a0*ab*.\n\nExactly the same method must be followed when a question involves a\ngreater number of terms. Thus by the Law of Duality the three terms A,\nB, C, give rise to eight conceivable combinations, namely\n\n  ABC        (\u03b1)         *a*BC       (\u03b5)\n  AB*c*      (\u03b2)         *a*B*c*     (\u03b6)\n  A*b*C      (\u03b3)         *ab*C       (\u03b7)\n  A*bc*      (\u03b4)         *abc*.      (\u03b8)\n\nThe development of the term A is formed by the first four of these; for\nB we must select (\u03b1), (\u03b2), (\u03b5), (\u03b6); C consists of (\u03b1), (\u03b3), (\u03b5), (\u03b7);\n*b* of (\u03b3), (\u03b4), (\u03b7), (\u03b8), and so on.\n\nNow if we want to investigate completely the meaning of the premises\n\n  A = AB      (1)\n  B = BC      (2)\n\nwe examine each of the eight combinations as regards each premise; (\u03b3)\nand (\u03b4) are contradicted by (1), and (\u03b2) and (\u03b6) by (2), so that there\nremain only\n\n   ABC        (\u03b1)\n  *a*BC       (\u03b5)\n  *ab*C       (\u03b7)\n  *abc*.      (\u03b8)\n\nTo describe any term under the conditions of the premises (1) and (2),\nwe have simply to draw out the proper combinations from this list;\nthus, A is represented only by ABC, that is to say\n\n                  A  =  ABC,\n  similarly      *c* = *abc*.\n\nFor B we have two alternatives thus stated,\n\n  B = ABC \ua58c *a*BC;\n\nand for *b* we have\n\n  *b* = *ab*C \ua58c *abc*.\n\nWhen we have a problem involving four distinct terms we need to\ndouble the number of combinations, and as we add each new term the\ncombinations become twice as numerous. Thus\n\n  A, B              produce four       combinations\n  A, B, C,             \"    eight           \"\n  A, B, C, D           \"    sixteen         \"\n  A, B, C, D, E        \"    thirty-two      \"\n  A, B, C, D, E, F     \"    sixty-four      \"\n\nand so on.\n\nI propose to call any such series of combinations the *Logical\nAlphabet*. It holds in logical science a position the importance\nof which cannot be exaggerated, and as we proceed from logical to\nmathematical considerations, it will become apparent that there is\na close connection between these combinations and the fundamental\ntheorems of mathematical science. For the convenience of the reader\nwho may wish to employ the *Alphabet* in logical questions, I have\nhad printed on the next page a complete series of the combinations up\nto those of six terms. At the very commencement, in the first column,\nis placed a single letter X, which might seem to be superfluous. This\nletter serves to denote that it is always some higher class which is\ndivided up. Thus the combination AB really means ABX, or that part of\nsome larger class, say X, which has the qualities of A and B present.\nThe letter X is omitted in the greater part of the table merely for\nthe sake of brevity and clearness. In a later chapter on Combinations\nit will become apparent that the introduction of this unit class is\nrequisite in order to complete the analogy with the Arithmetical\nTriangle there described.\n\nThe reader ought to bear in mind that though the Logical Alphabet seems\nto give mere lists of combinations, these combinations are intended in\nevery case to constitute the development of a term of a proposition.\nThus the four combinations AB, A*b*, *a*B, *ab* really mean that any\nclass X is described by the following proposition,\n\n  X\u00a0=\u00a0XAB\u00a0\ua58c\u00a0XA*b*\u00a0\ua58c\u00a0X*a*B\u00a0\ua58c\u00a0X*ab*.\n\nIf we select the A\u2019s, we obtain the following proposition\n\n  AX = XAB \ua58c XA*b*.\n\nThus whatever group of combinations we treat must be conceived as part\nof a higher class, *summum genus* or universe symbolised in the term\nX; but, bearing this in mind, it is needless to complicate our formul\u00e6\nby always introducing the letter. All inference consists in passing\nfrom propositions to propositions, and combinations *per se* have no\nmeaning. They are consequently to be regarded in all cases as forming\nparts of propositions.\n\n\nTHE LOGICAL ALPHABET.\n\n  I.   II.    III.    IV.     V.        VI.         VII.\n  X    AX     AB      ABC     ABCD      ABCDE       ABCDEF\n      *a*X    A*b*    AB*c*   ABC*d*    ABCD*e*     ABCDE*f*\n             *a*B     A*b*C   AB*c*D    ABC*d*E     ABCD*e*F\n             *ab*     A*bc*   AB*cd*    ABC*de*     ABCD*ef*\n                     *a*BC    A*b*CD    AB*c*DE     ABC*d*EF\n                     *a*B*c*  A*b*C*d*  AB*c*D*e*   ABC*d*E*f*\n                     *ab*C    A*bc*D    AB*cd*E     ABC*de*F\n                     *abc*    Ab*cd*    AB*cde*     ABC*def*\n                             *a*BCD     A*b*CDE     AB*c*DEF\n                             *a*BC*d*   A*b*CD*e*   AB*c*DE*f*\n                             *a*B*c*D   A*b*C*d*E   AB*c*D*e*F\n                             *a*B*cd*   A*b*C*de*   AB*c*D*ef*\n                             *ab*CD     A*bc*DE     AB*cd*EF\n                             *ab*C*d*   A*bc*D*e*   AB*cd*E*f*\n                             *abc*D     A*bcd*E     AB*cde*F\n                             *abcd*     A*bcde*     AB*cdef*\n                                       *a*BCDE      A*b*CDEF\n                                       *a*BCD*e*    A*b*CDE*f*\n                                       *a*BC*d*E    A*b*CD*e*F\n                                       *a*BC*de*    A*b*CD*ef*\n                                       *a*B*c*DE    A*b*C*d*EF\n                                       *a*B*c*D*e*  A*b*C*d*E*f*\n                                       *a*B*cd*E    A*b*C*de*F\n                                       *a*B*cde*    A*b*C*def*\n                                       *ab*CDE      A*bc*DEF\n                                       *ab*CD*e*    A*bc*DE*f*\n                                       *ab*C*d*E    A*bc*D*e*F\n                                       *ab*Cd*e*    A*bc*D*ef*\n                                       *abc*DE      A*bcd*EF\n                                       *abc*D*e*    A*bcd*E*f*\n                                       *abcd*E      A*bcde*F\n                                       *abcde*      A*bcdef*\n                                                   *a*BCDEF\n                                                   *a*BCDE*f*\n                                                   *a*BCD*e*F\n                                                   *a*BCD*ef*\n                                                   *a*BC*d*EF\n                                                   *a*BC*d*E*f*\n                                                   *a*BC*de*F\n                                                   *a*BC*def*\n                                                   *a*B*c*DEF\n                                                   *a*B*c*DE*f*\n                                                   *a*B*c*D*e*F\n                                                   *a*B*c*D*ef*\n                                                   *a*B*cd*EF\n                                                   *a*B*cd*E*f*\n                                                   *a*B*cde*F\n                                                   *a*B*cdef*\n                                                   *ab*CDEF\n                                                   *ab*CDE*f*\n                                                   *ab*CD*e*F\n                                                   *ab*CD*ef*\n                                                   *ab*C*d*EF\n                                                   *ab*C*d*E*f*\n                                                   *ab*C*de*F\n                                                   *ab*C*def*\n                                                   *abc*DEF\n                                                   *abc*DE*f*\n                                                   *abc*D*e*F\n                                                   *abc*D*ef*\n                                                   *abcd*EF\n                                                   *abcd*E*f*\n                                                   *abcde*F\n                                                   *abcdef*\n\nIn a theoretical point of view we may conceive that the Logical\nAlphabet is infinitely extended. Every new quality or circumstance\nwhich can belong to an object, subdivides each combination or class,\nso that the number of such combinations, when unrestricted by logical\nconditions, is represented by an infinitely high power of two. The\nextremely rapid increase in the number of subdivisions obliges us to\nconfine our attention to a few qualities at a time.\n\nWhen contemplating the properties of this Alphabet I am often inclined\nto think that Pythagoras perceived the deep logical importance of\nduality; for while unity was the symbol of identity and harmony, he\ndescribed the number two as the origin of contrasts, or the symbol\nof diversity, division and separation. The number four, or the\n*Tetractys*, was also regarded by him as one of the chief elements of\nexistence, for it represented the generating virtue whence come all\ncombinations. In one of the golden verses ascribed to Pythagoras, he\nconjures his pupil to be virtuous:[77]\n\n  \u201cBy him who stampt *The Four* upon the Mind,\n  *The Four*, the fount of Nature\u2019s endless stream.\u201d\n\n  [77] Whewell, *History of the Inductive Sciences*, vol.\u00a0i. p.\u00a0222.\n\nNow four and the higher powers of duality do represent in this logical\nsystem the numbers of combinations which can be generated in the\nabsence of logical restrictions. The followers of Pythagoras may have\nshrouded their master\u2019s doctrines in mysterious and superstitious\nnotions, but in many points these doctrines seem to have some basis in\nlogical philosophy.\n\n\n*The Logical Slate.*\n\nTo a person who has once comprehended the extreme significance and\nutility of the Logical Alphabet the indirect process of inference\nbecomes reduced to the repetition of a few uniform operations of\nclassification, selection, and elimination of contradictories. Logical\ndeduction, even in the most complicated questions, becomes a matter\nof mere routine, and the amount of labour required is the only\nimpediment, when once the meaning of the premises is rendered clear.\nBut the amount of labour is often found to be considerable. The mere\nwriting down of sixty-four combinations of six letters each is no small\ntask, and, if we had a problem of five premises, each of the sixty-four\ncombinations would have to be examined in connection with each premise.\nThe requisite comparison is often of a very tedious character, and\nconsiderable chance of error intervenes.\n\nI have given much attention, therefore, to lessening both the manual\nand mental labour of the process, and I shall describe several devices\nwhich may be adopted for saving trouble and risk of mistake.\n\nIn the first place, as the same sets of combinations occur over and\nover again in different problems, we may avoid the labour of writing\nthem out by having the sets of letters ready printed upon small sheets\nof writing-paper. It has also been suggested by a correspondent that,\nif any one series of combinations were marked upon the margin of a\nsheet of paper, and a slit cut between each pair of combinations, it\nwould be easy to fold down any particular combination, and thus strike\nit out of view. The combinations consistent with the premises would\nthen remain in a broken series. This method answers sufficiently well\nfor occasional use.\n\nA more convenient mode, however, is to have the series of letters shown\non p.\u00a094, engraved upon a common school writing slate, of such a size,\nthat the letters may occupy only about a third of the space on the\nleft hand side of the slate. The conditions of the problem can then be\nwritten down on the unoccupied part of the slate, and the proper series\nof combinations being chosen, the contradictory combinations can be\nstruck out with the pencil. I have used a slate of this kind, which I\ncall a *Logical Slate*, for more than twelve years, and it has saved me\nmuch trouble. It is hardly possible to apply this process to problems\nof more than six terms, owing to the large number of combinations which\nwould require examination.\n\n\n*Abstraction of Indifferent Circumstances.*\n\nThere is a simple but highly important process of inference which\nenables us to abstract, eliminate or disregard all circumstances\nindifferently present and absent. Thus if I were to state that \u201ca\ntriangle is a three-sided rectilinear figure, either large or not\nlarge,\u201d these two alternatives would be superfluous, because, by the\nLaw of Duality, I know that everything must be either large or not\nlarge. To add the qualification gives no new knowledge, since the\nexistence of the two alternatives will be understood in the absence of\nany information to the contrary. Accordingly, when two alternatives\ndiffer only as regards a single component term which is positive in one\nand negative in the other, we may reduce them to one term by striking\nout their indifferent part. It is really a process of substitution\nwhich enables us to do this; for having any proposition of the form\n\n  A = ABC \ua58c AB*c*,      (1)\n\nwe know by the Law of Duality that\n\n  AB = ABC \ua58c AB*c*.     (2)\n\nAs the second member of this is identical with the second member of (1)\nwe may substitute, obtaining\n\n  A = AB.\n\nThis process of reducing useless alternatives may be applied again and\nagain; for it is plain that\n\n  A = AB (CD \ua58c C*d* \ua58c *c*D \ua58c *cd*)\n\ncommunicates no more information than that A is B. Abstraction\nof indifferent terms is in fact the converse process to that of\ndevelopment described in p.\u00a089; and it is one of the most important\noperations in the whole sphere of reasoning.\n\nThe reader should observe that in the proposition\n\n  AC = BC\n\nwe cannot abstract C and infer\n\n  A = B;\n\nbut from\n\n  AC \ua58c A*c* = BC \ua58c B*c*\n\nwe may abstract all reference to the term C.\n\nIt ought to be carefully remarked, however, that alternatives which\nseem to be without meaning often imply important knowledge. Thus if\nI say that \u201ca triangle is a three-sided rectilinear figure, with or\nwithout three equal angles,\u201d the last alternatives really express a\nproperty of triangles, namely, that some triangles have three equal\nangles, and some do not have them. If we put P = \u201cSome,\u201d meaning by the\nindefinite adjective \u201cSome,\u201d one or more of the undefined properties of\ntriangles with three equal angles, and take\n\n  A = triangle\n  B = three-sided rectilinear figure\n  C = with three equal angles,\n\nthen the knowledge implied is expressed in the two propositions\n\n   PA = PBC\n  *p*A = *p*B*c*.\n\nThese may also be thrown into the form of one proposition, namely,\n\n  A = PBC \ua58c *p*B*c*;\n\nbut these alternatives cannot be reduced, and the proposition is quite\ndifferent from\n\n  A = BC \ua58c B*c*.\n\n\n*Illustrations of the Indirect Method.*\n\nA great variety of arguments and logical problems might be introduced\nhere to show the comprehensive character and powers of the Indirect\nMethod. We can treat either a single premise or a series of premises.\n\nTake in the first place a simple definition, such as \u201ca triangle is a\nthree-sided rectilinear figure.\u201d Let\n\n  A = triangle\n  B = three-sided\n  C = rectilinear figure,\n\nthen the definition is of the form\n\n  A = BC.\n\nIf we take the series of eight combinations of three letters in the\nLogical Alphabet (p.\u00a094) and strike out those which are inconsistent\nwith the definition, we have the following result:--\n\n   ABC\n  *a*B*c*\n  *ab*C\n  *abc.*\n\nFor the description of the class C we have\n\n  C = ABC \ua58c *ab*C,\n\nthat is, \u201ca rectilinear figure is either a triangle and three-sided, or\nnot a triangle and not three-sided.\u201d\n\nFor the class *b* we have\n\n  *b* = *ab*C \ua58c *abc*.\n\nTo the second side of this we may apply the process of simplification\nby abstraction described in the last section; for by the Law of Duality\n\n  *ab* = *ab*C \ua58c *abc*;\n\nand as we have two propositions identical in the second side of each we\nmay substitute, getting\n\n  *b* = *ab*,\n\nor what is not three-sided is not a triangle (whether it be rectilinear\nor not).\n\n\n*Second Example.*\n\nLet us treat by this method the following argument:--\n\n  \u201cBlende is not an elementary substance; elementary substances\n     are those which are undecomposable; blende, therefore, is\n     decomposable.\u201d\n\nTaking our letters thus--\n\n  A = blende,\n  B = elementary substance,\n  C = undecomposable,\n\nthe premises are of the forms\n\n  A = A*b*,              (1)\n  B = C.                 (2)\n\nNo immediate substitution can be made; but if we take the\ncontrapositive of (2) (see p.\u00a086), namely\n\n  *b* = *c*,             (3)\n\nwe can substitute in (1) obtaining the conclusion\n\n  A = A*c*.\n\nBut the same result may be obtained by taking the eight combinations\nof A, B, C, of the Logical Alphabet; it will be found that only three\ncombinations, namely,\n\n   A*bc*\n  *a*BC\n  *abc*,\n\nare consistent with the premises, whence it results that\n\n  A = A*bc*,\n\nor by the process of Ellipsis before described (p.\u00a057)\n\n  A = A*c*.\n\n\n*Third Example.*\n\nAs a somewhat more complex example I take the argument thus stated, one\nwhich could not be thrown into the syllogistic form:--\n\n  \u201cAll metals except gold and silver are opaque; therefore what is not\n     opaque is either gold or silver or is not-metal.\u201d\n\nThere is more implied in this statement than is distinctly asserted,\nthe full meaning being as follows:\n\n  All metals not gold or silver are opaque,      (1)\n  Gold is not opaque but is a metal,             (2)\n  Silver is not opaque but is a metal,           (3)\n  Gold is not silver.                            (4)\n\nTaking our letters thus--\n\n  A = metal      C = silver\n  B = gold       D = opaque,\n\nwe may state the premises in the forms\n\n  A*bc* = A*bc*D        (1)\n  B = AB*d*             (2)\n  C = AC*d*             (3)\n  B = B*c*.             (4)\n\nTo obtain a complete solution of the question we take the sixteen\ncombinations of A, B, C, D, and striking out those which are\ninconsistent with the premises, there remain only\n\n   AB*cd*\n   A*b*C*d*\n   A*bc*D\n  *abc*D\n  *abcd*.\n\nThe expression for not-opaque things consists of the three combinations\ncontaining *d*, thus\n\n       *d*\u00a0=\u00a0AB*cd*\u00a0\ua58c\u00a0A*b*C*d*\u00a0\ua58c\u00a0*abcd*,\n  or   *d*\u00a0=\u00a0A*d*\u00a0(B*c*\u00a0\ua58c\u00a0*b*C)\u00a0\ua58c\u00a0*abcd*.\n\nIn ordinary language, what is not-opaque is either metal which is\ngold, and then not-silver, or silver and then not-gold, or else it is\nnot-metal and neither gold nor silver.\n\n\n*Fourth Example.*\n\nA good example for the illustration of the Indirect Method is to be\nfound in De Morgan\u2019s *Formal Logic* (p. 123), the premises being\nsubstantially as follows:--\n\nFrom A follows B, and from C follows D; but B and D are inconsistent\nwith each other; therefore A and C are inconsistent.\n\nThe meaning no doubt is that where A is, B will be found, or that\nevery A is a B, and similarly every C is a D; but B and D cannot occur\ntogether. The premises therefore appear to be of the forms\n\n  A = AB,          (1)\n  C = CD,          (2)\n  B = B*d*.        (3)\n\nOn examining the series of sixteen combinations, only five are found to\nbe consistent with the above conditions, namely,\n\n   AB*cd*\n  *a*B*cd*\n  *ab*CD\n  *abc*D\n  *abcd*.\n\nIn these combinations the only A which appears is joined to *c*, and\nsimilarly C is joined to *a*, or A is inconsistent with C.\n\n\n*Fifth Example.*\n\nA more complex argument, also given by De Morgan,[78] contains five\nterms, and is as stated below, except that the letters are altered.\n\n  Every A is one only of the two B or C; D is both B and C, except\n    when B is E, and then it is neither; therefore no A is D.\n\n  [78] *Formal Logic*, p.\u00a0124. As Professor Croom Robertson has pointed\n  out to me, the second and third premises may be thrown into a single\n  proposition, D\u00a0= D*e*BC\u00a0\ua58c\u00a0DE*bc*.\n\nThe meaning of the above premises is difficult to interpret, but seems\nto be capable of expression in the following symbolic forms--\n\n   A = AB*c* \ua58c A*b*C,      (1)\n  De = D*e*BC,             (2)\n  DE = DE*bc*.             (3)\n\nAs five terms enter into these premises it is requisite to treat their\nthirty-two combinations, and it will be found that fourteen of them\nremain consistent with the premises, namely\n\n  AB*cd*E      *a*BCD*e*     *ab*C*d*E\n  AB*cde*      *a*BC*d*E     *ab*C*de*\n  A*b*C*d*E    *a*BC*de*     *abc*DE\n  A*b*C*de*    *a*B*cd*E     *abcd*E\n               *a*B*cde*     *abcde*.\n\nIf we examine the first four combinations, all of which contain A, we\nfind that they none of them contain D; or again, if we select those\nwhich contain D, we have only two, thus--\n\n  D = *a*BCD*e* \ua58c *abc*DE.\n\nHence it is clear that no A is D, and *vice vers\u00e2* no D is A. We might\ndraw many other conclusions from the same premises; for instance--\n\n  DE = *abc*DE,\n\nor D and E never meet but in the absence of A, B, and C.\n\n\n*Fallacies analysed by the Indirect Method.*\n\nIt has been sufficiently shown, perhaps, that we can by the Indirect\nMethod of Inference extract the whole truth from a series of\npropositions, and exhibit it anew in any required form of conclusion.\nBut it may also need to be shown by examples that so long as we follow\ncorrectly the almost mechanical rules of the method, we cannot fall\ninto any of the fallacies or paralogisms which are often committed in\nordinary discussion. Let us take the example of a fallacious argument,\npreviously treated by the Method of Direct Inference (p.\u00a062),\n\n  Granite is not a sedimentary rock,    (1)\n  Basalt is not a sedimentary rock,     (2)\n\nand let us ascertain whether any precise conclusion can be drawn\nconcerning the relation of granite and basalt. Taking as before\n\n  A = granite,\n  B = sedimentary rock,\n  C = basalt,\n\nthe premises become\n\n  A = A*b*,       (1)\n  C = C*b*.       (2)\n\nOf the eight conceivable combinations of A, B, C, five agree with these\nconditions, namely\n\n  A*b*C     *a*B*c*\n  A*bc*     *ab*C\n            *abc*.\n\nSelecting the combinations which contain A, we find the description of\ngranite to be\n\n  A\u00a0=\u00a0A*b*C\u00a0\ua58c\u00a0A*bc*\u00a0=\u00a0A*b*(C\u00a0\ua58c\u00a0*c*),\n\nthat is, granite is not a sedimentary rock, and is either basalt or\nnot-basalt. If we want a description of basalt the answer is of like\nform\n\n  C\u00a0=\u00a0A*b*C\u00a0\ua58c\u00a0*ab*C\u00a0=\u00a0*b*C(A\u00a0\ua58c\u00a0*a*),\n\nthat is basalt is not a sedimentary rock, and is either granite or\nnot-granite. As it is already perfectly evident that basalt must be\neither granite or not, and *vice vers\u00e2*, the premises fail to give us\nany information on the point, that is to say the Method of Indirect\nInference saves us from falling into any fallacious conclusions. This\nexample sufficiently illustrates both the fallacy of Negative premises\nand that of Undistributed Middle of the old logic.\n\nThe fallacy called the Illicit Process of the Major Term is also\nincapable of commission in following the rules of the method. Our\nexample was (p.\u00a065)\n\n  All planets are subject to gravity,    (1)\n  Fixed stars are not planets.           (2)\n\nThe false conclusion is that \u201cfixed stars are not subject to gravity.\u201d\nThe terms are\n\n  A = planet\n  B = fixed star\n  C = subject to gravity.\n\nAnd the premises are A = AC, (1) B = *a*B. (2)\n\nThe combinations which remain uncontradicted on comparison with these\npremises are\n\n   A*b*C     *a*B*c*\n  *a*BC      *ab*C\n             *abc*.\n\nFor fixed star we have the description\n\n  B = *a*BC \ua58c *a*B*c*,\n\nthat is, \u201ca fixed star is not a planet, but is either subject or not,\nas the case may be, to gravity.\u201d Here we have no conclusion concerning\nthe connection of fixed stars and gravity.\n\n\n*The Logical Abacus.*\n\nThe Indirect Method of Inference has now been sufficiently described,\nand a careful examination of its powers will show that it is capable of\ngiving a full analysis and solution of every question involving only\nlogical relations. The chief difficulty of the method consists in the\ngreat number of combinations which may have to be examined; not only\nmay the requisite labour become formidable, but a considerable chance\nof mistake arises. I have therefore given much attention to modes\nof facilitating the work, and have succeeded in reducing the method\nto an almost mechanical form. It soon appeared obvious that if the\nconceivable combinations of the Logical Alphabet, for any number of\nletters, instead of being printed in fixed order on a piece of paper\nor slate, were marked upon light movable pieces of wood, mechanical\narrangements could readily be devised for selecting any required class\nof the combinations. The labour of comparison and rejection might thus\nbe immensely reduced. This idea was first carried out in the Logical\nAbacus, which I have found useful in the lecture-room for exhibiting\nthe complete solution of logical problems. A minute description of\nthe construction and use of the Abacus, together with figures of the\nparts, has already been given in my essay called *The Substitution of\nSimilars*,[79] and I will here give only a general description.\n\n  [79] Pp. 55\u201359, 81\u201386.\n\nThe Logical Abacus consists of a common school black-board placed in a\nsloping position and furnished with four horizontal and equi-distant\nledges. The combinations of the letters shown in the first four columns\nof the Logical Alphabet are printed in somewhat large type, so that\neach letter is about an inch from the neighbouring one, but the letters\nare placed one above the other instead of being in horizontal lines\nas in p.\u00a094. Each combination of letters is separately fixed to the\nsurface of a thin slip of wood one inch broad and about one-eighth\ninch thick. Short steel pins are then driven in an inclined position\ninto the wood. When a letter is a large capital representing a positive\nterm, the pin is fixed in the upper part of its space; when the letter\nis a small italic representing a negative term, the pin is fixed in\nthe lower part of the space. Now, if one of the series of combinations\nbe ranged upon a ledge of the black-board, the sharp edge of a flat\nrule can be inserted beneath the pins belonging to any one letter--say\nA, so that all the combinations marked A can be lifted out and placed\nupon a separate ledge. Thus we have represented the act of thought\nwhich separates the class A from what is not-A. The operation can be\nrepeated; out of the A\u2019s we can in like manner select those which are\nB\u2019s, obtaining the AB\u2019s; and in like manner we may select any other\nclasses such as the *a*B\u2019s, the *ab*\u2019s, or the *abc*\u2019s.\n\nIf now we take the series of eight combinations of the letters A, B,\nC, *a*, *b*, *c*, and wish to analyse the argument anciently called\nBarbara, having the premises\n\n  A = AB         (1)\n  B = BC,        (2)\n\nwe proceed as follows--We raise the combinations marked *a*, leaving\nthe A\u2019s behind; out of these A\u2019s we move to a lower ledge such as\nare *b*\u2019s, and to the remaining AB\u2019s we join the *a*\u2019s which have\nbeen raised. The result is that we have divided all the combinations\ninto two classes, namely, the A*b*\u2019s which are incapable of existing\nconsistently with premise (1), and the combinations which are\nconsistent with the premise. Turning now to the second premise, we\nraise out of those which agree with (1) the *b*\u2019s, then we lower\nthe B*c*\u2019s; lastly we join the *b*\u2019s to the BC\u2019s. We now find our\ncombinations arranged as below.\n\n  +---+-----+-----+-----+-----+-----+-----+-----+\n  | A |     |     |     | *a* |     | *a* | *a* |\n  | B |     |     |     |  B  |     | *b* | *b* |\n  | C |     |     |     |  C  |     |  C  | *c* |\n  +---+-----+-----+-----+-----+-----+-----+-----+\n  |   |  A  |  A  |  A  |     | *a* |     |     |\n  |   |  B  | *b* | *b* |     |  B  |     |     |\n  |   | *c* |  C  | *c* |     | *c* |     |     |\n  +---+-----+-----+-----+-----+-----+-----+-----+\n\nThe lower line contains all the combinations which are inconsistent\nwith either premise; we have carried out in a mechanical manner that\nexclusion of self-contradictories which was formerly done upon the\nslate or upon paper. Accordingly, from the combinations remaining in\nthe upper line we can draw any inference which the premises yield. If\nwe raise the A\u2019s we find only one, and that is C, so that A must be C.\nIf we select the *c*\u2019s we again find only one, which is *a* and also\n*b*; thus we prove that not-C is not-A and not-B.\n\nWhen a disjunctive proposition occurs among the premises the requisite\nmovements become rather more complicated. Take the disjunctive argument\n\n  A is either B or C or D,\n  A is not C and not D,\n  Therefore A is B.\n\nThe premises are represented accurately as follows:--\n\n  A = AB \ua58c AC \ua58c AD       (1)\n  A = A*c*               (2)\n  A = A*d*.              (3)\n\nAs there are four terms, we choose the series of sixteen combinations\nand place them on the highest ledge of the board but one. We raise\nthe *a*\u2019s and out of the A\u2019s, which remain, we lower the *b*\u2019s. But\nwe are not to reject all the A*b*\u2019s as contradictory, because by the\nfirst premise A\u2019s may be either B\u2019s or C\u2019s or D\u2019s. Accordingly out\nof the A*b*\u2019s we must select the *c*\u2019s, and out of these again the\n*d*\u2019s, so that only A*bcd* will remain to be rejected finally. Joining\nall the other fifteen combinations together again, and proceeding to\npremise (2), we raise the *a*\u2019s and lower the AC\u2019s, and thus reject\nthe combinations inconsistent with (2); similarly we reject the AD\u2019s\nwhich are inconsistent with (3). It will be found that there remain,\nin addition to all the eight combinations containing *a*, only one\ncontaining A, namely\n\n  AB*cd*,\n\nwhence it is apparent that A must be B, the ordinary conclusion of the\nargument.\n\nIn my \u201cSubstitution of Similars\u201d (pp.\u00a056\u201359) I have described the\nworking upon the Abacus of two other logical problems, which it would\nbe tedious to repeat in this place.\n\n\n*The Logical Machine.*\n\nAlthough the Logical Abacus considerably reduced the labour of using\nthe Indirect Method, it was not free from the possibility of error.\nI thought moreover that it would afford a conspicuous proof of the\ngenerality and power of the method if I could reduce it to a purely\nmechanical form. Logicians had long been accustomed to speak of Logic\nas an Organon or Instrument, and even Lord Bacon, while he rejected\nthe old syllogistic logic, had insisted, in the second aphorism of his\n\u201cNew Instrument,\u201d that the mind required some kind of systematic aid.\nIn the kindred science of mathematics mechanical assistance of one kind\nor another had long been employed. Orreries, globes, mechanical clocks,\nand such like instruments, are really aids to calculation and are of\nconsiderable antiquity. The Arithmetical Abacus is still in common use\nin Russia and China. The calculating machine of Pascal is more than two\ncenturies old, having been constructed in 1642\u201345. M. Thomas of Colmar\nmanufactures an arithmetical machine on Pascal\u2019s principles which\nis employed by engineers and others who need frequently to multiply\nor divide. To Babbage and Scheutz is due the merit of embodying the\nCalculus of Differences in a machine, which thus became capable of\ncalculating the most complicated tables of figures. It seemed strange\nthat in the more intricate science of quantity mechanism should be\napplicable, whereas in the simple science of qualitative reasoning, the\nsyllogism was only called an instrument by a figure of speech. It is\ntrue that Swift satirically described the Professors of Laputa as in\npossession of a thinking machine, and in 1851 Mr. Alfred Smee actually\nproposed the construction of a Relational machine and a Differential\nmachine, the first of which would be a mechanical dictionary and the\nsecond a mode of comparing ideas; but with these exceptions I have not\nyet met with so much as a suggestion of a reasoning machine. It may be\nadded that Mr. Smee\u2019s designs, though highly ingenious, appear to be\nimpracticable, and in any case they do not attempt the performance of\nlogical inference.[80]\n\n  [80] See his work called *The Process of Thought adapted to Words\n  and Language, together with a Description of the Relational and\n  Differential Machines*. Also *Philosophical Transactions*, [1870]\n  vol.\u00a0160, p. 518.\n\nThe Logical Abacus soon suggested the notion of a Logical Machine,\nwhich, after two unsuccessful attempts, I succeeded in constructing\nin a comparatively simple and effective form. The details of the\nLogical Machine have been fully described by the aid of plates in the\nPhilosophical Transactions,[81] and it would be needless to repeat the\naccount of the somewhat intricate movements of the machine in this\nplace.\n\n  [81] *Philosophical Transactions* [1870], vol.\u00a0160, p.\u00a0497.\n  *Proceedings of the Royal Society*, vol.\u00a0xviii. p.\u00a0166, Jan. 20,\n  1870. *Nature*, vol, i. p. 343.\n\nThe general appearance of the machine is shown in a plate facing\nthe title-page of this volume. It somewhat resembles a very small\nupright piano or organ, and has a keyboard containing twenty-one keys.\nThese keys are of two kinds, sixteen of them representing the terms\nor letters A, *a*, B, *b*, C, *c*, D, *d*, which have so often been\nemployed in our logical notation. When letters occur on the left-hand\nside of a proposition, formerly called the subject, each is represented\nby a key on the left-hand half of the keyboard; but when they occur on\nthe right-hand side, or as it used to be called the predicate of the\nproposition, the letter-keys on the right-hand side of the keyboard are\nthe proper representatives. The five other keys may be called operation\nkeys, to distinguish them from the letter or term keys. They stand for\nthe stops, copula, and disjunctive conjunctions of a proposition. The\nmiddle key of all is the copula, to be pressed when the verb *is* or\nthe sign = is met. The key to the extreme right-hand is called the Full\nStop, because it should be pressed when a proposition is completed,\nin fact in the proper place of the full stop. The key to the extreme\nleft-hand is used to terminate an argument or to restore the machine to\nits initial condition; it is called the Finis key. The last keys but\none on the right and left complete the whole series, and represent the\nconjunction *or* in its unexclusive meaning, or the sign \ua58c which I have\nemployed, according as it occurs in the right or left hand side of the\nproposition. The whole keyboard is arranged as shown on the next page--\n\n  +-+-----------------------------------+-+-----------------------------------+---+\n  | |                                   |C|                                   |   |\n  |F|   Left-hand side of Proposition.  |o|  Right-hand side of Proposition.  |F S|\n  |i|                                   |p|                                   |u t|\n  |n+---+---+---+---+---+---+---+---+---+u+---+---+---+---+---+---+---+---+---+l o|\n  |i|   |   |   |   |   |   |   |   |   |l|   |   |   |   |   |   |   |   |   |l p|\n  |s|\ua58c|*d*| D |*c*| C |*b*| B |*a*| A |a| A |*a*| B |*b*| C |*c*| D |*d*|\ua58c|  .|\n  |.|Or |   |   |   |   |   |   |   |   |.|   |   |   |   |   |   |   |   | Or|   |\n  +-+---+---+---+---+---+---+---+---+---+-+---+---+---+---+---+---+---+---+---+---+\n\nTo work the machine it is only requisite to press the keys in\nsuccession as indicated by the letters and signs of a symbolical\nproposition. All the premises of an argument are supposed to be reduced\nto the simple notation which has been employed in the previous pages.\nTaking then such a simple proposition as\n\n  A = AB,\n\nwe press the keys A (left), copula, A (right), B (right), and full stop.\n\nIf there be a second premise, for instance\n\n  B = BC,\n\nwe press in like manner the keys--\n\n  B (left), copula, B (right), C (right), full stop.\n\nThe process is exactly the same however numerous the premises may be.\nWhen they are completed the operator will see indicated on the face of\nthe machine the exact combinations of letters which are consistent with\nthe premises according to the principles of thought.\n\nAs shown in the figure opposite the title-page, the machine exhibits in\nfront a Logical Alphabet of sixteen combinations, exactly like that of\nthe Abacus, except that the letters of each combination are separated\nby a certain interval. After the above problem has been worked upon the\nmachine the Logical Alphabet will have been modified so as to present\nthe following appearance--\n\n  +-------------------------------------------------------+\n  |                                                       |\n  +---+---+--+--+--+--+--+--+---+---+--+--+---+---+---+---+\n  | A | A |  |  |  |  |  |  |*a*|*a*|  |  |*a*|*a*|*a*|*a*|\n  +---+---+--+--+--+--+--+--+---+---+--+--+---+---+---+---+\n  |                                                       |\n  +---+---+--+--+--+--+--+--+---+---+--+--+---+---+---+---+\n  | B | B |  |  |  |  |  |  | B | B |  |  |*b*|*b*|*b*|*b*|\n  +---+---+--+--+--+--+--+--+---+---+--+--+---+---+---+---+\n  |                                                       |\n  +---+---+--+--+--+--+--+--+---+---+--+--+---+---+---+---+\n  | C | C |  |  |  |  |  |  | C | C |  |  | C | C |*c*|*c*|\n  +---+---+--+--+--+--+--+--+---+---+--+--+---+---+---+---+\n  |                                                       |\n  +---+---+--+--+--+--+--+--+---+---+--+--+---+---+---+---+\n  | D |*d*|  |  |  |  |  |  | D |*d*|  |  | D |*d*| D |*d*|\n  +---+---+--+--+--+--+--+--+---+---+--+--+---+---+---+---+\n  |                                                       |\n  +-------------------------------------------------------+\n\nThe operator will readily collect the various conclusions in the manner\ndescribed in previous pages, as, for instance that A is always C, that\nnot-C is not-B and not-A; and not-B is not-A but either C or not-C. The\nresults are thus to be read off exactly as in the case of the Logical\nSlate, or the Logical Abacus.\n\nDisjunctive propositions are to be treated in an exactly similar\nmanner. Thus, to work the premises\n\n  A = AB \ua58c AC\n  B \ua58c C = BD \ua58c CD,\n\nit is only necessary to press in succession the keys\n\n  A (left), copula, A (right), B, \ua58c, A, C, full stop.\n  B (left), \ua58c, C, copula, B (right), D, \ua58c, C, D, full stop.\n\nThe combinations then remaining will be as follows\n\n  ABCD        *a*BCD        *abc*D\n  AB*c*D      *a*B*c*D      *abcd.*\n  A*c*CD      *ab*CD\n\nOn pressing the left-hand key A, all the possible combinations which do\nnot contain A will disappear, and the description of A may be gathered\nfrom what remain, namely that it is always D. The full-stop key\nrestores all combinations consistent with the premises and any other\nselection may be made, as say not-D, which will be found to be always\nnot-A, not-B, and not-C.\n\nAt the end of every problem, when no further questions need be\naddressed to the machine, we press the Finis key, which has the effect\nof bringing into view the whole of the conceivable combinations of\nthe alphabet. This key in fact obliterates the conditions impressed\nupon the machine by moving back into their ordinary places those\ncombinations which had been rejected as inconsistent with the premises.\nBefore beginning any new problem it is requisite to observe that the\nwhole sixteen combinations are visible. After the Finis key has been\nused the machine represents a mind endowed with powers of thought,\nbut wholly devoid of knowledge. It would not in that condition give\nany answer but such as would consist in the primary laws of thought\nthemselves. But when any proposition is worked upon the keys, the\nmachine analyses and digests the meaning of it and becomes charged with\nthe knowledge embodied in that proposition. Accordingly it is able\nto return as an answer any description of a term or class so far as\nfurnished by that proposition in accordance with the Laws of Thought.\nThe machine is thus the embodiment of a true logical system. The\ncombinations are classified, selected or rejected, just as they should\nbe by a reasoning mind, so that at each step in a problem, the Logical\nAlphabet represents the proper condition of a mind exempt from mistake.\nIt cannot be asserted indeed that the machine entirely supersedes the\nagency of conscious thought; mental labour is required in interpreting\nthe meaning of grammatical expressions, and in correctly impressing\nthat meaning on the machine; it is further required in gathering\nthe conclusion from the remaining combinations. Nevertheless the\ntrue process of logical inference is really accomplished in a purely\nmechanical manner.\n\nIt is worthy of remark that the machine can detect any\nself-contradiction existing between the premises presented to it;\nshould the premises be self-contradictory it will be found that one or\nmore of the letter-terms disappears entirely from the Logical Alphabet.\nThus if we work the two propositions, A is B, and A is not-B, and then\ninquire for a description of A, the machine will refuse to give it\nby exhibiting no combination at all containing A. This result is in\nagreement with the law, which I have explained, that every term must\nhave its negative (p.\u00a074). Accordingly, whenever any one of the letters\nA, B, C, D, *a*, *b*, *c*, *d*, wholly disappears from the alphabet,\nit may be safely inferred that some act of self-contradiction has been\ncommitted.\n\nIt ought to be carefully observed that the logical machine cannot\nreceive a simple identity of the form A = B except in the double form\nof A\u00a0=\u00a0B and B\u00a0=\u00a0A. To work the proposition A\u00a0=\u00a0B, it is therefore\nnecessary to press the keys--\n\n  A (left), copula, B (right), full stop;\n  B (left), copula, A (right), full stop.\n\nThe same double operation will be necessary whenever the proposition is\nnot of the kind called a partial identity (p.\u00a040). Thus AB\u00a0= CD, AB\u00a0=\nAC, A\u00a0= B\u00a0\ua58c\u00a0C, A\u00a0\ua58c\u00a0B\u00a0= C\u00a0\ua58c\u00a0D, all require to be read from both ends\nseparately.\n\nThe proper rule for using the machine may in fact be given in the\nfollowing way:--(1) *Read each proposition as it stands, and play\nthe corresponding keys*: (2) *Convert the proposition and read and\nplay the keys again in the transposed order of the terms.* So long\nas this rule is observed the true result must always be obtained.\nThere can be no mistake. But it will be found that in the case of\npartial identities, and some other similar forms of propositions, the\ntransposed reading has no effect upon the combinations of the Logical\nAlphabet. One reading is in such cases all that is practically needful.\nAfter some experience has been gained in the use of the machine, the\nworker naturally saves himself the trouble of the second reading when\npossible.\n\nIt is no doubt a remarkable fact that a simple identity cannot\nbe impressed upon the machine except in the form of two partial\nidentities, and this may be thought by some logicians to militate\nagainst the equational mode of representing propositions.\n\nBefore leaving the subject I may remark that these mechanical devices\nare not likely to possess much practical utility. We do not require in\ncommon life to be constantly solving complex logical questions. Even in\nmathematical calculation the ordinary rules of arithmetic are generally\nsufficient, and a calculating machine can only be used with advantage\nin peculiar cases. But the machine and abacus have nevertheless two\nimportant uses.\n\nIn the first place I hope that the time is not very far distant when\nthe predominance of the ancient Aristotelian Logic will be a matter\nof history only, and when the teaching of logic will be placed on a\nfooting more worthy of its supreme importance. It will then be found\nthat the solution of logical questions is an exercise of mind at least\nas valuable and necessary as mathematical calculation. I believe that\nthese mechanical devices, or something of the same kind, will then\nbecome useful for exhibiting to a class of students a clear and visible\nanalysis of logical problems of any degree of complexity, the nature\nof each step being rendered plain to the eyes of the students. I often\nused the machine or abacus for this purpose in my class lectures while\nI was Professor of Logic at Owens College.\n\nSecondly, the more immediate importance of the machine seems to consist\nin the unquestionable proof which it affords that correct views of the\nfundamental principles of reasoning have now been attained, although\nthey were unknown to Aristotle and his followers. The time must come\nwhen the inevitable results of the admirable investigations of the late\nDr. Boole must be recognised at their true value, and the plain and\npalpable form in which the machine presents those results will, I hope,\nhasten the time. Undoubtedly Boole\u2019s life marks an era in the science\nof human reason. It may seem strange that it had remained for him first\nto set forth in its full extent the problem of logic, but I am not\naware that anyone before him had treated logic as a symbolic method\nfor evolving from any premises the description of any class whatsoever\nas defined by those premises. In spite of several serious errors into\nwhich he fell, it will probably be allowed that Boole discovered the\ntrue and general form of logic, and put the science substantially into\nthe form which it must hold for evermore. He thus effected a reform\nwith which there is hardly anything comparable in the history of logic\nbetween his time and the remote age of Aristotle.\n\nNevertheless, Boole\u2019s quasi-mathematical system could hardly be\nregarded as a final and unexceptionable solution of the problem. Not\nonly did it require the manipulation of mathematical symbols in a very\nintricate and perplexing manner, but the results when obtained were\ndevoid of demonstrative force, because they turned upon the employment\nof unintelligible symbols, acquiring meaning only by analogy. I\nhave also pointed out that he imported into his system a condition\nconcerning the exclusive nature of alternatives (p.\u00a070), which is\nnot necessarily true of logical terms. I shall have to show in the\nnext chapter that logic is really the basis of the whole science of\nmathematical reasoning, so that Boole inverted the true order of proof\nwhen he proposed to infer logical truths by algebraic processes. It is\nwonderful evidence of his mental power that by methods fundamentally\nfalse he should have succeeded in reaching true conclusions and\nwidening the sphere of reason.\n\nThe mechanical performance of logical inference affords a demonstration\nboth of the truth of Boole\u2019s results and of the mistaken nature of his\nmode of deducing them. Conclusions which he could obtain only by pages\nof intricate calculation, are exhibited by the machine after one or\ntwo minutes of manipulation. And not only are those conclusions easily\nreached, but they are demonstratively true, because every step of the\nprocess involves nothing more obscure than the three fundamental Laws\nof Thought.\n\n\n*The Order of Premises.*\n\nBefore quitting the subject of deductive reasoning, I may remark that\nthe order in which the premises of an argument are placed is a matter\nof logical indifference. Much discussion has taken place at various\ntimes concerning the arrangement of the premises of a syllogism; and it\nhas been generally held, in accordance with the opinion of Aristotle,\nthat the so-called major premise, containing the major term, or the\npredicate of the conclusion, should stand first. This distinction\nhowever falls to the ground in our system, since the proposition is\nreduced to an identical form, in which there is no distinction of\nsubject and predicate. In a strictly logical point of view the order\nof statement is wholly devoid of significance. The premises are\nsimultaneously coexistent, and are not related to each other according\nto the properties of space and time. Just as the qualities of the same\nobject are neither before nor after each other in nature (p.\u00a033), and\nare only thought of in some one order owing to the limited capacity of\nmind, so the premises of an argument are neither before nor after each\nother, and are only thought of in succession because the mind cannot\ngrasp many ideas at once. The combinations of the logical alphabet\nare exactly the same in whatever order the premises be treated on\nthe logical slate or machine. Some difference may doubtless exist as\nregards convenience to human memory. The mind may take in the results\nof an argument more easily in one mode of statement than another,\nalthough there is no real difference in the logical results. But in\nthis point of view I think that Aristotle and the old logicians were\nclearly wrong. It is more easy to gather the conclusion that \u201call A\u2019s\nare C\u2019s\u201d from \u201call A\u2019s are B\u2019s and all B\u2019s are C\u2019s,\u201d than from the same\npropositions in inverted order, \u201call B\u2019s are C\u2019s and all A\u2019s are B\u2019s.\u201d\n\n\n*The Equivalence of Propositions*.\n\nOne great advantage which arises from the study of this Indirect\nMethod of Inference consists in the clear notion which we gain of\nthe Equivalence of Propositions. The older logicians showed how from\ncertain simple premises we might draw an inference, but they failed\nto point out whether that inference contained the whole, or only a\npart, of the information embodied in the premises. Any one proposition\nor group of propositions may be classed with respect to another\nproposition or group of propositions, as\n\n  1. Equivalent,\n  2. Inferrible,\n  3. Consistent,\n  4. Contradictory.\n\nTaking the proposition \u201cAll men are mortals\u201d as the original, then\n\u201cAll immortals are not men\u201d is its equivalent; \u201cSome mortals are men\u201d\nis inferrible, or capable of inference, but is not equivalent; \u201cAll\nnot-men are not mortals\u201d cannot be inferred, but is consistent, that\nis, may be true at the same time; \u201cAll men are immortals\u201d is of course\ncontradictory.\n\nOne sufficient test of equivalence is capability of mutual inference.\nThus from\n\n  All electrics = all non-conductors,\n\nI can infer\n\n  All non-electrics = all conductors,\n\nand *vice vers\u00e2* from the latter I can pass back to the former. In\nshort, A\u00a0=\u00a0B is equivalent to *a*\u00a0=\u00a0*b*. Again, from the union of the\ntwo propositions, A = AB and B\u00a0=\u00a0AB, I get A\u00a0=\u00a0B, and from this I\nmight as easily deduce the two with which I started. In this case one\nproposition is equivalent to two other propositions. There are in fact\nno less than four modes in which we may express the identity of two\nclasses A and B, namely,\n\n  FIRST MODE.    SECOND MODE.    THIRD MODE.    FOURTH MODE.\n\n    A = B         *a* = *b*       A = AB }      *a* = *ab* }\n                                  B = AB }      *b* = *ab* }\n\nThe Indirect Method of Inference furnishes a universal and clear\ncriterion as to the relationship of propositions. The import of a\nstatement is always to be measured by the combinations of terms which\nit destroys. Hence two propositions are equivalent when they remove\nthe same combinations from the Logical Alphabet, and neither more nor\nless. A proposition is inferrible but not equivalent to another when\nit removes some but not all the combinations which the other removes,\nand none except what this other removes. Again, propositions are\nconsistent provided that they jointly allow each term and the negative\nof each term to remain somewhere in the Logical Alphabet. If after all\nthe combinations inconsistent with two propositions are struck out,\nthere still appears each of the letters A, *a*, B, *b*, C, *c*, D, *d*,\nwhich were there before, then no inconsistency between the propositions\nexists, although they may not be equivalent or even inferrible.\nFinally, contradictory propositions are those which taken together\nremove any one or more letter-terms from the Logical Alphabet.\n\nWhat is true of single propositions applies also to groups of\npropositions, however large or complicated; that is to say, one group\nmay be equivalent, inferrible, consistent, or contradictory as regards\nanother, and we may similarly compare one proposition with a group of\npropositions.\n\nTo give in this place illustrations of all the four kinds of relation\nwould require much space: as the examples given in previous sections or\nchapters may serve more or less to explain the relations of inference,\nconsistency, and contradiction, I will only add a few instances of\nequivalent propositions or groups.\n\nIn the following list each proposition or group of propositions is\nexactly equivalent in meaning to the corresponding one in the other\ncolumn, and the truth of this statement may be tested by working out\nthe combinations of the alphabet, which ought to be found exactly the\nsame in the case of each pair of equivalents.\n\n         A  =  A*b*  . . . . . . .  B   = *a*B\n         A  = *b*  . . . . . . . . *a*  =  B\n         A  =  BC  . . . . . . . . *a*  = *b* \ua58c *c*\n         A  =  AB \ua58c AC . . . . . . *b*  = *ab* \ua58c A*b*C\n      A \ua58cB  =  C \ua58c D . . . . . . . *ab* = *cd*\n    A \ua58c *c* =  B \ua58c *d* . . . . . . *a*C = *b*D\n         A  =  AB*c* \ua58c A*b*C . . .{ A   =  AB \ua58c AC\n                                  { AB  =  AB*c*\n\n          A = B }                   { A = B\n          B = C } . . . . . . . . . { A = C\n\n          A = AB }                  { A = AC\n          B = BC }. . . . . . . . . { B = A \ua58c *a*BC\n\nAlthough in these and many other cases the equivalents of certain\npropositions can readily be given, yet I believe that no uniform and\ninfallible process can be pointed out by which the exact equivalents\nof premises can be ascertained. Ordinary deductive inference usually\ngives us only a portion of the contained information. It is true that\nthe combinations consistent with a set of premises may always be thrown\ninto the form of a proposition which must be logically equivalent to\nthose premises; but the difficulty consists in detecting the other\nforms of propositions which will be equivalent to the premises. The\ntask is here of a different character from any which we have yet\nattempted. It is in reality an inverse process, and is just as much\nmore troublesome and uncertain than the direct process, as seeking is\ncompared with hiding. Not only may several different answers equally\napply, but there is no method of discovering any of those answers\nexcept by repeated trial. The problem which we have here met is really\nthat of induction, the inverse of deduction; and, as I shall soon show,\ninduction is always tentative, and, unless conducted with peculiar\nskill and insight, must be exceedingly laborious in cases of complexity.\n\nDe Morgan was unfortunately led by this equivalence of propositions\ninto the most serious error of his ingenious system of Logic. He held\nthat because the proposition \u201cAll A\u2019s are all B\u2019s,\u201d is but another\nexpression for the two propositions \u201cAll A\u2019s are B\u2019s\u201d and \u201cAll B\u2019s\nare A\u2019s,\u201d it must be a composite and not really an elementary form\nof proposition.[82] But on taking a general view of the equivalence\nof propositions such an objection seems to have no weight. Logicians\nhave, with few exceptions, persistently upheld the original error of\nAristotle in rejecting from their science the one simple relation of\nidentity on which all more complex logical relations must really rest.\n\n  [82] *Syllabus of a proposed system of Logic*, \u00a7\u00a7 57, 121, &c.\n  *Formal Logic*, p.\u00a066.\n\n\n*The Nature of Inference.*\n\nThe question, What is Inference? is involved, even to the present day,\nin as much uncertainty as that ancient question, What is Truth? I shall\nin more than one part of this work endeavour to show that inference\nnever does more than explicate, unfold, or develop the information\ncontained in certain premises or facts. Neither in deductive nor\ninductive reasoning can we add a tittle to our implicit knowledge,\nwhich is like that contained in an unread book or a sealed letter. Sir\nW. Hamilton has well said, \u201cReasoning is the showing out explicitly\nthat a proposition not granted or supposed, is implicitly contained in\nsomething different, which is granted or supposed.\u201d[83]\n\n  [83] Lectures on Metaphysics, vol.\u00a0iv. p.\u00a0369.\n\nProfessor Bowen has explained[84] with much clearness that the\nconclusion of an argument states explicitly what is virtually or\nimplicitly thought. \u201cThe process of reasoning is not so much a mode of\nevolving a new truth, as it is of establishing or proving an old one,\nby showing how much was admitted in the concession of the two premises\ntaken together.\u201d It is true that the whole meaning of these statements\nrests upon that of such words as \u201cexplicit,\u201d \u201cimplicit,\u201d \u201cvirtual.\u201d\nThat is implicit which is wrapped up, and we render it explicit when\nwe unfold it. Just as the conception of a circle involves a hundred\nimportant geometrical properties, all following from what we know, if\nwe have acuteness to unfold the results, so every fact and statement\ninvolves more meaning than seems at first sight. Reasoning explicates\nor brings to conscious possession what was before unconscious. It does\nnot create, nor does it destroy, but it transmutes and throws the same\nmatter into a new form.\n\n  [84] Bowen, *Treatise on Logic*, Cambridge, U.S., 1866; p.\u00a0362.\n\nThe difficult question still remains, Where does novelty of form begin?\nIs it a case of inference when we pass from \u201cSincerity is the parent of\ntruth\u201d to \u201cThe parent of truth is sincerity?\u201d The old logicians would\nhave called this change *conversion*, one case of immediate inference.\nBut as all identity is necessarily reciprocal, and the very meaning\nof such a proposition is that the two terms are identical in their\nsignification, I fail to see any difference between the statements\nwhatever. As well might we say that *x*\u00a0=\u00a0*y* and *y*\u00a0=\u00a0*x* are\ndifferent equations.\n\nAnother point of difficulty is to decide when a change is merely\ngrammatical and when it involves a real logical transformation. Between\na *table of wood* and a *wooden table* there is no logical difference\n(p.\u00a031), the adjective being merely a convenient substitute for the\nprepositional phrase. But it is uncertain to my mind whether the\nchange from \u201cAll men are mortal\u201d to \u201cNo men are not mortal\u201d is purely\ngrammatical. Logical change may perhaps be best described as consisting\nin the determination of a relation between certain classes of objects\nfrom a relation between certain other classes. Thus I consider it a\ntruly logical inference when we pass from \u201cAll men are mortal\u201d to \u201cAll\nimmortals are not-men,\u201d because the classes *immortals* and *not-men*\nare different from *mortals* and *men*, and yet the propositions\ncontain at the bottom the very same truth, as shown in the combinations\nof the Logical Alphabet.\n\nThe passage from the qualitative to the quantitative mode of expressing\na proposition is another kind of change which we must discriminate\nfrom true logical inference. We state the same truth when we say that\n\u201cmortality belongs to all men,\u201d as when we assert that \u201call men are\nmortals.\u201d Here we do not pass from class to class, but from one kind\nof term, the abstract, to another kind, the concrete. But inference\nprobably enters when we pass from either of the above propositions to\nthe assertion that the class of immortal men is zero, or contains no\nobjects.\n\nIt is of course a question of words to what processes we shall or shall\nnot apply the name \u201cinference,\u201d and I have no wish to continue the\ntrifling discussions which have already taken place upon the subject.\nWhat we need to do is to define accurately the sense in which we use\nthe word \u201cinference,\u201d and to distinguish the relation of inferrible\npropositions from other possible relations. It seems to be sufficient\nto recognise four modes in which two apparently different propositions\nmay be related. Thus two propositions may be--\n\n1. *Tautologous* or *identical*, involving the same relation between\nthe same terms and classes, and only differing in the order of\nstatement; thus \u201cVictoria is the Queen of England\u201d is tautologous with\n\u201cThe Queen of England is Victoria.\u201d\n\n2. *Grammatically related*, when the classes or objects are the same\nand similarly related, and the only difference is in the words; thus\n\u201cVictoria is the Queen of England\u201d is grammatically equivalent to\n\u201cVictoria is England\u2019s Queen.\u201d\n\n3. *Equivalents* in qualitative and quantitative form, the classes\nbeing the same, but viewed in a different manner.\n\n4. *Logically inferrible*, one from the other, or it may be\n*equivalent*, when the classes and relations are different, but involve\nthe same knowledge of the possible combinations.\n\n\n\n\nCHAPTER VII.\n\nINDUCTION.\n\n\nWe enter in this chapter upon the second great department of logical\nmethod, that of Induction or the Inference of general from particular\ntruths. It cannot be said that the Inductive process is of greater\nimportance than the Deductive process already considered, because the\nlatter process is absolutely essential to the existence of the former.\nEach is the complement and counterpart of the other. The principles\nof thought and existence which underlie them are at the bottom the\nsame, just as subtraction of numbers necessarily rests upon the same\nprinciples as addition. Induction is, in fact, the inverse operation of\ndeduction, and cannot be conceived to exist without the corresponding\noperation, so that the question of relative importance cannot arise.\nWho thinks of asking whether addition or subtraction is the more\nimportant process in arithmetic? But at the same time much difference\nin difficulty may exist between a direct and inverse operation; the\nintegral calculus, for instance, is infinitely more difficult than the\ndifferential calculus of which it is the inverse. Similarly, it must\nbe allowed that inductive investigations are of a far higher degree of\ndifficulty and complexity than any questions of deduction; and it is\nthis fact no doubt which led some logicians, such as Francis Bacon,\nLocke, and J.\u00a0S. Mill, to erroneous opinions concerning the exclusive\nimportance of induction.\n\nHitherto we have been engaged in considering how from certain\nconditions, laws, or identities governing the combinations of\nqualities, we may deduce the nature of the combinations agreeing\nwith those conditions. Our work has been to unfold the results of\nwhat is contained in any statements, and the process has been one of\n*Synthesis*. The terms or combinations of which the character has been\ndetermined have usually, though by no means always, involved more\nqualities, and therefore, by the relation of extension and intension,\nfewer objects than the terms in which they were described. The truths\ninferred were thus usually less general than the truths from which they\nwere inferred.\n\nIn induction all is inverted. The truths to be ascertained are more\ngeneral than the data from which they are drawn. The process by which\nthey are reached is *analytical*, and consists in separating the\ncomplex combinations in which natural phenomena are presented to us,\nand determining the relations of separate qualities. Given events\nobeying certain unknown laws, we have to discover the laws obeyed.\nInstead of the comparatively easy task of finding what effects will\nfollow from a given law, the effects are now given and the law is\nrequired. We have to interpret the will by which the conditions of\ncreation were laid down.\n\n\n*Induction an Inverse Operation*\n\nI have already asserted that induction is the inverse operation of\ndeduction, but the difference is one of such great importance that I\nmust dwell upon it. There are many cases in which we can easily and\ninfallibly do a certain thing but may have much trouble in undoing\nit. A person may walk into the most complicated labyrinth or the\nmost extensive catacombs, and turn hither and thither at his will;\nit is when he wishes to return that doubt and difficulty commence.\nIn entering, any path served him; in leaving, he must select certain\ndefinite paths, and in this selection he must either trust to memory\nof the way he entered or else make an exhaustive trial of all possible\nways. The explorer entering a new country makes sure his line of return\nby barking the trees.\n\nThe same difficulty arises in many scientific processes. Given any\ntwo numbers, we may by a simple and infallible process obtain their\nproduct; but when a large number is given it is quite another\nmatter to determine its factors. Can the reader say what two numbers\nmultiplied together will produce the number 8,616,460,799? I think it\nunlikely that anyone but myself will ever know; for they are two large\nprime numbers, and can only be rediscovered by trying in succession\na long series of prime divisors until the right one be fallen upon.\nThe work would probably occupy a good computer for many weeks, but it\ndid not occupy me many minutes to multiply the two factors together.\nSimilarly there is no direct process for discovering whether any number\nis a prime or not; it is only by exhaustively trying all inferior\nnumbers which could be divisors, that we can show there is none, and\nthe labour of the process would be intolerable were it not performed\nsystematically once for all in the process known as the Sieve of\nEratosthenes, the results being registered in tables of prime numbers.\n\nThe immense difficulties which are encountered in the solution of\nalgebraic equations afford another illustration. Given any algebraic\nfactors, we can easily and infallibly arrive at the product; but given\na product it is a matter of infinite difficulty to resolve it into\nfactors. Given any series of quantities however numerous, there is very\nlittle trouble in making an equation which shall have those quantities\nas roots. Let *a*, *b*, *c*, *d*, &c., be the quantities; then\n\n  (*x* - *a*)(*x* - *b*)(*x* - *c*)(*x* - d) ... = 0\n\nis the equation required, and we only need to multiply out the\nexpression on the left hand by ordinary rules. But having given a\ncomplex algebraic expression equated to zero, it is a matter of\nexceeding difficulty to discover all the roots. Mathematicians have\nexhausted their highest powers in carrying the complete solution up to\nthe fourth degree. In every other mathematical operation the inverse\nprocess is far more difficult than the direct process, subtraction than\naddition, division than multiplication, evolution than involution;\nbut the difficulty increases vastly as the process becomes more\ncomplex. Differentiation, the direct process, is always capable of\nperformance by fixed rules, but as these rules produce considerable\nvariety of results, the inverse process of integration presents\nimmense difficulties, and in an infinite majority of cases surpasses\nthe present resources of mathematicians. There are no infallible and\ngeneral rules for its accomplishment; it must be done by trial, by\nguesswork, or by remembering the results of differentiation, and using\nthem as a guide.\n\nComing more nearly to our own immediate subject, exactly the same\ndifficulty exists in determining the law which certain things obey.\nGiven a general mathematical expression, we can infallibly ascertain\nits value for any required value of the variable. But I am not aware\nthat mathematicians have ever attempted to lay down the rules of a\nprocess by which, having given certain numbers, one might discover a\nrational or precise formula from which they proceed. The reader may\ntest his power of detecting a law, by contemplation of its results, if\nhe, not being a mathematician, will attempt to point out the law obeyed\nby the following numbers:\n\n  1/6, 1/30, 1/42, 1/30, 5/66, 691/2730, 7/6, 3617/510, 43867/798, etc.\n\nThese numbers are sometimes in low terms, but unexpectedly spring up\nto high terms; in absolute magnitude they are very variable. They seem\nto set all regularity and method at defiance, and it is hardly to be\nsupposed that anyone could, from contemplation of the numbers, have\ndetected the relations between them. Yet they are derived from the\nmost regular and symmetrical laws of relation, and are of the highest\nimportance in mathematical analysis, being known as the numbers of\nBernoulli.\n\nCompare again the difficulty of decyphering with that of cyphering.\nAnyone can invent a secret language, and with a little steady labour\ncan translate the longest letter into the character. But to decypher\nthe letter, having no key to the signs adopted, is a wholly different\nmatter. As the possible modes of secret writing are infinite in number\nand exceedingly various in kind, there is no direct mode of discovery\nwhatever. Repeated trial, guided more or less by knowledge of the\ncustomary form of cypher, and resting entirely on the principles of\nprobability and logical induction, is the only resource. A peculiar\ntact or skill is requisite for the process, and a few men, such as\nWallis or Wheatstone, have attained great success.\n\nInduction is the decyphering of the hidden meaning of natural\nphenomena. Given events which happen in certain definite combinations,\nwe are required to point out the laws which govern those combinations.\nAny laws being supposed, we can, with ease and certainty, decide\nwhether the phenomena obey those laws. But the laws which may exist\nare infinite in variety, so that the chances are immensely against\nmere random guessing. The difficulty is much increased by the fact\nthat several laws will usually be in operation at the same time, the\neffects of which are complicated together. The only modes of discovery\nconsist either in exhaustively trying a great number of supposed laws,\na process which is exhaustive in more senses than one, or else in\ncarefully contemplating the effects, endeavouring to remember cases\nin which like effects followed from known laws. In whatever manner we\naccomplish the discovery, it must be done by the more or less conscious\napplication of the direct process of deduction.\n\nThe Logical Alphabet illustrates induction as well as deduction. In\nconsidering the Indirect Process of Inference we found that from\ncertain propositions we could infallibly determine the combinations\nof terms agreeing with those premises. The inductive problem is just\nthe inverse. Having given certain combinations of terms, we need to\nascertain the propositions with which the combinations are consistent,\nand from which they may have proceeded. Now, if the reader contemplates\nthe following combinations,\n\n   ABC       *ab*C\n  *a*BC      *abc*,\n\nhe will probably remember at once that they belong to the premises\nA\u00a0=\u00a0AB, B\u00a0=\u00a0BC (p.\u00a092). If not, he will require a few trials before he\nmeets with the right answer, and every trial will consist in assuming\ncertain laws and observing whether the deduced results agree with the\ndata. To test the facility with which he can solve this inductive\nproblem, let him casually strike out any of the combinations of the\nfourth column of the Logical Alphabet, (p.\u00a094), and say what laws\nthe remaining combinations obey, observing that every one of the\nletter-terms and their negatives ought to appear in order to avoid\nself-contradiction in the premises (pp.\u00a074, 111). Let him say, for\ninstance, what laws are embodied in the combinations\n\n  ABC       *a*BC\n  A*bc*     *ab*C.\n\nThe difficulty becomes much greater when more terms enter into the\ncombinations. It would require some little examination to ascertain the\ncomplete conditions fulfilled in the combinations\n\n   AC*e*      *ab*C*e*\n  *a*BC*e*    *abc*E.\n  *a*B*cd*E\n\nThe reader may discover easily enough that the principal laws are\nC\u00a0=\u00a0*e*, and A\u00a0=\u00a0A*e*; but he would hardly discover without some\ntrouble the remaining law, namely, that BD\u00a0=\u00a0BD*e*.\n\nThe difficulties encountered in the inductive investigations of\nnature, are of an exactly similar kind. We seldom observe any law in\nuninterrupted and undisguised operation. The acuteness of Aristotle and\nthe ancient Greeks did not enable them to detect that all terrestrial\nbodies tend to fall towards the centre of the earth. A few nights of\nobservation might have convinced an astronomer viewing the solar system\nfrom its centre, that the planets travelled round the sun; but the\nfact that our place of observation is one of the travelling planets,\nso complicates the apparent motions of the other bodies, that it\nrequired all the sagacity of Copernicus to prove the real simplicity of\nthe planetary system. It is the same throughout nature; the laws may\nbe simple, but their combined effects are not simple, and we have no\nclue to guide us through their intricacies. \u201cIt is the glory of God,\u201d\nsaid Solomon, \u201cto conceal a thing, but the glory of a king to search\nit out.\u201d The laws of nature are the invaluable secrets which God has\nhidden, and it is the kingly prerogative of the philosopher to search\nthem out by industry and sagacity.\n\n\n*Inductive Problems for Solution by the Reader.*\n\nIn the first edition (vol.\u00a0ii. p.\u00a0370) I gave a logical problem\ninvolving six terms, and requested readers to discover the laws\ngoverning the combinations given. I received satisfactory replies\nfrom readers both in the United States and in England. I formed\nthe combinations deductively from four laws of correction, but my\ncorrespondents found that three simpler laws, equivalent to the four\nmore complex ones, were the best answer; these laws are as follows:\n*a*\u00a0=\u00a0*ac*, *b*\u00a0=\u00a0*cd*, *d*\u00a0=\u00a0E*f*.\n\nIn case other readers should like to test their skill in the inductive\nor inverse problem, I give below several series of combinations forming\nproblems of graduated difficulty.\n\n  PROBLEM I.\n\n       A  B *c*\n       A *b* C\n      *a* B  C\n\n  PROBLEM II.\n\n       A  B  C\n       A *b* C\n      *a* B  C\n      *a* B *c*\n\n  PROBLEM III.\n\n       A   B   C\n       A  *b*  C\n      *a*  B   C\n      *a*  B  *c*\n      *a* *b* *c*\n\n  PROBLEM IV.\n\n       A   B   C   D\n       A  *b* *c*  D\n      *a*  B  *c* *d*\n      *a* *b*  C  *d*\n\n  PROBLEM V.\n\n       A   B   C   D\n       A   B   C  *d*\n       A   B  *c* *d*\n       A  *b*  C   D\n       A  *b* *c*  D\n      *a*  B   C   D\n      *a*  B  *c*  D\n      *a*  B  *c* *d*\n      *a* *b*  C  *d*\n\n  PROBLEM VI.\n\n       A   B   C   D   E\n       A   B   C  *d* *e*\n       A   B  *c*  D   E\n       A   B  *c* *d* *e*\n       A  *b*  C   D   E\n      *a*  B   C   D   E\n      *a*  B   C  *d* *e*\n      *a* *b*  C   D   E\n      *a* *b* *c* *d* *e*\n\n  PROBLEM VII.\n\n       A  *b* *c*  D  *e*\n      *a*  B   C  *d*  E\n      *a* *b*  C  *d*  E\n\n  PROBLEM VIII.\n\n       A   B   C   D   E\n       A   B   C   D  *e*\n       A   B   C  *d* *e*\n       A   B  *c* *d* *e*\n       A  *b*  C   D   E\n       A  *b* *c* *d*  E\n       A  *b* *c* *d* *e*\n      *a*  B   C   D  *e*\n      *a*  B   C  *d* *e*\n      *a*  B  *c*  D  *e*\n      *a* *b*  C   D  *e*\n      *a* *b*  C  *d*  E\n      *a* *b* *c*  D  *e*\n      *a* *b* *c* *d*  E\n\n  PROBLEM IX.\n\n       A   B  *c*  D   E   F\n       A   B  *c*  D  *e*  F\n       A  *b*  C   D  *e* *f*\n       A  *b* *c*  D   E  *f*\n       A  *b* *c*  D  *e* *f*\n       A  *b* *c* *d*  E   F\n       A  *b* *c* *d* *e*  F\n      *a*  B  *c*  D   E   F\n      *a*  B  *c*  D  *e*  F\n      *a*  B  *c* *d*  E   F\n      *a* *b*  C   D   E   F\n      *a* *b*  C   D  *e*  F\n      *a* *b*  C   D  *e* *f*\n      *a* *b* *c*  D  *e* *f*\n      *a* *b* *c*  D   E  *f*\n      *a* *b* *c* *d* *e*  F\n\n  PROBLEM X.\n\n       A   B   C   D  *e*  F\n       A   B  *c*  D   E  *f*\n       A  *b*  C   D   E   F\n       A  *b*  C   D  *e*  F\n       A  *b* *c*  D  *e*  F\n      *a*  B   C   D   E  *f*\n      *a*  B  *c*  D   E  *f*\n      *a* *b*  C   D  *e*  F\n      *a* *b*  C  *d* *e*  F\n      *a* *b* *c*  D  *e* *f*\n      *a* *b* *c* *d* *e* *f*\n\n\n*Induction of Simple Identities*.\n\nMany important laws of nature are expressible in the form of simple\nidentities, and I can at once adduce them as examples to illustrate\nwhat I have said of the difficulty of the inverse process of induction.\nTwo phenomena are conjoined. Thus all gravitating matter is exactly\ncoincident with all matter possessing inertia; where one property\nappears, the other likewise appears. All crystals of the cubical\nsystem, are all the crystals which do not doubly refract light. All\nexogenous plants are, with some exceptions, those which have two\ncotyledons or seed-leaves.\n\nA little reflection will show that there is no direct and infallible\nprocess by which such complete coincidences may be discovered.\nNatural objects are aggregates of many qualities, and any one of\nthose qualities may prove to be in close connection with some others.\nIf each of a numerous group of objects is endowed with a hundred\ndistinct physical or chemical qualities, there will be no less than\n(1/2)(100\u00a0\u00d7\u00a099) or 4950 pairs of qualities, which may be connected,\nand it will evidently be a matter of great intricacy and labour to\nascertain exactly which qualities are connected by any simple law.\n\nOne principal source of difficulty is that the finite powers of the\nhuman mind are not sufficient to compare by a single act any large\ngroup of objects with another large group. We cannot hold in the\nconscious possession of the mind at any one moment more than five or\nsix different ideas. Hence we must treat any more complex group by\nsuccessive acts of attention. The reader will perceive by an almost\nindividual act of comparison that the words *Roma* and *Mora* contain\nthe same letters. He may perhaps see at a glance whether the same is\ntrue of *Causal* and *Casual*, and of *Logica* and *Caligo*. To assure\nhimself that the letters in *Astronomers* make *No more stars*, that\n*Serpens in akuleo* is an anagram of *Joannes Keplerus*, or *Great gun\ndo us a sum* an anagram of *Augustus de Morgan*, it will certainly be\nnecessary to break up the act of comparison into several successive\nacts. The process will acquire a double character, and will consist in\nascertaining that each letter of the first group is among the letters\nof the second group, and *vice vers\u00e2*, that each letter of the second\nis among those of the first group. In the same way we can only prove\nthat two long lists of names are identical, by showing that each name\nin one list occurs in the other, and *vice vers\u00e2*.\n\nThis process of comparison really consists in establishing two partial\nidentities, which are, as already shown (p.\u00a058), equivalent in\nconjunction to one simple identity. We first ascertain the truth of the\ntwo propositions A\u00a0=\u00a0AB, B\u00a0=\u00a0AB, and we then rise by substitution to\nthe single law A\u00a0=\u00a0B.\n\nThere is another process, it is true, by which we may get to exactly\nthe same result; for the two propositions A\u00a0=\u00a0AB, *a*\u00a0=\u00a0*ab* are also\nequivalent to the simple identity A\u00a0=\u00a0B. If then we can show that\nall objects included under A are included under B, and also that all\nobjects not included under A are not included under B, our purpose is\neffected. By this process we should usually compare two lists if we are\nallowed to mark them. For each name in the first list we should strike\noff one in the second, and if, when the first list is exhausted, the\nsecond list is also exhausted, it follows that all names absent from\nthe first must be absent from the second, and the coincidence must be\ncomplete.\n\nThese two modes of proving an identity are so closely allied that it\nis doubtful how far we can detect any difference in their powers and\ninstances of application. The first method is perhaps more convenient\nwhen the phenomena to be compared are rare. Thus we prove that all the\nmusical concords coincide with all the more simple numerical ratios, by\nshowing that each concord arises from a simple ratio of undulations,\nand then showing that each simple ratio gives rise to one of the\nconcords. To examine all the possible cases of discord or complex ratio\nof undulation would be impossible. By a happy stroke of induction Sir\nJohn Herschel discovered that all crystals of quartz which cause the\nplane of polarization of light to rotate are precisely those crystals\nwhich have plagihedral faces, that is, oblique faces on the corners of\nthe prism unsymmetrical with the ordinary faces. This singular relation\nwould be proved by observing that all plagihedral crystals possessed\nthe power of rotation, and *vice vers\u00e2* all crystals possessing this\npower were plagihedral. But it might at the same time be noticed that\nall ordinary crystals were devoid of the power. There is no reason\nwhy we should not detect any of the four propositions A\u00a0=\u00a0AB, B\u00a0=\u00a0AB,\n*a*\u00a0=\u00a0*ab*, *b*\u00a0=\u00a0*ab*, all of which follow from A\u00a0=\u00a0B (p.\u00a0115).\n\nSometimes the terms of the identity may be singular objects; thus we\nobserve that diamond is a combustible gem, and being unable to discover\nany other that is, we affirm--\n\n  Diamond = combustible gem.\n\nIn a similar manner we ascertain that\n\n  Mercury = metal liquid at ordinary temperatures,\n  Substance of least density = substance of least atomic weight.\n\nTwo or three objects may occasionally enter into the induction, as when\nwe learn that\n\n  Sodium \ua58c potassium = metal of less density than water,\n\n  Venus \ua58c Mercury \ua58c Mars = major planet devoid of satellites.\n\n\n*Induction of Partial Identities*.\n\nWe found in the last section that the complete identity of two classes\nis almost always discovered not by direct observation of the fact,\nbut by first establishing two partial identities. There are also a\nmultitude of cases in which the partial identity of one class with\nanother is the only relation to be discovered. Thus the most common\nof all inductive inferences consists in establishing the fact that\nall objects having the properties of A have also those of B, or that\nA\u00a0=\u00a0AB. To ascertain the truth of a proposition of this kind it is\nmerely necessary to assemble together, mentally or physically, all the\nobjects included under A, and then observe whether B is present in\neach of them, or, which is the same, whether it would be impossible\nto select from among them any not-B. Thus, if we mentally assemble\ntogether all the heavenly bodies which move with apparent rapidity,\nthat is to say, the planets, we find that they all possess the property\nof not scintillating. We cannot analyse any vegetable substance without\ndiscovering that it contains carbon and hydrogen, but it is not true\nthat all substances containing carbon and hydrogen are vegetable\nsubstances.\n\nThe great mass of scientific truths consists of propositions of\nthis form A\u00a0=\u00a0AB. Thus in astronomy we learn that all the planets\nare spheroidal bodies; that they all revolve in one direction round\nthe sun; that they all shine by reflected light; that they all obey\nthe law of gravitation. But of course it is not to be asserted that\nall bodies obeying the law of gravitation, or shining by reflected\nlight, or revolving in a particular direction, or being spheroidal\nin form, are planets. In other sciences we have immense numbers of\npropositions of the same form, as, for instance, all substances in\nbecoming gaseous absorb heat; all metals are elements; they are all\ngood conductors of heat and electricity; all the alkaline metals are\nmonad elements; all foraminifera are marine organisms; all parasitic\nanimals are non-mammalian; lightning never issues from stratous clouds;\npumice never occurs where only Labrador felspar is present; milkmaids\ndo not suffer from small-pox; and, in the works of Darwin, scientific\nimportance may attach even to such an apparently trifling observation\nas that \u201cwhite tom-cats having blue eyes are deaf.\u201d\n\nThe process of inference by which all such truths are obtained may\nreadily be exhibited in a precise symbolic form. We must have one\npremise specifying in a disjunctive form all the possible individuals\nwhich belong to a class; we resolve the class, in short, into its\nconstituents. We then need a number of propositions, each of which\naffirms that one of the individuals possesses a certain property. Thus\nthe premises must be of the forms\n\n  A\u00a0=\u00a0B\u00a0\ua58c\u00a0C\u00a0\ua58c\u00a0D\u00a0\ua58c\u00a0....\u00a0\ua58c\u00a0P\u00a0\ua58c\u00a0Q\n                B = BX\n                C = CX\n                ... ...\n                ... ...\n                Q = QX.\n\nNow, if we substitute for each alternative of the first premise its\ndescription as found among the succeeding premises, we obtain\n\n  A\u00a0=\u00a0BX\u00a0\ua58c\u00a0CX\u00a0\ua58c\u00a0....\u00a0\ua58c\u00a0PX\u00a0\ua58c\u00a0QX\n\nor\n\n  A\u00a0=\u00a0(B\u00a0\ua58c\u00a0C\u00a0\ua58c\u00a0....\u00a0\ua58c\u00a0Q)X\n\nBut for the aggregate of alternatives we may now substitute their\nequivalent as given in the first premise, namely A, so that we get the\nrequired result:\n\n  A = AX.\n\nWe should have reached the same result if the first premise had been of\nthe form\n\n  A\u00a0=\u00a0AB\u00a0\ua58c\u00a0AC\u00a0\ua58c\u00a0....\u00a0\ua58c\u00a0AQ.\n\nWe can always prove a proposition, if we find it more convenient, by\nproving its equivalent. To assert that all not-B\u2019s are not-A\u2019s, is\nexactly the same as to assert that all A\u2019s are B\u2019s. Accordingly we may\nascertain that A\u00a0=\u00a0AB by first ascertaining that *b*\u00a0=\u00a0*ab*. If we\nobserve, for instance, that all substances which are not solids are\nalso not capable of double refraction, it follows necessarily that all\ndouble refracting substances are solids. We may convince ourselves that\nall electric substances are nonconductors of electricity, by reflecting\nthat all good conductors do not, and in fact cannot, retain electric\nexcitation. When we come to questions of probability it will be found\ndesirable to prove, as far as possible, both the original proposition\nand its equivalent, as there is then an increased area of observation.\n\nThe number of alternatives which may arise in the division of a class\nvaries greatly, and may be any number from two upwards. Thus it is\nprobable that every substance is either magnetic or diamagnetic, and no\nsubstance can be both at the same time. The division then must be made\nin the form\n\n  A = AB*c* \ua58c A*b*C.\n\nIf now we can prove that all magnetic substances are capable of\npolarity, say B\u00a0=\u00a0BD, and also that all diamagnetic substances are\ncapable of polarity, C\u00a0=\u00a0CD, it follows by substitution that all\nsubstances are capable of polarity, or A = AD. We commonly divide\nthe class substance into the three subclasses, solid, liquid, and\ngas; and if we can show that in each of these forms it obeys Carnot\u2019s\nthermodynamic law, it follows that all substances obey that law.\nSimilarly we may show that all vertebrate animals possess red blood,\nif we can show separately that fish, reptiles, birds, marsupials, and\nmammals possess red blood, there being, as far as is known, only five\nprincipal subclasses of vertebrata.\n\nOur inductions will often be embarrassed by exceptions, real or\napparent. We might affirm that all gems are incombustible were not\ndiamonds undoubtedly combustible. Nothing seems more evident than that\nall the metals are opaque until we examine them in fine films, when\ngold and silver are found to be transparent. All plants absorb carbonic\nacid except certain fungi; all the bodies of the planetary system\nhave a progressive motion from west to east, except the satellites of\nUranus and Neptune. Even some of the profoundest laws of matter are not\nquite universal; all solids expand by heat except india-rubber, and\npossibly a few other substances; all liquids which have been tested\nexpand by heat except water below 4\u00b0\u00a0C. and fused bismuth; all gases\nhave a coefficient of expansion increasing with the temperature, except\nhydrogen. In a later chapter I shall consider how such anomalous cases\nmay be regarded and classified; here we have only to express them in a\nconsistent manner by our notation.\n\nLet us take the case of the transparency of metals, and assign the\nterms thus:--\n\n  A = metal    D = iron\n  B = gold     E, F, &c. = copper, lead, &c.\n  C = silver   X = opaque.\n\nOur premises will be\n\n  A = B \ua58c C \ua58c D \ua58c E, &c.\n  B = B*x*\n  C = C*x*\n  D = DX\n  E = EX,\n\nand so on for the rest of the metals. Now evidently\n\n  A*bc*\u00a0=\u00a0(D\u00a0\ua58c\u00a0E\u00a0\ua58c\u00a0F\u00a0\ua58c\u00a0...)*bc*,\n\nand by substitution as before we shall obtain\n\n  A*bc* = A*bc*X,\n\nor in words, \u201cAll metals not gold nor silver are opaque;\u201d at the same\ntime we have\n\n  A(B\u00a0\ua58c\u00a0C)\u00a0=\u00a0AB\u00a0\ua58c\u00a0AC\u00a0=\u00a0AB*x*\u00a0\ua58c\u00a0AC*x*\u00a0=\u00a0A(B\u00a0\ua58c\u00a0C)*x*,\n\nor \u201cMetals which are either gold or silver are not opaque.\u201d\n\nIn some cases the problem of induction assumes a much higher degree of\ncomplexity. If we examine the properties of crystallized substances\nwe may find some properties which are common to all, as cleavage or\nfracture in definite planes; but it would soon become requisite to\nbreak up the class into several minor ones. We should divide crystals\naccording to the seven accepted systems--and we should then find\nthat crystals of each system possess many common properties. Thus\ncrystals of the Regular or Cubical system expand equally by heat,\nconduct heat and electricity with uniform rapidity, and are of like\nelasticity in all directions; they have but one index of refraction\nfor light; and every facet is repeated in like relation to each of\nthe three axes. Crystals of the system having one principal axis\nwill be found to possess the various physical powers of conduction,\nrefraction, elasticity, &c., uniformly in directions perpendicular\nto the principal axis; in other directions their properties vary\naccording to complicated laws. The remaining systems in which the\ncrystals possess three unequal axes, or have inclined axes, exhibit\nstill more complicated results, the effects of the crystal upon light,\nheat, electricity, &c., varying in all directions. But when we pursue\ninduction into the intricacies of its application to nature we really\nenter upon the subject of classification, which we must take up again\nin a later part of this work.\n\n\n*Solution of the Inverse or Inductive Problem, involving Two Classes*.\n\nIt is now plain that Induction consists in passing back from a series\nof combinations to the laws by which such combinations are governed.\nThe natural law that all metals are conductors of electricity really\nmeans that in nature we find three classes of objects, namely--\n\n  1. Metals, conductors;\n  2. Not-metals, conductors;\n  3. Not-metals, not-conductors.\n\nIt comes to the same thing if we say that it excludes the existence\nof the class, \u201cmetals not-conductors.\u201d In the same way every other\nlaw or group of laws will really mean the exclusion from existence\nof certain combinations of the things, circumstances or phenomena\ngoverned by those laws. Now in logic, strictly speaking, we treat not\nthe phenomena, nor the laws, but the general forms of the laws; and a\nlittle consideration will show that for a finite number of things the\npossible number of forms or kinds of law governing them must also be\nfinite. Using general terms, we know that A and B can be present or\nabsent in four ways and no more--thus:\n\n  AB, A*b*, *a*B, *ab*;\n\ntherefore every possible law which can exist concerning the relation\nof A and B must be marked by the exclusion of one or more of the above\ncombinations. The number of possible laws then cannot exceed the\nnumber of selections which we can make from these four combinations.\nSince each combination may be present or absent, the number of cases\nto be considered is 2\u00a0\u00d7\u00a02\u00a0\u00d7\u00a02\u00a0\u00d7\u00a02, or sixteen; and these cases are all\nshown in the following table, in which the sign 0 indicates absence or\nnon-existence of the combination shown at the left-hand column in the\nsame line, and the mark 1 its presence:--\n\n        1  2  3  4  5  6  7  8  9  10 11 12 13 14 15 16\n                          *  *     *     *     *  *  *\n   AB   0  0  0  0  0  0  0  0  1  1  1  1  1  1  1  1\n   A*b* 0  0  0  0  1  1  1  1  0  0  0  0  1  1  1  1\n  *a*B  0  0  1  1  0  0  1  1  0  0  1  1  0  0  1  1\n  *ab*  0  1  0  1  0  1  0  1  0  1  0  1  0  1  0  1\n\nThus in column sixteen we find that all the conceivable combinations\nare present, which means that there are no special laws in existence\nin such a case, and that the combinations are governed only by the\nuniversal Laws of Identity and Difference. The example of metals and\nconductors of electricity would be represented by the twelfth column;\nand every other mode in which two things or qualities might present\nthemselves is shown in one or other of the columns. More than half\nthe cases may indeed be at once rejected, because they involve the\nentire absence of a term or its negative. It has been shown to be a\nlogical principle that every term must have its negative (p.\u00a0111),\nand when this is not the case, inconsistency between the conditions\nof combination must exist. Thus if we laid down the two following\npropositions, \u201cGraphite conducts electricity,\u201d and \u201cGraphite does not\nconduct electricity,\u201d it would amount to asserting the impossibility\nof graphite existing at all; or in general terms, A is B and A is\nnot B result in destroying altogether the combinations containing A,\na case shown in the fourth column of the above table. We therefore\nrestrict our attention to those cases which may be represented in\nnatural phenomena when at least two combinations are present, and which\ncorrespond to those columns of the table in which each of A, *a*,\nB, *b* appears. These cases are shown in the columns marked with an\nasterisk.\n\nWe find that seven cases remain for examination, thus characterised--\n\n  Four cases exhibiting three combinations,\n  Two cases exhibiting two combinations,\n  One case exhibiting four combinations.\n\nIt has already been pointed out that a proposition of the form A =\nAB destroys one combination, A*b*, so that this is the form of law\napplying to the twelfth column. But by changing one or more of the\nterms in A\u00a0=\u00a0AB into its negative, or by interchanging A and B, *a* and\n*b*, we obtain no less than eight different varieties of the one form;\nthus--\n\n  12th case.    8th case.    15th case.    14th case.\n   A  =  AB     A =  A*b*    *a* = *a*B    *a* = *ab*\n  *b* = *ab*    B = *a*B     *b* =  A*b*    B  = AB\n\nThe reader of the preceding sections will see that each proposition\nin the lower line is logically equivalent to, and is in fact the\ncontrapositive of, that above it (p.\u00a083). Thus the propositions\nA\u00a0=\u00a0A*b* and B\u00a0=\u00a0*a*B both give the same combinations, shown in the\neighth column of the table, and trial shows that the twelfth, eighth,\nfifteenth and fourteenth columns are thus accounted for. We come to\nthis conclusion then--*The general form of proposition* A\u00a0=\u00a0AB *admits\nof four logically distinct varieties, each capable of expression in two\nmodes*.\n\nIn two columns of the table, namely the seventh and tenth, we observe\nthat two combinations are missing. Now a simple identity A\u00a0=\u00a0B renders\nimpossible both A*b* and *a*B, accounting for the tenth case; and if we\nchange B into *b* the identity A\u00a0=\u00a0*b* accounts for the seventh case.\nThere may indeed be two other varieties of the simple identity, namely\n*a*\u00a0=\u00a0*b* and *a*\u00a0=\u00a0B; but it has already been shown repeatedly that\nthese are equivalent respectively to A\u00a0=\u00a0B and A\u00a0=\u00a0*b* (p.\u00a0115). As\nthe sixteenth column has already been accounted for as governed by no\nspecial conditions, we come to the following general conclusion:--The\nlaws governing the combinations of two terms must be capable of\nexpression either in a partial identity or a simple identity; the\npartial identity is capable of only four logically distinct varieties,\nand the simple identity of two. Every logical relation between two\nterms must be expressed in one of these six forms of law, or must be\nlogically equivalent to one of them.\n\nIn short, we may conclude that in treating of partial and complete\nidentity, we have exhaustively treated the modes in which two terms or\nclasses of objects can be related. Of any two classes it can be said\nthat one must either be included in the other, or must be identical\nwith it, or a like relation must exist between one class and the\nnegative of the other. We have thus completely solved the inverse\nlogical problem concerning two terms.[85]\n\n  [85] The contents of this and the following section nearly correspond\n  with those of a paper read before the Manchester Literary and\n  Philosophical Society on December 26th, 1871. See Proceedings of the\n  Society, vol.\u00a0xi. pp.\u00a065\u201368, and Memoirs, Third Series, vol.\u00a0v. pp.\n  119\u2013130.\n\n\n*The Inverse Logical Problem involving Three Classes.*\n\nNo sooner do we introduce into the problem a third term C, than the\ninvestigation assumes a far more complex character, so that some\nreaders may prefer to pass over this section. Three terms and their\nnegatives may be combined, as we have frequently seen, in eight\ndifferent combinations, and the effect of laws or logical conditions\nis to destroy any one or more of these combinations. Now we may make\nselections from eight things in 2^{8} or 256 ways; so that we have no\nless than 256 different cases to treat, and the complete solution is\nat least fifty times as troublesome as with two terms. Many series of\ncombinations, indeed, are contradictory, as in the simpler problem,\nand may be passed over, the test of consistency being that each of the\nletters A, B, C, *a*, *b*, *c*, shall appear somewhere in the series of\ncombinations.\n\nMy mode of solving the problem was as follows:--Having written out the\nwhole of the 256 series of combinations, I examined them separately and\nstruck out such as did not fulfil the test of consistency. I then chose\nsome form of proposition involving two or three terms, and varied it\nin every possible manner, both by the circular interchange of letters\n(A, B, C into B, C, A and then into C, A, B), and by the substitution\nfor any one or more of the terms of the corresponding negative terms.\nFor instance, the proposition AB = ABC can be first varied by circular\ninterchange so as to give BC = BCA and then CA\u00a0=\u00a0CAB. Each of these\nthree can then be thrown into eight varieties by negative change. Thus\nAB = ABC gives *a*B\u00a0=\u00a0*a*BC, A*b*\u00a0=\u00a0A*b*C, AB\u00a0=\u00a0AB*c*, *ab*\u00a0=\u00a0*ab*C,\nand so on. Thus there may possibly exist no less than twenty-four\nvarieties of the law having the general form AB\u00a0=\u00a0ABC, meaning that\nwhatever has the properties of A and B has those also of C. It by no\nmeans follows that some of the varieties may not be equivalent to\nothers; and trial shows, in fact, that AB\u00a0=\u00a0ABC is exactly the same\nin meaning as A*c*\u00a0=\u00a0A*bc* or B*c*\u00a0=\u00a0B*ca*. Thus the law in question\nhas but eight varieties of distinct logical meaning. I now ascertain\nby actual deductive reasoning which of the 256 series of combinations\nresult from each of these distinct laws, and mark them off as soon as\nfound. I then proceed to some other form of law, for instance A = ABC,\nmeaning that whatever has the qualities of A has those also of B and\nC. I find that it admits of twenty-four variations, all of which are\nfound to be logically distinct; the combinations being worked out, I am\nable to mark off twenty-four more of the list of 256 series. I proceed\nin this way to work out the results of every form of law which I can\nfind or invent. If in the course of this work I obtain any series of\ncombinations which had been previously marked off, I learn at once that\nthe law giving these combinations is logically equivalent to some law\npreviously treated. It may be safely inferred that every variety of the\napparently new law will coincide in meaning with some variety of the\nformer expression of the same law. I have sufficiently verified this\nassumption in some cases, and have never found it lead to error. Thus\nas AB\u00a0=\u00a0ABC is equivalent to A*c*\u00a0=\u00a0A*bc*, so we find that *ab*\u00a0=\u00a0*ab*C\nis equivalent to *ac*\u00a0=\u00a0*ac*B.\n\nAmong the laws treated were the two A = AB and A = B which involve only\ntwo terms, because it may of course happen that among three things two\nonly are in special logical relation, and the third independent; and\nthe series of combinations representing such cases of relation are sure\nto occur in the complete enumeration. All single propositions which\nI could invent having been treated, pairs of propositions were next\ninvestigated. Thus we have the relations, \u201cAll A\u2019s are B\u2019s, and all\nB\u2019s are C\u2019s,\u201d of which the old logical syllogism is the development.\nWe may also have \u201call A\u2019s are all B\u2019s, and all B\u2019s are C\u2019s,\u201d or even\n\u201call A\u2019s are all B\u2019s, and all B\u2019s are all C\u2019s.\u201d All such premises admit\nof variations, greater or less in number, the logical distinctness\nof which can only be determined by trial in detail. Disjunctive\npropositions either singly or in pairs were also treated, but were\noften found to be equivalent to other propositions of a simpler form;\nthus A\u00a0=\u00a0ABC\u00a0\ua58c\u00a0A*bc* is exactly the same in meaning as AB\u00a0=\u00a0AC.\n\nThis mode of exhaustive trial bears some analogy to that ancient\nmathematical process called the Sieve of Eratosthenes. Having taken\na long series of the natural numbers, Eratosthenes is said to have\ncalculated out in succession all the multiples of every number, and\nto have marked them off, so that at last the prime numbers alone\nremained, and the factors of every number were exhaustively discovered.\nMy problem of 256 series of combinations is the logical analogue, the\nchief points of difference being that there is a limit to the number of\ncases, and that prime numbers have no analogue in logic, since every\nseries of combinations corresponds to a law or group of conditions.\nBut the analogy is perfect in the point that they are both inverse\nprocesses. There is no mode of ascertaining that a number is prime but\nby showing that it is not the product of any assignable factors. So\nthere is no mode of ascertaining what laws are embodied in any series\nof combinations but trying exhaustively the laws which would give them.\nJust as the results of Eratosthenes\u2019 method have been worked out to\na great extent and registered in tables for the convenience of other\nmathematicians, I have endeavoured to work out the inverse logical\nproblem to the utmost extent which is at present practicable or useful.\n\nI have thus found that there are altogether fifteen conditions or\nseries of conditions which may govern the combinations of three\nterms, forming the premises of fifteen essentially different kinds\nof arguments. The following table contains a statement of these\nconditions, together with the numbers of combinations which are\ncontradicted or destroyed by each, and the numbers of logically\ndistinct variations of which the law is capable. There might be\nalso added, as a sixteenth case, that case where no special logical\ncondition exists, so that all the eight combinations remain.\n\n  +---------+-------------------------------+-----------+------------+\n  |         |                               | Number of |  Number of |\n  |Reference|  Propositions expressing the  | distinct  |combinations|\n  | Number. |  general type of the logical  |  logical  |contradicted|\n  |         |         conditions.           |variations.|  by each.  |\n  +---------+-------------------------------+-----------+------------+\n  |     I.  |  A = B                        |      6    |     4      |\n  |    II.  |  A = AB                       |     12    |     2      |\n  |   III.  |  A = B, B = C                 |      4    |     6      |\n  |    IV.  |  A = B, B = BC                |     24    |     5      |\n  |     V.  |  A = AB, B = BC               |     24    |     4      |\n  |    VI.  |  A = BC                       |     24    |     4      |\n  |   VII.  |  A = ABC                      |     24    |     3      |\n  |  VIII.  |  AB = ABC                     |      8    |     1      |\n  |    IX.  |  A = AB, *a*B = *a*B*c*       |     24    |     3      |\n  |     X.  |  A = ABC, *ab* = *ab*C        |      8    |     4      |\n  |    XI.  |  AB = ABC, *ab* = *abc*       |      4    |     2      |\n  |   XII.  |  AB = AC                      |     12    |     2      |\n  |  XIII.  |  A = BC \ua58c A*bc*             |      8    |     3      |\n  |   XIV.  |  A = BC \ua58c *bc*              |      2    |     4      |\n  |    XV.  |  A = ABC, *a* = B*c* \ua58c *b*C |      8    |     5      |\n  +---------+-------------------------------+-----------+------------+\n\nThere are sixty-three series of combinations derived from\nself-contradictory premises, which with 192, the sum of the numbers of\ndistinct logical variations stated in the third column of the table,\nand with the one case where there are no conditions or laws at all,\nmake up the whole conceivable number of 256 series.\n\nWe learn from this table, for instance, that two propositions of\nthe form A\u00a0=\u00a0AB, B\u00a0=\u00a0BC, which are such as constitute the premises\nof the old syllogism Barbara, exclude as impossible four of the\neight combinations in which three terms may be united, and that\nthese propositions are capable of taking twenty-four variations by\ntranspositions of the terms or the introduction of negatives. This\ntable then presents the results of a complete analysis of all the\npossible logical relations arising in the case of three terms, and the\nold syllogism forms but one out of fifteen typical forms. Generally\nspeaking, every form can be converted into apparently different\npropositions; thus the fourth type A\u00a0=\u00a0B, B\u00a0=\u00a0BC may appear in the\nform A\u00a0=\u00a0ABC, *a* = *ab*, or again in the form of three propositions\nA\u00a0=\u00a0AB, B\u00a0=\u00a0BC, *a*B\u00a0=\u00a0*a*B*c*; but all these sets of premises yield\nidentically the same series of combinations, and are therefore of\nequivalent logical meaning. The fifth type, or Barbara, can also be\nthrown into the equivalent forms A\u00a0=\u00a0ABC, *a*B\u00a0=\u00a0*a*BC and A\u00a0=\u00a0AC,\nB\u00a0=\u00a0A\u00a0\ua58c\u00a0*a*BC. In other cases I have obtained the very same logical\nconditions in four modes of statements. As regards mere appearance and\nform of statement, the number of possible premises would be very great,\nand difficult to exhibit exhaustively.\n\nThe most remarkable of all the types of logical condition is the\nfourteenth, namely, A\u00a0=\u00a0BC\u00a0\ua58c\u00a0*bc*. It is that which expresses the\ndivision of a genus into two doubly marked species, and might be\nillustrated by the example--\u201cComponent of the physical universe =\nmatter, gravitating, or not-matter (ether), not-gravitating.\u201d It is\ncapable of only two distinct logical variations, namely, A\u00a0=\u00a0BC\u00a0\ua58c\u00a0*bc*\nand A\u00a0=\u00a0B*c*\u00a0\ua58c\u00a0*b*C. By transposition or negative change of the letters\nwe can indeed obtain six different expressions of each of these\npropositions; but when their meanings are analysed, by working out the\ncombinations, they are found to be logically equivalent to one or other\nof the above two. Thus the proposition A\u00a0=\u00a0BC\u00a0\ua58c\u00a0*bc* can be written in\nany of the following five other modes,\n\n  *a*\u00a0=\u00a0*b*C\u00a0\ua58c\u00a0B*c*, B\u00a0=\u00a0CA\u00a0\ua58c\u00a0*ca*, *b*\u00a0=\u00a0*c*A\u00a0\ua58c\u00a0C*a*,\n          C\u00a0=\u00a0AB\u00a0\ua58c\u00a0*ab*, *c*\u00a0=\u00a0*a*B\u00a0\ua58c\u00a0A*b*.\n\nI do not think it needful to publish at present the complete table of\n193 series of combinations and the premises corresponding to each. Such\na table enables us by mere inspection to learn the laws obeyed by any\nset of combinations of three things, and is to logic what a table of\nfactors and prime numbers is to the theory of numbers, or a table of\nintegrals to the higher mathematics. The table already given (p.\u00a0140)\nwould enable a person with but little labour to discover the law of any\ncombinations. If there be seven combinations (one contradicted) the law\nmust be of the eighth type, and the proper variety will be apparent.\nIf there be six combinations (two contradicted), either the second,\neleventh, or twelfth type applies, and a certain number of trials will\ndisclose the proper type and variety. If there be but two combinations\nthe law must be of the third type, and so on.\n\nThe above investigations are complete as regards the possible logical\nrelations of two or three terms. But when we attempt to apply the\nsame kind of method to the relations of four or more terms, the labour\nbecomes impracticably great. Four terms give sixteen combinations\ncompatible with the laws of thought, and the number of possible\nselections of combinations is no less than 2^{16} or 65,536. The\nfollowing table shows the extraordinary manner in which the number of\npossible logical relations increases with the number of terms involved.\n\n  +---------+-------------+---------------------------------------+\n  |Number of|  Number of  |Number of possible selections of combi-|\n  |  terms. |  possible   |  nations corresponding to consistent  |\n  |         |combinations.|   or inconsistent logical relations.  |\n  +-----------------------+---------------------------------------+\n  |    2    |       4     |                                    16 |\n  |    3    |       8     |                                   256 |\n  |    4    |      16     |                                65,536 |\n  |    5    |      32     |                         4,294,967,296 |\n  |    6    |      64     |            18,446,744,073,709,551,616 |\n  +---------+-------------+---------------------------------------+\n\nSome years of continuous labour would be required to ascertain the\ntypes of laws which may govern the combinations of only four things,\nand but a small part of such laws would be exemplified or capable of\npractical application in science. The purely logical inverse problem,\nwhereby we pass from combinations to their laws, is solved in the\npreceding pages, as far as it is likely to be for a long time to come;\nand it is almost impossible that it should ever be carried more than a\nsingle step further.\n\nIn the first edition, vol i. p.\u00a0158, I stated that I had not been\nable to discover any mode of calculating the number of cases in which\ninconsistency would be implied in the selection of combinations from\nthe Logical Alphabet. The logical complexity of the problem appeared\nto be so great that the ordinary modes of calculating numbers of\ncombinations failed, in my opinion, to give any aid, and exhaustive\nexamination of the combinations in detail seemed to be the only method\napplicable. This opinion, however, was mistaken, for both Mr. R.\u00a0B.\nHayward, of Harrow, and Mr. W.\u00a0H. Brewer have calculated the numbers\nof inconsistent cases both for three and for four terms, without much\ndifficulty. In the case of four terms they find that there are 1761\ninconsistent selections and 63,774 consistent, which with one case\nwhere no condition exists, make up the total of 65,536 possible\nselections.\n\nThe inconsistent cases are distributed in the manner shown in the\nfollowing table:--\n\n  +--------------+---------------------------------------------------+\n  |  Number of   |                                                   |\n  | Combinations | 0   1    2    3    4    5    6   7  8  9  10, &c. |\n  |  remaining.  |                                                   |\n  +--------------+---------------------------------------------------+\n  |  Number of   |                                                   |\n  | Inconsistent | 1  16  112  352  536  448  224  64  8  0   0, &c. |\n  |    Cases.    |                                                   |\n  +--------------+---------------------------------------------------+\n\nWhen more than eight combinations of the Logical Alphabet (p.\u00a094,\ncolumn V.) remain unexcluded, there cannot be inconsistency. The whole\nnumbers of ways of selecting 0, 1, 2, &c., combinations out of 16 are\ngiven in the 17th line of the Arithmetical Triangle given further on in\nthe Chapter on Combinations and Permutations, the sum of the numbers in\nthat line being 65,536.\n\n\n*Professor Clifford on the Types of Compound Statement involving Four\nClasses.*\n\nIn the first edition (vol.\u00a0i. p.\u00a0163), I asserted that some years of\nlabour would be required to ascertain even the precise number of types\nof law governing the combinations of four classes of things. Though I\nstill believe that some years\u2019 labour would be required to work out the\ntypes themselves, it is clearly a mistake to suppose that the *numbers*\nof such types cannot be calculated with a reasonable amount of labour,\nProfessor W.\u00a0K. Clifford having actually accomplished the task. His\nsolution of the numerical problem involves the use of a complete new\nsystem of nomenclature and is far too intricate to be fully described\nhere. I can only give a brief abstract of the results, and refer\nreaders, who wish to follow out the reasoning, to the Proceedings of\nthe Literary and Philosophical Society of Manchester, for the 9th\nJanuary, 1877, vol.\u00a0xvi., p.\u00a088, where Professor Clifford\u2019s paper is\nprinted in full.\n\nBy a *simple statement* Professor Clifford means the denial of the\nexistence of any single combination or *cross-division*, of the\nclasses, as in ABCD\u00a0=\u00a00, or A*b*C*d* =\u00a00. The denial of two or more\nsuch combinations is called a *compound statement*, and is further said\nto be *twofold*, *threefold*, &c., according to the number denied. Thus\nABC\u00a0=\u00a00 is a twofold compound statement in regard to four classes,\nbecause it involves both ABCD\u00a0=\u00a00 and ABC*d*\u00a0=\u00a00. When two compound\nstatements can be converted into one another by interchange of the\nclasses, A, B, C, D, with each other or with their complementary\nclasses, *a*, *b*, *c*, *d*, they are called *similar*, and all similar\nstatements are said to belong to the same *type*.\n\nTwo statements are called *complementary* when they deny between them\nall the sixteen combinations without both denying any one; or, which\nis the same thing, when each denies just those combinations which\nthe other permits to exist. It is obvious that when two statements\nare similar, the complementary statements will also be similar,\nand consequently for every type of *n*-fold statement, there is a\ncomplementary type of (16--*n*)-fold statement. It follows that we need\nonly enumerate the types as far as the eighth order; for the types\nof more-than-eight-fold statement will already have been given as\ncomplementary to types of lower orders.\n\nOne combination, ABCD, may be converted into another A*b*C*d* by\ninterchanging one or more of the classes with the complementary\nclasses. The number of such changes is called the *distance*, which in\nthe above case is 2. In two similar compound statements the distances\nof the combinations denied must be the same; but it does not follow\nthat when all the distances are the same, the statements are similar.\nThere is, however, only one example of two dissimilar statements having\nthe same distances. When the distance is 4, the two combinations\nare said to be *obverse* to one another, and the statements denying\nthem are called *obverse statements*, as in ABCD\u00a0=\u00a00 and *abcd*\u00a0=\u00a00\nor again A*b*C*d*\u00a0=\u00a00 and *a*B*c*D\u00a0=\u00a00. When any one combination is\ngiven, called the *origin*, all the others may be grouped in respect\nof their relations to it as follows:--Four are at distance *one* from\nit, and may be called *proximates*; six are at distance *two*, and may\nbe called *mediates*; four are at distance *three*, and may be called\n*ultimates*; finally the obverse is at distance *four*.\n\n    Origin and               Six              Obverse and\n  four proximates.        mediates.         four ultimates.\n                           *ab*CD\n                              |\n        *a*BCD        A*bc*D  |  A*b*C*d*       A*bcd*\n         |                 \\  |  /                |\n         |                  \\ | /                 |\n         |                   \\|/                  |\n  ABC*d*--ABCD--A*b*CD        +         *abc*D--*abcd*--*a*B*cd*\n         |                   /|\\                  |\n         |                  / | \\                 |\n         |                 /  |  \\                |\n        AB*c*D      *a*B*c*D  |  *a*BC*d*      *ab*C*d*.\n                              |\n                            AB*cd*\n\nIt will be seen that the four proximates are respectively obverse to\nthe four ultimates, and that the mediates form three pairs of obverses.\nEvery proximate or ultimate is distant 1 and 3 respectively from such a\npair of mediates.\n\nAided by this system of nomenclature Professor Clifford proceeds to an\nexhaustive enumeration of types, in which it is impossible to follow\nhim. The results are as follows:--\n\n  1-fold statements         1 type }\n  2   \"      \"              4 types}\n  3   \"      \"              6   \"  }\n  4   \"      \"             19   \"  } 159\n  5   \"      \"             27   \"  }\n  6   \"      \"             47   \"  }\n  7   \"      \"             55   \"  }\n  8-fold statements        78   \"\n\nNow as each seven-fold or less-than-seven-fold statement is\ncomplementary to a nine-fold or more-than-nine-fold statement, it\nfollows that the complete number of types will be 159\u00a0\u00d7\u00a02\u00a0+\u00a078\u00a0= 396.\n\nIt appears then that the types of statement concerning four classes\nare only about 26 times as numerous as those concerning three classes,\nfifteen in number, although the number of possible combinations is 256\ntimes as great.\n\nProfessor Clifford informs me that the knowledge of the possible\ngroupings of subdivisions of classes which he obtained by this inquiry\nhas been of service to him in some applications of hyper-elliptic\nfunctions to which he has subsequently been led. Professor Cayley has\nsince expressed his opinion that this line of investigation should\nbe followed out, owing to the bearing of the theory of compound\ncombinations upon the higher geometry.[86] It seems likely that many\nunexpected points of connection will in time be disclosed between the\nsciences of logic and mathematics.\n\n  [86] *Proceedings of the Manchester Literary and Philosophical\n  Society*, 6th February, 1877, vol.\u00a0xvi., p.\u00a0113.\n\n\n*Distinction between Perfect and Imperfect Induction.*\n\nWe cannot proceed with advantage before noticing the extreme difference\nwhich exists between cases of perfect and those of imperfect induction.\nWe call an induction *perfect* when all the objects or events which\ncan possibly come under the class treated have been examined. But in\nthe majority of cases it is impossible to collect together, or in any\nway to investigate, the properties of all portions of a substance or\nof all the individuals of a race. The number of objects would often\nbe practically infinite, and the greater part of them might be beyond\nour reach, in the interior of the earth, or in the most distant parts\nof the Universe. In all such cases induction is *imperfect*, and is\naffected by more or less uncertainty. As some writers have fallen into\nmuch error concerning the functions and relative importance of these\ntwo branches of reasoning, I shall have to point out that--\n\n  1. Perfect Induction is a process absolutely requisite, both in the\n  performance of imperfect induction and in the treatment of large\n  bodies of facts of which our knowledge is complete.\n\n  2. Imperfect Induction is founded on Perfect Induction, but involves\n  another process of inference of a widely different character.\n\nIt is certain that if I can draw any inference at all concerning\nobjects not examined, it must be done on the data afforded by the\nobjects which have been examined. If I judge that a distant star obeys\nthe law of gravity, it must be because all other material objects\nsufficiently known to me obey that law. If I venture to assert that\nall ruminant animals have cloven hoofs, it is because all ruminant\nanimals which have come under my notice have cloven hoofs. On the\nother hand, I cannot safely say that all cryptogamous plants possess\na purely cellular structure, because some cryptogamous plants, which\nhave been examined by botanists, have a partially vascular structure.\nThe probability that a new cryptogam will be cellular only can be\nestimated, if at all, on the ground of the comparative numbers of\nknown cryptogams which are and are not cellular. Thus the first step\nin every induction will consist in accurately summing up the number\nof instances of a particular phenomenon which have fallen under our\nobservation. Adams and Leverrier, for instance, must have inferred\nthat the undiscovered planet Neptune would obey Bode\u2019s law, because\n*all the planets known at that time obeyed it*. On what principles the\npassage from the known to the apparently unknown is warranted, must be\ncarefully discussed in the next section, and in various parts of this\nwork.\n\nIt would be a great mistake, however, to suppose that Perfect Induction\nis in itself useless. Even when the enumeration of objects belonging\nto any class is complete, and admits of no inference to unexamined\nobjects, the statement of our knowledge in a general proposition is a\nprocess of so much importance that we may consider it necessary. In\nmany cases we may render our investigations exhaustive; all the teeth\nor bones of an animal; all the cells in a minute vegetable organ; all\nthe caves in a mountain side; all the strata in a geological section;\nall the coins in a newly found hoard, may be so completely scrutinized\nthat we may make some general assertion concerning them without fear\nof mistake. Every bone might be proved to contain phosphate of lime;\nevery cell to enclose a nucleus; every cave to hide remains of extinct\nanimals; every stratum to exhibit signs of marine origin; every coin\nto be of Roman manufacture. These are cases where our investigation\nis limited to a definite portion of matter, or a definite area on the\nearth\u2019s surface.\n\nThere is another class of cases where induction is naturally and\nnecessarily limited to a definite number of alternatives. Of the\nregular solids we can say without the least doubt that no one has\nmore than twenty faces, thirty edges, and twenty corners; for by the\nprinciples of geometry we learn that there cannot exist more than five\nregular solids, of each of which we easily observe that the above\nstatements are true. In the theory of numbers, an endless variety of\nperfect inductions might be made; we can show that no number less than\nsixty possesses so many divisors, and the like is true of 360; for it\ndoes not require a great amount of labour to ascertain and count all\nthe divisors of numbers up to sixty or 360. I can assert that between\n60,041 and 60,077 no prime number occurs, because the exhaustive\nexamination of those who have constructed tables of prime numbers\nproves it to be so.\n\nIn matters of human appointment or history, we can frequently have\na complete limitation of the number of instances to be included in\nan induction. We might show that the propositions of the third book\nof Euclid treat only of circles; that no part of the works of Galen\nmentions the fourth figure of the syllogism; that none of the other\nkings of England reigned so long as George III.; that Magna Charta has\nnot been repealed by any subsequent statute; that the price of corn in\nEngland has never been so high since 1847 as it was in that year; that\nthe price of the English funds has never been lower than it was on the\n23rd of January, 1798, when it fell to 47-1/4.\n\nIt has been urged against this process of Perfect Induction that it\ngives no new information, and is merely a summing up in a brief form\nof a multitude of particulars. But mere abbreviation of mental labour\nis one of the most important aids we can enjoy in the acquisition\nof knowledge. The powers of the human mind are so limited that\nmultiplicity of detail is alone sufficient to prevent its progress\nin many directions. Thought would be practically impossible if every\nseparate fact had to be separately thought and treated. Economy of\nmental power may be considered one of the main conditions on which our\nelevated intellectual position depends. Mathematical processes are for\nthe most part but abbreviations of the simpler acts of addition and\nsubtraction. The invention of logarithms was one of the most striking\nadditions ever made to human power: yet it was a mere abbreviation of\noperations which could have been done before had a sufficient amount\nof labour been available. Similar additions to our power will, it\nis hoped, be made from time to time; for the number of mathematical\nproblems hitherto solved is but an indefinitely small fraction of those\nwhich await solution, because the labour they have hitherto demanded\nrenders them impracticable. So it is throughout all regions of thought.\nThe amount of our knowledge depends upon our power of bringing it\nwithin practicable compass. Unless we arrange and classify facts and\ncondense them into general truths, they soon surpass our powers of\nmemory, and serve but to confuse. Hence Perfect Induction, even as a\nprocess of abbreviation, is absolutely essential to any high degree of\nmental achievement.\n\n\n*Transition from Perfect to Imperfect Induction.*\n\nIt is a question of profound difficulty on what grounds we are\nwarranted in inferring the future from the present, or the nature\nof undiscovered objects from those which we have examined with our\nsenses. We pass from Perfect to Imperfect Induction when once we\nallow our conclusion to apply, at all events apparently, beyond the\ndata on which it was founded. In making such a step we seem to gain\na net addition to our knowledge; for we learn the nature of what was\nunknown. We reap where we have never sown. We appear to possess the\ndivine power of creating knowledge, and reaching with our mental arms\nfar beyond the sphere of our own observation. I shall have, indeed, to\npoint out certain methods of reasoning in which we do pass altogether\nbeyond the sphere of the senses, and acquire accurate knowledge which\nobservation could never have given; but it is not imperfect induction\nthat accomplishes such a task. Of imperfect induction itself, I venture\nto assert that it never makes any real addition to our knowledge, in\nthe meaning of the expression sometimes accepted. As in other cases\nof inference, it merely unfolds the information contained in past\nobservations; it merely renders explicit what was implicit in previous\nexperience. It transmutes, but certainly does not create knowledge.\n\nThere is no fact which I shall more constantly keep before the reader\u2019s\nmind in the following pages than that the results of imperfect\ninduction, however well authenticated and verified, are never more than\nprobable. We never can be sure that the future will be as the present.\nWe hang ever upon the will of the Creator: and it is only so far as He\nhas created two things alike, or maintains the framework of the world\nunchanged from moment to moment, that our most careful inferences can\nbe fulfilled. All predictions, all inferences which reach beyond their\ndata, are purely hypothetical, and proceed on the assumption that new\nevents will conform to the conditions detected in our observation of\npast events. No experience of finite duration can give an exhaustive\nknowledge of the forces which are in operation. There is thus a\ndouble uncertainty; even supposing the Universe as a whole to proceed\nunchanged, we do not really know the Universe as a whole. We know only\na point in its infinite extent, and a moment in its infinite duration.\nWe cannot be sure, then, that our observations have not escaped some\nfact, which will cause the future to be apparently different from the\npast; nor can we be sure that the future really will be the outcome of\nthe past. We proceed then in all our inferences to unexamined objects\nand times on the assumptions--\n\n  1. That our past observation gives us a complete knowledge of what\n  exists.\n\n  2. That the conditions of things which did exist will continue to be\n  the conditions which will exist.\n\nWe shall often need to illustrate the character of our knowledge of\nnature by the simile of a ballot-box, so often employed by mathematical\nwriters in the theory of probability. Nature is to us like an infinite\nballot-box, the contents of which are being continually drawn, ball\nafter ball, and exhibited to us. Science is but the careful observation\nof the succession in which balls of various character present\nthemselves; we register the combinations, notice those which seem to\nbe excluded from occurrence, and from the proportional frequency of\nthose which appear we infer the probable character of future drawings.\nBut under such circumstances certainty of prediction depends on two\nconditions:--\n\n  1. That we acquire a perfect knowledge of the comparative numbers of\n  balls of each kind within the box.\n\n  2. That the contents of the ballot-box remain unchanged.\n\nOf the latter assumption, or rather that concerning the constitution\nof the world which it illustrates, the logician or physicist can\nhave nothing to say. As the Creation of the Universe is necessarily\nan act passing all experience and all conception, so any change in\nthat Universe, or, it may be, a termination of it, must likewise be\ninfinitely beyond the bounds of our mental faculties. No science\nno reasoning upon the subject, can have any validity; for without\nexperience we are without the basis and materials of knowledge. It\nis the fundamental postulate accordingly of all inference concerning\nthe future, that there shall be no arbitrary change in the subject\nof inference; of the probability or improbability of such a change I\nconceive that our faculties can give no estimate.\n\nThe other condition of inductive inference--that we acquire an\napproximately complete knowledge of the combinations in which events\ndo occur, is in some degree within our power. There are branches\nof science in which phenomena seem to be governed by conditions of\na most fixed and general character. We have ground in such cases\nfor believing that the future occurrence of such phenomena can be\ncalculated and predicted. But the whole question now becomes one\nof probability and improbability. We seem to leave the region of\nlogic to enter one in which the number of events is the ground of\ninference. We do not really leave the region of logic; we only leave\nthat where certainty, affirmative or negative, is the result, and the\nagreement or disagreement of qualities the means of inference. For the\nfuture, number and quantity will commonly enter into our processes of\nreasoning; but then I hold that number and quantity are but portions\nof the great logical domain. I venture to assert that number is wholly\nlogical, both in its fundamental nature and in its developments.\nQuantity in all its forms is but a development of number. That which is\nmathematical is not the less logical; if anything it is more logical,\nin the sense that it presents logical results in a higher degree of\ncomplexity and variety.\n\nBefore proceeding then from Perfect to Imperfect Induction I must\ndevote a portion of this work to treating the logical conditions\nof number. I shall then employ number to estimate the variety of\ncombinations in which natural phenomena may present themselves, and\nthe probability or improbability of their occurrence under definite\ncircumstances. It is in later parts of the work that I must endeavour\nto establish the notions which I have set forth upon the subject of\nImperfect Induction, as applied in the investigation of Nature, which\nnotions maybe thus briefly stated:--\n\n  1. Imperfect Induction entirely rests upon Perfect Induction for its\n  materials.\n\n  2. The logical process by which we seem to pass directly from\n  examined to unexamined cases consists in an inverse application of\n  deductive inference, so that all reasoning may be said to be either\n  directly or inversely deductive.\n\n  3. The result is always of a hypothetical character, and is never\n  more than probable.\n\n  4. No net addition is ever made to our knowledge by reasoning; what\n  we know of future events or unexamined objects is only the unfolded\n  contents of our previous knowledge, and it becomes less probable as\n  it is more boldly extended to remote cases.\n\n\n\n\nBOOK II.\n\nNUMBER, VARIETY, AND PROBABILITY.\n\n\n\n\nCHAPTER VIII.\n\nPRINCIPLES OF NUMBER.\n\n\nNot without reason did Pythagoras represent the world as ruled by\nnumber. Into almost all our acts of thought number enters, and in\nproportion as we can define numerically we enjoy exact and useful\nknowledge of the Universe. The science of numbers, too, has hitherto\npresented the widest and most practicable training in logic. So free\nand energetic has been the study of mathematical forms, compared\nwith the forms of logic, that mathematicians have passed far in\nadvance of pure logicians. Occasionally, in recent times, they have\ncondescended to apply their algebraic instrument to a reflex treatment\nof the primary logical science. It is thus that we owe to profound\nmathematicians, such as John Herschel, Whewell, De Morgan, or Boole,\nthe regeneration of logic in the present century. I entertain no\ndoubt that it is in maintaining a close alliance with quantitative\nreasoning that we must look for further progress in our comprehension\nof qualitative inference.\n\nI cannot assent, indeed, to the common notion that certainty begins and\nends with numerical determination. Nothing is more certain than logical\ntruth. The laws of identity and difference are the tests of all that is\ncertain throughout the range of thought, and mathematical reasoning is\ncogent only when it conforms to these conditions, of which logic is the\nfirst development. And if it be erroneous to suppose that all certainty\nis mathematical, it is equally an error to imagine that all which is\nmathematical is certain. Many processes of mathematical reasoning are\nof most doubtful validity. There are points of mathematical doctrine\nwhich must long remain matter of opinion; for instance, the best form\nof the definition and axiom concerning parallel lines, or the true\nnature of a limit. In the use of symbolic reasoning questions occur on\nwhich the best mathematicians may differ, as Bernoulli and Leibnitz\ndiffered irreconcileably concerning the existence of the logarithms of\nnegative quantities.[87] In fact we no sooner leave the simple logical\nconditions of number, than we find ourselves involved in a mazy and\nmysterious science of symbols.\n\n  [87] Montucla. *Histoire des Math\u00e9matiques*, vol.\u00a0iii. p.\u00a0373.\n\nMathematical science enjoys no monopoly, and not even a supremacy,\nin certainty of results. It is the boundless extent and variety of\nquantitative questions that delights the mathematical student. When\nsimple logic can give but a bare answer Yes or No, the algebraist\nraises a score of subtle questions, and brings out a crowd of curious\nresults. The flower and the fruit, all that is attractive and\ndelightful, fall to the share of the mathematician, who too often\ndespises the plain but necessary stem from which all has arisen. In\nno region of thought can a reasoner cast himself free from the prior\nconditions of logical correctness. The mathematician is only strong and\ntrue as long as he is logical, and if number rules the world, it is\nlogic which rules number.\n\nNearly all writers have hitherto been strangely content to look upon\nnumerical reasoning as something apart from logical inference. A long\ndivorce has existed between quality and quantity, and it has not\nbeen uncommon to treat them as contrasted in nature and restricted\nto independent branches of thought. For my own part, I believe that\nall the sciences meet somewhere. No part of knowledge can stand\nwholly disconnected from other parts of the universe of thought; it\nis incredible, above all, that the two great branches of abstract\nscience, interlacing and co-operating in every discourse, should\nrest upon totally distinct foundations. I assume that a connection\nexists, and care only to inquire, What is its nature? Does the science\nof quantity rest upon that of quality; or, *vice vers\u00e2*, does the\nscience of quality rest upon that of quantity? There might conceivably\nbe a third view, that they both rest upon some still deeper set of\nprinciples.\n\nIt is generally supposed that Boole adopted the second view, and\ntreated logic as an application of algebra, a special case of\nanalytical reasoning which admits only two quantities, unity and zero.\nIt is not easy to ascertain clearly which of these views really was\naccepted by Boole. In his interesting biographical sketch of Boole,[88]\nthe Rev. R. Harley protests against the statement that Boole\u2019s logical\ncalculus imported the conditions of number and quantity into logic.\nHe says: \u201cLogic is never identified or confounded with mathematics;\nthe two systems of thought are kept perfectly distinct, each being\nsubject to its own laws and conditions. The symbols are the same for\nboth systems, but they have not the same interpretation.\u201d The Rev. J.\nVenn, again, in his review of Boole\u2019s logical system,[89] holds that\nBoole\u2019s processes are at bottom logical, not mathematical, though\nstated in a highly generalized form and with a mathematical dress. But\nit is quite likely that readers of Boole should be misled. Not only\nhave his logical works an entirely mathematical appearance, but I find\non p.\u00a012 of his *Laws of Thought* the following unequivocal statement:\n\u201cThat logic, as a science, is susceptible of very wide applications\nis admitted; but it is equally certain that its ultimate forms and\nprocesses are mathematical.\u201d A few lines below he adds, \u201cIt is not of\nthe essence of mathematics to be conversant with the ideas of number\nand quantity.\u201d\n\n  [88] *British Quarterly Review*, No. lxxxvii, July 1866.\n\n  [89] *Mind*, October 1876, vol.\u00a0i. p.\u00a0484.\n\nThe solution of the difficulty is that Boole used the term mathematics\nin a wider sense than that usually attributed to it. He probably\nadopted the third view, so that his mathematical *Laws of Thought* are\nthe common basis both of logic and of quantitative mathematics. But\nI do not care to pursue the subject because I think that, in either\ncase Boole was wrong. In my opinion logic is the superior science, the\ngeneral basis of mathematics as well as of all other sciences. Number\nis but logical discrimination, and algebra a highly developed logic.\nThus it is easy to understand the deep analogy which Boole pointed out\nbetween the forms of algebraic and logical deduction. Logic resembles\nalgebra as the mould resembles that which is cast in it. Boole mistook\nthe cast for the mould. Considering that logic imposes its own laws\nupon every branch of mathematical science, it is no wonder that we\nconstantly meet with the traces of logical laws in mathematical\nprocesses.\n\n\n*The Nature of Number.*\n\nNumber is but another name for *diversity*. Exact identity is unity,\nand with difference arises plurality. An abstract notion, as was\npointed out (p.\u00a028), possesses a certain *oneness*. The quality of\n*justice*, for instance, is one and the same in whatever just acts it\nis manifested. In justice itself there are no marks of difference by\nwhich to discriminate justice from justice. But one just act can be\ndiscriminated from another just act by circumstances of time and place,\nand we can count many acts thus discriminated each from each. In like\nmanner pure gold is simply pure gold, and is so far one and the same\nthroughout. But besides its intrinsic qualities, gold occupies space\nand must have shape and size. Portions of gold are always mutually\nexclusive and capable of discrimination, in respect that they must be\neach without the other. Hence they may be numbered.\n\nPlurality arises when and only when we detect difference. For instance,\nin counting a number of gold coins I must count each coin once, and not\nmore than once. Let C denote a coin, and the mark above it the order of\ncounting. Then I must count the coins\n\n  C\u2032 + C\u2033 + C\u2034 + C\u2033\u2033 + ....\n\nIf I were to count them as follows\n\n  C\u2032 + C\u2033 + C\u2034 + C\u2034 + C\u2033\u2033 + ...,\n\nI should make the third coin into two, and should imply the existence\nof difference where there is no difference.[90] C\u2034 and C\u2034 are but\nthe names of one coin named twice over. But according to one of the\nconditions of logical symbols, which I have called the Law of Unity\n(p.\u00a072), the same name repeated has no effect, and\n\n  A \ua58c A = A.\n\n  [90] *Pure Logic*, Appendix, p.\u00a082, \u00a7 192.\n\nWe must apply the Law of Unity, and must reduce all identical\nalternatives before we can count with certainty and use the processes\nof numerical calculation. Identical alternatives are harmless in\nlogic, but are wholly inadmissible in number. Thus logical science\nascertains the nature of the mathematical unit, and the definition may\nbe given in these terms--*A unit is any object of thought which can be\ndiscriminated from every other object treated as a unit in the same\nproblem.*\n\nIt has often been said that units are units in respect of being\nperfectly similar to each other; but though they may be perfectly\nsimilar in some respects, they must be different in at least one point,\notherwise they would be incapable of plurality. If three coins were\nso similar that they occupied the same space at the same time, they\nwould not be three coins, but one coin. It is a property of space that\nevery point is discriminable from every other point, and in time every\nmoment is necessarily distinct from any other moment before or after.\nHence we frequently count in space or time, and Locke, with some other\nphilosophers, has held that number arises from repetition in time.\nBeats of a pendulum may be so perfectly similar that we can discover no\ndifference except that one beat is before and another after. Time alone\nis here the ground of difference and is a sufficient foundation for the\ndiscrimination of plurality; but it is by no means the only foundation.\nThree coins are three coins, whether we count them successively or\nregard them all simultaneously. In many cases neither time nor space\nis the ground of difference, but pure quality alone enters. We can\ndiscriminate the weight, inertia, and hardness of gold as three\nqualities, though none of these is before nor after the other, neither\nin space nor time. Every means of discrimination may be a source of\nplurality.\n\nOur logical notation may be used to express the rise of number.\nThe symbol A stands for one thing or one class, and in itself must\nbe regarded as a unit, because no difference is specified. But the\ncombinations AB and A*b* are necessarily *two*, because they cannot\nlogically coalesce, and there is a mark B which distinguishes one\nfrom the other. A logical definition of the number *four* is given in\nthe combinations ABC, AB*c*, A*b*C, A*bc*, where there is a double\ndifference. As Puck says--\n\n  \u201cYet but three? Come one more;\n  Two of both kinds makes up four.\u201d\n\nI conceive that all numbers might be represented as arising out of\nthe combinations of the Logical Alphabet, more or less of each series\nbeing struck out by various logical conditions. The number three, for\ninstance, arises from the condition that A must be either B or C, so\nthat the combinations are ABC, AB*c*, A*b*C.\n\n\n*Of Numerical Abstraction.*\n\nThere will now be little difficulty in forming a clear notion of\nthe nature of numerical abstraction. It consists in abstracting the\ncharacter of the difference from which plurality arises, retaining\nmerely the fact. When I speak of *three men* I need not at once specify\nthe marks by which each may be known from each. Those marks must exist\nif they are really three men and not one and the same, and in speaking\nof them as many I imply the existence of the requisite differences.\nAbstract number, then, is *the empty form of difference*; the abstract\nnumber *three* asserts the existence of marks without specifying their\nkind.\n\nNumerical abstraction is thus seen to be a different process from\nlogical abstraction (p.\u00a027), for in the latter process we drop out\nof notice the very existence of difference and plurality. In forming\nthe abstract notion *hardness*, we ignore entirely the diverse\ncircumstances in which the quality may appear. It is the concrete\nnotion *three hard objects*, which asserts the existence of hardness\nalong with sufficient other undefined qualities, to mark out *three*\nsuch objects. Numerical thought is indeed closely interwoven with\nlogical thought. We cannot use a concrete term in the plural, as\n*men*, without implying that there are marks of difference. But when we\nuse an abstract term, we deal with unity.\n\nThe origin of the great generality of number is now apparent. Three\nsounds differ from three colours, or three riders from three horses;\nbut they agree in respect of the variety of marks by which they can be\ndiscriminated. The symbols 1\u00a0+\u00a01\u00a0+\u00a01 are thus the empty marks asserting\nthe existence of discrimination. But in dropping out of sight the\ncharacter of the differences we give rise to new agreements on which\nmathematical reasoning is founded. Numerical abstraction is so far from\nbeing incompatible with logical abstraction that it is the origin of\nour widest acts of generalization.\n\n\n*Concrete and Abstract Number.*\n\nThe common distinction between concrete and abstract number can now be\neasily stated. In proportion as we specify the logical characters of\nthe things numbered, we render them concrete. In the abstract number\nthree there is no statement of the points in which the *three* objects\nagree; but in *three coins*, *three men*, or *three horses*, not only\nare the objects numbered but their nature is restricted. Concrete\nnumber thus implies the same consciousness of difference as abstract\nnumber, but it is mingled with a groundwork of similarity expressed in\nthe logical terms. There is identity so far as logical terms enter;\ndifference so far as the terms are merely numerical.\n\nThe reason of the important Law of Homogeneity will now be apparent.\nThis law asserts that in every arithmetical calculation the logical\nnature of the things numbered must remain unaltered. The specified\nlogical agreement of the things must not be affected by the unspecified\nnumerical differences. A calculation would be palpably absurd which,\nafter commencing with length, gave a result in hours. It is equally\nabsurd, in a purely arithmetical point of view, to deduce areas from\nthe calculation of lengths, masses from the combination of volume\nand density, or momenta from mass and velocity. It must remain for\nsubsequent consideration to decide in what sense we may truly say that\ntwo linear feet multiplied by two linear feet give four superficial\nfeet; arithmetically it is absurd, because there is a change of unit.\n\nAs a general rule we treat in each calculation only objects of one\nnature. We do not, and cannot properly add, in the same sum yards of\ncloth and pounds of sugar. We cannot even conceive the result of adding\narea to velocity, or length to density, or weight to value. The units\nadded must have a basis of homogeneity, or must be reducible to some\ncommon denominator. Nevertheless it is possible, and in fact common, to\ntreat in one complex calculation the most heterogeneous quantities, on\nthe condition that each kind of object is kept distinct, and treated\nnumerically only in conjunction with its own kind. Different units,\nso far as their logical differences are specified, must never be\nsubstituted one for the other. Chemists continually use equations which\nassert the equivalence of groups of atoms. Ordinary fermentation is\nrepresented by the formula\n\n  C^{6}\u00a0H^{12}\u00a0O^{6}\u00a0=\u00a02C^{2}\u00a0H^{6}\u00a0O\u00a0+\u00a02CO^{2}.\n\nThree kinds of units, the atoms respectively of carbon, hydrogen, and\noxygen, are here intermingled, but there is really a separate equation\nin regard to each kind. Mathematicians also employ compound equations\nof the same kind; for in, *a*\u00a0+\u00a0*b*\u00a0\u221a -\u00a01 = *c*\u00a0+\u00a0*d*\u00a0\u221a\u00a0-\u00a01,\nit is impossible by ordinary addition to add *a* to *b*\u00a0\u221a\u00a0-\u00a01.\nHence we really have the separate equations *a*\u00a0=\u00a0*b*, and\n*c*\u00a0\u221a\u00a0-\u00a01\u00a0= *d*\u00a0\u221a\u00a0-\u00a01. Similarly an equation between\ntwo quaternions is equivalent to four equations between ordinary\nquantities, whence indeed the name *quaternion*.\n\n\n*Analogy of Logical and Numerical Terms.*\n\nIf my assertion is correct that number arises out of logical\nconditions, we ought to find number obeying all the laws of logic.\nIt is almost superfluous to point out that this is the case with the\nfundamental laws of identity and difference, and it only remains to\nshow that mathematical symbols do really obey the special conditions\nof logical symbols which were formerly pointed out (p.\u00a032). Thus the\nLaw of Commutativeness, is equally true of quality and quantity. As in\nlogic we have\n\n  AB = BA,\n\nso in mathematics it is familiarly known that\n\n  2\u00a0\u00d7\u00a03\u00a0=\u00a03\u00a0\u00d7\u00a02, or *x*\u00a0\u00d7\u00a0*y*\u00a0=\u00a0*y*\u00a0\u00d7\u00a0*x*.\n\nThe properties of space are as indifferent in multiplication as we\nfound them in pure logical thought.\n\nSimilarly, as in logic\n\n  triangle or square = square or triangle,\n\n  or generally       A \ua58c B  =  B \ua58c A,\n  so in quantity     2 \u00a0+ \u00a03\u00a0 =\u00a0 3\u00a0 + \u00a02,\n  or generally      *x*\u00a0+\u00a0*y*\u00a0=\u00a0*y*\u00a0+\u00a0*x*.\n\nThe symbol \ua58c is not identical with +, but it is thus far analogous.\n\nHow far, now, is it true that mathematical symbols obey the Law of\nSimplicity expressed in the form\n\n  AA = A,\n\nor the example\n\n  Round round = round?\n\nApparently there are but two numbers which obey this law; for it is\ncertain that\n\n  *x* \u00d7 *x* = *x*\n\nis true only in the two cases when *x*\u00a0=\u00a01, or *x*\u00a0=\u00a00.\n\nIn reality all numbers obey the law, for 2\u00a0\u00d7\u00a02\u00a0=\u00a02 is not really\nanalogous to AA\u00a0=\u00a0A. According to the definition of a unit already\ngiven, each unit is discriminated from each other in the same problem,\nso that in 2\u2032\u00a0\u00d7\u00a02\u2033, the first *two* involves a different discrimination\nfrom the second *two*. I get four kinds of things, for instance, if I\nfirst discriminate \u201cheavy and light\u201d and then \u201ccubical and spherical,\u201d\nfor we now have the following classes--\n\n  heavy, cubical.         light, cubical.\n  heavy, spherical.       light, spherical.\n\nBut suppose that my two classes are in both cases discriminated by the\nsame difference of light and heavy, then we have\n\n  heavy heavy = heavy,\n  heavy light = 0,\n  light heavy = 0,\n  light light = light.\n\nThus, (heavy or light) \u00d7 (heavy or light) = (heavy or light).\n\nIn short, *twice two is two* unless we take care that the second two\nhas a different meaning from the first. But under similar circumstances\nlogical terms give the like result, and it is not true that A\u2032A\u2033\u00a0=\u00a0A\u2032,\nwhen A\u2033 is different in meaning from A\u2032.\n\nIn a similar manner it may be shown that the Law of Unity\n\n  A \ua58c A = A.\n\nholds true alike of logical and mathematical terms. It is absurd indeed\nto say that\n\n  *x* + *x* = *x*\n\nexcept in the one case when *x*\u00a0=\u00a0absolute zero. But this contradiction\n*x*\u00a0+\u00a0*x*\u00a0=\u00a0*x* arises from the fact that we have already defined\nthe units in one x as differing from those in the other. Under such\ncircumstances the Law of Unity does not apply. For if in\n\n  A\u2032 \ua58c A\u2033 = A\u2032\n\nwe mean that A\u2033 is in any way different from A\u2032 the assertion of\nidentity is evidently false.\n\nThe contrast then which seems to exist between logical and mathematical\nsymbols is only apparent. It is because the Laws of Simplicity and\nUnity must always be observed in the operation of counting that those\nlaws seem no further to apply. This is the understood condition under\nwhich we use all numerical symbols. Whenever I write the symbol 5 I\nreally mean\n\n  1\u00a0+\u00a01\u00a0+\u00a01\u00a0+\u00a01\u00a0+\u00a01,\n\nand it is perfectly understood that each of these units is distinct\nfrom each other. If requisite I might mark them thus\n\n  1\u2032+ 1\u2033 + 1\u2034 + 1\u2033\u2033 + 1\u2033\u2034.\n\n\nWere this not the case and were the units really\n\n  1\u2032 + 1\u2033 + 1\u2033 + 1\u2034 + 1\u2033\u2033,\n\nthe Law of Unity would, as before remarked, apply, and\n\n  1\u2033 + 1\u2033 = 1\u2033.\n\nMathematical symbols then obey all the laws of logical symbols, but\ntwo of these laws seem to be inapplicable simply because they are\npresupposed in the definition of the mathematical unit. Logic thus lays\ndown the conditions of number, and the science of arithmetic developed\nas it is into all the wondrous branches of mathematical calculus is but\nan outgrowth of logical discrimination.\n\n\n*Principle of Mathematical Inference.*\n\nThe universal principle of all reasoning, as I have asserted, is that\nwhich allows us to substitute like for like. I have now to point out\nhow in the mathematical sciences this principle is involved in each\nstep of reasoning. It is in these sciences indeed that we meet with the\nclearest cases of substitution, and it is the simplicity with which the\nprinciple can be applied which probably led to the comparatively early\nperfection of the sciences of geometry and arithmetic. Euclid, and\nthe Greek mathematicians from the first, recognised *equality* as the\nfundamental relation of quantitative thought, but Aristotle rejected\nthe exactly analogous, but far more general relation of identity, and\nthus crippled the formal science of logic as it has descended to the\npresent day.\n\nGeometrical reasoning starts from the axiom that \u201cthings equal to the\nsame thing are equal to each other.\u201d Two equalities enable us to infer\na third equality; and this is true not only of lines and angles, but\nof areas, volumes, numbers, intervals of time, forces, velocities,\ndegrees of intensity, or, in short, anything which is capable of being\nequal or unequal. Two stars equally bright with the same star must be\nequally bright with each other, and two forces equally intense with a\nthird force are equally intense with each other. It is remarkable that\nEuclid has not explicitly stated two other axioms, the truth of which\nis necessarily implied. The second axiom should be that \u201cTwo things of\nwhich one is equal and the other unequal to a third common thing, are\nunequal to each other.\u201d An equality and inequality, in short, give an\ninequality, and this is equally true with the first axiom of all kinds\nof quantity. If Venus, for instance, agrees with Mars in density, but\nMars differs from Jupiter, then Venus differs from Jupiter. A third\naxiom must exist to the effect that \u201cThings unequal to the same thing\nmay or may not be equal to each other.\u201d *Two inequalities give no\nground of inference whatever.* If we only know, for instance, that\nMercury and Jupiter differ in density from Mars, we cannot say whether\nor not they agree between themselves. As a fact they do not agree;\nbut Venus and Mars on the other hand both differ from Jupiter and yet\nclosely agree with each other. The force of the axioms can be most\nclearly illustrated by drawing equal and unequal lines.[91]\n\n  [91] *Elementary Lessons in Logic* (Macmillan), p.\u00a0123. It is pointed\n  out in the preface to this Second Edition, that the views here given\n  were partially stated by Leibnitz.\n\nThe general conclusion then must be that where there is equality there\nmay be inference, but where there is not equality there cannot be\ninference. A plain induction will lead us to believe that *equality is\nthe condition of inference concerning quantity*. All the three axioms\nmay in fact be summed up in one, to the effect, that \u201c*in whatever\nrelation one quantity stands to another, it stands in the same relation\nto the equal of that other*.\u201d\n\nThe active power is always the substitution of equals, and it is an\naccident that in a pair of equalities we can make the substitution\nin two ways. From *a*\u00a0=\u00a0*b*\u00a0=\u00a0*c* we can infer *a*\u00a0=\u00a0*c*, either by\nsubstituting in *a*\u00a0=\u00a0*b* the value of *b* as given in *b*\u00a0=\u00a0*c*,\nor else by substituting in *b*\u00a0=\u00a0*c* the value of *b* as given in\n*a*\u00a0=\u00a0*b*. In *a*\u00a0=\u00a0*b*\u00a0~\u00a0*d* we can make but the one substitution of\n*a* for *b*. In *e*\u00a0~\u00a0*f*\u00a0~\u00a0*g* we can make no substitution and get no\ninference.\n\nIn mathematics the relations in which terms may stand to each other are\nfar more varied than in pure logic, yet our principle of substitution\nalways holds true. We may say in the most general manner that *In\nwhatever relation one quantity stands to another, it stands in the same\nrelation to the equal of that other.* In this axiom we sum up a number\nof axioms which have been stated in more or less detail by algebraists.\nThus, \u201cIf equal quantities be added to equal quantities, the sums will\nbe equal.\u201d To explain this, let\n\n  *a* = *b*,        *c* = *d*.\n\nNow *a* + *c*, whatever it means, must be identical with itself, so that\n\n  *a* + *c* = *a* + *c*.\n\nIn one side of this equation substitute for the quantities their\nequivalents, and we have the axiom proved\n\n  *a* + *c* = *b* + *d*.\n\nThe similar axiom concerning subtraction is equally evident, for\nwhatever *a* - *c* may mean it is equal to *a*\u00a0-\u00a0*c*, and therefore by\nsubstitution to *b*\u00a0-\u00a0*d*. Again, \u201cif equal quantities be multiplied by\nthe same or equal quantities, the products will be equal,\u201d For evidently\n\n  *ac* = *ac*,\n\nand if for *c* in one side we substitute its equal *d*, we have\n\n  *ac* = *ad*,\n\nand a second similar substitution gives us\n\n  *ac* = *bd*.\n\nWe might prove a like axiom concerning division in an exactly\nsimilar manner. I might even extend the list of axioms and say that\n\u201cEqual powers of equal numbers are equal.\u201d For certainly, whatever\n*a*\u00a0\u00d7\u00a0*a*\u00a0\u00d7\u00a0*a* may mean, it is equal to *a*\u00a0\u00d7\u00a0*a*\u00a0\u00d7\u00a0*a*; hence by our\nusual substitution it is equal to *b*\u00a0\u00d7\u00a0*b*\u00a0\u00d7\u00a0*b*. The same will be\ntrue of roots of numbers and ^{c}\u221a*a*\u00a0=\u00a0^{d}\u221a*b* provided that\nthe roots are so taken that the root of *a* shall really be related\nto *a* as the root of *b* is to *b*. The ambiguity of meaning of an\noperation thus fails in any way to shake the universality of the\nprinciple. We may go further and assert that, not only the above common\nrelations, but all other known or conceivable mathematical relations\nobey the same principle. Let Q*a* denote in the most general manner\nthat we do something with the quantity *a*; then if *a*\u00a0=\u00a0*b* it\nfollows that\n\n  Q*a* = Q*b*.\n\nThe reader will also remember that one of the most frequent operations\nin mathematical reasoning is to substitute for a quantity its equal,\nas known either by assumed, natural, or self-evident conditions.\nWhenever a quantity appears twice over in a problem, we may apply\nwhat we learn of its relations in one place to its relations in the\nother. All reasoning in mathematics, as in other branches of science,\nthus involves the principle of treating equals equally, or similars\nsimilarly. In whatever way we employ quantitative reasoning in the\nremaining parts of this work, we never can desert the simple principle\non which we first set out.\n\n\n*Reasoning by Inequalities.*\n\nI have stated that all the processes of mathematical reasoning may\nbe deduced from the principle of substitution. Exceptions to this\nassertion may seem to exist in the use of inequalities. The greater of\na greater is undoubtedly a greater, and what is less than a less is\ncertainly less. Snowdon is higher than the Wrekin, and Ben Nevis than\nSnowdon; therefore Ben Nevis is higher than the Wrekin. But a little\nconsideration discloses sufficient reason for believing that even in\nsuch cases, where equality does not apparently enter, the force of the\nreasoning entirely depends upon underlying and implied equalities.\n\nIn the first place, two statements of mere difference do not give\nany ground of inference. We learn nothing concerning the comparative\nheights of St. Paul\u2019s and Westminster Abbey from the assertions that\nthey both differ in height from St. Peter\u2019s at Rome. We need something\nmore than inequality; we require one identity in addition, namely the\nidentity in direction of the two differences. Thus we cannot employ\ninequalities in the simple way in which we do equalities, and, when we\ntry to express what other conditions are requisite, we find ourselves\nlapsing into the use of equalities or identities.\n\nIn the second place, every argument by inequalities may be represented\nin the form of equalities. We express that *a* is greater than *b* by\nthe equation\n\n  *a* = *b* + *p*,        (1)\n\nwhere *p* is an intrinsically positive quantity, denoting the\ndifference of *a* and *b*. Similarly we express that *b* is greater\nthan *c* by the equation\n\n  *b* = *c* + *q*,        (2)\n\nand substituting for *b* in (1) its value in (2) we have\n\n  *a* = *c* + *q* + *p*.        (3)\n\nNow as *p* and *q* are both positive, it follows that *a* is greater\nthan *c*, and we have the exact amount of excess specified. It will be\neasily seen that the reasoning concerning that which is less than a\nless will result in an equation of the form\n\n  *c* = *a* - *r* - *s*.\n\nEvery argument by inequalities may then be thrown into the form of an\nequality; but the converse is not true. We cannot possibly prove that\ntwo quantities are equal by merely asserting that they are both greater\nor both less than another quantity. From *e* > *f* and *g* > *f*, or\n*e* < *f* and *g* < *f*, we can infer no relation between *e* and *g*.\nAnd if the reader take the equations *x*\u00a0=\u00a0*y*\u00a0=\u00a03 and attempt to prove\nthat therefore *x*\u00a0=\u00a03, by throwing them into inequalities, he will\nfind it impossible to do so.\n\nFrom these considerations I gather that reasoning in arithmetic or\nalgebra by so-called inequalities, is only an imperfectly expressed\nreasoning by equalities, and when we want to exhibit exactly and\nclearly the conditions of reasoning, we are obliged to use equalities\nexplicitly. Just as in pure logic a negative proposition, as expressing\nmere difference, cannot be the means of inference, so inequality can\nnever really be the true ground of inference. I do not deny that\naffirmation and negation, agreement and difference, equality and\ninequality, are pairs of equally fundamental relations, but I assert\nthat inference is possible only where affirmation, agreement, or\nequality, some species of identity in fact, is present, explicitly or\nimplicitly.\n\n\n*Arithmetical Reasoning.*\n\nIt may seem somewhat inconsistent that I assert number to arise out of\ndifference or discrimination, and yet hold that no reasoning can be\ngrounded on difference. Number, of course, opens a most wide sphere\nfor inference, and a little consideration shows that this is due to\nthe unlimited series of identities which spring up out of numerical\nabstraction. If six people are sitting on six chairs, there is no\nresemblance between the chairs and the people in logical character.\nBut if we overlook all the qualities both of a chair and a person and\nmerely remember that there are marks by which each of six chairs may\nbe discriminated from the others, and similarly with the people, then\nthere arises a resemblance between the chairs and the people, and this\nresemblance in number may be the ground of inference. If on another\noccasion the chairs are filled by people again, we may infer that these\npeople resemble the others in number though they need not resemble them\nin any other points.\n\nGroups of units are what we really treat in arithmetic. The number\n*five* is really 1\u00a0+\u00a01\u00a0+\u00a01\u00a0+\u00a01\u00a0+\u00a01, but for the sake of conciseness we\nsubstitute the more compact sign 5, or the name *five*. These names\nbeing arbitrarily imposed in any one manner, an infinite variety of\nrelations spring up between them which are not in the least arbitrary.\nIf we define *four* as 1\u00a0+\u00a01\u00a0+\u00a01\u00a0+\u00a01, and *five* as 1\u00a0+\u00a01\u00a0+\u00a01\u00a0+\u00a01\u00a0+\u00a01,\nthen of course it follows that *five*\u00a0=\u00a0*four*\u00a0+\u00a01; but it would be\nequally possible to take this latter equality as a definition, in\nwhich case one of the former equalities would become an inference. It\nis hardly requisite to decide how we define the names of numbers,\nprovided we remember that out of the infinitely numerous relations\nof one number to others, some one relation expressed in an equality\nmust be a definition of the number in question and the other relations\nimmediately become necessary inferences.\n\nIn the science of number the variety of classes which can be formed is\naltogether infinite, and statements of perfect generality may be made\nsubject only to difficulty or exception at the lower end of the scale.\nEvery existing number for instance belongs to the class *m*\u00a0+\u00a07; that\nis, every number must be the sum of another number and seven, except of\ncourse the first six or seven numbers, negative quantities not being\nhere taken into account. Every number is the half of some other, and so\non. The subject of generalization, as exhibited in mathematical truths,\nis an infinitely wide one. In number we are only at the first step of\nan extensive series of generalizations. As number is general compared\nwith the particular things numbered, so we have general symbols for\nnumbers, and general symbols for relations between undetermined\nnumbers. There is an unlimited hierarchy of successive generalizations.\n\n\n*Numerically Definite Reasoning.*\n\nIt was first discovered by De Morgan that many arguments are valid\nwhich combine logical and numerical reasoning, although they cannot be\nincluded in the ancient logical formulas. He developed the doctrine of\nthe \u201cNumerically Definite Syllogism,\u201d fully explained in his *Formal\nLogic* (pp.\u00a0141\u2013170). Boole also devoted considerable attention to the\ndetermination of what he called \u201cStatistical Conditions,\u201d meaning the\nnumerical conditions of logical classes. In a paper published among the\nMemoirs of the Manchester Literary and Philosophical Society, Third\nSeries, vol.\u00a0IV. p.\u00a0330 (Session 1869\u201370), I have pointed out that we\ncan apply arithmetical calculation to the Logical Alphabet. Having\ngiven certain logical conditions and the numbers of objects in certain\nclasses, we can either determine the numbers of objects in other\nclasses governed by those conditions, or can show what further data\nare required to determine them. As an example of the kind of questions\ntreated in numerical logic, and the mode of treatment, I give the\nfollowing problem suggested by De Morgan, with my mode of representing\nits solution.\n\n\u201cFor every man in the house there is a person who is aged; some of the\nmen are not aged. It follows that some persons in the house are not\nmen.\u201d[92]\n\n  [92] *Syllabus of a Proposed System of Logic*, p.\u00a029.\n\n  Now let    A = person in house,\n             B = male,\n             C = aged.\n\nBy enclosing a logical symbol in brackets, let us denote the number of\nobjects belonging to the class indicated by the symbol. Thus let\n\n    (A) = number of persons in house,\n   (AB) = number of male persons in house,\n  (ABC) = number of aged male persons in house,\n\nand so on. Now if we use *w* and *w*\u2032 to denote unknown numbers,\nthe conditions of the problem may be thus stated according to my\ninterpretation of the words--\n\n  (AB) = (AC) - *w*,        (1)\n\nthat is to say, the number of persons in the house who are aged is at\nleast equal to, and may exceed, the number of male persons in the house;\n\n  (AB*c*) = *w*\u2032,        (2)\n\nthat is to say, the number of male persons in the house who are not\naged is some unknown positive quantity.\n\nIf we develop the terms in (1) by the Law of Duality (pp.\u00a074, 81, 89),\nwe obtain\n\n  (ABC)\u00a0+\u00a0(AB*c*)\u00a0=\u00a0(ABC)\u00a0+\u00a0(A*b*C)\u00a0-\u00a0*w*.\n\nSubtracting the common term (ABC) from each side and substituting for\n(AB*c*) its value as given in (2), we get at once\n\n  (A*b*C) = *w* + *w*\u2032,\n\nand adding (A*bc*) to each side, we have\n\n  (A*b*)\u00a0=\u00a0(A*bc*)\u00a0+\u00a0*w*\u00a0+\u00a0*w*\u2032.\n\nThe meaning of this result is that the number of persons in the house\nwho are not men is at least equal to *w*\u00a0+\u00a0*w*\u2032, and exceeds it by the\nnumber of persons in the house who are neither men nor aged (A*bc*).\n\nIt should be understood that this solution applies only to the terms of\nthe example quoted above, and not to the general problem for which De\nMorgan intended it to serve as an illustration.\n\nAs a second instance, let us take the following question:--The\nwhole number of voters in a borough is *a*; the number against whom\nobjections have been lodged by liberals is *b*; and the number against\nwhom objections have been lodged by conservatives is *c*; required the\nnumber, if any, who have been objected to on both sides. Taking\n\n  A = voter,\n  B = objected to by liberals,\n  C = objected to by conservatives,\n\nthen we require the value of (ABC). Now the following equation is\nidentically true--\n\n  (ABC)\u00a0=\u00a0(AB)\u00a0+\u00a0(AC)\u00a0+\u00a0(A*bc*)\u00a0-\u00a0(A).        (1)\n\nFor if we develop all the terms on the second side we obtain\n\n  (ABC)\u00a0=\u00a0(ABC)\u00a0+\u00a0(AB*c*) + (ABC)\u00a0+\u00a0(A*b*C)\u00a0+\u00a0(A*bc*)\n        - (ABC)\u00a0-\u00a0(AB*c*)\u00a0-\u00a0(A*b*C)\u00a0-\u00a0(A*bc*);\n\nand striking out the corresponding positive and negative terms, we have\nleft only (ABC)\u00a0=\u00a0(ABC). Since then (1) is necessarily true, we have\nonly to insert the known values, and we have\n\n  (ABC)\u00a0=\u00a0*b*\u00a0+\u00a0*c*\u00a0-\u00a0*a*\u00a0+\u00a0(A*bc*).\n\nHence the number who have received objections from both sides is equal\nto the excess, if any, of the whole number of objections over the\nnumber of voters together with the number of voters who have received\nno objection (A*bc*).\n\nThe following problem illustrates the expression for the common part of\nany three classes:--The number of paupers who are blind males, is equal\nto the excess, if any, of the sum of the whole number of blind persons,\nadded to the whole number of male persons, added to the number of those\nwho being paupers are neither blind nor males, above the sum of the\nwhole number of paupers added to the number of those who, not being\npaupers, are blind, and to the number of those who, not being paupers,\nare male.\n\nThe reader is requested to prove the truth of the above statement, (1)\nby his own unaided common sense; (2) by the Aristotelian Logic; (3) by\nthe method of numerical logic just expounded; and then to decide which\nmethod is most satisfactory.\n\n\n*Numerical meaning of Logical Conditions.*\n\nIn many cases classes of objects may exist under special logical\nconditions, and we must consider how these conditions can be\ninterpreted numerically. Every logical proposition gives rise to a\ncorresponding numerical equation. Sameness of qualities occasions\nsameness of numbers. Hence if\n\n  A = B\n\ndenotes the identity of the qualities of A and B, we may conclude that\n\n  (A) = (B).\n\nIt is evident that exactly those objects, and those objects only, which\nare comprehended under A must be comprehended under B. It follows that\nwherever we can draw an equation of qualities, we can draw a similar\nequation of numbers. Thus, from\n\n  A = B = C\n\nwe infer\n\n  A = C;\n\nand similarly from\n\n  (A) = (B) = (C),\n\nmeaning that the numbers of A\u2019s and C\u2019s are equal to the number of B\u2019s,\nwe can infer\n\n  (A) = (C).\n\nBut, curiously enough, this does not apply to negative propositions and\ninequalities. For if\n\n  A = B ~ D\n\nmeans that A is identical with B, which differs from D, it does not\nfollow that\n\n  (A) = (B) ~ (D).\n\nTwo classes of objects may differ in qualities, and yet they may agree\nin number. This point strongly confirms me in the opinion which I have\nalready expressed, that all inference really depends upon equations,\nnot differences.\n\nThe Logical Alphabet thus enables us to make a complete analysis of any\nnumerical problem, and though the symbolical statement may sometimes\nseem prolix, I conceive that it really represents the course which the\nmind must follow in solving the question. Although thought may outstrip\nthe rapidity with which the symbols can be written down, yet the mind\ndoes not really follow a different course from that indicated by the\nsymbols. For a fuller explanation of this natural system of Numerically\nDefinite Reasoning, with more abundant illustrations and an analysis\nof De Morgan\u2019s Numerically Definite Syllogism, I must refer the\nreader to the paper[93] in the Memoirs of the Manchester Literary and\nPhilosophical Society, already mentioned, portions of which, however,\nhave been embodied in the present section.\n\n  [93] It has been pointed out to me by Mr. C.\u00a0J. Monroe, that section\n  14 (p.\u00a0339) of this paper is erroneous, and ought to be cancelled.\n  The problem concerning the number of paupers illustrates the answer\n  which should have been obtained. Mr. A.\u00a0J. Ellis, F.R.S., had\n  previously observed that my solution in the paper of De Morgan\u2019s\n  problem about \u201cmen in the house\u201d did not answer the conditions\n  intended by De Morgan, and I therefore give in the text a more\n  satisfactory solution.\n\nThe reader may be referred, also, to Boole\u2019s writings upon the\nsubject in the *Laws of Thought*, chap. xix. p. 295, and in a paper\non \u201cPropositions Numerically Definite,\u201d communicated by De Morgan, in\n1868, to the Cambridge Philosophical Society, and printed in their\n*Transactions*, vol.\u00a0xi. part ii.\n\n\n\n\nCHAPTER IX.\n\nTHE VARIETY OF NATURE, OR THE DOCTRINE OF COMBINATIONS AND PERMUTATIONS.\n\n\nNature may be said to be evolved from the monotony of non-existence\nby the creation of diversity. It is plausibly asserted that we are\nconscious only so far as we experience difference. Life is change, and\nperfectly uniform existence would be no better than non-existence.\nCertain it is that life demands incessant novelty, and that nature,\nthough it probably never fails to obey the same fixed laws, yet\npresents to us an apparently unlimited series of varied combinations\nof events. It is the work of science to observe and record the kinds\nand comparative numbers of such combinations of phenomena, occurring\nspontaneously or produced by our interference. Patient and skilful\nexamination of the records may then disclose the laws imposed on matter\nat its creation, and enable us more or less successfully to predict, or\neven to regulate, the future occurrence of any particular combination.\n\nThe Laws of Thought are the first and most important of all the laws\nwhich govern the combinations of phenomena, and, though they be binding\non the mind, they may also be regarded as verified in the external\nworld. The Logical Alphabet develops the utmost variety of things and\nevents which may occur, and it is evident that as each new quality is\nintroduced, the number of combinations is doubled. Thus four qualities\nmay occur in 16 combinations; five qualities in 32; six qualities in\n64; and so on. In general language, if n be the number of qualities,\n2^{n} is the number of varieties of things which may be formed from\nthem, if there be no conditions but those of logic. This number,\nit need hardly be said, increases after the first few terms, in an\nextraordinary manner, so that it would require 302 figures to express\nthe number of combinations in which 1,000 qualities might conceivably\npresent themselves.\n\nIf all the combinations allowed by the Laws of Thought occurred\nindifferently in nature, then science would begin and end with those\nlaws. To observe nature would give us no additional knowledge, because\nno two qualities would in the long run be oftener associated than any\nother two. We could never predict events with more certainty than we\nnow predict the throws of dice, and experience would be without use.\nBut the universe, as actually created, presents a far different and\nmuch more interesting problem. The most superficial observation shows\nthat some things are constantly associated with other things. The more\nmature our examination, the more we become convinced that each event\ndepends upon the prior occurrence of some other series of events.\nAction and reaction are gradually discovered to underlie the whole\nscene, and an independent or casual occurrence does not exist except\nin appearance. Even dice as they fall are surely determined in their\ncourse by prior conditions and fixed laws. Thus the combinations of\nevents which can really occur are found to be comparatively restricted,\nand it is the work of science to detect these restricting conditions.\n\nIn the English alphabet, for instance, we have twenty-six letters. Were\nthe combinations of such letters perfectly free, so that any letter\ncould be indifferently sounded with any other, the number of words\nwhich could be formed without any repetition would be 2^{26}\u00a0-\u00a01, or\n67,108,863, equal in number to the combinations of the twenty-seventh\ncolumn of the Logical Alphabet, excluding one for the case in which\nall the letters would be absent. But the formation of our vocal organs\nprevents us from using the far greater part of these conjunctions of\nletters. At least one vowel must be present in each word; more than two\nconsonants cannot usually be brought together; and to produce words\ncapable of smooth utterance a number of other rules must be observed.\nTo determine exactly how many words might exist in the English language\nunder these circumstances, would be an exceedingly complex problem,\nthe solution of which has never been attempted. The number of existing\nEnglish words may perhaps be said not to exceed one hundred thousand,\nand it is only by investigating the combinations presented in the\ndictionary, that we can learn the Laws of Euphony or calculate the\npossible number of words. In this example we have an epitome of the\nwork and method of science. The combinations of natural phenomena are\nlimited by a great number of conditions which are in no way brought to\nour knowledge except so far as they are disclosed in the examination of\nnature.\n\nIt is often a very difficult matter to determine the numbers\nof permutations or combinations which may exist under various\nrestrictions. Many learned men puzzled themselves in former centuries\nover what were called Protean verses, or verses admitting many\nvariations in accordance with the Laws of Metre. The most celebrated of\nthese verses was that invented by Bernard Bauhusius, as follows:[94]--\n\n  \u201cTot tibi sunt dotes, Virgo, quot sidera c\u0153lo.\u201d\n\n  [94] Montucla, *Histoire*, &c., vol.\u00a0iii. p.\u00a0388.\n\nOne author, Ericius Puteanus, filled forty-eight pages of a work in\nreckoning up its possible transpositions, making them only 1022. Other\ncalculators gave 2196, 3276, 2580 as their results. Wallis assigned\n3096, but without much confidence in the accuracy of his result.[95]\nIt required the skill of James Bernoulli to decide that the number of\ntranspositions was 3312, under the condition that the sense and metre\nof the verse shall be perfectly preserved.\n\n  [95] Wallis, *Of Combinations*, &c., p.\u00a0119.\n\nIn approaching the consideration of the great Inductive problem,\nit is very necessary that we should acquire correct notions as\nto the comparative numbers of combinations which may exist under\ndifferent circumstances. The doctrine of combinations is that part of\nmathematical science which applies numerical calculation to determine\nthe numbers of combinations under various conditions. It is a part of\nthe science which really lies at the base not only of other sciences,\nbut of other branches of mathematics. The forms of algebraical\nexpressions are determined by the principles of combination, and\nHindenburg recognised this fact in his Combinatorial Analysis. The\ngreatest mathematicians have, during the last three centuries,\ngiven their best powers to the treatment of this subject; it was\nthe favourite study of Pascal; it early attracted the attention of\nLeibnitz, who wrote his curious essay, *De Arte Combinatoria*, at\ntwenty years of age; James Bernoulli, one of the very profoundest\nmathematicians, devoted no small part of his life to the investigation\nof the subject, as connected with that of Probability; and in his\ncelebrated work, *De Arte Conjectandi*, he has so finely described the\nimportance of the doctrine of combinations, that I need offer no excuse\nfor quoting his remarks at full length.\n\n\u201cIt is easy to perceive that the prodigious variety which appears\nboth in the works of nature and in the actions of men, and which\nconstitutes the greatest part of the beauty of the universe, is owing\nto the multitude of different ways in which its several parts are\nmixed with, or placed near, each other. But, because the number of\ncauses that concur in producing a given event, or effect, is oftentimes\nso immensely great, and the causes themselves are so different one\nfrom another, that it is extremely difficult to reckon up all the\ndifferent ways in which they may be arranged or combined together, it\noften happens that men, even of the best understandings and greatest\ncircumspection, are guilty of that fault in reasoning which the writers\non logic call *the insufficient or imperfect enumeration of parts or\ncases*: insomuch that I will venture to assert, that this is the chief,\nand almost the only, source of the vast number of erroneous opinions,\nand those too very often in matters of great importance, which we are\napt to form on all the subjects we reflect upon, whether they relate to\nthe knowledge of nature, or the merits and motives of human actions.\n\n\u201cIt must therefore be acknowledged, that that art which affords a cure\nto this weakness, or defect, of our understandings, and teaches us so\nto enumerate all the possible ways in which a given number of things\nmay be mixed and combined together, that we may be certain that we have\nnot omitted any one arrangement of them that can lead to the object\nof our inquiry, deserves to be considered as most eminently useful and\nworthy of our highest esteem and attention. And this is the business\nof *the art or doctrine of combinations*. Nor is this art or doctrine\nto be considered merely as a branch of the mathematical sciences. For\nit has a relation to almost every species of useful knowledge that the\nmind of man can be employed upon. It proceeds indeed upon mathematical\nprinciples, in calculating the number of the combinations of the things\nproposed: but by the conclusions that are obtained by it, the sagacity\nof the natural philosopher, the exactness of the historian, the skill\nand judgment of the physician, and the prudence and foresight of the\npolitician may be assisted; because the business of all these important\nprofessions is but *to form reasonable conjectures* concerning the\nseveral objects which engage their attention, and all wise conjectures\nare the results of a just and careful examination of the several\ndifferent effects that may possibly arise from the causes that are\ncapable of producing them.\u201d[96]\n\n  [96] James Bernoulli, *De Arte Conjectandi*, translated by Baron\n  Maseres. London, 1795, pp.\u00a035, 36.\n\n\n*Distinction of Combinations and Permutations.*\n\nWe must first consider the deep difference which exists between\nCombinations and Permutations, a difference involving important logical\nprinciples, and influencing the form of mathematical expressions.\nIn *permutation* we recognise varieties of order, treating AB as a\ndifferent group from BA. In *combination* we take notice only of the\npresence or absence of a certain thing, and pay no regard to its place\nin order of time or space. Thus the four letters *a*, *e*, *m*, *n*\ncan form but one combination, but they occur in language in several\npermutations, as *name*, *amen*, *mean*, *mane*.\n\nWe have hitherto been dealing with purely logical questions, involving\nonly combination of qualities. I have fully pointed out in more than\none place that, though our symbols could not but be written in order\nof place and read in order of time, the relations expressed had no\nregard to place or time (pp.\u00a033, 114). The Law of Commutativeness, in\nfact, expresses the condition that in logic we deal with combinations,\nand the same law is true of all the processes of algebra. In some\ncases, order may be a matter of indifference; it makes no difference,\nfor instance, whether gunpowder is a mixture of sulphur, carbon, and\nnitre, or carbon, nitre, and sulphur, or nitre, sulphur, and carbon,\nprovided that the substances are present in proper proportions and\nwell mixed. But this indifference of order does not usually extend to\nthe events of physical science or the operations of art. The change of\nmechanical energy into heat is not exactly the same as the change from\nheat into mechanical energy; thunder does not indifferently precede and\nfollow lightning; it is a matter of some importance that we load, cap,\npresent, and fire a rifle in this precise order. Time is the condition\nof all our thoughts, space of all our actions, and therefore both in\nart and science we are to a great extent concerned with permutations.\nLanguage, for instance, treats different permutations of letters as\nhaving different meanings.\n\nPermutations of things are far more numerous than combinations of those\nthings, for the obvious reason that each distinct thing is regarded\ndifferently according to its place. Thus the letters A, B, C, will\nmake different permutations according as A stands first, second, or\nthird; having decided the place of A, there are two places between\nwhich we may choose for B; and then there remains but one place for\nC. Accordingly the permutations of these letters will be altogether\n3\u00a0\u00d7\u00a02\u00a0\u00d7\u00a01 or 6 in number. With four things or letters, A, B, C, D,\nwe shall have four choices of place for the first letter, three for\nthe second, two for the third, and one for the fourth, so that there\nwill be altogether, 4\u00a0\u00d7\u00a03\u00a0\u00d7\u00a02\u00a0\u00d7\u00a01, or 24 permutations. The same simple\nrule applies in all cases; beginning with the whole number of things\nwe multiply at each step by a number decreased by a unit. In general\nlanguage, if *n* be the number of things in a combination, the number\nof permutations is\n\n*n* (*n*\u00a0-\u00a01)(*n*\u00a0-\u00a02) .... 4\u00a0.\u00a03 . 2\u00a0.\u00a01.\n\nIf we were to re-arrange the names of the days of the week, the\npossible arrangements out of which we should have to choose the new\norder, would be no less than 7\u00a0.\u00a06\u00a0.\u00a05\u00a0.\u00a04\u00a0.\u00a03\u00a0.\u00a02\u00a0.\u00a01, or 5040, or,\nexcluding the existing order, 5039.\n\nThe reader will see that the numbers which we reach in questions\nof permutation, increase in a more extraordinary manner even than\nin combination. Each new object or term doubles the number of\ncombinations, but increases the permutations by a factor continually\ngrowing. Instead of 2\u00a0\u00d7\u00a02\u00a0\u00d7\u00a02\u00a0\u00d7\u00a02\u00a0\u00d7 .... we have 2\u00a0\u00d7\u00a03\u00a0\u00d7\u00a04\u00a0\u00d7\u00a05\u00a0\u00d7 ....\nand the products of the latter expression immensely exceed those of the\nformer. These products of increasing factors are frequently employed,\nas we shall see, in questions both of permutation and combination. They\nare technically called *factorials*, that is to say, the product of all\ninteger numbers, from unity up to any number *n* is the *factorial*\nof *n*, and is often indicated symbolically by *n*!. I give below the\nfactorials up to that of twelve:--\n\n           24 = 1 . 2 . 3 . 4\n          120 = 1 . 2 ... 5\n          720 = 1 . 2 ... 6\n        5,040 = 7!\n       40,320 = 8!\n      362,880 = 9!\n    3,628,800 = 10!\n   39,916,800 = 11!\n  479,001,600 = 12!\n\nThe factorials up to 36! are given in Rees\u2019s \u2018Cyclop\u00e6dia,\u2019 art.\n*Cipher*, and the logarithms of factorials up to 265! are to be\nfound at the end of the table of logarithms published under the\nsuperintendence of the Society for the Diffusion of Useful Knowledge\n(p.\u00a0215). To express the factorial 265! would require 529 places of\nfigures.\n\nMany writers have from time to time remarked upon the extraordinary\nmagnitude of the numbers with which we deal in this subject. Tacquet\ncalculated[97] that the twenty-four [sic] letters of the alphabet may\nbe arranged in more than 620 thousand trillions of orders; and Schott\nestimated[98] that if a thousand millions of men were employed for the\nsame number of years in writing out these arrangements, and each man\nfilled each day forty pages with forty arrangements in each, they would\nnot have accomplished the task, as they would have written only 584\nthousand trillions instead of 620 thousand trillions.\n\n  [97] *Arithmetic\u00e6 Theoria.* Ed. Amsterd. 1704. p.\u00a0517.\n\n  [98] Rees\u2019s *Cyclop\u00e6dia*, art. *Cipher*.\n\nIn some questions the number of permutations may be restricted\nand reduced by various conditions. Some things in a group may\nbe undistinguishable from others, so that change of order will\nproduce no difference. Thus if we were to permutate the letters of\nthe name *Ann*, according to our previous rule, we should obtain\n3\u00a0\u00d7\u00a02\u00a0\u00d7\u00a01, or 6 orders; but half of these arrangements would be\nidentical with the other half, because the interchange of the two\n*n*\u2019s has no effect. The really different orders will therefore be\n(3\u00a0.\u00a02\u00a0.\u00a01)/(1\u00a0.\u00a02) or 3, namely *Ann*, *Nan*, *Nna*. In the word\n*utility* there are two *i*\u2019s and two *t*\u2019s, in respect of both of\nwhich pairs the numbers of permutations must be halved. Thus we obtain\n(7\u00a0.\u00a06\u00a0.\u00a05\u00a0.\u00a04\u00a0.\u00a03\u00a0.\u00a02\u00a0.\u00a01)/(1\u00a0.\u00a02\u00a0.\u00a01\u00a0.\u00a02) or 1260, as the number of\npermutations. The simple rule evidently is--when some things or letters\nare undistinguished, proceed in the first place to calculate all the\npossible permutations as if all were different, and then divide by\nthe numbers of possible permutations of those series of things which\nare not distinguished, and of which the permutations have therefore\nbeen counted in excess. Thus since the word *Utilitarianism* contains\nfourteen letters, of which four are *i*\u2019s, two *a*\u2019s, and two *t*\u2019s,\nthe number of distinct arrangements will be found by dividing the\nfactorial of 14, by the factorials of 4, 2, and 2, the result being\n908,107,200. From the letters of the word *Mississippi* we can get in\nlike manner 11!/(4!\u00a0\u00d7\u00a04!\u00a0\u00d7\u00a02!) or 34,650 permutations, which is not\nthe one-thousandth part of what we should obtain were all the letters\ndifferent.\n\n\n*Calculation of Number of Combinations.*\n\nAlthough in many questions both of art and science we need to calculate\nthe number of permutations on account of their own interest, it far\nmore frequently happens in scientific subjects that they possess but\nan indirect interest. As I have already pointed out, we almost always\ndeal in the logical and mathematical sciences with *combinations*, and\nvariety of order enters only through the inherent imperfections of our\nsymbols and modes of calculation. Signs must be used in some order,\nand we must withdraw our attention from this order before the signs\ncorrectly represent the relations of things which exist neither before\nnor after each other. Now, it often happens that we cannot choose all\nthe combinations of things, without first choosing them subject to the\naccidental variety of order, and we must then divide by the number of\npossible variations of order, that we may get to the true number of\npure combinations.\n\nSuppose that we wish to determine the number of ways in which we can\nselect a group of three letters out of the alphabet, without allowing\nthe same letter to be repeated. At the first choice we can take any\none of 26 letters; at the next step there remain 25 letters, any one\nof which may be joined with that already taken; at the third step\nthere will be 24 choices, so that apparently the whole number of ways\nof choosing is 26\u00a0\u00d7\u00a025\u00a0\u00d7\u00a024. But the fact that one choice succeeded\nanother has caused us to obtain the same combinations of letters in\ndifferent orders; we should get, for instance, *a*, *p*, *r* at one\ntime, and *p*, *r*, *a* at another, and every three distinct letters\nwill appear six times over, because three things can be arranged in\nsix permutations. To get the number of combinations, then, we must\ndivide the whole number of ways of choosing, by six, the number of\npermutations of three things, obtaining (26\u00a0\u00d7\u00a025\u00a0\u00d7\u00a024)/(1\u00a0\u00d7\u00a02\u00a0\u00d7\u00a03) or\n2,600.\n\nIt is apparent that we need the doctrine of combinations in order\nthat we may in many questions counteract the exaggerating effect of\nsuccessive selection. If out of a senate of 30 persons we have to\nchoose a committee of 5, we may choose any of 30 first, any of 29 next,\nand so on, in fact there will be 30\u00a0\u00d7\u00a029\u00a0\u00d7\u00a028\u00a0\u00d7\u00a027\u00a0\u00d7\u00a026 selections;\nbut as the actual character of the members of the committee will not\nbe affected by the accidental order of their selection, we divide by\n1\u00a0\u00d7\u00a02\u00a0\u00d7\u00a03\u00a0\u00d7\u00a04\u00a0\u00d7\u00a05, and the possible number of different committees will\nbe 142,506. Similarly if we want to calculate the number of ways in\nwhich the eight major planets may come into conjunction, it is evident\nthat they may meet either two at a time or three at a time, or four or\nmore at a time, and as nothing is said as to the relative order or\nplace in the conjunction, we require the number of combinations. Now\na selection of 2 out of 8 is possible in (8\u00a0.\u00a07)/(1\u00a0.\u00a02) or 28 ways;\nof 3 out of 8 in (8\u00a0.\u00a07\u00a0.\u00a06)/(1\u00a0.\u00a02\u00a0.\u00a03) or 56 ways; of 4 out of 8 in\n(8\u00a0.\u00a07\u00a0.\u00a06\u00a0.\u00a05)/(1\u00a0.\u00a02\u00a0.\u00a03\u00a0.\u00a04) or 70 ways; and it may be similarly\nshown that for 5, 6, 7, and 8 planets, meeting at one time, the numbers\nof ways are 56, 28, 8, and 1. Thus we have solved the whole question\nof the variety of conjunctions of eight planets; and adding all the\nnumbers together, we find that 247 is the utmost possible number of\nmodes of meeting.\n\nIn general algebraic language, we may say that a group of *m* things\nmay be chosen out of a total number of *n* things, in a number of\ncombinations denoted by the formula\n\n  (*n*\u00a0.\u00a0(*n*-1)(*n*-2)(*n*-3)\u00a0....\u00a0(*n*\u00a0-\u00a0*m*\u00a0+\u00a01))/(1\u00a0.\u00a02\u00a0.\u00a03\u00a0.\u00a04 .... *m*)\n\nThe extreme importance and significance of this formula seems to have\nbeen first adequately recognised by Pascal, although its discovery\nis attributed by him to a friend, M. de Gani\u00e8res.[99] We shall find\nit perpetually recurring in questions both of combinations and\nprobability, and throughout the formul\u00e6 of mathematical analysis traces\nof its influence may be noticed.\n\n  [99] *\u0152uvres Compl\u00e8tes de Pascal* (1865), vol.\u00a0iii. p.\u00a0302. Montucla\n  states the name as De Grui\u00e8res, *Histoire des Math\u00e9matiques*,\n  vol.\u00a0iii. p. 389.\n\n\n*The Arithmetical Triangle.*\n\nThe Arithmetical Triangle is a name long since given to a series\nof remarkable numbers connected with the subject we are treating.\nAccording to Montucla[100] \u201cthis triangle is in the theory of\ncombinations and changes of order, almost what the table of Pythagoras\nis in ordinary arithmetic, that is to say, it places at once under the\neyes the numbers required in a multitude of cases of this theory.\u201d As\nearly as 1544 Stifels had noticed the remarkable properties of these\nnumbers and the mode of their evolution. Briggs, the inventor of the\ncommon system of logarithms, was so struck with their importance that\nhe called them the Abacus Panchrestus. Pascal, however, was the first\nwho wrote a distinct treatise on these numbers, and gave them the name\nby which they are still known. But Pascal did not by any means exhaust\nthe subject, and it remained for James Bernoulli to demonstrate fully\nthe importance of the *figurate numbers*, as they are also called. In\nhis treatise *De Arte Conjectandi*, he points out their application\nin the theory of combinations and probabilities, and remarks of the\nArithmetical Triangle, \u201cIt not only contains the clue to the mysterious\ndoctrine of combinations, but it is also the ground or foundation of\nmost of the important and abstruse discoveries that have been made in\nthe other branches of the mathematics.\u201d[101]\n\n  [100] *Histoire des Math\u00e9matiques*, vol.\u00a0iii. p.\u00a0378.\n\n  [101] Bernoulli, *De Arte Conjectandi*, translated by Francis\n  Maseres. London, 1795, p.\u00a075.\n\nThe numbers of the triangle can be calculated in a very easy manner by\nsuccessive additions. We commence with unity at the apex; in the next\nline we place a second unit to the right of this; to obtain the third\nline of figures we move the previous line one place to the right, and\nadd them to the same figures as they were before removal; we can then\nrepeat the same process *ad infinitum*. The fourth line of figures,\nfor instance, contains 1, 3, 3, 1; moving them one place and adding as\ndirected we obtain:--\n\n  Fourth line ...  1  3  3  1\n                      1  3  3  1\n                   --------------\n  Fifth line ....  1  4  6  4  1\n                      1  4  6  4  1\n                   ----------------\n  Sixth line ....  1  5 10 10  5  1\n\nCarrying out this simple process through ten more steps we obtain the\nfirst seventeen lines of the Arithmetical Triangle as printed on the\nnext page. Theoretically speaking the Triangle must be regarded as\ninfinite in extent, but the numbers increase so rapidly that it soon\nbecomes impracticable to continue the table. The longest table of the\nnumbers which I have found is in Fortia\u2019s \u201cTrait\u00e9 des Progressions\u201d\n(p.\u00a080), where they are given up to the fortieth line and the ninth\ncolumn.\n\nTHE ARITHMETICAL TRIANGLE.\n\nLine. First Column.\n1 1 Second Column.\n2 1 1 Third Column.\n3 1 2 1 Fourth Column.\n4 1 3 3 1 Fifth Column.\n5 1 4 6 4 1 Sixth Column.\n6 1 5 10 10 5 1 Seventh Column.\n7 1 6 15 20 15 6 1 Eighth Column.\n8 1 7 21 35 35 21 7 1 Ninth Column.\n9 1 8 28 56 70 56 28 8 1 Tenth Column.\n10 1 9 36 84 126 126 84 36 9 1 Eleventh Column.\n11 1 10 45 120 210 252 210 120 45 10 1 Twelfth Column.\n12 1 11 55 165 330 462 462 330 165 55 11 1 Thirteenth Column.\n13 1 12 66 220 495 792 924 792 495 220 66 12 1 Fourteenth Column.\n14 1 13 78 286 715 1287 1716 1716 1287 715 286 78 13 1 Fifteenth Column.\n15 1 14 91 364 1001 2002 3003 3432 3003 2002 1001 364 91 14 1 Sixteenth Column.\n16 1 15 105 455 1365 3003 5005 6435 6435 5005 3003 1365 455 105 15 1 Seventeenth Col.\n17 1 16 120 560 1820 4368 8008 11440 12870 11440 8008 4368 1820 560 120 16 1\n\nExamining these numbers, we find that they are connected by an\nunlimited series of relations, a few of the more simple of which may be\nnoticed. Each vertical column of numbers exactly corresponds with an\noblique series descending from left to right, so that the triangle is\nperfectly symmetrical in its contents. The first column contains only\n*units*; the second column contains the *natural numbers*, 1, 2, 3,\n&c.; the third column contains a remarkable series of numbers, 1, 3,\n6, 10, 15, &c., which have long been called *the triangular numbers*,\nbecause they correspond with the numbers of balls which may be arranged\nin a triangular form, thus--\n\n[Illustration]\n\nThe fourth column contains the *pyramidal numbers*, so called because\nthey correspond to the numbers of equal balls which can be piled in\nregular triangular pyramids. Their differences are the triangular\nnumbers. The numbers of the fifth column have the pyramidal numbers\nfor their differences, but as there is no regular figure of which\nthey express the contents, they have been arbitrarily called the\n*trianguli-triangular numbers*. The succeeding columns have, in a\nsimilar manner, been said to contain the *trianguli-pyramidal*, the\n*pyramidi-pyramidal* numbers, and so on.[102]\n\n  [102] Wallis\u2019s *Algebra*, Discourse of Combinations, &c., p.\u00a0109.\n\nFrom the mode of formation of the table, it follows that the\ndifferences of the numbers in each column will be found in the\npreceding column to the left. Hence the *second differences*, or the\n*differences of differences*, will be in the second column to the left\nof any given column, the third differences in the third column, and so\non. Thus we may say that unity which appears in the first column is the\n*first difference* of the numbers in the second column; the *second\ndifference* of those in the third column; the *third difference* of\nthose in the fourth, and so on. The triangle is seen to be a complete\nclassification of all numbers according as they have unity for any of\ntheir differences.\n\nSince each line is formed by adding the previous line to itself, it\nis evident that the sum of the numbers in each horizontal line must be\ndouble the sum of the numbers in the line next above. Hence we know,\nwithout making the additions, that the successive sums must be 1, 2,\n4, 8, 16, 32, 64, &c., the same as the numbers of combinations in the\nLogical Alphabet. Speaking generally, the sum of the numbers in the\n*n*th line will be 2^{*n*\u00a0-\u00a01}.\n\nAgain, if the whole of the numbers down to any line be added together,\nwe shall obtain a number less by unity than some power of 2; thus,\nthe first line gives 1 or 2^{1}\u00a0-\u00a01; the first two lines give 3 or\n2^{2}\u00a0-\u00a01; the first three lines 7 or 2^{3}\u00a0-\u00a01; the first six lines\ngive 63 or 2^{6}\u00a0-\u00a01; or, speaking in general language, the sum of the\nfirst *n* lines is 2^{*n*}\u00a0-\u00a01. It follows that the sum of the numbers\nin any one line is equal to the sum of those in all the preceding\nlines increased by a unit. For the sum of the *n*th line is, as\nalready shown, 2^{*n*\u00a0-\u00a01}, and the sum of the first *n*\u00a0-\u00a01 lines is\n2^{*n*\u00a0-\u00a01}\u00a0-\u00a01, or less by a unit.\n\nThis account of the properties of the figurate numbers does not\napproach completeness; a considerable, probably an unlimited, number of\nless simple and obvious relations might be traced out. Pascal, after\ngiving many of the properties, exclaims[103]: \u201cMais j\u2019en laisse bien\nplus que je n\u2019en donne; c\u2019est une chose \u00e9trange combien il est fertile\nen propri\u00e9t\u00e9s! Chacun peut s\u2019y exercer.\u201d The arithmetical triangle may\nbe considered a natural classification of numbers, exhibiting, in the\nmost complete manner, their evolution and relations in a certain point\nof view. It is obvious that in an unlimited extension of the triangle,\neach number, with the single exception of the number *two*, has at\nleast two places.\n\n  [103] *\u0152uvres Compl\u00e8tes*, vol.\u00a0iii. p.\u00a0251.\n\nThough the properties above explained are highly curious, the greatest\nvalue of the triangle arises from the fact that it contains a complete\nstatement of the values of the formula (p.\u00a0182), for the numbers of\ncombinations of *m* things out of *n*, for all possible values of *m*\nand *n*. Out of seven things one may be chosen in seven ways, and\nseven occurs in the eighth line of the second column. The combinations\nof two things chosen out of seven are (7\u00a0\u00d7\u00a06)/(1\u00a0\u00d7\u00a02) or 21, which\nis the third number in the eighth line. The combinations of three\nthings out of seven are (7\u00a0\u00d7\u00a06\u00a0\u00d7\u00a05)/(1\u00a0\u00d7\u00a02\u00a0\u00d7\u00a03) or 35, which appears\nfourth in the eighth line. In a similar manner, in the fifth, sixth,\nseventh, and eighth columns of the eighth line I find it stated in\nhow many ways I can select combinations of 4, 5, 6, and 7 things out\nof 7. Proceeding to the ninth line, I find in succession the number\nof ways in which I can select 1, 2, 3, 4, 5, 6, 7, and 8 things, out\nof 8 things. In general language, if I wish to know in how many ways\n*m* things can be selected in combinations out of *n* things, I must\nlook in the *n*\u00a0+\u00a01^{th} line, and take the *m*\u00a0+\u00a01^{th} number, as\nthe answer. In how many ways, for instance, can a subcommittee of\nfive be chosen out of a committee of nine. The answer is 126, and\nis the sixth number in the tenth line; it will be found equal to\n(9\u00a0.\u00a08\u00a0.\u00a07\u00a0.\u00a06\u00a0.\u00a05)/(1\u00a0.\u00a02\u00a0.\u00a03\u00a0.\u00a04\u00a0.\u00a05), which our formula (p.\u00a0182)\ngives.\n\nThe full utility of the figurate numbers will be more apparent when\nwe reach the subject of probabilities, but I may give an illustration\nor two in this place. In how many ways can we arrange four pennies as\nregards head and tail? The question amounts to asking in how many ways\nwe can select 0, 1, 2, 3, or 4 heads, out of 4 heads, and the *fifth*\nline of the triangle gives us the complete answer, thus--\n\n  We can select No head and 4 tails in 1 way.\n         \"       1 head and 3 tails in 4 ways.\n         \"       2 heads and 2 tails in 6 ways.\n         \"       3 heads and 1 tail in 4 ways.\n         \"       4 heads and 0 tail in 1 way.\n\nThe total number of different cases is 16, or 2^{4}, and when we come\nto the next chapter, it will be found that these numbers give us the\nrespective probabilities of all throws with four pennies.\n\nI gave in p.\u00a0181 a calculation of the number of ways in which eight\nplanets can meet in conjunction; the reader will find all the numbers\ndetailed in the ninth line of the arithmetical triangle. The sum of the\nwhole line is 2^{8} or 256; but we must subtract a unit for the case\nwhere no planet appears, and 8 for the 8 cases in which only one planet\nappears; so that the total number of conjunctions is 2^{8} -1\u00a0-\u00a08\nor 247. If an organ has eleven stops we find in the twelfth line the\nnumbers of ways in which we can draw them, 1, 2, 3, or more at a time.\nThus there are 462 ways of drawing five stops at once, and as many of\ndrawing six stops. The total number of ways of varying the sound is\n2048, including the single case in which no stop at all is drawn.\n\nOne of the most important scientific uses of the arithmetical triangle\nconsists in the information which it gives concerning the comparative\nfrequency of divergencies from an average. Suppose, for the sake of\nargument, that all persons were naturally of the equal stature of five\nfeet, but enjoyed during youth seven independent chances of growing one\ninch in addition. Of these seven chances, one, two, three, or more,\nmay happen favourably to any individual; but, as it does not matter\nwhat the chances are, so that the inch is gained, the question really\nturns upon the number of combinations of 0, 1, 2, 3, &c., things out of\nseven. Hence the eighth line of the triangle gives us a complete answer\nto the question, as follows:--\n\nOut of every 128 people--\n\n                                      Feet  Inches.\nOne person would have the stature of    5      0\n 7 persons       \"           \"          5      1\n21 persons       \"           \"          5      2\n35 persons       \"           \"          5      3\n35 persons       \"           \"          5      4\n21 persons       \"           \"          5      5\n 7 persons       \"           \"          5      6\n 1 person        \"           \"          5      7\n\nBy taking a proper line of the triangle, an answer may be had under\nany more natural supposition. This theory of comparative frequency of\ndivergence from an average, was first adequately noticed by Quetelet,\nand has lately been employed in a very interesting and bold manner by\nMr. Francis Galton,[104] in his remarkable work on \u201cHereditary Genius.\u201d\nWe shall afterwards find that the theory of error, to which is made the\nultimate appeal in cases of quantitative investigation, is founded upon\nthe comparative numbers of combinations as displayed in the triangle.\n\n  [104] See also Galton\u2019s Lecture at the Royal Institution, 27th\n  February, 1874; Catalogue of the Special Loan Collection of\n  Scientific Instruments, South Kensington, Nos. 48, 49; and Galton,\n  *Philosophical Magazine*, January 1875.\n\n\n*Connection between the Arithmetical Triangle and the Logical Alphabet.*\n\nThere exists a close connection between the arithmetical triangle\ndescribed in the last section, and the series of combinations of\nletters called the Logical Alphabet. The one is to mathematical science\nwhat the other is to logical science. In fact the figurate numbers, or\nthose exhibited in the triangle, are obtained by summing up the logical\ncombinations. Accordingly, just as the total of the numbers in each\nline of the triangle is twice as great as that for the preceding line\n(p.\u00a0186), so each column of the Alphabet (p.\u00a094) contains twice as many\ncombinations as the preceding one. The like correspondence also exists\nbetween the sums of all the lines of figures down to any particular\nline, and of the combinations down to any particular column.\n\nBy examining any column of the Logical Alphabet we find that the\ncombinations naturally group themselves according to the figurate\nnumbers. Take the combinations of the letters A, B, C, D; they consist\nof all the ways in which I can choose four, three, two, one, or none of\nthe four letters, filling up the vacant spaces with negative terms.\n\nThere is one combination, ABCD, in which all the positive letters are\npresent; there are four combinations in each of which three positive\nletters are present; six in which two are present; four in which only\none is present; and, finally, there is the single case, *abcd*, in\nwhich all positive letters are absent. These numbers, 1, 4, 6, 4, 1,\nare those of the fifth line of the arithmetical triangle, and a like\ncorrespondence will be found to exist in each column of the Logical\nAlphabet.\n\nNumerical abstraction, it has been asserted, consists in overlooking\nthe kind of difference, and retaining only a consciousness of its\nexistence (p.\u00a0158). While in logic, then, we have to deal with each\ncombination as a separate kind of thing, in arithmetic we distinguish\nonly the classes which depend upon more or less positive terms being\npresent, and the numbers of these classes immediately produce the\nnumbers of the arithmetical triangle.\n\nIt may here be pointed out that there are two modes in which we\ncan calculate the whole number of combinations of certain things.\nEither we may take the whole number at once as shown in the Logical\nAlphabet, in which case the number will be some power of two, or else\nwe may calculate successively, by aid of permutations, the number of\ncombinations of none, one, two, three things, and so on. Hence we\narrive at a necessary identity between two series of numbers. In the\ncase of four things we shall have\n\n  2\u00a0=\u00a01\u00a0+\u00a04/1\u00a0+\u00a0(4\u00a0.\u00a03)/(1\u00a0.\u00a02)\u00a0+ (4\u00a0.\u00a03\u00a0.\u00a02)/(1\u00a0.\u00a02\u00a0.\u00a03)\u00a0+\n  (4\u00a0.\u00a03\u00a0.\u00a02\u00a0.\u00a01)/(1\u00a0.\u00a02\u00a0.\u00a03\u00a0.\u00a04).\n\nIn a general form of expression we shall have\n\n  2\u00a0=\u00a01\u00a0+\u00a0*n*/1\u00a0+ (*n*\u00a0.\u00a0(*n*\u00a0-\u00a01))/(1\u00a0.\u00a02)\u00a0+ (*n*\n  (*n*\u00a0-\u00a01)(*n*\u00a0-\u00a02))/(1\u00a0.\u00a02\u00a0.\u00a03)\u00a0+\u00a0&c.,\n\nthe terms being continued until they cease to have any value. Thus we\narrive at a proof of simple cases of the Binomial Theorem, of which\neach column of the Logical Alphabet is an exemplification. It may be\nshown that all other mathematical expansions likewise arise out of\nsimple processes of combination, but the more complete consideration of\nthis subject must be deferred to another work.\n\n\n*Possible Variety of Nature and Art.*\n\nWe cannot adequately understand the difficulties which beset us in\ncertain branches of science, unless we have some clear idea of the vast\nnumbers of combinations or permutations which may be possible under\ncertain conditions. Thus only can we learn how hopeless it would be\nto attempt to treat nature in detail, and exhaust the whole number of\nevents which might arise. It is instructive to consider, in the first\nplace, how immensely great are the numbers of combinations with which\nwe deal in many arts and amusements.\n\nIn dealing a pack of cards, the number of hands, of thirteen cards\neach, which can be produced is evidently 52\u00a0\u00d7\u00a051\u00a0\u00d7\u00a050\u00a0\u00d7\u00a0...\u00a0\u00d7\u00a040\ndivided by 1\u00a0\u00d7\u00a02\u00a0\u00d7\u00a03\u00a0...\u00a0\u00d7\u00a013. or 635,013,559,600. But in whist\nfour hands are simultaneously held, and the number of distinct\ndeals becomes so vast that it would require twenty-eight figures to\nexpress it. If the whole population of the world, say one thousand\nmillions of persons, were to deal cards day and night, for a hundred\nmillion of years, they would not in that time have exhausted one\nhundred-thousandth part of the possible deals. Even with the same hands\nof cards the play may be almost infinitely varied, so that the complete\nvariety of games at whist which may exist is almost incalculably great.\nIt is in the highest degree improbable that any one game of whist was\never exactly like another, except it were intentionally so.\n\nThe end of novelty in art might well be dreaded, did we not find that\nnature at least has placed no attainable limit, and that the deficiency\nwill lie in our inventive faculties. It would be a cheerless time\nindeed when all possible varieties of melody were exhausted, but it\nis readily shown that if a peal of twenty-four bells had been rung\ncontinuously from the so-called beginning of the world to the present\nday, no approach could have been made to the completion of the possible\nchanges. Nay, had every single minute been prolonged to 10,000 years,\nstill the task would have been unaccomplished.[105] As regards ordinary\nmelodies, the eight notes of a single octave give more than 40,000\npermutations, and two octaves more than a million millions. If we were\nto take into account the semitones, it would become apparent that it\nis impossible to exhaust the variety of music. When the late Mr. J.\u00a0S.\nMill, in a depressed state of mind, feared the approaching exhaustion\nof musical melodies, he had certainly not bestowed sufficient study on\nthe subject of permutations.\n\n  [105] Wallis, *Of Combinations*, p.\u00a0116, quoting Vossius.\n\nSimilar considerations apply to the possible number of natural\nsubstances, though we cannot always give precise numerical results. It\nwas recommended by Hatchett[106] that a systematic examination of all\nalloys of metals should be carried out, proceeding from the binary ones\nto more complicated ternary or quaternary ones. He can hardly have been\naware of the extent of his proposed inquiry. If we operate only upon\nthirty of the known metals, the number of binary alloys would be 435,\nof ternary alloys 4060, of quaternary 27,405, without paying regard\nto the varying proportions of the metals, and only regarding the kind\nof metal. If we varied all the ternary alloys by quantities not less\nthan one per cent., the number of these alloys would be 11,445,060.\nAn exhaustive investigation of the subject is therefore out of the\nquestion, and unless some laws connecting the properties of the alloy\nand its components can be discovered, it is not apparent how our\nknowledge of them can ever be more than fragmentary.\n\n  [106] *Philosophical Transactions* (1803), vol.\u00a0xciii. p.\u00a0193.\n\nThe possible variety of definite chemical compounds, again, is\nenormously great. Chemists have already examined many thousands\nof inorganic substances, and a still greater number of organic\ncompounds;[107] they have nevertheless made no appreciable impression\non the number which may exist. Taking the number of elements at\nsixty-one, the number of compounds containing different selections of\nfour elements each would be more than half a million (521,855). As the\nsame elements often combine in many different proportions, and some of\nthem, especially carbon, have the power of forming an almost endless\nnumber of compounds, it would hardly be possible to assign any limit\nto the number of chemical compounds which may be formed. There are\nbranches of physical science, therefore, of which it is unlikely that\nscientific men, with all their industry, can ever obtain a knowledge in\nany appreciable degree approaching to completeness.\n\n  [107] Hofmann\u2019s *Introduction to Chemistry*, p.\u00a036.\n\n\n*Higher Orders of Variety.*\n\nThe consideration of the facts already given in this chapter will\nnot produce an adequate notion of the possible variety of existence,\nunless we consider the comparative numbers of combinations of different\norders. By a combination of a higher order, I mean a combination of\ngroups, which are themselves groups. The immense numbers of compounds\nof carbon, hydrogen, and oxygen, described in organic chemistry, are\ncombinations of a second order, for the atoms are groups of groups.\nThe wave of sound produced by a musical instrument may be regarded as\na combination of motions; the body of sound proceeding from a large\norchestra is therefore a complex aggregate of sounds, each in itself\na complex combination of movements. All literature may be said to be\ndeveloped out of the difference of white paper and black ink. From the\nunlimited number of marks which might be chosen we select twenty-six\nconventional letters. The pronounceable combinations of letters are\nprobably some trillions in number. Now, as a sentence is a selection\nof words, the possible sentences must be inconceivably more numerous\nthan the words of which it may be composed. A book is a combination\nof sentences, and a library is a combination of books. A library,\ntherefore, may be regarded as a combination of the fifth order, and the\npowers of numerical expression would be severely tasked in attempting\nto express the number of distinct libraries which might be constructed.\nThe calculation, of course, would not be possible, because the union\nof letters in words, of words in sentences, and of sentences in books,\nis governed by conditions so complex as to defy analysis. I wish only\nto point out that the infinite variety of literature, existing or\npossible, is all developed out of one fundamental difference. Galileo\nremarked that all truth is contained in the compass of the alphabet. He\nought to have said that it is all contained in the difference of ink\nand paper.\n\nOne consequence of successive combination is that the simplest marks\nwill suffice to express any information. Francis Bacon proposed for\nsecret writing a biliteral cipher, which resolves all letters of the\nalphabet into permutations of the two letters *a* and *b*. Thus A\nwas *aaaaa*, B *aaaab*, X *babab*, and so on.[108] In a similar way,\nas Bacon clearly saw, any one difference can be made the ground of a\ncode of signals; we can express, as he says, *omnia per omnia*. The\nMorse alphabet uses only a succession of long and short marks, and\nother systems of telegraphic language employ right and left strokes.\nA single lamp obscured at various intervals, long or short, may be\nmade to spell out any words, and with two lamps, distinguished by\ncolour, position, or any other circumstance, we could at once represent\nBacon\u2019s biliteral alphabet. Babbage ingeniously suggested that every\nlighthouse in the world should be made to spell out its own name or\nnumber perpetually, by flashes or obscurations of various duration\nand succession. A system like that of Babbage is now being applied\nto lighthouses in the United Kingdom by Sir W. Thomson and Dr. John\nHopkinson.\n\n  [108] *Works*, edited by Shaw, vol.\u00a0i. pp.\u00a0141\u2013145, quoted in Rees\u2019s\n  *Encyclop\u00e6dia*, art. *Cipher*.\n\nLet us calculate the numbers of combinations of different orders which\nmay arise out of the presence or absence of a single mark, say A. In\nthese figures\n\n  +---+---+   +---+---+   +---+---+   +---+---+\n  | A | A |   | A |   |   |   | A |   |   |   |\n  +---+---+   +---+---+   +---+---+   +---+---+\n\nwe have four distinct varieties. Form them into a group of a higher\norder, and consider in how many ways we may vary that group by omitting\none or more of the component parts. Now, as there are four parts,\nand any one may be present or absent, the possible varieties will\nbe 2\u00a0\u00d7\u00a02\u00a0\u00d7\u00a02\u00a0\u00d7\u00a02, or 16 in number. Form these into a new whole, and\nproceed again to create variety by omitting any one or more of the\nsixteen. The number of possible changes will now be 2\u00a0.\u00a02\u00a0. 2\u00a0.\u00a02\u00a0.\n2\u00a0.\u00a02\u00a0. 2\u00a0.\u00a02\u00a0. 2\u00a0.\u00a02\u00a0. 2\u00a0.\u00a02\u00a0. 2\u00a0.\u00a02\u00a0. 2\u00a0.\u00a02, or 2^{16}, and we can\nrepeat the process again and again. We are imagining the creation of\nobjects, whose numbers are represented by the successive orders of the\npowers of *two*.\n\nAt the first step we have 2; at the next 2^{2}, or 4; at the third\n(2^{2})^{2}, or 16, numbers of very moderate amount. Let the reader\ncalculate the next term, ((2^{2})^{2})^{2}, and he will be surprised\nto find it leap up to 65,536. But at the next step he has to calculate\nthe value of 65,536 *two*\u2019s multiplied together, and it is so great\nthat we could not possibly compute it, the mere expression of the\nresult requiring 19,729 places of figures. But go one step more and we\npass the bounds of all reason. The sixth order of the powers of *two*\nbecomes so great, that we could not even express the number of figures\nrequired in writing it down, without using about 19,729 figures for\nthe purpose. The successive orders of the powers of two have then the\nfollowing values so far as we can succeed in describing them:--\n\n  First order                           2\n  Second order                          4\n  Third order                          16\n  Fourth order                     65,536\n  Fifth order, number expressed by 19,729 figures.\n  Sixth order, number expressed by\n    figures, to express the number\n    of which figures would require\n    about                          19,729 figures.\n\nIt may give us some notion of infinity to remember that at this sixth\nstep`, having long surpassed all bounds of intuitive conception, we\nmake no approach to a limit. Nay, were we to make a hundred such steps,\nwe should be as far away as ever from actual infinity.\n\nIt is well worth observing that our powers of expression rapidly\novercome the possible multitude of finite objects which may exist in\nany assignable space. Archimedes showed long ago, in one of the most\nremarkable writings of antiquity, the *Liber de Arcn\u00e6 Numero*, that\nthe grains of sand in the world could be numbered, or rather, that\nif numbered, the result could readily be expressed in arithmetical\nnotation. Let us extend his problem, and ascertain whether we could\nexpress the number of atoms which could exist in the visible universe.\nThe most distant stars which can now be seen by telescopes--those of\nthe sixteenth magnitude--are supposed to have a distance of about\n33,900,000,000,000,000 miles. Sir W. Thomson has shown reasons for\nsupposing that there do not exist more than from 3\u00a0\u00d7\u00a010^{24} to 10^{26}\nmolecules in a cubic centimetre of a solid or liquid substance.[109]\nAssuming these data to be true, for the sake of argument, a simple\ncalculation enables us to show that the almost inconceivably vast\nsphere of our stellar system if entirely filled with solid matter,\nwould not contain more than about 68\u00a0\u00d7\u00a010^{90} atoms, that is to say,\na number requiring for its expression 92 places of figures. Now, this\nnumber would be immensely less than the fifth order of the powers of\ntwo.\n\n  [109] *Nature*, vol.\u00a0i. p.\u00a0553.\n\nIn the variety of logical relations, which may exist between a certain\nnumber of logical terms, we also meet a case of higher combinations.\nWe have seen (p.\u00a0142) that with only six terms the number of possible\nselections of combinations is 18,446,744,073,709,551,616. Considering\nthat it is the most common thing in the world to use an argument\ninvolving six objects or terms, it may excite some surprise that the\ncomplete investigation of the relations in which six such terms may\nstand to each other, should involve an almost inconceivable number of\ncases. Yet these numbers of possible logical relations belong only to\nthe second order of combinations.\n\n\n\n\nCHAPTER X.\n\nTHE THEORY OF PROBABILITY.\n\n\nThe subject upon which we now enter must not be regarded as an isolated\nand curious branch of speculation. It is the necessary basis of the\njudgments we make in the prosecution of science, or the decisions we\ncome to in the conduct of ordinary affairs. As Butler truly said,\n\u201cProbability is the very guide of life.\u201d Had the science of numbers\nbeen studied for no other purpose, it must have been developed for\nthe calculation of probabilities. All our inferences concerning the\nfuture are merely probable, and a due appreciation of the degree of\nprobability depends upon a comprehension of the principles of the\nsubject. I am convinced that it is impossible to expound the methods of\ninduction in a sound manner, without resting them upon the theory of\nprobability. Perfect knowledge alone can give certainty, and in nature\nperfect knowledge would be infinite knowledge, which is clearly beyond\nour capacities. We have, therefore, to content ourselves with partial\nknowledge--knowledge mingled with ignorance, producing doubt.\n\nA great difficulty in this subject consists in acquiring a precise\nnotion of the matter treated. What is it that we number, and measure,\nand calculate in the theory of probabilities? Is it belief, or opinion,\nor doubt, or knowledge, or chance, or necessity, or want of art? Does\nprobability exist in the things which are probable, or in the mind\nwhich regards them as such? The etymology of the name lends us no\nassistance: for, curiously enough, *probable* is ultimately the same\nword as *provable*, a good instance of one word becoming differentiated\nto two opposite meanings.\n\nChance cannot be the subject of the theory, because there is really\nno such thing as chance, regarded as producing and governing events.\nThe word chance signifies *falling*, and the notion of falling is\ncontinually used as a simile to express uncertainty, because we can\nseldom predict how a die, a coin, or a leaf will fall, or when a bullet\nwill hit the mark. But everyone sees, after a little reflection, that\nit is in our knowledge the deficiency lies, not in the certainty of\nnature\u2019s laws. There is no doubt in lightning as to the point it shall\nstrike; in the greatest storm there is nothing capricious; not a grain\nof sand lies upon the beach, but infinite knowledge would account for\nits lying there; and the course of every falling leaf is guided by the\nprinciples of mechanics which rule the motions of the heavenly bodies.\n\nChance then exists not in nature, and cannot coexist with knowledge;\nit is merely an expression, as Laplace remarked, for our ignorance\nof the causes in action, and our consequent inability to predict the\nresult, or to bring it about infallibly. In nature the happening of\nan event has been pre-determined from the first fashioning of the\nuniverse. *Probability belongs wholly to the mind.* This is proved by\nthe fact that different minds may regard the very same event at the\nsame time with widely different degrees of probability. A steam-vessel,\nfor instance, is missing and some persons believe that she has sunk\nin mid-ocean; others think differently. In the event itself there can\nbe no such uncertainty; the steam-vessel either has sunk or has not\nsunk, and no subsequent discussion of the probable nature of the event\ncan alter the fact. Yet the probability of the event will really vary\nfrom day to day, and from mind to mind, according as the slightest\ninformation is gained regarding the vessels met at sea, the weather\nprevailing there, the signs of wreck picked up, or the previous\ncondition of the vessel. Probability thus belongs to our mental\ncondition, to the light in which we regard events, the occurrence\nor non-occurrence of which is certain in themselves. Many writers\naccordingly have asserted that probability is concerned with degree or\nquantity of belief. De Morgan says,[110] \u201cBy degree of probability\nwe really mean or ought to mean degree of belief.\u201d The late Professor\nDonkin expressed the meaning of probability as \u201cquantity of belief;\u201d\nbut I have never felt satisfied with such definitions of probability.\nThe nature of *belief* is not more clear to my mind than the notion\nwhich it is used to define. But an all-sufficient objection is, that\n*the theory does not measure what the belief is, but what it ought to\nbe*. Few minds think in close accordance with the theory, and there\nare many cases of evidence in which the belief existing is habitually\ndifferent from what it ought to be. Even if the state of belief in any\nmind could be measured and expressed in figures, the results would be\nworthless. The value of the theory consists in correcting and guiding\nour belief, and rendering our states of mind and consequent actions\nharmonious with our knowledge of exterior conditions.\n\n  [110] *Formal Logic*, p.\u00a0172.\n\nThis objection has been clearly perceived by some of those who still\nused quantity of belief as a definition of probability. Thus De\nMorgan adds--\u201cBelief is but another name for imperfect knowledge.\u201d\nDonkin has well said that the quantity of belief is \u201calways relative\nto a particular state of knowledge or ignorance; but it must be\nobserved that it is absolute in the sense of not being relative to\nany individual mind; since, the same information being presupposed,\nall minds *ought* to distribute their belief in the same way.\u201d[111]\nBoole seemed to entertain a like view, when he described the theory as\nengaged with \u201cthe equal distribution of ignorance;\u201d[112] but we may\njust as well say that it is engaged with the equal distribution of\nknowledge.\n\n  [111] *Philosophical Magazine*, 4th Series, vol.\u00a0i. p.\u00a0355.\n\n  [112] *Transactions of the Royal Society of Edinburgh*, vol.\u00a0xxi.\n  part 4.\n\nI prefer to dispense altogether with this obscure word belief, and to\nsay that the theory of probability deals with *quantity of knowledge*,\nan expression of which a precise explanation and measure can presently\nbe given. An event is only probable when our knowledge of it is diluted\nwith ignorance, and exact calculation is needed to discriminate how\nmuch we do and do not know. The theory has been described by some\nwriters as professing *to evolve knowledge out of ignorance*; but as\nDonkin admirably remarked, it is really \u201ca method of avoiding the\nerection of belief upon ignorance.\u201d It defines rational expectation\nby measuring the comparative amounts of knowledge and ignorance, and\nteaches us to regulate our actions with regard to future events in a\nway which will, in the long run, lead to the least disappointment. It\nis, as Laplace happily said, *good sense reduced to calculation*. This\ntheory appears to me the noblest creation of intellect, and it passes\nmy conception how two such men as Auguste Comte and J.\u00a0S. Mill could be\nfound depreciating it and vainly questioning its validity. To eulogise\nthe theory ought to be as needless as to eulogise reason itself.\n\n\n*Fundamental Principles of the Theory.*\n\nThe calculation of probabilities is really founded, as I conceive, upon\nthe principle of reasoning set forth in preceding chapters. We must\ntreat equals equally, and what we know of one case may be affirmed of\nevery case resembling it in the necessary circumstances. The theory\nconsists in putting similar cases on a par, and distributing equally\namong them whatever knowledge we possess. Throw a penny into the air,\nand consider what we know with regard to its way of falling. We know\nthat it will certainly fall upon a side, so that either head or tail\nwill be uppermost; but as to whether it will be head or tail, our\nknowledge is equally divided. Whatever we know concerning head, we know\nalso concerning tail, so that we have no reason for expecting one more\nthan the other. The least predominance of belief to either side would\nbe irrational; it would consist in treating unequally things of which\nour knowledge is equal.\n\nThe theory does not require, as some writers have erroneously supposed,\nthat we should first ascertain by experiment the equal facility of\nthe events we are considering. So far as we can examine and measure\nthe causes in operation, events are removed out of the sphere of\nprobability. The theory comes into play where ignorance begins, and the\nknowledge we possess requires to be distributed over many cases. Nor\ndoes the theory show that the coin will fall as often on the one side\nas the other. It is almost impossible that this should happen, because\nsome inequality in the form of the coin, or some uniform manner in\nthrowing it up, is almost sure to occasion a slight preponderance\nin one direction. But as we do not previously know in which way a\npreponderance will exist, we have no reason for expecting head more\nthan tail. Our state of knowledge will be changed should we throw up\nthe coin many times and register the results. Every throw gives us some\nslight information as to the probable tendency of the coin, and in\nsubsequent calculations we must take this into account. In other cases\nexperience might show that we had been entirely mistaken; we might\nexpect that a die would fall as often on each of the six sides as on\neach other side in the long run; trial might show that the die was a\nloaded one, and falls most often on a particular face. The theory would\nnot have misled us: it treated correctly the information we had, which\nis all that any theory can do.\n\nIt may be asked, as Mill asks, Why spend so much trouble in calculating\nfrom imperfect data, when a little trouble would enable us to render a\nconclusion certain by actual trial? Why calculate the probability of a\nmeasurement being correct, when we can try whether it is correct? But I\nshall fully point out in later parts of this work that in measurement\nwe never can attain perfect coincidence. Two measurements of the\nsame base line in a survey may show a difference of some inches, and\nthere may be no means of knowing which is the better result. A third\nmeasurement would probably agree with neither. To select any one of the\nmeasurements, would imply that we knew it to be the most nearly correct\none, which we do not. In this state of ignorance, the only guide is\nthe theory of probability, which proves that in the long run the mean\nof divergent results will come most nearly to the truth. In all other\nscientific operations whatsoever, perfect knowledge is impossible, and\nwhen we have exhausted all our instrumental means in the attainment of\ntruth, there is a margin of error which can only be safely treated by\nthe principles of probability.\n\nThe method which we employ in the theory consists in calculating the\nnumber of all the cases or events concerning which our knowledge is\nequal. If we have the slightest reason for suspecting that one event\nis more likely to occur than another, we should take this knowledge\ninto account. This being done, we must determine the whole number of\nevents which are, so far as we know, equally likely. Thus, if we have\nno reason for supposing that a penny will fall more often one way than\nanother, there are two cases, head and tail, equally likely. But if\nfrom trial or otherwise we know, or think we know, that of 100 throws\n55 will give tail, then the probability is measured by the ratio of 55\nto 100.\n\nThe mathematical formul\u00e6 of the theory are exactly the same as those\nof the theory of combinations. In this latter theory we determine in\nhow many ways events may be joined together, and we now proceed to use\nthis knowledge in calculating the number of ways in which a certain\nevent may come about. It is the comparative numbers of ways in which\nevents can happen which measure their comparative probabilities. If\nwe throw three pennies into the air, what is the probability that two\nof them will fall tail uppermost? This amounts to asking in how many\npossible ways can we select two tails out of three, compared with the\nwhole number of ways in which the coins can be placed. Now, the fourth\nline of the Arithmetical Triangle (p.\u00a0184) gives us the answer. The\nwhole number of ways in which we can select or leave three things is\neight, and the possible combinations of two things at a time is three;\nhence the probability of two tails is the ratio of three to eight. From\nthe numbers in the triangle we may similarly draw all the following\nprobabilities:--\n\n  One combination gives 0 tail. Probability 1/8.\n  Three combinations gives 1 tail. Probability 3/8.\n  Three combinations give 2 tails. Probability 3/8.\n  One combination gives 3 tails. Probability 1/8.\n\nWe can apply the same considerations to the imaginary causes of the\ndifference of stature, the combinations of which were shown in p.\u00a0188.\nThere are altogether 128 ways in which seven causes can be present or\nabsent. Now, twenty-one of these combinations give an addition of two\ninches, so that the probability of a person under the circumstances\nbeing five feet two inches is 21/128. The probability of five feet\nthree inches is 35/128; of five feet one inch 7/128; of five feet\n1/128, and so on. Thus the eighth line of the Arithmetical Triangle\ngives all the probabilities arising out of the combinations of seven\ncauses.\n\n\n*Rules for the Calculation of Probabilities.*\n\nI will now explain as simply as possible the rules for calculating\nprobabilities. The principal rule is as follows:--\n\nCalculate the number of events which may happen independently of each\nother, and which, as far as is known, are equally probable. Make this\nnumber the denominator of a fraction, and take for the numerator the\nnumber of such events as imply or constitute the happening of the\nevent, whose probability is required.\n\nThus, if the letters of the word *Roma* be thrown down casually in a\nrow, what is the probability that they will form a significant Latin\nword? The possible arrangements of four letters are 4\u00a0\u00d7\u00a03\u00a0\u00d7\u00a02\u00a0\u00d7\u00a01,\nor 24 in number (p.\u00a0178), and if all the arrangements be examined,\nseven of these will be found to have meaning, namely *Roma*, *ramo*,\n*oram*, *mora*, *maro*, *armo*, and *amor*. Hence the probability of a\nsignificant result is 7/24.\n\nWe must distinguish comparative from absolute probabilities. In drawing\na card casually from a pack, there is no reason to expect any one card\nmore than any other. Now, there are four kings and four queens in a\npack, so that there are just as many ways of drawing one as the other,\nand the probabilities are equal. But there are thirteen diamonds, so\nthat the probability of a king is to that of a diamond as four to\nthirteen. Thus the probabilities of each are proportional to their\nrespective numbers of ways of happening. Again, I can draw a king in\nfour ways, and not draw one in forty-eight, so that the probabilities\nare in this proportion, or, as is commonly said, the *odds* against\ndrawing a king are forty-eight to four. The odds are seven to seventeen\nin favour, or seventeen to seven against the letters R,o,m,a,\naccidentally forming a significant word. The odds are five to three\nagainst two tails appearing in three throws of a penny. Conversely,\nwhen the odds of an event are given, and the probability is required,\n*take the odds in favour of the event for numerator, and the sum of the\nodds for denominator*.\n\nIt is obvious that an event is certain when all the combinations of\ncauses which can take place produce that event. If we represent the\nprobability of such event according to our rule, it gives the ratio\nof some number to itself, or unity. An event is certain not to happen\nwhen no possible combination of causes gives the event, and the ratio\nby the same rule becomes that of 0 to some number. Hence it follows\nthat in the theory of probability certainty is expressed by 1, and\nimpossibility by 0; but no mystical meaning should be attached to these\nsymbols, as they merely express the fact that *all* or *no* possible\ncombinations give the event.\n\nBy a *compound event*, we mean an event which may be decomposed into\ntwo or more simpler events. Thus the firing of a gun may be decomposed\ninto pulling the trigger, the fall of the hammer, the explosion of\nthe cap, &c. In this example the simple events are not *independent*,\nbecause if the trigger is pulled, the other events will under proper\nconditions necessarily follow, and their probabilities are therefore\nthe same as that of the first event. Events are *independent* when\nthe happening of one does not render the other either more or less\nprobable than before. Thus the death of a person is neither more nor\nless probable because the planet Mars happens to be visible. When\nthe component events are independent, a simple rule can be given for\ncalculating the probability of the compound event, thus--*Multiply\ntogether the fractions expressing the probabilities of the independent\ncomponent events.*\n\nThe probability of throwing tail twice with a penny is 1/2\u00a0\u00d7\u00a01/2,\nor 1/4; the probability of throwing it three times running is\n1/2\u00a0\u00d7\u00a01/2\u00a0\u00d7\u00a01/2, or 1/8; a result agreeing with that obtained in\nan apparently different manner (p.\u00a0202). In fact, when we multiply\ntogether the denominators, we get the whole number of ways of happening\nof the compound event, and when we multiply the numerators, we get the\nnumber of ways favourable to the required event.\n\nProbabilities may be added to or subtracted from each other under the\nimportant condition that the events in question are exclusive of each\nother, so that not more than one of them can happen. It might be argued\nthat, since the probability of throwing head at the first trial is\n1/2, and at the second trial also 1/2, the probability of throwing it\nin the first two throws is 1/2\u00a0+\u00a01/2, or certainty. Not only is this\nresult evidently absurd, but a repetition of the process would lead\nus to a probability of 1-1/2 or of any greater number, results which\ncould have no meaning whatever. The probability we wish to calculate is\nthat of one head in two throws, but in our addition we have included\nthe case in which two heads appear. The true result is 1/2\u00a0+\u00a01/2\u00a0\u00d7\u00a01/2\nor 3/4, or the probability of head at the first throw, added to the\nexclusive probability that if it does not come at the first, it will\ncome at the second. The greatest difficulties of the theory arise\nfrom the confusion of exclusive and unexclusive alternatives. I may\nremind the reader that the possibility of unexclusive alternatives was\na point previously discussed (p.\u00a068), and to the reasons then given\nfor considering alternation as logically unexclusive, may be added\nthe existence of these difficulties in the theory of probability. The\nerroneous result explained above really arose from overlooking the\nfact that the expression \u201chead first throw or head second throw\u201d might\ninclude the case of head at both throws.\n\n\n*The Logical Alphabet in questions of Probability.*\n\nWhen the probabilities of certain simple events are given, and it is\nrequired to deduce the probabilities of compound events, the Logical\nAlphabet may give assistance, provided that there are no special\nlogical conditions so that all the combinations are possible. Thus,\nif there be three events, A, B, C, of which the probabilities are, \u03b1,\n\u03b2, \u03b3, then the negatives of those events, expressing the absence of\nthe events, will have the probabilities 1\u00a0-\u00a0\u03b1, 1\u00a0-\u00a0\u03b2, 1\u00a0-\u00a0\u03b3. We have\nonly to insert these values for the letters of the combinations and\nmultiply, and we obtain the probability of each combination. Thus the\nprobability of ABC is \u03b1\u03b2\u03b3; of A*bc*, \u03b1(1\u00a0-\u00a0\u03b2)(1\u00a0-\u00a0\u03b3).\n\nWe can now clearly distinguish between the probabilities of exclusive\nand unexclusive events. Thus, if A and B are events which may happen\ntogether like rain and high tide, or an earthquake and a storm, the\nprobability of A or B happening is not the sum of their separate\nprobabilities. For by the Laws of Thought we develop A\u00a0\ua58c \u00a0B into\nAB\u00a0\ua58c\u00a0A*b* \ua58c *a*B, and substituting \u03b1 and \u03b2, the probabilities of A\nand B respectively, we obtain \u03b1\u00a0.\u00a0\u03b2\u00a0+\u00a0\u03b1\u00a0.\u00a0(1\u00a0-\u00a0\u03b2)\u00a0+ (1\u00a0-\u00a0\u03b1)\u00a0.\u00a0\u03b2 or\n\u03b1\u00a0+\u00a0\u03b2\u00a0-\u00a0\u03b1\u00a0.\u00a0\u03b2. But if events are *incompossible* or incapable of\nhappening together, like a clear sky and rain, or a new moon and a full\nmoon, then the events are not really A or B, but A not-B, or B not-A,\nor in symbols A*b*\u00a0\ua58c\u00a0*a*B. Now if we take \u03bc = probability of A*b* and \u03bd\n= probability of *a*B, then we may add simply, and the probability of\nA*b*\u00a0\ua58c\u00a0*a*B is \u03bc\u00a0+\u00a0\u03bd.\n\nLet the reader carefully observe that if the combination AB cannot\nexist, the probability of A*b* is not the product of the probabilities\nof A and *b*. When certain combinations are logically impossible, it\nis no longer allowable to substitute the probability of each term for\nthe term, because the multiplication of probabilities presupposes the\nindependence of the events. A large part of Boole\u2019s Laws of Thought\nis devoted to an attempt to overcome this difficulty and to produce\na General Method in Probabilities by which from certain logical\nconditions and certain given probabilities it would be possible to\ndeduce the probability of any other combinations of events under those\nconditions. Boole pursued his task with wonderful ingenuity and power,\nbut after spending much study on his work, I am compelled to adopt\nthe conclusion that his method is fundamentally erroneous. As pointed\nout by Mr. Wilbraham,[113] Boole obtained his results by an arbitrary\nassumption, which is only the most probable, and not the only possible\nassumption. The answer obtained is therefore not the real probability,\nwhich is usually indeterminate, but only, as it were, the most probable\nprobability. Certain problems solved by Boole are free from logical\nconditions and therefore may admit of valid answers. These, as I have\nshown,[114] may be solved by the combinations of the Logical Alphabet,\nbut the rest of the problems do not admit of a determinate answer, at\nleast by Boole\u2019s method.\n\n  [113] *Philosophical Magazine*, 4th Series, vol.\u00a0vii. p.\u00a0465;\n  vol.\u00a0viii. p. 91.\n\n  [114] *Memoirs of the Manchester Literary and Philosophical Society*,\n  3rd Series, vol.\u00a0iv. p.\u00a0347.\n\n\n*Comparison of the Theory with Experience.*\n\nThe Laws of Probability rest upon the fundamental principles of\nreasoning, and cannot be really negatived by any possible experience.\nIt might happen that a person should always throw a coin head\nuppermost, and appear incapable of getting tail by chance. The theory\nwould not be falsified, because it contemplates the possibility of\nthe most extreme runs of luck. Our actual experience might be counter\nto all that is probable; the whole course of events might seem to be\nin complete contradiction to what we should expect, and yet a casual\nconjunction of events might be the real explanation. It is just\npossible that some regular coincidences, which we attribute to fixed\nlaws of nature, are due to the accidental conjunction of phenomena\nin the cases to which our attention is directed. All that we can\nlearn from finite experience is capable, according to the theory of\nprobabilities, of misleading us, and it is only infinite experience\nthat could assure us of any inductive truths.\n\nAt the same time, the probability that any extreme runs of luck will\noccur is so excessively slight, that it would be absurd seriously\nto expect their occurrence. It is almost impossible, for instance,\nthat any whist player should have played in any two games where the\ndistribution of the cards was exactly the same, by pure accident\n(p.\u00a0191). Such a thing as a person always losing at a game of\npure chance, is wholly unknown. Coincidences of this kind are not\nimpossible, as I have said, but they are so unlikely that the lifetime\nof any person, or indeed the whole duration of history, does not give\nany appreciable probability of their being encountered. Whenever we\nmake any extensive series of trials of chance results, as in throwing\na die or coin, the probability is great that the results will agree\nnearly with the predictions yielded by theory. Precise agreement must\nnot be expected, for that, as the theory shows, is highly improbable.\nSeveral attempts have been made to test, in this way, the accordance of\ntheory and experience. Buffon caused the first trial to be made by a\nyoung child who threw a coin many times in succession, and he obtained\n1992 tails to 2048 heads. A pupil of De Morgan repeated the trial for\nhis own satisfaction, and obtained 2044 tails to 2048 heads. In both\ncases the coincidence with theory is as close as could be expected, and\nthe details may be found in De Morgan\u2019s \u201cFormal Logic,\u201d p.\u00a0185.\n\nQuetelet also tested the theory in a rather more complete manner, by\nplacing 20 black and 20 white balls in an urn and drawing a ball out\ntime after time in an indifferent manner, each ball being replaced\nbefore a new drawing was made. He found, as might be expected, that the\ngreater the number of drawings made, the more nearly were the white\nand black balls equal in number. At the termination of the experiment\nhe had registered 2066 white and 2030 black balls, the ratio being\n1\u00b702.[115]\n\n  [115] *Letters on the Theory of Probabilities*, translated by Downes,\n  1849, pp. 36, 37.\n\nI have made a series of experiments in a third manner, which seemed to\nme even more interesting, and capable of more extensive trial. Taking\na handful of ten coins, usually shillings, I threw them up time after\ntime, and registered the numbers of heads which appeared each time. Now\nthe probability of obtaining 10, 9, 8, 7, &c., heads is proportional\nto the number of combinations of 10, 9, 8, 7, &c., things out of 10\nthings. Consequently the results ought to approximate to the numbers in\nthe eleventh line of the Arithmetical Triangle. I made altogether 2048\nthrows, in two sets of 1024 throws each, and the numbers obtained are\ngiven in the following table:--\n\n+-------------------+-----------+---------+---------+----------+-----------+\n|Character of Throw.|Theoretical|  First  | Second  | Average. |Divergence.|\n|                   | Numbers.  | Series. | Series. |          |           |\n+-------------------+-----------+---------+---------+----------+-----------+\n|  10 Heads 0 Tail  |      1    |     3   |     1   |    2     |  +  1     |\n|   9   \"   1   \"   |     10    |    12   |    23   |   17-1/2 |  +  7-1/2 |\n|   8   \"   2   \"   |     45    |    57   |    73   |   65     |  + 20     |\n|   7   \"   3   \"   |    120    |   129   |   123   |  126     |  +  6     |\n|   6   \"   4   \"   |    210    |   181   |   190   |  185-1/2 |  - 25     |\n|   5   \"   5   \"   |    252    |   257   |   232   |  244-1/2 |  -  7-1/2 |\n|   4   \"   6   \"   |    210    |   201   |   197   |  199     |  - 11     |\n|   3   \"   7   \"   |    120    |   111   |   119   |  115     |  -  5     |\n|   2   \"   8   \"   |     45    |    52   |    50   |   51     |  +  6     |\n|   1   \"   9   \"   |     10    |    21   |    15   |   18     |  +  8     |\n|   0   \"  10   \"   |      1    |     0   |     1   |      1/2 |  -    1/2 |\n+-------------------+-----------+---------+---------+----------+-----------+\n|      Totals       |   1024    |  1024   |  1024   | 1024     |  -  1     |\n+-------------------+-----------+---------+---------+----------+-----------+\n\nThe whole number of single throws of coins amounted to 10\u00a0\u00d7\u00a02048, or\n20,480 in all, one half of which or 10,240 should theoretically give\nhead. The total number of heads obtained was actually 10,353, or 5222\nin the first series, and 5131 in the second. The coincidence with\ntheory is pretty close, but considering the large number of throws\nthere is some reason to suspect a tendency in favour of heads.\n\nThe special interest of this trial consists in the exhibition, in a\npractical form, of the results of Bernoulli\u2019s theorem, and the law\nof error or divergence from the mean to be afterwards more fully\nconsidered. It illustrates the connection between combinations and\npermutations, which is exhibited in the Arithmetical Triangle, and\nwhich underlies many important theorems of science.\n\n\n*Probable Deductive Arguments*.\n\nWith the aid of the theory of probabilities, we may extend the sphere\nof deductive argument. Hitherto we have treated propositions as\ncertain, and on the hypothesis of certainty have deduced conclusions\nequally certain. But the information on which we reason in ordinary\nlife is seldom or never certain, and almost all reasoning is really a\nquestion of probability. We ought therefore to be fully aware of the\nmode and degree in which deductive reasoning is affected by the theory\nof probability, and many persons may be surprised at the results which\nmust be admitted. Some controversial writers appear to consider, as De\nMorgan remarked,[116] that an inference from several equally probable\npremises is itself as probable as any of them, but the true result is\nvery different. If an argument involves many propositions, and each of\nthem is uncertain, the conclusion will be of very little force.\n\n  [116] *Encyclop\u00e6dia Metropolitana*, art. *Probabilities*, p.\u00a0396.\n\nThe validity of a conclusion may be regarded as a compound event,\ndepending upon the premises happening to be true; thus, to obtain the\nprobability of the conclusion, we must multiply together the fractions\nexpressing the probabilities of the premises. If the probability is\n1/2 that A is B, and also 1/2 that B is C, the conclusion that A is\nC, on the ground of these premises, is 1/2\u00a0\u00d7\u00a01/2 or 1/4. Similarly\nif there be any number of premises requisite to the establishment\nof a conclusion and their probabilities be *p*, *q*, *r*, &c., the\nprobability of the conclusion on the ground of these premises is\n*p*\u00a0\u00d7\u00a0*q*\u00a0\u00d7\u00a0*r*\u00a0\u00d7\u00a0... This product has but a small value, unless each\nof the quantities *p*, *q*, &c., be nearly unity.\n\nBut it is particularly to be noticed that the probability thus\ncalculated is not the whole probability of the conclusion, but that\nonly which it derives from the premises in question. Whately\u2019s[117]\nremarks on this subject might mislead the reader into supposing that\nthe calculation is completed by multiplying together the probabilities\nof the premises. But it has been fully explained by De Morgan[118] that\nwe must take into account the antecedent probability of the conclusion;\nA may be C for other reasons besides its being B, and as he remarks,\n\u201cIt is difficult, if not impossible, to produce a chain of argument of\nwhich the reasoner can rest the result on those arguments only.\u201d The\nfailure of one argument does not, except under special circumstances,\ndisprove the truth of the conclusion it is intended to uphold,\notherwise there are few truths which could survive the ill-considered\narguments adduced in their favour. As a rope does not necessarily break\nbecause one or two strands in it fail, so a conclusion may depend upon\nan endless number of considerations besides those immediately in view.\nEven when we have no other information we must not consider a statement\nas devoid of all probability. The true expression of complete doubt is\na ratio of equality between the chances in favour of and against it,\nand this ratio is expressed in the probability 1/2.\n\n  [117] *Elements of Logic*, Book III. sections 11 and 18.\n\n  [118] *Encyclop\u00e6dia Metropolitana*, art. *Probabilities*, p.\u00a0400.\n\nNow if A and C are wholly unknown things, we have no reason to believe\nthat A is C rather than A is not C. The antecedent probability is then\n1/2. If we also have the probabilities that A is B, 1/2 and that B is\nC, 1/2 we have no right to suppose that the probability of A being C\nis reduced by the argument in its favour. If the conclusion is true\non its own grounds, the failure of the argument does not affect it;\nthus its total probability is its antecedent probability, added to the\nprobability that this failing, the new argument in question establishes\nit. There is a probability 1/2 that we shall not require the special\nargument; a probability 1/2 that we shall, and a probability 1/4\nthat the argument does in that case establish it. Thus the complete\nresult is 1/2\u00a0+\u00a01/2\u00a0\u00d7\u00a01/4, or 5/8. In general language, if *a* be the\nprobability founded on a particular argument, and *c* the antecedent\nprobability of the event, the general result is 1\u00a0-\u00a0(1\u00a0-\u00a0*a*)(1\u00a0-\u00a0*c*),\nor *a*\u00a0+\u00a0*c*\u00a0-\u00a0*ac*.\n\nWe may put it still more generally in this way:--Let *a*, *b*, *c*, &c.\nbe the probabilities of a conclusion grounded on various arguments. It\nis only when all the arguments fail that our conclusion proves finally\nuntrue; the probabilities of each failing are respectively, 1\u00a0-\u00a0*a*,\n1\u00a0-\u00a0*b*, 1\u00a0-\u00a0*c*, &c.; the probability that they will all fail is\n(1\u00a0-\u00a0*a*)(1\u00a0-\u00a0*b*)(1\u00a0-\u00a0*c*) ...; therefore the probability that the\nconclusion will not fail is 1\u00a0-\u00a0(1\u00a0-\u00a0*a*)(1\u00a0-\u00a0*b*)(1\u00a0-\u00a0*c*) ... &c. It\nfollows that every argument in favour of a conclusion, however flimsy\nand slight, adds probability to it. When it is unknown whether an\noverdue vessel has foundered or not, every slight indication of a lost\nvessel will add some probability to the belief of its loss, and the\ndisproof of any particular evidence will not disprove the event.\n\nWe must apply these principles of evidence with great care, and\nobserve that in a great proportion of cases the adducing of a weak\nargument does tend to the disproof of its conclusion. The assertion\nmay have in itself great inherent improbability as being opposed to\nother evidence or to the supposed law of nature, and every reasoner\nmay be assumed to be dealing plainly, and putting forward the whole\nforce of evidence which he possesses in its favour. If he brings but\none argument, and its probability *a* is small, then in the formula\n1\u00a0-\u00a0(1\u00a0-\u00a0*a*)(1\u00a0-\u00a0*c*) both *a* and *c* are small, and the whole\nexpression has but little value. The whole effect of an argument thus\nturns upon the question whether other arguments remain, so that we\ncan introduce other factors (1\u00a0-\u00a0*b*), (1\u00a0-\u00a0*d*), &c., into the above\nexpression. In a court of justice, in a publication having an express\npurpose, and in many other cases, it is doubtless right to assume\nthat the whole evidence considered to have any value as regards the\nconclusion asserted, is put forward.\n\nTo assign the antecedent probability of any proposition, may be a\nmatter of difficulty or impossibility, and one with which logic and\nthe theory of probability have little concern. From the general body of\nscience in our possession, we must in each case make the best judgment\nwe can. But in the absence of all knowledge the probability should\nbe considered = 1/2, for if we make it less than this we incline to\nbelieve it false rather than true. Thus, before we possessed any means\nof estimating the magnitudes of the fixed stars, the statement that\nSirius was greater than the sun had a probability of exactly 1/2; it\nwas as likely that it would be greater as that it would be smaller; and\nso of any other star. This was the assumption which Michell made in\nhis admirable speculations.[119] It might seem, indeed, that as every\nproposition expresses an agreement, and the agreements or resemblances\nbetween phenomena are infinitely fewer than the differences (p.\u00a044),\nevery proposition should in the absence of other information be\ninfinitely improbable. But in our logical system every term may be\nindifferently positive or negative, so that we express under the same\nform as many differences as agreements. It is impossible therefore\nthat we should have any reason to disbelieve rather than to believe a\nstatement about things of which we know nothing. We can hardly indeed\ninvent a proposition concerning the truth of which we are absolutely\nignorant, except when we are entirely ignorant of the terms used. If I\nask the reader to assign the odds that a \u201cPlatythliptic Coefficient is\npositive\u201d he will hardly see his way to doing so, unless he regard them\nas even.\n\n  [119] *Philosophical Transactions* (1767). Abridg. vol.\u00a0xii. p.\u00a0435.\n\nThe assumption that complete doubt is properly expressed by 1/2 has\nbeen called in question by Bishop Terrot,[120] who proposes instead\nthe indefinite symbol 0/0; and he considers that \u201cthe *\u00e0 priori*\nprobability derived from absolute ignorance has no effect upon the\nforce of a subsequently admitted probability.\u201d But if we grant that the\nprobability may have any value between 0 and 1, and that every separate\nvalue is equally likely, then *n* and 1\u00a0-\u00a0*n* are equally likely, and\nthe average is always 1/2. Or we may take *p*\u00a0.\u00a0*dp* to express the\nprobability that our estimate concerning any proposition should lie\nbetween *p* and *p*\u00a0+\u00a0*dp*. The complete probability of the proposition\nis then the integral taken between the limits 1 and 0, or again 1/2.\n\n  [120] *Transactions of the Edinburgh Philosophical Society*,\n  vol.\u00a0xxi. p. 375.\n\n\n*Difficulties of the Theory.*\n\nThe theory of probability, though undoubtedly true, requires very\ncareful application. Not only is it a branch of mathematics in which\noversights are frequently committed, but it is a matter of great\ndifficulty in many cases, to be sure that the formula correctly\nrepresents the data of the problem. These difficulties often arise from\nthe logical complexity of the conditions, which might be, perhaps, to\nsome extent cleared up by constantly bearing in mind the system of\ncombinations as developed in the Indirect Logical Method. In the study\nof probabilities, mathematicians had unconsciously employed logical\nprocesses far in advance of those in possession of logicians, and the\nIndirect Method is but the full statement of these processes.\n\nIt is very curious how often the most acute and powerful intellects\nhave gone astray in the calculation of probabilities. Seldom was Pascal\nmistaken, yet he inaugurated the science with a mistaken solution.[121]\nLeibnitz fell into the extraordinary blunder of thinking that the\nnumber twelve was as probable a result in the throwing of two dice as\nthe number eleven.[122] In not a few cases the false solution first\nobtained seems more plausible to the present day than the correct\none since demonstrated. James Bernoulli candidly records two false\nsolutions of a problem which he at first thought self-evident; and he\nadds a warning against the risk of error, especially when we attempt\nto reason on this subject without a rigid adherence to methodical\nrules and symbols. Montmort was not free from similar mistakes.\nD\u2019Alembert constantly fell into blunders, and could not perceive,\nfor instance, that the probabilities would be the same when coins\nare thrown successively as when thrown simultaneously. Some men of\ngreat reputation, such as Ancillon, Moses Mendelssohn, Garve, Auguste\nComte,[123] Poinsot, and J.\u00a0S. Mill,[124] have so far misapprehended\nthe theory, as to question its value or even to dispute its validity.\nThe erroneous statements about the theory given in the earlier editions\nof Mill\u2019s *System of Logic* were partially withdrawn in the later\neditions.\n\n  [121] Montucla, *Histoire des Math\u00e9matiques*, vol.\u00a0iii. p.\u00a0386.\n\n  [122] Leibnitz *Opera*, Dutens\u2019 Edition, vol.\u00a0vi. part i. p.\u00a0217.\n  Todhunter\u2019s *History of the Theory of Probability*, p.\u00a048. To the\n  latter work I am indebted for many of the statements in the text.\n\n  [123] *Positive Philosophy*, translated by Martineau, vol.\u00a0ii. p.\u00a0120.\n\n  [124] *System of Logic*, bk. iii. chap. 18, 5th Ed. vol.\u00a0ii. p.\u00a061.\n\nMany persons have a fallacious tendency to believe that when a chance\nevent has happened several times together in an unusual conjunction, it\nis less likely to happen again. D\u2019Alembert seriously held that if head\nwas thrown three times running with a coin, tail would more probably\nappear at the next trial.[125] Bequelin adopted the same opinion, and\nyet there is no reason for it whatever. If the event be really casual,\nwhat has gone before cannot in the slightest degree influence it. As\na matter of fact, the more often a casual event takes place the more\nlikely it is to happen again; because there is some slight empirical\nevidence of a tendency. The source of the fallacy is to be found\nentirely in the feelings of surprise with which we witness an event\nhappening by chance, in a manner which seems to proceed from design.\n\n  [125] Montucla, *Histoire*, vol.\u00a0iii. p.\u00a0405; Todhunter, p.\u00a0263.\n\nMisapprehension may also arise from overlooking the difference between\npermutations and combinations. To throw ten heads in succession with a\ncoin is no more unlikely than to throw any other particular succession\nof heads and tails, but it is much less likely than five heads and five\ntails without regard to their order, because there are no less than\n252 different particular throws which will give this result, when we\nabstract the difference of order.\n\nDifficulties arise in the application of the theory from our habitual\ndisregard of slight probabilities. We are obliged practically to accept\ntruths as certain which are nearly so, because it ceases to be worth\nwhile to calculate the difference. No punishment could be inflicted\nif absolutely certain evidence of guilt were required, and as Locke\nremarks, \u201cHe that will not stir till he infallibly knows the business\nhe goes about will succeed, will have but little else to do but to\nsit still and perish.\u201d[126] There is not a moment of our lives when\nwe do not lie under a slight danger of death, or some most terrible\nfate. There is not a single action of eating, drinking, sitting down,\nor standing up, which has not proved fatal to some person. Several\nphilosophers have tried to assign the limit of the probabilities\nwhich we regard as zero; Buffon named 1/10,000, because it is the\nprobability, practically disregarded, that a man of 56 years of age\nwill die the next day. Pascal remarked that a man would be esteemed a\nfool for hesitating to accept death when three dice gave sixes twenty\ntimes running, if his reward in case of a different result was to be a\ncrown; but as the chance of death in question is only 1\u00a0\u00f7\u00a06^{60}, or\nunity divided by a number of 47 places of figures, we may be said to\nincur greater risks every day for less motives. There is far greater\nrisk of death, for instance, in a game of cricket or a visit to the\nrink.\n\n  [126] *Essay concerning Human Understanding*, bk. iv. ch. 14. \u00a7 1.\n\nNothing is more requisite than to distinguish carefully between the\ntruth of a theory and the truthful application of the theory to actual\ncircumstances. As a general rule, events in nature and art will present\na complexity of relations exceeding our powers of treatment. The\nintricate action of the mind often intervenes and renders complete\nanalysis hopeless. If, for instance, the probability that a marksman\nshall hit the target in a single shot be 1 in 10, we might seem to\nhave no difficulty in calculating the probability of any succession of\nhits; thus the probability of three successive hits would be one in a\nthousand. But, in reality, the confidence and experience derived from\nthe first successful shot would render a second success more probable.\nThe events are not really independent, and there would generally be\na far greater preponderance of runs of apparent luck, than a simple\ncalculation of probabilities could account for. In some persons,\nhowever, a remarkable series of successes will produce a degree of\nexcitement rendering continued success almost impossible.\n\nAttempts to apply the theory of probability to the results of judicial\nproceedings have proved of little value, simply because the conditions\nare far too intricate. As Laplace said, \u201cTant de passions, d\u2019int\u00e9r\u00eats\ndivers et de circonstances compliquent les questions relatives \u00e0 ces\nobjets, qu\u2019elles sont presque toujours insolubles.\u201d Men acting on a\njury, or giving evidence before a court, are subject to so many complex\ninfluences that no mathematical formulas can be framed to express the\nreal conditions. Jurymen or even judges on the bench cannot be regarded\nas acting independently, with a definite probability in favour of each\ndelivering a correct judgment. Each man of the jury is more or less\ninfluenced by the opinion of the others, and there are subtle effects\nof character and manner and strength of mind which defy analysis. Even\nin physical science we can in comparatively few cases apply the theory\nin a definite manner, because the data required are too complicated and\ndifficult to obtain. But such failures in no way diminish the truth and\nbeauty of the theory itself; in reality there is no branch of science\nin which our symbols can cope with the complexity of Nature. As Donkin\nsaid,--\n\n\u201cI do not see on what ground it can be doubted that every definite\nstate of belief concerning a proposed hypothesis, is in itself capable\nof being represented by a numerical expression, however difficult or\nimpracticable it may be to ascertain its actual value. It would be very\ndifficult to estimate in numbers the *vis viva* of all the particles of\na human body at any instant; but no one doubts that it is capable of\nnumerical expression.\u201d[127]\n\n  [127] *Philosophical Magazine*, 4th Series, vol.\u00a0i. p.\u00a0354.\n\nThe difficulty, in short, is merely relative to our knowledge and\nskill, and is not absolute or inherent in the subject. We must\ndistinguish between what is theoretically conceivable and what is\npracticable with our present mental resources. Provided that our\naspirations are pointed in a right direction, we must not allow them\nto be damped by the consideration that they pass beyond what can now\nbe turned to immediate use. In spite of its immense difficulties of\napplication, and the aspersions which have been mistakenly cast upon\nit, the theory of probabilities, I repeat, is the noblest, as it\nwill in course of time prove, perhaps the most fruitful branch of\nmathematical science. It is the very guide of life, and hardly can\nwe take a step or make a decision of any kind without correctly or\nincorrectly making an estimation of probabilities. In the next chapter\nwe proceed to consider how the whole cogency of inductive reasoning\nrests upon probabilities. The truth or untruth of a natural law, when\ncarefully investigated, resolves itself into a high or low degree of\nprobability, and this is the case whether or not we are capable of\nproducing precise numerical data.\n\n\n\n\nCHAPTER XI.\n\nPHILOSOPHY OF INDUCTIVE INFERENCE.\n\n\nWe have inquired into the nature of perfect induction, whereby we pass\nbackwards from certain observed combinations of events, to the logical\nconditions governing such combinations. We have also investigated the\ngrounds of that theory of probability, which must be our guide when we\nleave certainty behind, and dilute knowledge with ignorance. There is\nnow before us the difficult task of endeavouring to decide how, by the\naid of that theory, we can ascend from the facts to the laws of nature;\nand may then with more or less success anticipate the future course\nof events. All our knowledge of natural objects must be ultimately\nderived from observation, and the difficult question arises--How can\nwe ever know anything which we have not directly observed through one\nof our senses, the apertures of the mind? The utility of reasoning is\nto assure ourselves that, at a determinate time and place, or under\nspecified conditions, a certain phenomenon will be observed. When\nwe can use our senses and perceive that the phenomenon does occur,\nreasoning is superfluous. If the senses cannot be used, because the\nevent is in the future, or out of reach, how can reasoning take their\nplace? Apparently, at least, we must infer the unknown from the known,\nand the mind must itself create an addition to the sum of knowledge.\nBut I hold that it is quite impossible to make any real additions to\nthe contents of our knowledge, except through new impressions upon\nthe senses, or upon some seat of feeling. I shall attempt to show\nthat inference, whether inductive or deductive, is never more than\nan unfolding of the contents of our experience, and that it always\nproceeds upon the assumption that the future and the unperceived will\nbe governed by the same conditions as the past and the perceived, an\nassumption which will often prove to be mistaken.\n\nIn inductive as in deductive reasoning the conclusion never passes\nbeyond the premises. Reasoning adds no more to the implicit contents of\nour knowledge, than the arrangement of the specimens in a museum adds\nto the number of those specimens. Arrangement adds to our knowledge\nin a certain sense: it allows us to perceive the similarities and\npeculiarities of the specimens, and on the assumption that the museum\nis an adequate representation of nature, it enables us to judge of\nthe prevailing forms of natural objects. Bacon\u2019s first aphorism holds\nperfectly true, that man knows nothing but what he has observed,\nprovided that we include his whole sources of experience, and the whole\nimplicit contents of his knowledge. Inference but unfolds the hidden\nmeaning of our observations, and *the theory of probability shows how\nfar we go beyond our data in assuming that new specimens will resemble\nthe old ones*, or that the future may be regarded as proceeding\nuniformly with the past.\n\n\n*Various Classes of Inductive Truths.*\n\nIt will be desirable, in the first place, to distinguish between the\nseveral kinds of truths which we endeavour to establish by induction.\nAlthough there is a certain common and universal element in all our\nprocesses of reasoning, yet diversity arises in their application.\nSimilarity of condition between the events from which we argue, and\nthose to which we argue, must always be the ground of inference; but\nthis similarity may have regard either to time or place, or the simple\nlogical combination of events, or to any conceivable junction of\ncircumstances involving quality, time, and place. Having met with many\npieces of substance possessing ductility and a bright yellow colour,\nand having discovered, by perfect induction, that they all possess a\nhigh specific gravity, and a freedom from the corrosive action of\nacids, we are led to expect that every piece of substance, possessing\nlike ductility and a similar yellow colour, will have an equally high\nspecific gravity, and a like freedom from corrosion by acids. This\nis a case of the coexistence of qualities; for the character of the\nspecimens examined alters not with time nor place.\n\nIn a second class of cases, time will enter as a principal ground of\nsimilarity. When we hear a clock pendulum beat time after time, at\nequal intervals, and with a uniform sound, we confidently expect that\nthe stroke will continue to be repeated uniformly. A comet having\nappeared several times at nearly equal intervals, we infer that it will\nprobably appear again at the end of another like interval. A man who\nhas returned home evening after evening for many years, and found his\nhouse standing, may, on like grounds, expect that it will be standing\nthe next evening, and on many succeeding evenings. Even the continuous\nexistence of an object in an unaltered state, or the finding again of\nthat which we have hidden, is but a matter of inference depending on\nexperience.\n\nA still larger and more complex class of cases involves the relations\nof space, in addition to those of time and quality. Having observed\nthat every triangle drawn upon the diameter of a circle, with its\napex upon the circumference, apparently contains a right angle,\nwe may ascertain that all triangles in similar circumstances will\ncontain right angles. This is a case of pure space reasoning, apart\nfrom circumstances of time or quality, and it seems to be governed by\ndifferent principles of reasoning. I shall endeavour to show, however,\nthat geometrical reasoning differs but in degree from that which\napplies to other natural relations.\n\n\n*The Relation of Cause and Effect.*\n\nIn a very large part of the scientific investigations which must be\nconsidered, we deal with events which follow from previous events, or\nwith existences which succeed existences. Science, indeed, might arise\neven were material nature a fixed and changeless whole. Endow mind\nwith the power to travel about, and compare part with part, and it\ncould certainly draw inferences concerning the similarity of forms, the\ncoexistence of qualities, or the preponderance of a particular kind of\nmatter in a changeless world. A solid universe, in at least approximate\nequilibrium, is not inconceivable, and then the relation of cause and\neffect would evidently be no more than the relation of before and\nafter. As nature exists, however, it is a progressive existence, ever\nmoving and changing as time, the great independent variable, proceeds.\nHence it arises that we must continually compare what is happening now\nwith what happened a moment before, and a moment before that moment,\nand so on, until we reach indefinite periods of past time. A comet\nis seen moving in the sky, or its constituent particles illumine\nthe heavens with their tails of fire. We cannot explain the present\nmovements of such a body without supposing its prior existence, with\na definite amount of energy and a definite direction of motion; nor\ncan we validly suppose that our task is concluded when we find that it\ncame wandering to our solar system through the unmeasured vastness of\nsurrounding space. Every event must have a cause, and that cause again\na cause, until we are lost in the obscurity of the past, and are driven\nto the belief in one First Cause, by whom the course of nature was\ndetermined.\n\n\n*Fallacious Use of the Term Cause.*\n\nThe words Cause and Causation have given rise to infinite trouble\nand obscurity, and have in no slight degree retarded the progress of\nscience. From the time of Aristotle, the work of philosophy has been\ndescribed as the discovery of the causes of things, and Francis Bacon\nadopted the notion when he said \u201c*vere scire esse per causas scire*.\u201d\nEven now it is not uncommonly supposed that the knowledge of causes is\nsomething different from other knowledge, and consists, as it were, in\ngetting possession of the keys of nature. A single word may thus act\nas a spell, and throw the clearest intellect into confusion, as I have\noften thought that Locke was thrown into confusion when endeavouring to\nfind a meaning for the word *power*.[128] In Mill\u2019s *System of Logic*\nthe term *cause* seems to have re-asserted its old noxious power. Not\nonly does Mill treat the Laws of Causation as almost coextensive with\nscience, but he so uses the expression as to imply that when once we\npass within the circle of causation we deal with certainties.\n\n  [128] *Essay concerning Human Understanding*, bk. ii. chap. xxi.\n\nThe philosophical danger which attaches to the use of this word may\nbe thus described. A cause is defined as the necessary or invariable\nantecedent of an event, so that when the cause exists the effect will\nalso exist or soon follow. If then we know the cause of an event, we\nknow what will certainly happen; and as it is implied that science,\nby a proper experimental method, may attain to a knowledge of causes,\nit follows that experience may give us a certain knowledge of future\nevents. But nothing is more unquestionable than that finite experience\ncan never give us certain knowledge of the future, so that either\na cause is not an invariable antecedent, or else we can never gain\ncertain knowledge of causes. The first horn of this dilemma is hardly\nto be accepted. Doubtless there is in nature some invariably acting\nmechanism, such that from certain fixed conditions an invariable result\nalways emerges. But we, with our finite minds and short experience, can\nnever penetrate the mystery of those existences which embody the Will\nof the Creator, and evolve it throughout time. We are in the position\nof spectators who witness the productions of a complicated machine, but\nare not allowed to examine its intimate structure. We learn what does\nhappen and what does appear, but if we ask for the reason, the answer\nwould involve an infinite depth of mystery. The simplest bit of matter,\nor the most trivial incident, such as the stroke of two billiard balls,\noffers infinitely more to learn than ever the human intellect can\nfathom. The word cause covers just as much untold meaning as any of the\nwords *substance*, *matter*, *thought*, *existence*.\n\n\n*Confusion of Two Questions.*\n\nThe subject is much complicated, too, by the confusion of two distinct\nquestions. An event having happened, we may ask--\n\n  (1) Is there any cause for the event?\n  (2) Of what kind is that cause?\n\nNo one would assert that the mind possesses any faculty capable of\ninferring, prior to experience, that the occurrence of a sudden noise\nwith flame and smoke indicates the combustion of a black powder, formed\nby the mixture of black, white, and yellow powders. The greatest\nupholder of *\u00e0 priori* doctrines will allow that the particular aspect,\nshape, size, colour, texture, and other qualities of a cause must be\ngathered through the senses.\n\nThe question whether there is any cause at all for an event, is of\na totally different kind. If an explosion could happen without any\nprior existing conditions, it must be a new creation--a distinct\naddition to the universe. It may be plausibly held that we can imagine\nneither the creation nor annihilation of anything. As regards matter,\nthis has long been held true; as regards force, it is now almost\nuniversally assumed as an axiom that energy can neither come into\nnor go out of existence without distinct acts of Creative Will. That\nthere exists any instinctive belief to this effect, indeed, seems\ndoubtful. We find Lucretius, a philosopher of the utmost intellectual\npower and cultivation, gravely assuming that his raining atoms could\nturn aside from their straight paths in a self-determining manner,\nand by this spontaneous origination of energy determine the form of\nthe universe.[129] Sir George Airy, too, seriously discussed the\nmathematical conditions under which a perpetual motion, that is, a\nperpetual source of self-created energy, might exist.[130] The larger\npart of the philosophic world has long held that in mental acts there\nis free will--in short, self-causation. It is in vain to attempt to\nreconcile this doctrine with that of an intuitive belief in causation,\nas Sir W. Hamilton candidly allowed.\n\n  [129] *De Rerum Natura*, bk. ii. ll. 216\u2013293.\n\n  [130] *Cambridge Philosophical Transactions* (1830), vol.\u00a0iii. pp.\n  369\u2013372.\n\nIt is obvious, moreover, that to assert the existence of a cause for\nevery event cannot do more than remove into the indefinite past the\ninconceivable fact and mystery of creation. At any given moment matter\nand energy were equal to what they are at present, or they were not;\nif equal, we may make the same inquiry concerning any other moment,\nhowever long prior, and we are thus obliged to accept one horn of the\ndilemma--existence from infinity, or creation at some moment. This is\nbut one of the many cases in which we are compelled to believe in one\nor other of two alternatives, both inconceivable. My present purpose,\nhowever, is to point out that we must not confuse this supremely\ndifficult question with that into which inductive science inquires on\nthe foundation of facts. By induction we gain no certain knowledge;\nbut by observation, and the inverse use of deductive reasoning, we\nestimate the probability that an event which has occurred was preceded\nby conditions of specified character, or that such conditions will be\nfollowed by the event.\n\n\n*Definition of the Term Cause.*\n\nClear definitions of the word cause have been given by several\nphilosophers. Hobbes has said, \u201cA cause is the sum or aggregate of all\nsuch accidents, both in the agents and the patients, as concur in the\nproducing of the effect propounded; all which existing together, it\ncannot be understood but that the effect existeth with them; or that\nit can possibly exist if any of them be absent.\u201d Brown, in his *Essay\non Causation*, gave a nearly corresponding statement. \u201cA cause,\u201d he\nsays,[131] \u201cmay be defined to be the object or event which immediately\nprecedes any change, and which existing again in similar circumstances\nwill be always immediately followed by a similar change.\u201d Of the\nkindred word *power*, he likewise says:[132] \u201cPower is nothing more\nthan that invariableness of antecedence which is implied in the belief\nof causation.\u201d\n\n  [131] *Observations on the Nature and Tendency of the Doctrine of Mr.\n  Hume, concerning the Relation of Cause and Effect.* Second ed. p. 44.\n\n  [132] Ibid. p.\u00a097.\n\nThese definitions may be accepted with the qualification that our\nknowledge of causes in such a sense can be probable only. The work of\nscience consists in ascertaining the combinations in which phenomena\npresent themselves. Concerning every event we shall have to determine\nits probable conditions, or the group of antecedents from which it\nprobably follows. An antecedent is anything which exists prior to\nan event; a consequent is anything which exists subsequently to an\nantecedent. It will not usually happen that there is any probable\nconnection between an antecedent and consequent. Thus nitrogen is an\nantecedent to the lighting of a common fire; but it is so far from\nbeing a cause of the lighting, that it renders the combustion less\nactive. Daylight is an antecedent to all fires lighted during the day,\nbut it probably has no appreciable effect upon their burning. But\nin the case of any given event it is usually possible to discover a\ncertain number of antecedents which seem to be always present, and with\nmore or less probability we conclude that when they exist the event\nwill follow.\n\nLet it be observed that the utmost latitude is at present enjoyed in\nthe use of the term *cause*. Not only may a cause be an existent thing\nendowed with powers, as oxygen is the cause of combustion, gunpowder\nthe cause of explosion, but the very absence or removal of a thing may\nalso be a cause. It is quite correct to speak of the dryness of the\nEgyptian atmosphere, or the absence of moisture, as being the cause of\nthe preservation of mummies, and other remains of antiquity. The cause\nof a mountain elevation, Ingleborough for instance, is the excavation\nof the surrounding valleys by denudation. It is not so usual to speak\nof the existence of a thing at one moment as the cause of its existence\nat the next, but to me it seems the commonest case of causation which\ncan occur. The cause of motion of a billiard ball may be the stroke of\nanother ball; and recent philosophy leads us to look upon all motions\nand changes, as but so many manifestations of prior existing energy.\nIn all probability there is no creation of energy and no destruction,\nso that as regards both mechanical and molecular changes, the cause is\nreally the manifestation of existing energy. In the same way I see not\nwhy the prior existence of matter is not also a cause as regards its\nsubsequent existence. All science tends to show us that the existence\nof the universe in a particular state at one moment, is the condition\nof its existence at the next moment, in an apparently different\nstate. When we analyse the meaning which we can attribute to the word\n*cause*, it amounts to the existence of suitable portions of matter\nendowed with suitable quantities of energy. If we may accept Horne\nTooke\u2019s assertion, *cause* has etymologically the meaning of *thing\nbefore*. Though, indeed, the origin of the word is very obscure, its\nderivatives, the Italian *cosa*, and the French *chose*, mean simply\n*thing*. In the German equivalent *ursache*, we have plainly the\noriginal meaning of *thing before*, the *sache* denoting \u201cinteresting\nor important object,\u201d the English *sake*, and *ur* being the equivalent\nof the English *ere*, *before*. We abandon, then, both etymology and\nphilosophy, when we attribute to the *laws of causation* any meaning\nbeyond that of the *conditions* under which an event may be expected to\nhappen, according to our observation of the previous course of nature.\n\nI have no objection to use the words cause and causation, provided they\nare never allowed to lead us to imagine that our knowledge of nature\ncan attain to certainty. I repeat that if a cause is an invariable\nand necessary condition of an event, we can never know certainly\nwhether the cause exists or not. To us, then, a cause is not to be\ndistinguished from the group of positive or negative conditions which,\nwith more or less probability, precede an event. In this sense, there\nis no particular difference between knowledge of causes and our general\nknowledge of the succession of combinations, in which the phenomena of\nnature are presented to us, or found to occur in experimental inquiry.\n\n\n*Distinction of Inductive and Deductive Results.*\n\nWe must carefully avoid confusing together inductive investigations\nwhich terminate in the establishment of general laws, and those which\nseem to lead directly to the knowledge of future particular events.\nThat process only can be called induction which gives general laws,\nand it is by the subsequent employment of deduction that we anticipate\nparticular events. If the observation of a number of cases shows that\nalloys of metals fuse at lower temperatures than their constituent\nmetals, I may with more or less probability draw a general inference\nto that effect, and may thence deductively ascertain the probability\nthat the next alloy examined will fuse at a lower temperature than\nits constituents. It has been asserted, indeed, by Mill,[133] and\npartially admitted by Mr. Fowler,[134] that we can argue directly from\ncase to case, so that what is true of some alloys will be true of the\nnext. Professor Bain has adopted the same view of reasoning. He thinks\nthat Mill has extricated us from the dead lock of the syllogism and\neffected a total revolution in logic. He holds that reasoning from\nparticulars to particulars is not only the usual, the most obvious and\nthe most ready method, but that it is the type of reasoning which best\ndiscloses the real process.[135] Doubtless, this is the usual result of\nour reasoning, regard being had to degrees of probability; but these\nlogicians fail entirely to give any explanation of the process by which\nwe get from case to case.\n\n  [133] *System of Logic*, bk. II. chap, iii.\n\n  [134] *Inductive Logic*, pp.\u00a013, 14.\n\n  [135] Bain, *Deductive Logic*, pp.\u00a0208, 209.\n\nIt may be allowed that the knowledge of future particular events is\nthe main purpose of our investigations, and if there were any process\nof thought by which we could pass directly from event to event without\nascending into general truths, this method would be sufficient, and\ncertainly the briefest. It is true, also, that the laws of mental\nassociation lead the mind always to expect the like again in apparently\nlike circumstances, and even animals of very low intelligence must\nhave some trace of such powers of association, serving to guide them\nmore or less correctly, in the absence of true reasoning faculties.\nBut it is the purpose of logic, according to Mill, to ascertain\nwhether inferences have been correctly drawn, rather than to discover\nthem.[136] Even if we can, then, by habit, association, or any rude\nprocess of inference, infer the future directly from the past, it is\nthe work of logic to analyse the conditions on which the correctness\nof this inference depends. Even Mill would admit that such analysis\ninvolves the consideration of general truths,[137] and in this, as in\nseveral other important points, we might controvert Mill\u2019s own views\nby his own statements. It seems to me undesirable in a systematic work\nlike this to enter into controversy at any length, or to attempt to\nrefute the views of other logicians. But I shall feel bound to state,\nin a separate publication, my very deliberate opinion that many of\nMill\u2019s innovations in logical science, and especially his doctrine of\nreasoning from particulars to particulars, are entirely groundless and\nfalse.\n\n  [136] *System of Logic.* Introduction, \u00a7 4. Fifth ed. pp.\u00a08, 9.\n\n  [137] Ibid. bk. II. chap. iii. \u00a7 5, pp.\u00a0225, &c.\n\n\n*The Grounds of Inductive Inference.*\n\nI hold that in all cases of inductive inference we must invent\nhypotheses, until we fall upon some hypothesis which yields deductive\nresults in accordance with experience. Such accordance renders the\nchosen hypothesis more or less probable, and we may then deduce, with\nsome degree of likelihood, the nature of our future experience, on the\nassumption that no arbitrary change takes place in the conditions of\nnature. We can only argue from the past to the future, on the general\nprinciple set forth in this work, that what is true of a thing will\nbe true of the like. So far then as one object or event differs from\nanother, all inference is impossible, particulars as particulars can\nno more make an inference than grains of sand can make a rope. We\nmust always rise to something which is general or same in the cases,\nand assuming that sameness to be extended to new cases we learn their\nnature. Hearing a clock tick five thousand times without exception or\nvariation, we adopt the very probable hypothesis that there is some\ninvariably acting machine which produces those uniform sounds, and\nwhich will, in the absence of change, go on producing them. Meeting\ntwenty times with a bright yellow ductile substance, and finding it\nalways to be very heavy and incorrodible, I infer that there was some\nnatural condition which tended in the creation of things to associate\nthese properties together, and I expect to find them associated in the\nnext instance. But there always is the possibility that some unknown\nchange may take place between past and future cases. The clock may run\ndown, or be subject to a hundred accidents altering its condition.\nThere is no reason in the nature of things, so far as known to us, why\nyellow colour, ductility, high specific gravity, and incorrodibility,\nshould always be associated together, and in other cases, if not in\nthis, men\u2019s expectations have been deceived. Our inferences, therefore,\nalways retain more or less of a hypothetical character, and are so far\nopen to doubt. Only in proportion as our induction approximates to the\ncharacter of perfect induction, does it approximate to certainty. The\namount of uncertainty corresponds to the probability that other objects\nthan those examined may exist and falsity our inferences; the amount\nof probability corresponds to the amount of information yielded by our\nexamination; and the theory of probability will be needed to prevent us\nfrom over-estimating or under-estimating the knowledge we possess.\n\n\n*Illustrations of the Inductive Process.*\n\nTo illustrate the passage from the known to the apparently unknown, let\nus suppose that the phenomena under investigation consist of numbers,\nand that the following six numbers being exhibited to us, we are\nrequired to infer the character of the next in the series:--\n\n  5, 15, 35, 45, 65, 95.\n\nThe question first of all arises, How may we describe this series of\nnumbers? What is uniformly true of them? The reader cannot fail to\nperceive at the first glance that they all end in five, and the problem\nis, from the properties of these six numbers, to infer the properties\nof the next number ending in five. If we test their properties by the\nprocess of perfect induction, we soon perceive that they have another\ncommon property, namely that of being *divisible by five without\nremainder*. May we then assert that the next number ending in five is\nalso divisible by five, and, if so, upon what grounds? Or extending\nthe question, Is every number ending in five divisible by five? Does\nit follow that because six numbers obey a supposed law, therefore\n376,685,975 or any other number, however large, obeys the law? I answer\n*certainly not*. The law in question is undoubtedly true; but its truth\nis not proved by any finite number of examples. All that these six\nnumbers can do is to suggest to my mind the possible existence of such\na law; and I then ascertain its truth, by proving deductively from the\nrules of decimal numeration, that any number ending in five must be\nmade up of multiples of five, and must therefore be itself a multiple.\n\nTo make this more plain, let the reader now examine the numbers--\n\n  7, 17, 37, 47, 67, 97.\n\nThey all end in 7 instead of 5, and though not at equal intervals, the\nintervals are the same as in the previous case. After consideration,\nthe reader will perceive that these numbers all agree in being *prime\nnumbers*, or multiples of unity only. May we then infer that the next,\nor any other number ending in 7, is a prime number? Clearly not, for\non trial we find that 27, 57, 117 are not primes. Six instances,\nthen, treated empirically, lead us to a true and universal law in one\ncase, and mislead us in another case. We ought, in fact, to have no\nconfidence in any law until we have treated it deductively, and have\nshown that from the conditions supposed the results expected must\nensue. No one can show from the principles of number, that numbers\nending in 7 should be primes.\n\nFrom the history of the theory of numbers some good examples of false\ninduction can be adduced. Taking the following series of prime numbers,\n\n  41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, &c.,\n\nit will be found that they all agree in being values of the general\nexpression *x*^{2}\u00a0+\u00a0*x*\u00a0+\u00a041, putting for *x* in succession the\nvalues, 0, 1, 2, 3, 4, &c. We seem always to obtain a prime number, and\nthe induction is apparently strong, to the effect that this expression\nalways will give primes. Yet a few more trials disprove this false\nconclusion. Put *x*\u00a0=\u00a040, and we obtain 40\u00a0\u00d7\u00a040\u00a0+\u00a040\u00a0+\u00a041, or 41\u00a0\u00d7\u00a041.\nSuch a failure could never have happened, had we shown any deductive\nreason why *x*^{2}\u00a0+\u00a0*x*\u00a0+\u00a041 should give primes.\n\nThere can be no doubt that what here happens with forty instances,\nmight happen with forty thousand or forty million instances. An\napparent law never once failing up to a certain point may then suddenly\nbreak down, so that inductive reasoning, as it has been described by\nsome writers, can give no sure knowledge of what is to come. Babbage\npointed out, in his Ninth Bridgewater Treatise, that a machine could\nbe constructed to give a perfectly regular series of numbers through\na vast series of steps, and yet to break the law of progression\nsuddenly at any required point. No number of particular cases as\nparticulars enables us to pass by inference to any new case. It is\nhardly needful to inquire here what can be inferred from an infinite\nseries of facts, because they are never practically within our power;\nbut we may unhesitatingly accept the conclusion, that no finite number\nof instances can ever prove a general law, or can give us certain\nknowledge of even one other instance.\n\nGeneral mathematical theorems have indeed been discovered by the\nobservation of particular cases, and may again be so discovered. We\nhave Newton\u2019s own statement, to the effect that he was thus led to\nthe all-important Binomial Theorem, the basis of the whole structure\nof mathematical analysis. Speaking of a certain series of terms,\nexpressing the area of a circle or hyperbola, he says: \u201cI reflected\nthat the denominators were in arithmetical progression; so that\nonly the numerical co-efficients of the numerators remained to be\ninvestigated. But these, in the alternate areas, were the figures of\nthe powers of the number eleven, namely 11^{0}, 11^{1}, 11^{2}, 11^{3},\n11^{4}; that is, in the first 1; in the second 1, 1; in the third 1,\n2, 1; in the fourth 1, 3, 3, 1; in the fifth 1, 4, 6, 4, 1.[138] I\ninquired, therefore, in what manner all the remaining figures could\nbe found from the first two; and I found that if the first figure be\ncalled *m*, all the rest could be found by the continual multiplication\nof the terms of the formula\n\n  ((*m*\u00a0-\u00a00)/1)\u00a0\u00d7\u00a0((*m*\u00a0-\u00a01)/2)\u00a0\u00d7 ((*m*\u00a0-\u00a02)/3)\u00a0\u00d7\n  ((*m*\u00a0-\u00a03)/4)\u00a0\u00d7\u00a0&c.\u201d[139]\n\n  [138] These are the figurate numbers considered in pages 183, 187, &c.\n\n  [139] *Commercium Epistolicum.* *Epistola ad Oldenburgum*, Oct. 24,\n  1676. Horsley\u2019s *Works of Newton*, vol.\u00a0iv. p.\u00a0541. See De Morgan in\n  *Penny Cyclop\u00e6dia*, art. \u201cBinomial Theorem,\u201d p.\u00a0412.\n\nIt is pretty evident, from this most interesting statement, that\nNewton, having simply observed the succession of the numbers, tried\nvarious formul\u00e6 until he found one which agreed with them all. He\nwas so little satisfied with this process, however, that he verified\nparticular results of his new theorem by comparison with the results\nof common multiplication, and the rule for the extraction of the\nsquare root. Newton, in fact, gave no demonstration of his theorem;\nand the greatest mathematicians of the last century, James Bernoulli,\nMaclaurin, Landen, Euler, Lagrange, &c., occupied themselves with\ndiscovering a conclusive method of deductive proof.\n\nThere can be no doubt that in geometry also discoveries have been\nsuggested by direct observation. Many of the now trivial propositions\nof Euclid\u2019s Elements were probably thus discovered, by the ancient\nGreek geometers; and we have pretty clear evidence of this in the\nCommentaries of Proclus.[140] Galileo was the first to examine the\nremarkable properties of the cycloid, the curve described by a point in\nthe circumference of a wheel rolling on a plane. By direct observation\nhe ascertained that the area of the curve is apparently three times\nthat of the generating circle or wheel, but he was unable to prove this\nexactly, or to verify it by strict geometrical reasoning. Sir George\nAiry has recorded a curious case, in which he fell accidentally by\ntrial on a new geometrical property of the sphere.[141] But discovery\nin such cases means nothing more than suggestion, and it is always by\npure deduction that the general law is really established. As Proclus\nputs it, *we must pass from sense to consideration*.\n\n  [140] Bk. ii. chap. iv.\n\n  [141] *Philosophical Transactions* (1866), vol.\u00a0146, p.\u00a0334.\n\n[Illustration]\n\nGiven, for instance, the series of figures in the accompanying diagram,\nmeasurement will show that the curved lines approximate to semicircles,\nand the rectilinear figures to right-angled triangles. These figures\nmay seem to suggest to the mind the general law that angles inscribed\nin semicircles are right angles; but no number of instances, and no\npossible accuracy of measurement would really establish the truth of\nthat general law. Availing ourselves of the suggestion furnished by\nthe figures, we can only investigate deductively the consequences\nwhich flow from the definition of a circle, until we discover among\nthem the property of containing right angles. Persons have thought\nthat they had discovered a method of trisecting angles by plane\ngeometrical construction, because a certain complex arrangement of\nlines and circles had appeared to trisect an angle in every case tried\nby them, and they inferred, by a supposed act of induction, that it\nwould succeed in all other cases. De Morgan has recorded a proposed\nmode of trisecting the angle which could not be discriminated by the\nsenses from a true general solution, except when it was applied to\nvery obtuse angles.[142] In all such cases, it has always turned out\neither that the angle was not trisected at all, or that only certain\nparticular angles could be thus trisected. The trisectors were misled\nby some apparent or special coincidence, and only deductive proof could\nestablish the truth and generality of the result. In this particular\ncase, deductive proof shows that the problem attempted is impossible,\nand that angles generally cannot be trisected by common geometrical\nmethods.\n\n  [142] *Budget of Paradoxes*, p.\u00a0257.\n\n\n*Geometrical Reasoning.*\n\nThis view of the matter is strongly supported by the further\nconsideration of geometrical reasoning. No skill and care could ever\nenable us to verify absolutely any one geometrical proposition.\nRousseau, in his *Emile*, tells us that we should teach a child\ngeometry by causing him to measure and compare figures by\nsuperposition. While a child was yet incapable of general reasoning,\nthis would doubtless be an instructive exercise; but it never could\nteach geometry, nor prove the truth of any one proposition. All our\nfigures are rude approximations, and they may happen to seem unequal\nwhen they should be equal, and equal when they should be unequal.\nMoreover figures may from chance be equal in case after case, and yet\nthere may be no general reason why they should be so. The results of\ndeductive geometrical reasoning are absolutely certain, and are either\nexactly true or capable of being carried to any required degree of\napproximation. In a perfect triangle, the angles must be equal to one\nhalf-revolution precisely; even an infinitesimal divergence would be\nimpossible; and I believe with equal confidence, that however many are\nthe angles of a figure, provided there are no re-entrant angles, the\nsum of the angles will be precisely and absolutely equal to twice as\nmany right-angles as the figure has sides, less by four right-angles.\nIn such cases, the deductive proof is absolute and complete; empirical\nverification can at the most guard against accidental oversights.\n\nThere is a second class of geometrical truths which can only be\nproved by approximation; but, as the mind sees no reason why that\napproximation should not always go on, we arrive at complete\nconviction. We thus learn that the surface of a sphere is equal exactly\nto two-thirds of the whole surface of the circumscribing cylinder, or\nto four times the area of the generating circle. The area of a parabola\nis exactly two-thirds of that of the circumscribing parallelogram.\nThe area of the cycloid is exactly three times that of the generating\ncircle. These are truths that we could never ascertain, nor even verify\nby observation; for any finite amount of difference, less than what the\nsenses can discern, would falsify them.\n\nThere are geometrical relations again which we cannot assign\nexactly, but can carry to any desirable degree of approximation. The\nratio of the circumference to the diameter of a circle is that of\n3\u00b714159265358979323846.... to 1, and the approximation may be carried\nto any extent by the expenditure of sufficient labour. Mr. W. Shanks\nhas given the value of this natural constant, known as \u03c0, to the extent\nof 707 places of decimals.[143] Some years since, I amused myself\nby trying how near I could get to this ratio, by the careful use of\ncompasses, and I did not come nearer than 1 part in 540. We might\nimagine measurements so accurately executed as to give us eight or ten\nplaces correctly. But the power of the hands and senses must soon\nstop, whereas the mental powers of deductive reasoning can proceed\nto an unlimited degree of approximation. Geometrical truths, then,\nare incapable of verification; and, if so, they cannot even be learnt\nby observation. How can I have learnt by observation a proposition\nof which I cannot even prove the truth by observation, when I am in\npossession of it? All that observation or empirical trial can do is\nto suggest propositions, of which the truth may afterwards be proved\ndeductively.\n\n  [143] *Proceedings of the Royal Society* (1872\u20133), vol.\u00a0xxi. p.\u00a0319.\n\nIf Viviani\u2019s story is to be believed, Galileo endeavoured to satisfy\nhimself about the area of the cycloid by cutting out several large\ncycloids in pasteboard, and then comparing the areas of the curve and\nthe generating circle by weighing them. In every trial the curve seemed\nto be rather less than three times the circle, so that Galileo, we are\ntold, began to suspect that the ratio was not precisely 3 to 1. It is\nquite clear, however, that no process of weighing or measuring could\never prove truths like these, and it remained for Torricelli to show\nwhat his master Galileo had only guessed at.[144]\n\n  [144] *Life of Galileo*, Society for the Diffusion of Useful\n  Knowledge, p. 102.\n\nMuch has been said about the peculiar certainty of mathematical\nreasoning, but it is only certainty of deductive reasoning, and equal\ncertainty attaches to all correct logical deduction. If a triangle be\nright-angled, the square on the hypothenuse will undoubtedly equal the\nsum of the two squares on the other sides; but I can never be sure that\na triangle is right-angled: so I can be certain that nitric acid will\nnot dissolve gold, provided I know that the substances employed really\ncorrespond to those on which I tried the experiment previously. Here is\nlike certainty of inference, and like doubt as to the facts.\n\n\n*Discrimination of Certainty and Probability.*\n\nWe can never recur too often to the truth that our knowledge of the\nlaws and future events of the external world is only probable. The mind\nitself is quite capable of possessing certain knowledge, and it is well\nto discriminate carefully between what we can and cannot know with\ncertainty. In the first place, whatever feeling is actually present to\nthe mind is certainly known to that mind. If I see blue sky, I may be\nquite sure that I do experience the sensation of blueness. Whatever\nI do feel, I do feel beyond all doubt. We are indeed very likely to\nconfuse what we really feel with what we are inclined to associate with\nit, or infer inductively from it; but the whole of our consciousness,\nas far as it is the result of pure intuition and free from inference,\nis certain knowledge beyond all doubt.\n\nIn the second place, we may have certainty of inference; the\nfundamental laws of thought, and the rule of substitution (p.\u00a09),\nare certainly true; and if my senses could inform me that A was\nindistinguishable in colour from B, and B from C, then I should be\nequally certain that A was indistinguishable from C. In short, whatever\ntruth there is in the premises, I can certainly embody in their correct\nlogical result. But the certainty generally assumes a hypothetical\ncharacter. I never can be quite sure that two colours are exactly\nalike, that two magnitudes are exactly equal, or that two bodies\nwhatsoever are identical even in their apparent qualities. Almost all\nour judgments involve quantitative relations, and, as will be shown in\nsucceeding chapters, we can never attain exactness and certainty where\ncontinuous quantity enters. Judgments concerning discontinuous quantity\nor numbers, however, allow of certainty; I may establish beyond doubt,\nfor instance, that the difference of the squares of 17 and 13 is the\nproduct of 17\u00a0+\u00a013 and 17\u00a0-\u00a013, and is therefore 30\u00a0\u00d7\u00a04, or 120.\n\nInferences which we draw concerning natural objects are never certain\nexcept in a hypothetical point of view. It might seem to be certain\nthat iron is magnetic, or that gold is incapable of solution in\nnitric acid; but, if we carefully investigate the meanings of these\nstatements, they will be found to involve no certainty but that of\nconsciousness and that of hypothetical inference. For what do I mean\nby iron or gold? If I choose a remarkable piece of yellow substance,\ncall it gold, and then immerse it in a liquid which I call nitric acid,\nand find that there is no change called solution, then consciousness\nhas certainly informed me that, with my meaning of the terms, \u201cGold is\ninsoluble in nitric acid.\u201d I may further be certain of something else;\nfor if this gold and nitric acid remain what they were, I may be sure\nthere will be no solution on again trying the experiment. If I take\nother portions of gold and nitric acid, and am sure that they really\nare identical in properties with the former portions, I can be certain\nthat there will be no solution. But at this point my knowledge becomes\npurely hypothetical; for how can I be sure without trial that the gold\nand acid are really identical in nature with what I formerly called\ngold and nitric acid. How do I know gold when I see it? If I judge by\nthe apparent qualities--colour, ductility, specific gravity, &c., I\nmay be misled, because there may always exist a substance which to the\ncolour, ductility, specific gravity, and other specified qualities,\njoins others which we do not expect. Similarly, if iron is magnetic,\nas shown by an experiment with objects answering to those names, then\nall iron is magnetic, meaning all pieces of matter identical with my\nassumed piece. But in trying to identify iron, I am always open to\nmistake. Nor is this liability to mistake a matter of speculation\nonly.[145]\n\n  [145] Professor Bowen has excellently stated this view. *Treatise on\n  Logic.* Cambridge, U.S.A., 1866, p.\u00a0354.\n\nThe history of chemistry shows that the most confident inferences may\nhave been falsified by the confusion of one substance with another.\nThus strontia was never discriminated from baryta until Klaproth\nand Ha\u00fcy detected differences between some of their properties.\nAccordingly chemists must often have inferred concerning strontia what\nwas only true of baryta, and *vice vers\u00e2*. There is now no doubt that\nthe recently discovered substances, c\u00e6sium and rubidium, were long\nmistaken for potassium.[146] Other elements have often been confused\ntogether--for instance, tantalum and niobium; sulphur and selenium;\ncerium, lanthanum, and didymium; yttrium and erbium.\n\n  [146] Roscoe\u2019s *Spectrum Analysis*, 1st edit., p.\u00a098.\n\nEven the best known laws of physical science do not exclude false\ninference. No law of nature has been better established than that of\nuniversal gravitation, and we believe with the utmost confidence that\nany body capable of affecting the senses will attract other bodies,\nand fall to the earth if not prevented. Euler remarks that, although\nhe had never made trial of the stones which compose the church of\nMagdeburg, yet he had not the least doubt that all of them were heavy,\nand would fall if unsupported. But he adds, that it would be extremely\ndifficult to give any satisfactory explanation of this confident\nbelief.[147] The fact is, that the belief ought not to amount to\ncertainty until the experiment has been tried, and in the meantime a\nslight amount of uncertainty enters, because we cannot be sure that\nthe stones of the Magdeburg Church resemble other stones in all their\nproperties.\n\n  [147] Euler\u2019s *Letters to a German Princess*, translated by Hunter.\n  2nd ed., vol.\u00a0ii. pp.\u00a017, 18.\n\nIn like manner, not one of the inductive truths which men have\nestablished, or think they have established, is really safe from\nexception or reversal. Lavoisier, when laying the foundations of\nchemistry, met with so many instances tending to show the existence\nof oxygen in all acids, that he adopted a general conclusion to that\neffect, and devised the name oxygen accordingly. He entertained no\nappreciable doubt that the acid existing in sea salt also contained\noxygen;[148] yet subsequent experience falsified his expectations. This\ninstance refers to a science in its infancy, speaking relatively to\nthe possible achievements of men. But all sciences are and ever will\nremain in their infancy, relatively to the extent and complexity of\nthe universe which they undertake to investigate. Euler expresses no\nmore than the truth when he says that it would be impossible to fix\non any one thing really existing, of which we could have so perfect\na knowledge as to put us beyond the reach of mistake.[149] We may be\nquite certain that a comet will go on moving in a similar path *if*\nall circumstances remain the same as before; but if we leave out this\nextensive qualification, our predictions will always be subject to the\nchance of falsification by some unexpected event, such as the division\nof Biela\u2019s comet or the interference of an unknown gravitating body.\n\n  [148] Lavoisier\u2019s *Chemistry*, translated by Kerr. 3rd ed., pp.\u00a0114,\n  121, 123.\n\n  [149] Euler\u2019s *Letters*, vol.\u00a0ii. p.\u00a021.\n\nInductive inference might attain to certainty if our knowledge of the\nagents existing throughout the universe were complete, and if we were\nat the same time certain that the same Power which created the universe\nwould allow it to proceed without arbitrary change. There is always\na possibility of causes being in existence without our knowledge,\nand these may at any moment produce an unexpected effect. Even when\nby the theory of probabilities we succeed in forming some notion of\nthe comparative confidence with which we should receive inductive\nresults, it yet appears to me that we must make an assumption. Events\ncome out like balls from the vast ballot-box of nature, and close\nobservation will enable us to form some notion, as we shall see in the\nnext chapter, of the contents of that ballot-box. But we must still\nassume that, between the time of an observation and that to which our\ninferences relate, no change in the ballot-box has been made.\n\n\n\n\nCHAPTER XII.\n\nTHE INDUCTIVE OR INVERSE APPLICATION OF THE THEORY OF PROBABILITY.\n\n\nWe have hitherto considered the theory of probability only in its\nsimple deductive employment, in which it enables us to determine\nfrom given conditions the probable character of events happening\nunder those conditions. But as deductive reasoning when inversely\napplied constitutes the process of induction, so the calculation of\nprobabilities may be inversely applied; from the known character\nof certain events we may argue backwards to the probability of a\ncertain law or condition governing those events. Having satisfactorily\naccomplished this work, we may indeed calculate forwards to the\nprobable character of future events happening under the same\nconditions; but this part of the process is a direct use of deductive\nreasoning (p.\u00a0226).\n\nNow it is highly instructive to find that whether the theory of\nprobability be deductively or inductively applied, the calculation is\nalways performed according to the principles and rules of deduction.\nThe probability that an event has a particular condition entirely\ndepends upon the probability that if the condition existed the event\nwould follow. If we take up a pack of common playing cards, and observe\nthat they are arranged in perfect numerical order, we conclude beyond\nall reasonable doubt that they have been thus intentionally arranged\nby some person acquainted with the usual order of sequence. This\nconclusion is quite irresistible, and rightly so; for there are but\ntwo suppositions which we can make as to the reason of the cards being\nin that particular order:--\n\n1. They may have been intentionally arranged by some one who would\nprobably prefer the numerical order.\n\n2. They may have fallen into that order by chance, that is, by some\nseries of conditions which, being unknown to us, cannot be known to\nlead by preference to the particular order in question.\n\nThe latter supposition is by no means absurd, for any one order is as\nlikely as any other when there is no preponderating tendency. But we\ncan readily calculate by the doctrine of permutations the probability\nthat fifty-two objects would fall by chance into any one particular\norder. Fifty-two objects can be arranged in 52\u00a0\u00d7\u00a051\u00a0\u00d7\u00a0...\u00a0\u00d7\u00a03 \u00d7\u00a02\u00a0\u00d7\u00a01\nor about 8066\u00a0\u00d7\u00a0(10)^{64} possible orders, the number obtained\nrequiring 68 places of figures for its full expression. Hence it is\nexcessively unlikely that anyone should ever meet with a pack of cards\narranged in perfect order by accident. If we do meet with a pack so\narranged, we inevitably adopt the other supposition, that some person,\nhaving reasons for preferring that special order, has thus put them\ntogether.\n\nWe know that of the immense number of possible orders the numerical\norder is the most remarkable; it is useful as proving the perfect\nconstitution of the pack, and it is the intentional result of certain\ngames. At any rate, the probability that intention should produce that\norder is incomparably greater than the probability that chance should\nproduce it; and as a certain pack exists in that order, we rightly\nprefer the supposition which most probably leads to the observed result.\n\nBy a similar mode of reasoning we every day arrive, and validly arrive,\nat conclusions approximating to certainty. Whenever we observe a\nperfect resemblance between two objects, as, for instance, two printed\npages, two engravings, two coins, two foot-prints, we are warranted in\nasserting that they proceed from the same type, the same plate, the\nsame pair of dies, or the same boot. And why? Because it is almost\nimpossible that with different types, plates, dies, or boots some\napparent distinction of form should not be produced. It is impossible\nfor the hand of the most skilful artist to make two objects alike, so\nthat mechanical repetition is the only probable explanation of exact\nsimilarity.\n\nWe can often establish with extreme probability that one document\nis copied from another. Suppose that each document contains 10,000\nwords, and that the same word is incorrectly spelt in each. There is\nthen a probability of less than 1 in 10,000 that the same mistake\nshould be made in each. If we meet with a second error occurring in\neach document, the probability is less than 1 in 10,000 \u00d7\u00a09999, that\ntwo such coincidences should occur by chance, and the numbers grow\nwith extreme rapidity for more numerous coincidences. We cannot make\nany precise calculations without taking into account the character of\nthe errors committed, concerning the conditions of which we have no\naccurate means of estimating probabilities. Nevertheless, abundant\nevidence may thus be obtained as to the derivation of documents from\neach other. In the examination of many sets of logarithmic tables, six\nremarkable errors were found to be present in all but two, and it was\nproved that tables printed at Paris, Berlin, Florence, Avignon, and\neven in China, besides thirteen sets printed in England between the\nyears 1633 and 1822, were derived directly or indirectly from some\ncommon source.[150] With a certain amount of labour, it is possible\nto establish beyond reasonable doubt the relationship or genealogy of\nany number of copies of one document, proceeding possibly from parent\ncopies now lost. The relations between the manuscripts of the New\nTestament have been elaborately investigated in this manner, and the\nsame work has been performed for many classical writings, especially by\nGerman scholars.\n\n  [150] Lardner, *Edinburgh Review*, July 1834, p.\u00a0277.\n\n\n*Principle of the Inverse Method.*\n\nThe inverse application of the rules of probability entirely depends\nupon a proposition which may be thus stated, nearly in the words of\nLaplace.[151] *If an event can be produced by any one of a certain\nnumber of different causes, all equally probable \u00e0 priori, the\nprobabilities of the existence of these causes as inferred from the\nevent, are proportional to the probabilities of the event as derived\nfrom these causes.* In other words, the most probable cause of an\nevent which has happened is that which would most probably lead to the\nevent supposing the cause to exist; but all other possible causes are\nalso to be taken into account with probabilities proportional to the\nprobability that the event would happen if the cause existed. Suppose,\nto fix our ideas clearly, that E is the event, and C_{1} C_{2} C_{3}\nare the three only conceivable causes. If C_{1} exist, the probability\nis *p*_{1} that E would follow; if C_{2} or C_{3} exist, the like\nprobabilities are respectively *p*_{2} and *p*_{3}. Then as *p*_{1}\nis to *p*_{2}, so is the probability of C_{1} being the actual cause\nto the probability of C_{2} being it; and, similarly, as *p*_{2} is\nto *p*_{3}, so is the probability of C_{2} being the actual cause to\nthe probability of C_{3} being it. By a simple mathematical process we\narrive at the conclusion that the actual probability of C_{1} being the\ncause is\n\n  *p*_{1}/(*p*_{1}\u00a0+\u00a0*p*_{2}\u00a0+\u00a0*p*_{3});\n\n  [151] *M\u00e9moires par divers Savans*, tom. vi.; quoted by Todhunter in\n  his *History of the Theory of Probability*, p.\u00a0458.\n\nand the similar probabilities of the existence of C_{2} and C_{3} are,\n\n  *p*_{2}/(*p*_{1}\u00a0+\u00a0*p*_{2}\u00a0+\u00a0*p*_{3}) and\n  *p*_{3}/(*p*_{1}\u00a0+\u00a0*p*_{2}\u00a0+\u00a0*p*_{3}).\n\nThe sum of these three fractions amounts to unity, which correctly\nexpresses the certainty that one cause or other must be in operation.\n\nWe may thus state the result in general language. *If it is certain\nthat one or other of the supposed causes exists, the probability that\nany one does exist is the probability that if it exists the event\nhappens, divided by the sum of all the similar probabilities.* There\nmay seem to be an intricacy in this subject which may prove distasteful\nto some readers; but this intricacy is essential to the subject in\nhand. No one can possibly understand the principles of inductive\nreasoning, unless he will take the trouble to master the meaning of\nthis rule, by which we recede from an event to the probability of each\nof its possible causes.\n\nThis rule or principle of the indirect method is that which common\nsense leads us to adopt almost instinctively, before we have any\ncomprehension of the principle in its general form. It is easy to see,\ntoo, that it is the rule which will, out of a great multitude of cases,\nlead us most often to the truth, since the most probable cause of an\nevent really means that cause which in the greatest number of cases\nproduces the event. Donkin and Boole have given demonstrations of this\nprinciple, but the one most easy to comprehend is that of Poisson.\nHe imagines each possible cause of an event to be represented by a\ndistinct ballot-box, containing black and white balls, in such a ratio\nthat the probability of a white ball being drawn is equal to that of\nthe event happening. He further supposes that each box, as is possible,\ncontains the same total number of balls, black and white; then, mixing\nall the contents of the boxes together, he shows that if a white ball\nbe drawn from the aggregate ballot-box thus formed, the probability\nthat it proceeded from any particular ballot-box is represented by the\nnumber of white balls in that particular box, divided by the total\nnumber of white balls in all the boxes. This result corresponds to that\ngiven by the principle in question.[152]\n\n  [152] Poisson, *Recherches sur la Probabilit\u00e9 des Jugements*, Paris,\n  1837, pp. 82, 83.\n\nThus, if there be three boxes, each containing ten balls in all, and\nrespectively containing seven, four, and three white balls, then on\nmixing all the balls together we have fourteen white ones; and if\nwe draw a white ball, that is if the event happens, the probability\nthat it came out of the first box is 7/14; which is exactly equal to\n(7/10)/(7/10\u00a0+\u00a04/10\u00a0+\u00a03/10), the fraction given by the rule of the\nInverse Method.\n\n\n*Simple Applications of the Inverse Method.*\n\nIn many cases of scientific induction we may apply the principle of the\ninverse method in a simple manner. If only two, or at the most a few\nhypotheses, may be made as to the origin of certain phenomena, we may\nsometimes easily calculate the respective probabilities. It was thus\nthat Bunsen and Kirchhoff established, with a probability little short\nof certainty, that iron exists in the sun. On comparing the spectra\nof sunlight and of the light proceeding from the incandescent vapour\nof iron, it became apparent that at least sixty bright lines in the\nspectrum of iron coincided with dark lines in the sun\u2019s spectrum. Such\ncoincidences could never be observed with certainty, because, even if\nthe lines only closely approached, the instrumental imperfections of\nthe spectroscope would make them apparently coincident, and if one line\ncame within half a millimetre of another, on the map of the spectra,\nthey could not be pronounced distinct. Now the average distance of the\nsolar lines on Kirchhoff\u2019s map is 2\u00a0mm., and if we throw down a line,\nas it were, by pure chance on such a map, the probability is about\none-half that the new line will fall within 1/2\u00a0mm. on one side or the\nother of some one of the solar lines. To put it in another way, we may\nsuppose that each solar line, either on account of its real breadth, or\nthe defects of the instrument, possesses a breadth of 1/2\u00a0mm., and that\neach line in the iron spectrum has a like breadth. The probability then\nis just one-half that the centre of each iron line will come by chance\nwithin 1\u00a0mm. of the centre of a solar line, so as to appear to coincide\nwith it. The probability of casual coincidence of each iron line with\na solar line is in like manner 1/2. Coincidence in the case of each of\nthe sixty iron lines is a very unlikely event if it arises casually,\nfor it would have a probability of only (1/2)^{60} or less than 1\nin a trillion. The odds, in short, are more than a million million\nmillions to unity against such casual coincidence.[153] But on the\nother hypothesis, that iron exists in the sun, it is highly probable\nthat such coincidences would be observed; it is immensely more probable\nthat sixty coincidences would be observed if iron existed in the sun,\nthan that they should arise from chance. Hence by our principle it is\nimmensely probable that iron does exist in the sun.\n\n  [153] Kirchhoff\u2019s *Researches on the Solar Spectrum*. First part,\n  translated by Roscoe, pp.\u00a018, 19.\n\nAll the other interesting results, given by the comparison of spectra,\nrest upon the same principle of probability. The almost complete\ncoincidence between the spectra of solar, lunar, and planetary light\nrenders it practically certain that the light is all of solar origin,\nand is reflected from the surfaces of the moon and planets, suffering\nonly slight alteration from the atmospheres of some of the planets.\nA fresh confirmation of the truth of the Copernican theory is thus\nfurnished.\n\nHerschel proved in this way the connection between the direction of the\noblique faces of quartz crystals, and the direction in which the same\ncrystals rotate the plane of polarisation of light. For if it is found\nin a second crystal that the relation is the same as in the first, the\nprobability of this happening by chance is 1/2; the probability that\nin another crystal also the direction will be the same is 1/4, and so\non. The probability that in *n*\u00a0+\u00a01 crystals there would be casual\nagreement of direction is the nth power of 1/2. Thus, if in examining\nfourteen crystals the same relation of the two phenomena is discovered\nin each, the odds that it proceeds from uniform conditions are more\nthan 8000 to 1.[154] Since the first observations on this subject were\nmade in 1820, no exceptions have been observed, so that the probability\nof invariable connection is incalculably great.\n\n  [154] *Edinburgh Review*, No. 185, vol.\u00a0xcii. July 1850, p.\u00a032;\n  Herschel\u2019s *Essays*, p.\u00a0421; *Transactions of the Cambridge\n  Philosophical Society*, vol.\u00a0i. p.\u00a043.\n\nIt is exceedingly probable that the ancient Egyptians had exactly\nrecorded the eclipses occurring during long periods of time, for\nDiogenes Laertius mentions that 373 solar and 832 lunar eclipses had\nbeen observed, and the ratio between these numbers exactly expresses\nthat which would hold true of the eclipses of any long period, of say\n1200 or 1300 years, as estimated on astronomical grounds. It is evident\nthat an agreement between small numbers, or customary numbers, such\nas seven, one hundred, a myriad, &c., is much more likely to happen\nfrom chance, and therefore gives much less presumption of dependence.\nIf two ancient writers spoke of the sacrifice of oxen, they would in\nall probability describe it as a hecatomb, and there would be nothing\nremarkable in the coincidence. But it is impossible to point out any\nspecial reason why an old writer should select such numbers as 373 and\n832, unless they had been the results of observation.\n\nOn similar grounds, we must inevitably believe in the human origin\nof the flint flakes so copiously discovered of late years. For\nthough the accidental stroke of one stone against another may often\nproduce flakes, such as are occasionally found on the sea-shore, yet\nwhen several flakes are found in close company, and each one bears\nevidence, not of a single blow only, but of several successive blows,\nall conducing to form a symmetrical knife-like form, the probability\nof a natural and accidental origin becomes incredibly small, and the\ncontrary supposition, that they are the work of intelligent beings,\napproximately certain.[155]\n\n  [155] Evans\u2019 *Ancient Stone Implements of Great Britain*. London,\n  1872 (Longmans).\n\n\n*The Theory of Probability in Astronomy.*\n\nThe science of astronomy, occupied with the simple relations of\ndistance, magnitude, and motion of the heavenly bodies, admits more\neasily than almost any other science of interesting conclusions founded\non the theory of probability. More than a century ago, in 1767, Michell\nshowed the extreme probability of bonds connecting together systems\nof stars. He was struck by the unexpected number of fixed stars\nwhich have companions close to them. Such a conjunction might happen\ncasually by one star, although possibly at a great distance from the\nother, happening to lie on a straight line passing near the earth.\nBut the probabilities are so greatly against such an optical union\nhappening often in the expanse of the heavens, that Michell asserted\nthe existence of some connection between most of the double stars.\nIt has since been estimated by Struve, that the odds are 9570 to 1\nagainst any two stars of not less than the seventh magnitude falling\nwithin the apparent distance of four seconds of each other by chance,\nand yet ninety-one such cases were known when the estimation was made,\nand many more cases have since been discovered. There were also four\nknown triple stars, and yet the odds against the appearance of any one\nsuch conjunction are 173,524 to 1.[156] The conclusions of Michell have\nbeen entirely verified by the discovery that many double stars are\nconnected by gravitation.\n\n  [156] Herschel, *Outlines of Astronomy*, 1849, p.\u00a0565; but Todhunter,\n  in his *History of the Theory of Probability*, p.\u00a0335, states that\n  the calculations do not agree with those published by Struve.\n\nMichell also investigated the probability that the six brightest stars\nin the Pleiades should have come by accidents into such striking\nproximity. Estimating the number of stars of equal or greater\nbrightness at 1500, be found the odds to be nearly 500,000 to 1 against\ncasual conjunction. Extending the same kind of argument to other\nclusters, such as that of Pr\u00e6sepe, the nebula in the hilt of Perseus\u2019\nsword, he says:[157] \u201cWe may with the highest probability conclude,\nthe odds against the contrary opinion being many million millions\nto one, that the stars are really collected together in clusters\nin some places, where they form a kind of system, while in others\nthere are either few or none of them, to whatever cause this may be\nowing, whether to their mutual gravitation, or to some other law or\nappointment of the Creator.\u201d\n\n  [157] *Philosophical Transactions*, 1767, vol.\u00a0lvii. p.\u00a0431.\n\nThe calculations of Michell have been called in question by the late\nJames D. Forbes,[158] and Mr. Todhunter vaguely countenances his\nobjections,[159] otherwise I should not have thought them of much\nweight. Certainly Laplace accepts Michell\u2019s views,[160] and if Michell\nbe in error it is in the methods of calculation, not in the general\nvalidity of his reasoning and conclusions.\n\n  [158] *Philosophical Magazine*, 3rd Series, vol.\u00a0xxxvii. p.\u00a0401,\n  December 1850; also August 1849.\n\n  [159] *History*, &c., p.\u00a0334.\n\n  [160] *Essai Philosophique*, p.\u00a057.\n\nSimilar calculations might no doubt be applied to the peculiar drifting\nmotions which have been detected by Mr. R A. Proctor in some of the\nconstellations.[161] The odds are very greatly against any numerous\ngroup of stars moving together in any one direction by chance. On like\ngrounds, there can be no doubt that the sun has a considerable proper\nmotion because on the average the fixed stars show a tendency to move\napparently from one point of the heavens towards that diametrically\nopposite. The sun\u2019s motion in the contrary direction would explain\nthis tendency, otherwise we must believe that thousands of stars\naccidentally agree in their direction of motion, or are urged by some\ncommon force from which the sun is exempt. It may be said that the\nrotation of the earth is proved in like manner, because it is immensely\nmore probable that one body would revolve than that the sun, moon,\nplanets, comets, and the whole of the stars of the heavens should be\nwhirled round the earth daily, with a uniform motion superadded to\ntheir own peculiar motions. This appears to be mainly the reason which\nled Gilbert, one of the earliest English Copernicans, and in every\nway an admirable physicist, to admit the rotation of the earth, while\nFrancis Bacon denied it.\n\n  [161] *Proceedings of the Royal Society*; 20 January, 1870;\n  *Philosophical Magazine*, 4th Series, vol.\u00a0xxxix. p.\u00a0381.\n\nIn contemplating the planetary system, we are struck with the\nsimilarity in direction of nearly all its movements. Newton remarked\nupon the regularity and uniformity of these motions, and contrasted\nthem with the eccentricity and irregularity of the cometary\norbits.[162] Could we, in fact, look down upon the system from the\nnorthern side, we should see all the planets moving round from west\nto east, the satellites moving round their primaries, and the sun,\nplanets, and satellites rotating in the same direction, with some\nexceptions on the verge of the system. In the time of Laplace eleven\nplanets were known, and the directions of rotation were known for the\nsun, six planets, the satellites of Jupiter, Saturn\u2019s ring, and one of\nhis satellites. Thus there were altogether 43 motions all concurring,\nnamely:--\n\n  Orbital motions of eleven planets       11\n  Orbital motions of eighteen satellites  18\n  Axial rotations                         14\n                                          --\n                                          43\n\n  [162] *Principia*, bk. ii. General scholium.\n\nThe probability that 43 motions independent of each other would\ncoincide by chance is the 42nd power of 1/2, so that the odds are\nabout 4,400,000,000,000 to 1 in favour of some common cause for the\nuniformity of direction. This probability, as Laplace observes,[163]\nis higher than that of many historical events which we undoubtingly\nbelieve. In the present day, the probability is much increased by the\ndiscovery of additional planets, and the rotation of other satellites,\nand it is only slightly weakened by the fact that some of the outlying\nsatellites are exceptional in direction, there being considerable\nevidence of an accidental disturbance in the more distant parts of the\nsystem.\n\n  [163] *Essai Philosophique*, p.\u00a055. Laplace appears to count the\n  rings of Saturn as giving two independent movements.\n\nHardly less remarkable than the uniform direction of motion is the\nnear approximation of the orbits of the planets to a common plane.\nDaniel Bernoulli roughly estimated the probability of such an agreement\narising from accident as 1\u00a0\u00f7\u00a0(12)^{6} the greatest inclination of any\norbit to the sun\u2019s equator being 1-12th part of a quadrant. Laplace\ndevoted to this subject some of his most ingenious investigations. He\nfound the probability that the sum of the inclinations of the planetary\norbits would not exceed by accident the actual amount (\u00b7914187\nof a right angle for the ten planets known in 1801) to be (1/10)!\n(\u00b7914187)^{10} or about \u00b700000011235. This probability may be combined\nwith that derived from the direction of motion, and it then becomes\nimmensely probable that the constitution of the planetary system arose\nout of uniform conditions, or, as we say, from some common cause.[164]\n\n  [164] Lubbock, *Essay on Probability*, p.\u00a014. De Morgan, *Encyc.\n  Metrop.* art. *Probability*, p.\u00a0412. Todhunter\u2019s *History of the\n  Theory of Probability*, p.\u00a0543. Concerning the objections raised to\n  these conclusions by Boole, see the *Philosophical Magazine*, 4th\n  Series, vol.\u00a0ii. p.\u00a098. Boole\u2019s *Laws of Thought*, pp.\u00a0364\u2013375.\n\nIf the same kind of calculation be applied to the orbits of comets,\nthe result is very different.[165] Of the orbits which have been\ndetermined 48\u00b79 per cent. only are direct or in the same direction as\nthe planetary motions.[166] Hence it becomes apparent that comets do\nnot properly belong to the solar system, and it is probable that they\nare stray portions of nebulous matter which have accidentally become\nattached to the system by the attractive powers of the sun or Jupiter.\n\n  [165] Laplace, *Essai Philosophique*, pp.\u00a055, 56.\n\n  [166] Chambers\u2019 *Astronomy*, 2nd ed. pp.\u00a0346\u201349.\n\n\n*The General Inverse Problem.*\n\nIn the instances described in the preceding sections, we have been\noccupied in receding from the occurrence of certain similar events to\nthe probability that there must have been a condition or cause for\nsuch events. We have found that the theory of probability, although\nnever yielding a certain result, often enables us to establish an\nhypothesis beyond the reach of reasonable doubt. There is, however,\nanother method of applying the theory, which possesses for us even\ngreater interest, because it illustrates, in the most complete manner,\nthe theory of inference adopted in this work, which theory indeed it\nsuggested. The problem to be solved is as follows:--\n\n*An event having happened a certain number of times, and failed a\ncertain number of times, required the probability that it will happen\nany given number of times in the future under the same circumstances.*\n\nAll the *larger* planets hitherto discovered move in one direction\nround the sun; what is the probability that, if a new planet exterior\nto Neptune be discovered, it will move in the same direction? All\nknown permanent gases, except chlorine, are colourless; what is the\nprobability that, if some new permanent gas should be discovered, it\nwill be colourless? In the general solution of this problem, we wish to\ninfer the future happening of any event from the number of times that\nit has already been observed to happen. Now, it is very instructive to\nfind that there is no known process by which we can pass directly from\nthe data to the conclusion. It is always requisite to recede from the\ndata to the probability of some hypothesis, and to make that hypothesis\nthe ground of our inference concerning future events. Mathematicians,\nin fact, make every hypothesis which is applicable to the question in\nhand; they then calculate, by the inverse method, the probability of\nevery such hypothesis according to the data, and the probability that\nif each hypothesis be true, the required future event will happen. The\ntotal probability that the event will happen is the sum of the separate\nprobabilities contributed by each distinct hypothesis.\n\nTo illustrate more precisely the method of solving the problem, it\nis desirable to adopt some concrete mode of representation, and the\nballot-box, so often employed by mathematicians, will best serve\nour purpose. Let the happening of any event be represented by the\ndrawing of a white ball from a ballot-box, while the failure of an\nevent is represented by the drawing of a black ball. Now, in the\ninductive problem we are supposed to be ignorant of the contents of\nthe ballot-box, and are required to ground all our inferences on our\nexperience of those contents as shown in successive drawings. Rude\ncommon sense would guide us nearly to a true conclusion. Thus, if we\nhad drawn twenty balls one after another, replacing the ball after each\ndrawing, and the ball had in each case proved to be white, we should\nbelieve that there was a considerable preponderance of white balls in\nthe urn, and a probability in favour of drawing a white ball on the\nnext occasion. Though we had drawn white balls for thousands of times\nwithout fail, it would still be possible that some black balls lurked\nin the urn and would at last appear, so that our inferences could never\nbe certain. On the other hand, if black balls came at intervals, we\nshould expect that after a certain number of trials the black balls\nwould appear again from time to time with somewhat the same frequency.\n\nThe mathematical solution of the question consists in little more\nthan a close analysis of the mode in which our common sense proceeds.\nIf twenty white balls have been drawn and no black ball, my common\nsense tells me that any hypothesis which makes the black balls in\nthe urn considerable compared with the white ones is improbable; a\npreponderance of white balls is a more probable hypothesis, and as a\ndeduction from this more probable hypothesis, I expect a recurrence\nof white balls. The mathematician merely reduces this process of\nthought to exact numbers. Taking, for instance, the hypothesis that\nthere are 99 white and one black ball in the urn, he can calculate the\nprobability that 20 white balls would be drawn in succession in those\ncircumstances; he thus forms a definite estimate of the probability\nof this hypothesis, and knowing at the same time the probability of a\nwhite ball reappearing if such be the contents of the urn, he combines\nthese probabilities, and obtains an exact estimate that a white ball\nwill recur in consequence of this hypothesis. But as this hypothesis\nis only one out of many possible ones, since the ratio of white and\nblack balls may be 98 to 2, or 97 to 3, or 96 to 4, and so on, he has\nto repeat the estimate for every such possible hypothesis. To make the\nmethod of solving the problem perfectly evident, I will describe in the\nnext section a very simple case of the problem, originally devised for\nthe purpose by Condorcet, which was also adopted by Lacroix,[167] and\nhas passed into the works of De Morgan, Lubbock, and others.\n\n  [167] *Trait\u00e9 \u00e9l\u00e9mentaire du Calcul des Probabilit\u00e9s*, 3rd ed.\n  (1833), p. 148.\n\n\n*Simple Illustration of the Inverse Problem.*\n\nSuppose it to be known that a ballot-box contains only four black or\nwhite balls, the ratio of black and white balls being unknown. Four\ndrawings having been made with replacement, and a white ball having\nappeared on each occasion but one, it is required to determine the\nprobability that a white ball will appear next time. Now the hypotheses\nwhich can be made as to the contents of the urn are very limited in\nnumber, and are at most the following five:--\n\n  4 white and 0 black balls\n  3   \"    \"  1   \"     \"\n  2   \"    \"  2   \"     \"\n  1   \"    \"  3   \"     \"\n  0   \"    \"  4   \"     \"\n\nThe actual occurrence of black and white balls in the drawings puts the\nfirst and last hypothesis out of the question, so that we have only\nthree left to consider.\n\nIf the box contains three white and one black, the probability of\ndrawing a white each time is 3/4, and a black 1/4; so that the compound\nevent observed, namely, three white and one black, has the probability\n3/4\u00a0\u00d7\u00a03/4\u00a0\u00d7\u00a03/4\u00a0\u00d7\u00a01/4, by the rule already given (p.\u00a0204). But as it is\nindifferent in what order the balls are drawn, and the black ball might\ncome first, second, third, or fourth, we must multiply by four, to\nobtain the probability of three white and one black in any order, thus\ngetting 27/64.\n\nTaking the next hypothesis of two white and two black balls\nin the urn, we obtain for the same probability the quantity\n1/2\u00a0\u00d7\u00a01/2\u00a0\u00d7\u00a01/2\u00a0\u00d7\u00a01/2\u00a0\u00d7\u00a04, or 16/64, and from the third hypothesis of\none white and three black we deduce likewise 1/4\u00a0\u00d7\u00a01/4\u00a0\u00d7\u00a01/4\u00a0\u00d7\u00a03/4\u00a0\u00d7\u00a04,\nor 3/64. According, then, as we adopt the first, second, or third\nhypothesis, the probability that the result actually noticed would\nfollow is 27/64, 16/64, and 3/64. Now it is certain that one or\nother of these hypotheses must be the true one, and their absolute\nprobabilities are proportional to the probabilities that the observed\nevents would follow from them (pp.\u00a0242, 243). All we have to do, then,\nin order to obtain the absolute probability of each hypothesis, is to\nalter these fractions in a uniform ratio, so that their sum shall be\nunity, the expression of certainty. Now, since 27\u00a0+\u00a016\u00a0+\u00a03\u00a0=\u00a046, this\nwill be effected by dividing each fraction by 46, and multiplying by\n64. Thus the probabilities of the first, second, and third hypotheses\nare respectively--\n\n  27/46, 16/46, 3/46.\n\nThe inductive part of the problem is completed, since we have found\nthat the urn most likely contains three white and one black ball, and\nhave assigned the exact probability of each possible supposition. But\nwe are now in a position to resume deductive reasoning, and infer the\nprobability that the next drawing will yield, say a white ball. For if\nthe box contains three white and one black ball, the probability of\ndrawing a white one is certainly 3/4; and as the probability of the box\nbeing so constituted is 27/46, the compound probability that the box\nwill be so filled and will give a white ball at the next trial, is\n\n  27/46\u00a0\u00d7\u00a03/4 or 81/184.\n\nAgain, the probability is 16/46 that the box contains two white and two\nblack, and under those conditions the probability is 1/2 that a white\nball will appear; hence the probability that a white ball will appear\nin consequence of that condition, is\n\n  16/46\u00a0\u00d7\u00a01/2 or 32/184.\n\nFrom the third supposition we get in like manner the probability\n\n  3/46\u00a0\u00d7\u00a01/4 or 3/184.\n\nSince one and not more than one hypothesis can be true, we may add\ntogether these separate probabilities, and we find that\n\n  81/184\u00a0+\u00a032/184\u00a0+\u00a03/184\u00a0or\u00a0116/184\n\nis the complete probability that a white ball will be next drawn under\nthe conditions and data supposed.\n\n\n*General Solution of the Inverse Problem.*\n\nIn the instance of the inverse method described in the last section,\nthe balls supposed to be in the ballot-box were few, for the purpose of\nsimplifying the calculation. In order that our solution may apply to\nnatural phenomena, we must render our hypotheses as little arbitrary\nas possible. Having no *\u00e0 priori* knowledge of the conditions of the\nphenomena in question, there is no limit to the variety of hypotheses\nwhich might be suggested. Mathematicians have therefore had recourse\nto the most extensive suppositions which can be made, namely, that the\nballot-box contains an infinite number of balls; they have then varied\nthe proportion of white to black balls continuously, from the smallest\nto the greatest possible proportion, and estimated the aggregate\nprobability which results from this comprehensive supposition.\n\nTo explain their procedure, let us imagine that, instead of an infinite\nnumber, the ballot-box contains a large finite number of balls, say\n1000. Then the number of white balls might be 1 or 2 or 3 or 4, and so\non, up to 999. Supposing that three white and one black ball have been\ndrawn from the urn as before, there is a certain very small probability\nthat this would have occurred in the case of a box containing one white\nand 999 black balls; there is also a small probability that from such\na box the next ball would be white. Compound these probabilities, and\nwe have the probability that the next ball really will be white, in\nconsequence of the existence of that proportion of balls. If there be\ntwo white and 998 black balls in the box, the probability is greater\nand will increase until the balls are supposed to be in the proportion\nof those drawn. Now 999 different hypotheses are possible, and the\ncalculation is to be made for each of these, and their aggregate taken\nas the final result. It is apparent that as the number of balls in\nthe box is increased, the absolute probability of any one hypothesis\nconcerning the exact proportion of balls is decreased, but the\naggregate results of all the hypotheses will assume the character of a\nwider average.\n\nWhen we take the step of supposing the balls within the urn to be\ninfinite in number, the possible proportions of white and black balls\nalso become infinite, and the probability of any one proportion\nactually existing is infinitely small. Hence the final result that\nthe next ball drawn will be white is really the sum of an infinite\nnumber of infinitely small quantities. It might seem impossible to\ncalculate out a problem having an infinite number of hypotheses,\nbut the wonderful resources of the integral calculus enable this\nto be done with far greater facility than if we supposed any large\nfinite number of balls, and then actually computed the results. I\nwill not attempt to describe the processes by which Laplace finally\naccomplished the complete solution of the problem. They are to be found\ndescribed in several English works, especially De Morgan\u2019s *Treatise\non Probabilities*, in the *Encyclop\u00e6dia Metropolitana*, and Mr.\nTodhunter\u2019s *History of the Theory of Probability*. The abbreviating\npower of mathematical analysis was never more strikingly shown. But\nI may add that though the integral calculus is employed as a means\nof summing infinitely numerous results, we in no way abandon the\nprinciples of combinations already treated. We calculate the values of\ninfinitely numerous factorials, not, however, obtaining their actual\nproducts, which would lead to an infinite number of figures, but\nobtaining the final answer to the problem by devices which can only be\ncomprehended after study of the integral calculus.\n\nIt must be allowed that the hypothesis adopted by Laplace is in some\ndegree arbitrary, so that there was some opening for the doubt which\nBoole has cast upon it.[168] But it may be replied, (1) that the\nsupposition of an infinite number of balls treated in the manner of\nLaplace is less arbitrary and more comprehensive than any other that\ncan be suggested. (2) The result does not differ much from that\nwhich would be obtained on the hypothesis of any large finite number\nof balls. (3) The supposition leads to a series of simple formulas\nwhich can be applied with ease in many cases, and which bear all the\nappearance of truth so far as it can be independently judged by a sound\nand practiced understanding.\n\n  [168] *Laws of Thought*, pp.\u00a0368\u2013375.\n\n\n*Rules of the Inverse Method.*\n\nBy the solution of the problem, as described in the last section, we\nobtain the following series of simple rules.\n\n1. *To find the probability that an event which has not hitherto been\nobserved to fail will happen once more, divide the number of times the\nevent has been observed increased by one, by the same number increased\nby two.*\n\nIf there have been *m* occasions on which a certain event might have\nbeen observed to happen, and it has happened on all those occasions,\nthen the probability that it will happen on the next occasion of the\nsame kind (*m*\u00a0+\u00a01)/(*m*\u00a0+\u00a02). For instance, we may say that there are\nnine places in the planetary system where planets might exist obeying\nBode\u2019s law of distance, and in every place there is a planet obeying\nthe law more or less exactly, although no reason is known for the\ncoincidence. Hence the probability that the next planet beyond Neptune\nwill conform to the law is 10/11.\n\n2. *To find the, probability that an event which has not hitherto\nfailed will not fail for a certain number of new occasions, divide the\nnumber of times the event has happened increased by one, by the same\nnumber increased by one and the number of times it is to happen.*\n\nAn event having happened *m* times without fail, the probability that\nit will happen *n* more times is (*m*\u00a0+\u00a01)/(*m*\u00a0+\u00a0*n*\u00a0+\u00a01). Thus the\nprobability that three new planets would obey Bode\u2019s law is 10/13; but\nit must be allowed that this, as well as the previous result, would be\nmuch weakened by the fact that Neptune can barely be said to obey the\nlaw.\n\n*3. An event having happened and failed a certain number of times, to\nfind the probability that it will happen the next time, divide the\nnumber of times the event has happened increased by one, by the whole\nnumber of times the event has happened or failed increased by two.*\n\nIf an event has happened *m* times and failed *n* times,\nthe probability that it will happen on the next occasion is\n(*m*\u00a0+\u00a01)/(*m*\u00a0+\u00a0*n*\u00a0+\u00a02). Thus, if we assume that of the elements\ndiscovered up to the year 1873, 50 are metallic and 14 non-metallic,\nthen the probability that the next element discovered will be metallic\nis 51/66. Again, since of 37 metals which have been sufficiently\nexamined only four, namely, sodium, potassium, lanthanum, and\nlithium, are of less density than water, the probability that the\nnext metal examined or discovered will be less dense than water is\n(4\u00a0+\u00a01)/(37\u00a0+\u00a02) or 5/39.\n\nWe may state the results of the method in a more general manner\nthus,[169]--If under given circumstances certain events A, B, C, &c.,\nhave happened respectively *m*, *n*, *p*, &c., times, and one or other\nof these events must happen, then the probabilities of these events are\nproportional to *m*\u00a0+\u00a01, *n*\u00a0+\u00a01, *p*\u00a0+\u00a01, &c., so that the probability\nof A will be (*m*\u00a0+\u00a01)/(*m*\u00a0+\u00a01\u00a0+\u00a0*n*\u00a0+\u00a01\u00a0+\u00a0*p*\u00a0+\u00a01\u00a0+\u00a0&c.) But if new\nevents may happen in addition to those which have been observed, we\nmust assign unity for the probability of such new event. The odds then\nbecome 1 for a new event, *m*\u00a0+\u00a01 for A, *n*\u00a0+\u00a01 for B, and so on, and\nthe absolute probability of A is (*m*\u00a0+\u00a01)/(1\u00a0+\u00a0*m*\u00a0+\u00a01\u00a0+\u00a0*n*\u00a0+\u00a01\u00a0+\u00a0&c.)\n\n  [169] De Morgan\u2019s *Essay on Probabilities*, Cabinet Cyclop\u00e6dia, p.\u00a067.\n\nIt is interesting to trace out the variations of probability according\nto these rules. The first time a casual event happens it is 2 to 1\nthat it will happen again; if it does happen it is 3 to 1 that it\nwill happen a third time; and on successive occasions of the like\nkind the odds become 4, 5, 6, &c., to 1. The odds of course will be\ndiscriminated from the probabilities which are successively 2/3, 3/4,\n4/5, &c. Thus on the first occasion on which a person sees a shark,\nand notices that it is accompanied by a little pilot fish, the odds\nare 2 to 1, or the probability 2/3, that the next shark will be so\naccompanied.\n\nWhen an event has happened a very great number of times, its\nhappening once again approaches nearly to certainty. If we suppose\nthe sun to have risen one thousand million times, the probability\nthat it will rise again, on the ground of this knowledge merely, is\n(1,000,000,000\u00a0+\u00a01)/(1,000,000,000\u00a0+\u00a01\u00a0+\u00a01). But then the probability\nthat it will continue to rise for as long a period in the future is\nonly (1,000,000,000\u00a0+\u00a01)/(2,000,000,000\u00a0+\u00a01), or almost exactly 1/2.\nThe probability that it will continue so rising a thousand times\nas long is only about 1/1001. The lesson which we may draw from\nthese figures is quite that which we should adopt on other grounds,\nnamely, that experience never affords certain knowledge, and that\nit is exceedingly improbable that events will always happen as we\nobserve them. Inferences pushed far beyond their data soon lose\nany considerable probability. De Morgan has said,[170] \u201cNo finite\nexperience whatsoever can justify us in saying that the future shall\ncoincide with the past in all time to come, or that there is any\nprobability for such a conclusion.\u201d On the other hand, we gain the\nassurance that experience sufficiently extended and prolonged will\ngive us the knowledge of future events with an unlimited degree of\nprobability, provided indeed that those events are not subject to\narbitrary interference.\n\n  [170] *Essay on Probabilities*, p.\u00a0128.\n\nIt must be clearly understood that these probabilities are only such\nas arise from the mere happening of the events, irrespective of any\nknowledge derived from other sources concerning those events or the\ngeneral laws of nature. All our knowledge of nature is indeed founded\nin like manner upon observation, and is therefore only probable. The\nlaw of gravitation itself is only probably true. But when a number of\ndifferent facts, observed under the most diverse circumstances, are\nfound to be harmonized under a supposed law of nature, the probability\nof the law approximates closely to certainty. Each science rests upon\nso many observed facts, and derives so much support from analogies or\nconnections with other sciences, that there are comparatively few cases\nwhere our judgment of the probability of an event depends entirely\nupon a few antecedent events, disconnected from the general body of\nphysical science.\n\nEvents, again, may often exhibit a regularity of succession or\npreponderance of character, which the simple formula will not take into\naccount. For instance, the majority of the elements recently discovered\nare metals, so that the probability of the next discovery being that\nof a metal, is doubtless greater than we calculated (p.\u00a0258). At\nthe more distant parts of the planetary system, there are symptoms\nof disturbance which would prevent our placing much reliance on any\ninference from the prevailing order of the known planets to those\nundiscovered ones which may possibly exist at great distances. These\nand all like complications in no way invalidate the theoretic truth of\nthe formulas, but render their sound application much more difficult.\n\nErroneous objections have been raised to the theory of probability, on\nthe ground that we ought not to trust to our *\u00e0 priori* conceptions\nof what is likely to happen, but should always endeavour to obtain\nprecise experimental data to guide us.[171] This course, however,\nis perfectly in accordance with the theory, which is our best and\nonly guide, whatever data we possess. We ought to be always applying\nthe inverse method of probabilities so as to take into account all\nadditional information. When we throw up a coin for the first time, we\nare probably quite ignorant whether it tends more to fall head or tail\nupwards, and we must therefore assume the probability of each event as\n1/2. But if it shows head in the first throw, we now have very slight\nexperimental evidence in favour of a tendency to show head. The chance\nof two heads is now slightly greater than 1/4, which it appeared to\nbe at first,[172] and as we go on throwing the coin time after time,\nthe probability of head appearing next time constantly varies in a\nslight degree according to the character of our previous experience. As\nLaplace remarks, we ought always to have regard to such considerations\nin common life. Events when closely scrutinized will hardly ever prove\nto be quite independent, and the slightest preponderance one way or\nthe other is some evidence of connection, and in the absence of better\nevidence should be taken into account.\n\n  [171] J.\u00a0S. Mill, *System of Logic*, 5th edition, bk. iii. chap.\n  xviii. \u00a7 3.\n\n  [172] Todhunter\u2019s *History*, pp.\u00a0472, 598.\n\nThe grand object of seeking to estimate the probability of future\nevents from past experience, seems to have been entertained by James\nBernoulli and De Moivre, at least such was the opinion of Condorcet;\nand Bernoulli may be said to have solved one case of the problem.[173]\nThe English writers Bayes and Price are, however, undoubtedly the first\nwho put forward any distinct rules on the subject.[174] Condorcet and\nseveral other eminent mathematicians advanced the mathematical theory\nof the subject; but it was reserved to the immortal Laplace to bring\nto the subject the full power of his genius, and carry the solution of\nthe problem almost to perfection. It is instructive to observe that a\ntheory which arose from petty games of chance, the rules and the very\nnames of which are forgotten, gradually advanced, until it embraced the\nmost sublime problems of science, and finally undertook to measure the\nvalue and certainty of all our inductions.\n\n  [173] Todhunter\u2019s *History*, pp.\u00a0378, 379.\n\n  [174] *Philosophical Transactions*, [1763], vol.\u00a0liii. p.\u00a0370, and\n  [1764], vol.\u00a0liv. p.\u00a0296. Todhunter, pp.\u00a0294\u2013300.\n\n\n*Fortuitous Coincidences.*\n\nWe should have studied the theory of probability to very little\npurpose, if we thought that it would furnish us with an infallible\nguide. The theory itself points out the approximate certainty, that we\nshall sometimes be deceived by extraordinary fortuitous coincidences.\nThere is no run of luck so extreme that it may not happen, and it\nmay happen to us, or in our time, as well as to other persons or in\nother times. We may be forced by correct calculation to refer such\ncoincidences to a necessary cause, and yet we may be deceived. All\nthat the calculus of probability pretends to give, is *the result in\nthe long run*, as it is called, and this really means in *an infinity\nof cases*. During any finite experience, however long, chances may be\nagainst us. Nevertheless the theory is the best guide we can have. If\nwe always think and act according to its well-interpreted indications,\nwe shall have the best chance of escaping error; and if all persons,\nthroughout all time to come, obey the theory in like manner, they will\nundoubtedly thereby reap the greatest advantage.\n\nNo rule can be given for discriminating between coincidences which\nare casual and those which are the effects of law. By a fortuitous\nor casual coincidence, we mean an agreement between events, which\nnevertheless arise from wholly independent and different causes or\nconditions, and which will not always so agree. It is a fortuitous\ncoincidence, if a penny thrown up repeatedly in various ways always\nfalls on the same side; but it would not be fortuitous if there were\nany similarity in the motions of the hand, and the height of the\nthrow, so as to cause or tend to cause a uniform result. Now among the\ninfinitely numerous events, objects, or relations in the universe, it\nis quite likely that we shall occasionally notice casual coincidences.\nThere are seven intervals in the octave, and there is nothing very\nimprobable in the colours of the spectrum happening to be apparently\ndivisible into the same or similar series of seven intervals. It is\nhardly yet decided whether this apparent coincidence, with which Newton\nwas much struck, is well founded or not,[175] but the question will\nprobably be decided in the negative.\n\n  [175] Newton\u2019s *Opticks*, Bk. I., Part ii. Prop. 3; *Nature*, vol.\u00a0i.\n  p.\u00a0286.\n\nIt is certainly a casual coincidence which the ancients noticed between\nthe seven vowels, the seven strings of the lyre, the seven Pleiades,\nand the seven chiefs at Thebes.[176] The accidents connected with the\nnumber seven have misled the human intellect throughout the historical\nperiod. Pythagoras imagined a connection between the seven planets and\nthe seven intervals of the monochord. The alchemists were never tired\nof drawing inferences from the coincidence in numbers of the seven\nplanets and the seven metals, not to speak of the seven days of the\nweek.\n\n  [176] Aristotle\u2019s *Metaphysics*, xiii. 6. 3.\n\nA singular circumstance was pointed out concerning the dimensions\nof the earth, sun, and moon; the sun\u2019s diameter was almost exactly\n110 times as great as the earth\u2019s diameter, while in almost exactly\nthe same ratio the mean distance of the earth was greater than the\nsun\u2019s diameter, and the mean distance of the moon from the earth was\ngreater than the moon\u2019s diameter. The agreement was so close that it\nmight have proved more than casual, but its fortuitous character is\nnow sufficiently shown by the fact, that the coincidence ceases to be\nremarkable when we adopt the amended dimensions of the planetary system.\n\nA considerable number of the elements have atomic weights, which are\napparently exact multiples of that of hydrogen. If this be not a law to\nbe ultimately extended to all the elements, as supposed by Prout, it\nis a most remarkable coincidence. But, as I have observed, we have no\nmeans of absolutely discriminating accidental coincidences from those\nwhich imply a deep producing cause. A coincidence must either be very\nstrong in itself, or it must be corroborated by some explanation or\nconnection with other laws of nature. Little attention was ever given\nto the coincidence concerning the dimensions of the sun, earth, and\nmoon, because it was not very strong in itself, and had no apparent\nconnection with the principles of physical astronomy. Prout\u2019s Law\nbears more probability because it would bring the constitution of\nthe elements themselves in close connection with the atomic theory,\nrepresenting them as built up out of a simpler substance.\n\nIn historical and social matters, coincidences are frequently pointed\nout which are due to chance, although there is always a strong popular\ntendency to regard them as the work of design, or as having some hidden\nmeaning. If to 1794, the number of the year in which Robespierre fell,\nwe add the sum of its digits, the result is 1815, the year in which\nNapoleon fell; the repetition of the process gives 1830 the year\nin which Charles the Tenth abdicated. Again, the French Chamber of\nDeputies, in 1830, consisted of 402 members, of whom 221 formed the\nparty called \u201cLa queue de Robespierre,\u201d while the remainder, 181 in\nnumber, were named \u201cLes honn\u00eates gens.\u201d If we give to each letter a\nnumerical value corresponding to its place in the alphabet, it will be\nfound that the sum of the values of the letters in each name exactly\nindicates the number of the party.\n\nA number of such coincidences, often of a very curious character,\nmight be adduced, and the probability against the occurrence of each\nis enormously great. They must be attributed to chance, because they\ncannot be shown to have the slightest connection with the general\nlaws of nature; but persons are often found to be greatly influenced\nby such coincidences, regarding them as evidence of fatality, that\nis of a system of causation governing human affairs independently of\nthe ordinary laws of nature. Let it be remembered that there are an\ninfinite number of opportunities in life for some strange coincidence\nto present itself, so that it is quite to be expected that remarkable\nconjunctions will sometimes happen.\n\nIn all matters of judicial evidence, we must bear in mind the probable\noccurrence from time to time of unaccountable coincidences. The Roman\njurists refused for this reason to invalidate a testamentary deed, the\nwitnesses of which had sealed it with the same seal. For witnesses\nindependently using their own seals might be found to possess identical\nones by accident.[177] It is well known that circumstantial evidence of\napparently overwhelming completeness will sometimes lead to a mistaken\njudgment, and as absolute certainty is never really attainable, every\ncourt must act upon probabilities of a high amount, and in a certain\nsmall proportion of cases they must almost of necessity condemn the\ninnocent victims of a remarkable conjuncture of circumstances.[178]\nPopular judgments usually turn upon probabilities of far less amount,\nas when the palace of Nicomedia, and even the bedchamber of Diocletian,\nhaving been on fire twice within fifteen days, the people entirely\nrefused to believe that it could be the result of accident. The Romans\nbelieved that there was fatality connected with the name of Sextus.\n\n  \u201cSemper sub Sextis perdita Roma fuit.\u201d\n\n  [177] Possunt autem omnes testes et uno annulo signare testamentum\n  Quid enim si septem annuli una sculptura fuerint, secundum quod\n  Pomponio visum est?--*Justinian*, ii. tit. x. 5.\n\n  [178] See Wills on *Circumstantial Evidence*, p.\u00a0148.\n\nThe utmost precautions will not provide against all contingencies.\nTo avoid errors in important calculations, it is usual to have them\nrepeated by different computers; but a case is on record in which three\ncomputers made exactly the same calculations of the place of a star,\nand yet all did it wrong in precisely the same manner, for no apparent\nreason.[179]\n\n  [179] *Memoirs of the Royal Astronomical Society*, vol.\u00a0iv. p.\u00a0290,\n  quoted by Lardner, *Edinburgh Review*, July 1834, p.\u00a0278.\n\n\n*Summary of the Theory of Inductive Inference.*\n\nThe theory of inductive inference stated in this and the previous\nchapters, was suggested by the study of the Inverse Method of\nProbability, but it also bears much resemblance to the so-called\nDeductive Method described by Mill, in his celebrated *System of\nLogic*. Mill\u2019s views concerning the Deductive Method, probably form\nthe most original and valuable part of his treatise, and I should\nhave ascribed the doctrine entirely to him, had I not found that\nthe opinions put forward in other parts of his work are entirely\ninconsistent with the theory here upheld. As this subject is the most\nimportant and difficult one with which we have to deal, I will try to\nremedy the imperfect manner in which I have treated it, by giving a\nrecapitulation of the views adopted.\n\nAll inductive reasoning is but the inverse application of deductive\nreasoning. Being in possession of certain particular facts or events\nexpressed in propositions, we imagine some more general proposition\nexpressing the existence of a law or cause; and, deducing the\nparticular results of that supposed general proposition, we observe\nwhether they agree with the facts in question. Hypothesis is thus\nalways employed, consciously or unconsciously. The sole conditions to\nwhich we need conform in framing any hypothesis is, that we both have\nand exercise the power of inferring deductively from the hypothesis to\nthe particular results, which are to be compared with the known facts.\nThus there are but three steps in the process of induction:--\n\n(1) Framing some hypothesis as to the character of the general law.\n\n(2) Deducing consequences from that law.\n\n(3) Observing whether the consequences agree with the particular facts\nunder consideration.\n\nIn very simple cases of inverse reasoning, hypothesis may seem\naltogether needless. To take numbers again as a convenient\nillustration, I have only to look at the series,\n\n  1, 2, 4, 8, 16, 32, &c.,\n\nto know at once that the general law is that of geometrical\nprogression; I need no successive trial of various hypotheses, because\nI am familiar with the series, and have long since learnt from what\ngeneral formula it proceeds. In the same way a mathematician becomes\nacquainted with the integrals of a number of common formulas, so\nthat he need not go through any process of discovery. But it is none\nthe less true that whenever previous reasoning does not furnish the\nknowledge, hypotheses must be framed and tried (p.\u00a0124).\n\nThere naturally arise two cases, according as the nature of the\nsubject admits of certain or only probable deductive reasoning.\nCertainty, indeed, is but a singular case of probability, and the\ngeneral principles of procedure are always the same. Nevertheless,\nwhen certainty of inference is possible, the process is simplified.\nOf several mutually inconsistent hypotheses, the results of which can\nbe certainly compared with fact, but one hypothesis can ultimately\nbe entertained. Thus in the inverse logical problem, two logically\ndistinct conditions could not yield the same series of possible\ncombinations. Accordingly, in the case of two terms we had to choose\none of six different kinds of propositions (p.\u00a0136), and in the case\nof three terms, our choice lay among 192 possible distinct hypotheses\n(p.\u00a0140). Natural laws, however, are often quantitative in character,\nand the possible hypotheses are then infinite in variety.\n\nWhen deduction is certain, comparison with fact is needed only to\nassure ourselves that we have rightly selected the hypothetical\nconditions. The law establishes itself, and no number of particular\nverifications can add to its probability. Having once deduced from\nthe principles of algebra that the difference of the squares of two\nnumbers is equal to the product of their sum and difference, no number\nof particular trials of its truth will render it more certain. On the\nother hand, no finite number of particular verifications of a supposed\nlaw will render that law certain. In short, certainty belongs only to\nthe deductive process, and to the teachings of direct intuition; and\nas the conditions of nature are not given by intuition, we can only be\ncertain that we have got a correct hypothesis when, out of a limited\nnumber conceivably possible, we select that one which alone agrees with\nthe facts to be explained.\n\nIn geometry and kindred branches of mathematics, deductive reasoning\nis conspicuously certain, and it would often seem as if the\nconsideration of a single diagram yields us certain knowledge of a\ngeneral proposition. But in reality all this certainty is of a purely\nhypothetical character. Doubtless if we could ascertain that a supposed\ncircle was a true and perfect circle, we could be certain concerning a\nmultitude of its geometrical properties. But geometrical figures are\nphysical objects, and the senses can never assure us as to their exact\nforms. The figures really treated in Euclid\u2019s *Elements* are imaginary,\nand we never can verify in practice the conclusions which we draw with\ncertainty in inference; questions of degree and probability enter.\n\nPassing now to subjects in which deduction is only probable, it ceases\nto be possible to adopt one hypothesis to the exclusion of the others.\nWe must entertain at the same time all conceivable hypotheses, and\nregard each with the degree of esteem proportionate to its probability.\nWe go through the same steps as before.\n\n(1) We frame an hypothesis.\n\n(2) We deduce the probability of various series of possible\nconsequences.\n\n(3) We compare the consequences with the particular facts, and observe\nthe probability that such facts would happen under the hypothesis.\n\nThe above processes must be performed for every conceivable hypothesis,\nand then the absolute probability of each will be yielded by the\nprinciple of the inverse method (p.\u00a0242). As in the case of certainty\nwe accept that hypothesis which certainly gives the required results,\nso now we accept as most probable that hypothesis which most probably\ngives the results; but we are obliged to entertain at the same time\nall other hypotheses with degrees of probability proportionate to the\nprobabilities that they would give the same results.\n\nSo far we have treated only of the process by which we pass from\nspecial facts to general laws, that inverse application of deduction\nwhich constitutes induction. But the direct employment of deduction\nis often combined with the inverse. No sooner have we established\na general law, than the mind rapidly draws particular consequences\nfrom it. In geometry we may almost seem to infer that *because* one\nequilateral triangle is equiangular, therefore another is so. In\nreality it is not because one is that another is, but because all\nare. The geometrical conditions are perfectly general, and by what\nis sometimes called *parity of reasoning* whatever is true of one\nequilateral triangle, so far as it is equilateral, is true of all\nequilateral triangles.\n\nSimilarly, in all other cases of inductive inference, where we seem to\npass from some particular instances to a new instance, we go through\nthe same process. We form an hypothesis as to the logical conditions\nunder which the given instances might occur; we calculate inversely\nthe probability of that hypothesis, and compounding this with the\nprobability that a new instance would proceed from the same conditions,\nwe gain the absolute probability of occurrence of the new instance in\nvirtue of this hypothesis. But as several, or many, or even an infinite\nnumber of mutually inconsistent hypotheses may be possible, we must\nrepeat the calculation for each such conceivable hypothesis, and then\nthe complete probability of the future instance will be the sum of the\nseparate probabilities. The complication of this process is often very\nmuch reduced in practice, owing to the fact that one hypothesis may be\nalmost certainly true, and other hypotheses, though conceivable, may be\nso improbable as to be neglected without appreciable error.\n\nWhen we possess no knowledge whatever of the conditions from which\nthe events proceed, we may be unable to form any probable hypotheses\nas to their mode of origin. We have now to fall back upon the general\nsolution of the problem effected by Laplace, which consists in\nadmitting on an equal footing every conceivable ratio of favourable\nand unfavourable chances for the production of the event, and then\naccepting the aggregate result as the best which can be obtained. This\nsolution is only to be accepted in the absence of all better means,\nbut like other results of the calculus of probability, it comes to our\naid where knowledge is at an end and ignorance begins, and it prevents\nus from over-estimating the knowledge we possess. The general results\nof the solution are in accordance with common sense, namely, that\nthe more often an event has happened the more probable, as a general\nrule, is its subsequent recurrence. With the extension of experience\nthis probability increases, but at the same time the probability is\nslight that events will long continue to happen as they have previously\nhappened.\n\nWe have now pursued the theory of inductive inference, as far as\ncan be done with regard to simple logical or numerical relations.\nThe laws of nature deal with time and space, which are infinitely\ndivisible. As we passed from pure logic to numerical logic, so we must\nnow pass from questions of discontinuous, to questions of continuous\nquantity, encountering fresh considerations of much difficulty. Before,\ntherefore, we consider how the great inductions and generalisations\nof physical science illustrate the views of inductive reasoning just\nexplained, we must break off for a time, and review the means which we\npossess of measuring and comparing magnitudes of time, space, mass,\nforce, momentum, energy, and the various manifestations of energy in\nmotion, heat, electricity, chemical change, and the other phenomena of\nnature.\n\n\n\n\nBOOK III.\n\nMETHODS OF MEASUREMENT.\n\n\n\n\nCHAPTER XIII.\n\nTHE EXACT MEASUREMENT OF PHENOMENA.\n\n\nAs physical science advances, it becomes more and more accurately\nquantitative. Questions of simple logical fact after a time resolve\nthemselves into questions of degree, time, distance, or weight. Forces\nhardly suspected to exist by one generation, are clearly recognised\nby the next, and precisely measured by the third generation. But\none condition of this rapid advance is the invention of suitable\ninstruments of measurement. We need what Francis Bacon called\n*Instanti\u00e6 citantes*, or *evocantes*, methods of rendering minute\nphenomena perceptible to the senses; and we also require *Instanti\u00e6\nradii* or *curriculi*, that is measuring instruments. Accordingly,\nthe introduction of a new instrument often forms an epoch in the\nhistory of science. As Davy said, \u201cNothing tends so much to the\nadvancement of knowledge as the application of a new instrument. The\nnative intellectual powers of men in different times are not so much\nthe causes of the different success of their labours, as the peculiar\nnature of the means and artificial resources in their possession.\u201d\n\nIn the absence indeed of advanced theory and analytical power, a\nvery precise instrument would be useless. Measuring apparatus and\nmathematical theory should advance *pari passu*, and with just such\nprecision as the theorist can anticipate results, the experimentalist\nshould be able to compare them with experience. The scrupulously\naccurate observations of Flamsteed were the proper complement to the\nintense mathematical powers of Newton.\n\nEvery branch of knowledge commences with quantitative notions of a\nvery rude character. After we have far progressed, it is often amusing\nto look back into the infancy of the science, and contrast present\nwith past methods. At Greenwich Observatory in the present day, the\nhundredth part of a second is not thought an inconsiderable portion\nof time. The ancient Chald\u00e6ans recorded an eclipse to the nearest\nhour, and the early Alexandrian astronomers thought it superfluous to\ndistinguish between the edge and centre of the sun. By the introduction\nof the astrolabe, Ptolemy and the later Alexandrian astronomers could\ndetermine the places of the heavenly bodies within about ten minutes\nof arc. Little progress then ensued for thirteen centuries, until\nTycho Brahe made the first great step towards accuracy, not only by\nemploying better instruments, but even more by ceasing to regard an\ninstrument as correct. Tycho, in fact, determined the errors of his\ninstruments, and corrected his observations. He also took notice of\nthe effects of atmospheric refraction, and succeeded in attaining an\naccuracy often sixty times as great as that of Ptolemy. Yet Tycho and\nHevelius often erred several minutes in the determination of a star\u2019s\nplace, and it was a great achievement of R\u0153mer and Flamsteed to reduce\nthis error to seconds. Bradley, the modern Hipparchus, carried on the\nimprovement, his errors in right ascension, according to Bessel, being\nunder one second of time, and those of declination under four seconds\nof arc. In the present day the average error of a single observation\nis probably reduced to the half or quarter of what it was in Bradley\u2019s\ntime; and further extreme accuracy is attained by the multiplication\nof observations, and their skilful combination according to the theory\nof error. Some of the more important constants, for instance that of\nnutation, have been determined within the tenth part of a second of\nspace.[180]\n\n  [180] Baily, *British Association Catalogue of Stars*, pp.\u00a07, 23.\n\nIt would be a matter of great interest to trace out the dependence of\nthis progress upon the introduction of new instruments. The astrolabe\nof Ptolemy, the telescope of Galileo, the pendulum of Galileo and\nHuyghens, the micrometer of Horrocks, and the telescopic sights and\nmicrometer of Gascoygne and Picard, R\u0153mer\u2019s transit instrument,\nNewton\u2019s and Hadley\u2019s quadrant, Dollond\u2019s achromatic lenses, Harrison\u2019s\nchronometer, and Ramsden\u2019s dividing engine--such were some of the\nprincipal additions to astronomical apparatus. The result is, that we\nnow take note of quantities, 300,000 or 400,000 times as small as in\nthe time of the Chald\u00e6ans.\n\nIt would be interesting again to compare the scrupulous accuracy of a\nmodern trigonometrical survey with Eratosthenes\u2019 rude but ingenious\nguess at the difference of latitude between Alexandria and Syene--or\nwith Norwood\u2019s measurement of a degree of latitude in 1635. \u201cSometimes\nI measured, sometimes I paced,\u201d said Norwood; \u201cand I believe I am\nwithin a scantling of the truth.\u201d Such was the germ of those elaborate\ngeodesical measurements which have made the dimensions of the globe\nknown to us within a few hundred yards.\n\nIn other branches of science, the invention of an instrument has\nusually marked, if it has not made, an epoch. The science of heat might\nbe said to commence with the construction of the thermometer, and it\nhas recently been advanced by the introduction of the thermo-electric\npile. Chemistry has been created chiefly by the careful use of the\nbalance, which forms a unique instance of an instrument remaining\nsubstantially in the form in which it was first applied to scientific\npurposes by Archimedes. The balance never has been and probably never\ncan be improved, except in details of construction. The torsion\nbalance, introduced by Coulomb towards the end of last century, has\nrapidly become essential in many branches of investigation. In the\nhands of Cavendish and Baily, it gave a determination of the earth\u2019s\ndensity; applied in the galvanometer, it gave a delicate measure of\nelectrical forces, and is indispensable in the thermo-electric pile.\nThis balance is made by simply suspending any light rod by a thin wire\nor thread attached to the middle point. And we owe to it almost all the\nmore delicate investigations in the theories of heat, electricity, and\nmagnetism.\n\nThough we can now take note of the millionth of an inch in space,\nand the millionth of a second in time, we must not overlook the fact\nthat in other operations of science we are yet in the position of the\nChald\u00e6ans. Not many years have elapsed since the magnitudes of the\nstars, meaning the amounts of light they send to the observer\u2019s eye,\nwere guessed at in the rudest manner, and the astronomer adjudged a\nstar to this or that order of magnitude by a rough comparison with\nother stars of the same order. To Sir John Herschel we owe an attempt\nto introduce a uniform method of measurement and expression, bearing\nsome relation to the real photometric magnitudes of the stars.[181]\nPrevious to the researches of Bunsen and Roscoe on the chemical action\nof light, we were devoid of any mode of measuring the energy of light;\neven now the methods are tedious, and it is not clear that they give\nthe energy of light so much as one of its special effects. Many natural\nphenomena have hardly yet been made the subject of measurement at all,\nsuch as the intensity of sound, the phenomena of taste and smell, the\nmagnitude of atoms, the temperature of the electric spark or of the\nsun\u2019s photosphere.\n\n  [181] *Outlines of Astronomy*, 4th ed. sect. 781, p.\u00a0522. *Results of\n  Observations at the Cape of Good Hope*, &c., p.\u00a037.\n\nTo suppose, then, that quantitative science treats only of exactly\nmeasurable quantities, is a gross if it be a common mistake. Whenever\nwe are treating of an event which either happens altogether or does\nnot happen at all, we are engaged with a non-quantitative phenomenon,\na matter of fact, not of degree; but whenever a thing may be greater\nor less, or twice or thrice as great as another, whenever, in short,\nratio enters even in the rudest manner, there science will have a\nquantitative character. There can be little doubt, indeed, that\nevery science as it progresses will become gradually more and more\nquantitative. Numerical precision is the soul of science, as Herschel\nsaid, and as all natural objects exist in space, and involve molecular\nmovements, measurable in velocity and extent, there is no apparent\nlimit to the ultimate extension of quantitative science. But the reader\nmust not for a moment suppose that, because we depend more and more\nupon mathematical methods, we leave logical methods behind us. Number,\nas I have endeavoured to show, is logical in its origin, and quantity\nis but a development of number, or analogous thereto.\n\n\n*Division of the Subject.*\n\nThe general subject of quantitative investigation will have to be\ndivided into several parts. We shall firstly consider the means at\nour disposal for measuring phenomena, and thus rendering them more or\nless amenable to mathematical treatment. This task will involve an\nanalysis of the principles on which accurate methods of measurement are\nfounded, forming the subject of the remainder of the present chapter.\nAs measurement, however, only yields ratios, we have in the next\nchapter to consider the establishment of unit magnitudes, in terms of\nwhich our results may be expressed. As every phenomenon is usually the\nsum of several distinct quantities depending upon different causes,\nwe have next to investigate in Chapter XV. the methods by which we\nmay disentangle complicated effects, and refer each part of the joint\neffect to its separate cause.\n\nIt yet remains for us in subsequent chapters to treat of quantitative\ninduction, properly so called. We must follow out the inverse logical\nmethod, as it presents itself in problems of a far higher degree of\ndifficulty than those which treat of objects related in a simple\nlogical manner, and incapable of merging into each other by addition\nand subtraction.\n\n\n*Continuous Quantity.*\n\nThe phenomena of nature are for the most part manifested in quantities\nwhich increase or decrease continuously. When we inquire into the\nprecise meaning of continuous quantity, we find that it can only be\ndescribed as that which is divisible without limit. We can divide\na millimetre into ten, or a hundred, or a thousand, or ten thousand\nparts, and mentally at any rate we can carry on the division *ad\ninfinitum*. Any finite space, then, must be conceived as made up of an\ninfinite number of parts each infinitely small. We cannot entertain the\nsimplest geometrical notions without allowing this. The conception of a\nsquare involves the conception of a side and diagonal, which, as Euclid\nbeautifully proves in the 117th proposition of his tenth book, have no\ncommon measure,[182] meaning no finite common measure. Incommensurable\nquantities are, in fact, those which have for their only common measure\nan infinitely small quantity. It is somewhat startling to find, too,\nthat in theory incommensurable quantities will be infinitely more\nfrequent than commensurable. Let any two lines be drawn haphazard;\nit is infinitely unlikely that they will be commensurable, so that\nthe commensurable quantities, which we are supposed to deal with in\npractice, are but singular cases among an infinitely greater number of\nincommensurable cases.\n\n  [182] See De Morgan, *Study of Mathematics*, in U.K.S. Library, p.\u00a081.\n\nPractically, however, we treat all quantities as made up of the\nleast quantities which our senses, assisted by the best measuring\ninstruments, can perceive. So long as microscopes were uninvented, it\nwas sufficient to regard an inch as made up of a thousand thousandths\nof an inch; now we must treat it as composed of a million millionths.\nWe might apparently avoid all mention of infinitely small quantities,\nby never carrying our approximations beyond quantities which the\nsenses can appreciate. In geometry, as thus treated, we should never\nassert two quantities to be equal, but only to be *apparently* equal.\nLegendre really adopts this mode of treatment in the twentieth\nproposition of the first book of his Geometry; and it is practically\nadopted throughout the physical sciences, as we shall afterwards\nsee. But though our fingers, and senses, and instruments must stop\nsomewhere, there is no reason why the mind should not go on. We can\nsee that a proof which is only carried through a few steps in fact,\nmight be carried on without limit, and it is this consciousness of no\nstopping-place, which renders Euclid\u2019s proof of his 117th proposition\nso impressive. Try how we will to circumvent the matter, we cannot\nreally avoid the consideration of the infinitely small and the\ninfinitely great. The same methods of approximation which seem confined\nto the finite, mentally extend themselves to the infinite.\n\nOne result of these considerations is, that we cannot possibly adjust\ntwo quantities in absolute equality. The suspension of Mahomet\u2019s coffin\nbetween two precisely equal magnets is theoretically conceivable but\npractically impossible. The story of the *Merchant of Venice* turns\nupon the infinite improbability that an exact quantity of flesh could\nbe cut. Unstable equilibrium cannot exist in nature, for it is that\nwhich is destroyed by an infinitely small displacement. It might be\npossible to balance an egg on its end practically, because no egg has\na surface of perfect curvature. Suppose the egg shell to be perfectly\nsmooth, and the feat would become impossible.\n\n\n*The Fallacious Indications of the Senses.*\n\nI may briefly remind the reader how little we can trust to our\nunassisted senses in estimating the degree or magnitude of any\nphenomenon. The eye cannot correctly estimate the comparative\nbrightness of two luminous bodies which differ much in brilliancy;\nfor we know that the iris is constantly adjusting itself to the\nintensity of the light received, and thus admits more or less light\naccording to circumstances. The moon which shines with almost dazzling\nbrightness by night, is pale and nearly imperceptible while the eye is\nyet affected by the vastly more powerful light of day. Much has been\nrecorded concerning the comparative brightness of the zodiacal light at\ndifferent times, but it would be difficult to prove that these changes\nare not due to the varying darkness at the time, or the different\nacuteness of the observer\u2019s eye. For a like reason it is exceedingly\ndifficult to establish the existence of any change in the form or\ncomparative brightness of nebul\u00e6; the appearance of a nebula greatly\ndepends upon the keenness of sight of the observer, or the accidental\ncondition of freshness or fatigue of his eye. The same is true of\nlunar observations; and even the use of the best telescope fails to\nremove this difficulty. In judging of colours, again, we must remember\nthat light of any given colour tends to dull the sensibility of the eye\nfor light of the same colour.\n\nNor is the eye when unassisted by instruments a much better judge\nof magnitude. Our estimates of the size of minute bright points,\nsuch as the fixed stars, are completely falsified by the effects of\nirradiation. Tycho calculated from the apparent size of the star-discs,\nthat no one of the principal fixed stars could be contained within the\narea of the earth\u2019s orbit. Apart, however, from irradiation or other\ndistinct causes of error our visual estimates of sizes and shapes are\noften astonishingly incorrect. Artists almost invariably draw distant\nmountains in ludicrous disproportion to nearer objects, as a comparison\nof a sketch with a photograph at once shows. The extraordinary apparent\ndifference of size of the sun or moon, according as it is high in the\nheavens or near the horizon, should be sufficient to make us cautious\nin accepting the plainest indications of our senses, unassisted by\ninstrumental measurement. As to statements concerning the height of the\naurora and the distance of meteors, they are to be utterly distrusted.\nWhen Captain Parry says that a ray of the aurora shot suddenly\ndownwards between him and the land which was only 3,000 yards distant,\nwe must consider him subject to an illusion of sense.[183]\n\n  [183] Loomis, *On the Aurora Borealis*. Smithsonian Transactions,\n  quoting Parry\u2019s Third Voyage, p.\u00a061.\n\nIt is true that errors of observation are more often errors of judgment\nthan of sense. That which is actually seen must be so far truly seen;\nand if we correctly interpret the meaning of the phenomenon, there\nwould be no error at all. But the weakness of the bare senses as\nmeasuring instruments, arises from the fact that they import varying\nconditions of unknown amount, and we cannot make the requisite\ncorrections and allowances as in the case of a solid and invariable\ninstrument.\n\nBacon has excellently stated the insufficiency of the senses for\nestimating the magnitudes of objects, or detecting the degrees in which\nphenomena present themselves. \u201cThings escape the senses,\u201d he says,\n\u201cbecause the object is not sufficient in quantity to strike the sense:\nas all minute bodies; because the percussion of the object is too\ngreat to be endured by the senses: as the form of the sun when looking\ndirectly at it in mid-day; because the time is not proportionate to\nactuate the sense: as the motion of a bullet in the air, or the quick\ncircular motion of a firebrand, which are too fast, or the hour-hand of\na common clock, which is too slow; from the distance of the object as\nto place: as the size of the celestial bodies, and the size and nature\nof all distant bodies; from prepossession by another object: as one\npowerful smell renders other smells in the same room imperceptible;\nfrom the interruption of interposing bodies: as the internal parts of\nanimals; and because the object is unfit to make an impression upon\nthe sense: as the air or the invisible and untangible spirit which is\nincluded in every living body.\u201d\n\n\n*Complexity of Quantitative Questions.*\n\nOne remark which we may well make in entering upon quantitative\nquestions, has regard to the great variety and extent of phenomena\npresented to our notice. So long as we deal only with a simply logical\nquestion, that question is merely, Does a certain event happen? or,\nDoes a certain object exist? No sooner do we regard the event or object\nas capable of more and less, than the question branches out into many.\nWe must now ask, How much is it compared with its cause? Does it change\nwhen the amount of the cause changes? If so, does it change in the same\nor opposite direction? Is the change in simple proportion to that of\nthe cause? If not, what more complex law of connection holds true? This\nlaw determined satisfactorily in one series of circumstances may be\nvaried under new conditions, and the most complex relations of several\nquantities may ultimately be established.\n\nIn every question of physical science there is thus a series of steps\nthe first one or two of which are usually made with ease while the\nsucceeding ones demand more and more careful measurement. We cannot\nlay down any invariable series of questions which must be asked from\nnature. The exact character of the questions will vary according\nto the nature of the case, but they will usually be of an evident\nkind, and we may readily illustrate them by examples. Suppose that\nwe are investigating the solution of some salt in water. The first\nis a purely logical question: Is there solution, or is there not?\nAssuming the answer to be in the affirmative, we next inquire, Does\nthe solubility vary with the temperature, or not? In all probability\nsome variation will exist, and we must have an answer to the further\nquestion, Does the quantity dissolved increase, or does it diminish\nwith the temperature? In by far the greatest number of cases salts and\nsubstances of all kinds dissolve more freely the higher the temperature\nof the water; but there are a few salts, such as calcium sulphate,\nwhich follow the opposite rule. A considerable number of salts resemble\nsodium sulphate in becoming more soluble up to a certain temperature,\nand then varying in the opposite direction. We next require to assign\nthe amount of variation as compared with that of the temperature,\nassuming at first that the increase of solubility is proportional to\nthe increase of temperature. Common salt is an instance of very slight\nvariation, and potassium nitrate of very considerable increase with\ntemperature. Accurate observations will probably show, however, that\nthe simple law of proportionate variation is only approximately true,\nand some more complicated law involving the second, third, or higher\npowers of the temperature may ultimately be established. All these\ninvestigations have to be carried out for each salt separately, since\nno distinct principles by which we may infer from one substance to\nanother have yet been detected. There is still an indefinite field\nfor further research open; for the solubility of salts will probably\nvary with the pressure under which the medium is placed; the presence\nof other salts already dissolved may have effects yet unknown. The\nresearches already effected as regards the solvent power of water must\nbe repeated with alcohol, ether, carbon bisulphide, and other media,\nso that unless general laws can be detected, this one phenomenon of\nsolution can never be exhaustively treated. The same kind of questions\nrecur as regards the solution or absorption of gases in liquids, the\npressure as well as the temperature having then a most decided effect,\nand Professor Roscoe\u2019s researches on the subject present an excellent\nexample of the successive determination of various complicated\nlaws.[184]\n\n  [184] Watts\u2019 *Dictionary of Chemistry*, vol.\u00a0ii. p.\u00a0790.\n\nThere is hardly a branch of physical science in which similar\ncomplications are not ultimately encountered. In the case of gravity,\nindeed, we arrive at the final law, that the force is the same for\nall kinds of matter, and varies only with the distance of action.\nBut in other subjects the laws, if simple in their ultimate nature,\nare disguised and complicated in their apparent results. Thus the\neffect of heat in expanding solids, and the reverse effect of forcible\nextension or compression upon the temperature of a body, will vary\nfrom one substance to another, will vary as the temperature is already\nhigher or lower, and, will probably follow a highly complex law, which\nin some cases gives negative or exceptional results. In crystalline\nsubstances the same researches have to be repeated in each distinct\naxial direction.\n\nIn the sciences of pure observation, such as those of astronomy,\nmeteorology, and terrestrial magnetism, we meet with many interesting\nseries of quantitative determinations. The so-called fixed stars, as\nGiordano Bruno divined, are not really fixed, and may be more truly\ndescribed as vast wandering orbs, each pursuing its own path through\nspace. We must then determine separately for each star the following\nquestions:--\n\n1. Does it move?\n\n2. In what direction?\n\n3. At what velocity?\n\n4. Is this velocity variable or uniform?\n\n5. If variable, according to what law?\n\n6. Is the direction uniform?\n\n7. If not, what is the form of the apparent path?\n\n8. Does it approach or recede? 9. What is the form of the real path?\n\nThe successive answers to such questions in the case of certain binary\nstars, have afforded a proof that the motions are due to a central\nforce coinciding in law with gravity, and doubtless identical with it.\nIn other cases the motions are usually so small that it is exceedingly\ndifficult to distinguish them with certainty. And the time is yet\nfar off when any general results as regards stellar motions can be\nestablished.\n\nThe variation in the brightness of stars opens an unlimited field for\ncurious observation. There is not a star in the heavens concerning\nwhich we might not have to determine:--\n\n1. Does it vary in brightness?\n\n2. Is the brightness increasing or decreasing?\n\n3. Is the variation uniform?\n\n4. If not, according to what law does it vary?\n\nIn a majority of cases the change will probably be found to have a\nperiodic character, in which case several other questions will arise,\nsuch as--\n\n5. What is the length of the period?\n\n6. Are there minor periods?\n\n7. What is the law of variation within the period?\n\n8. Is there any change in the amount of variation?\n\n9. If so, is it a secular, *i.e.* a continually growing change, or does\nit give evidence of a greater period?\n\nAlready the periodic changes of a certain number of stars have been\ndetermined with accuracy, and the lengths of the periods vary from less\nthan three days up to intervals of time at least 250 times as great.\nPeriods within periods have also been detected.\n\nThere is, perhaps, no subject in which more complicated quantitative\nconditions have to be determined than terrestrial magnetism. Since\nthe time when the declination of the compass was first noticed, as\nsome suppose by Columbus, we have had successive discoveries from\ntime to time of the progressive change of declination from century to\ncentury; of the periodic character of this change; of the difference\nof the declination in various parts of the earth\u2019s surface; of the\nvarying laws of the change of declination; of the dip or inclination\nof the needle, and the corresponding laws of its periodic changes; the\nhorizontal and perpendicular intensities have also been the subject of\nexact measurement, and have been found to vary with place and time,\nlike the directions of the needle; daily and yearly periodic changes\nhave also been detected, and all the elements are found to be subject\nto occasional storms or abnormal perturbations, in which the eleven\nyear period, now known to be common to many planetary relations,\nis apparent. The complete solution of these motions of the compass\nneedle involves nothing less than a determination of its position\nand oscillations in every part of the world at any epoch, the like\ndetermination for another epoch, and so on, time after time, until the\nperiods of all changes are ascertained. This one subject offers to men\nof science an almost inexhaustible field for interesting quantitative\nresearch, in which we shall doubtless at some future time discover the\noperation of causes now most mysterious and unaccountable.\n\n\n*The Methods of Accurate Measurement.*\n\nIn studying the modes by which physicists have accomplished very exact\nmeasurements, we find that they are very various, but that they may\nperhaps be reduced under the following three classes:--\n\n1. The increase or decrease, in some determinate ratio, of the quantity\nto be measured, so as to bring it within the scope of our senses, and\nto equate it with the standard unit, or some determinate multiple or\nsub-multiple of this unit.\n\n2. The discovery of some natural conjunction of events which will\nenable us to compare directly the multiples of the quantity with those\nof the unit, or a quantity related in a definite ratio to that unit.\n\n3. Indirect measurement, which gives us not the quantity itself, but\nsome other quantity connected with it by known mathematical relations.\n\n\n*Conditions of Accurate Measurement.*\n\nSeveral conditions are requisite in order that a measurement may be\nmade with great accuracy, and that the results may be closely accordant\nwhen several independent measurements are made.\n\nIn the first place the magnitude must be exactly defined by sharp\nterminations, or precise marks of inconsiderable thickness. When a\nboundary is vague and graduated, like the penumbra in a lunar eclipse,\nit is impossible to say where the end really is, and different people\nwill come to different results. We may sometimes overcome this\ndifficulty to a certain extent, by observations repeated in a special\nmanner, as we shall afterwards see; but when possible, we should choose\nopportunities for measurement when precise definition is easy. The\nmoment of occultation of a star by the moon can be observed with great\naccuracy, because the star disappears with perfect suddenness; but\nthere are other astronomical conjunctions, eclipses, transits, &c.,\nwhich occupy a certain length of time in happening, and thus open the\nway to differences of opinion. It would be impossible to observe with\nprecision the movements of a body possessing no definite points of\nreference. The colours of the complete spectrum shade into each other\nso continuously that exact determinations of refractive indices would\nhave been impossible, had we not the dark lines of the solar spectrum\nas precise points for measurement, or various kinds of homogeneous\nlight, such as that of sodium, possessing a nearly uniform length of\nvibration.\n\nIn the second place, we cannot measure accurately unless we have the\nmeans of multiplying or dividing a quantity without considerable error,\nso that we may correctly equate one magnitude with the multiple or\nsubmultiple of the other. In some cases we operate upon the quantity\nto be measured, and bring it into accurate coincidence with the actual\nstandard, as when in photometry we vary the distance of our luminous\nbody, until its illuminating power at a certain point is equal to\nthat of a standard lamp. In other cases we repeat the unit until it\nequals the object, as in surveying land, or determining a weight by\nthe balance. The requisites of accuracy now are:--(1) That we can\nrepeat unit after unit of exactly equal magnitude; (2) That these\ncan be joined together so that the aggregate shall really be the sum\nof the parts. The same conditions apply to subdivision, which may be\nregarded as a multiplication of subordinate units. In order to measure\nto the thousandth of an inch, we must be able to add thousandth after\nthousandth without error in the magnitude of these spaces, or in their\nconjunction.\n\n\n*Measuring Instruments.*\n\nTo consider the mechanical construction of scientific instruments, is\nno part of my purpose in this book. I wish to point out merely the\ngeneral purpose of such instruments, and the methods adopted to carry\nout that purpose with great precision. In the first place we must\ndistinguish between the instrument which effects a comparison between\ntwo quantities, and the standard magnitude which often forms one of\nthe quantities compared. The astronomer\u2019s clock, for instance, is\nno standard of the efflux of time; it serves but to subdivide, with\napproximate accuracy, the interval of successive passages of a star\nacross the meridian, which it may effect perhaps to the tenth part of\na second, or 1/864000 part of the whole. The moving globe itself is\nthe real standard clock, and the transit instrument the finger of the\nclock, while the stars are the hour, minute, and second marks, none\nthe less accurate because they are disposed at unequal intervals. The\nphotometer is a simple instrument, by which we compare the relative\nintensity of rays of light falling upon a given spot. The galvanometer\nshows the comparative intensity of electric currents passing through a\nwire. The calorimeter gauges the quantity of heat passing from a given\nobject. But no such instruments furnish the standard unit in terms of\nwhich our results are to be expressed. In one peculiar case alone does\nthe same instrument combine the unit of measurement and the means of\ncomparison. A theodolite, mural circle, sextant, or other instrument\nfor the measurement of angular magnitudes has no need of an additional\nphysical unit; for the circle itself, or complete revolution, is\nthe natural unit to which all greater or lesser amounts of angular\nmagnitude are referred.\n\nThe result of every measurement is to make known the purely numerical\nratio existing between the magnitude to be measured, and a certain\nother magnitude, which should, when possible, be a fixed unit or\nstandard magnitude, or at least an intermediate unit of which the\nvalue can be ascertained in terms of the ultimate standard. But though\na ratio is the required result, an equation is the mode in which the\nratio is determined and expressed. In every measurement we equate\nsome multiple or submultiple of one quantity, with some multiple or\nsubmultiple of another, and equality is always the fact which we\nascertain by the senses. By the eye, the ear, or the touch, we judge\nwhether there is a discrepancy or not between two lights, two sounds,\ntwo intervals of time, two bars of metal. Often indeed we substitute\none sense for the other, as when the efflux of time is judged by\nthe marks upon a moving slip of paper, so that equal intervals of\ntime are represented by equal lengths. There is a tendency to reduce\nall comparisons to the comparison of space magnitudes, but in every\ncase one of the senses must be the ultimate judge of coincidence or\nnon-coincidence.\n\nSince the equation to be established may exist between any multiples or\nsubmultiples of the quantities compared, there naturally arise several\ndifferent modes of comparison adapted to different cases. Let *p* be\nthe magnitude to be measured, and *q* that in terms of which it is to\nbe expressed. Then we wish to find such numbers *x* and *y*, that the\nequation *p\u00a0=\u00a0(x/y)q* may be true. This equation may be presented in\nfour forms, namely:--\n\n   First Form.    Second Form.    Third Form.    Fourth Form.\n  *p\u00a0=\u00a0(x/y)q*    *p(y/x)\u00a0=\u00a0q*     *py\u00a0=\u00a0qx*     *p/x\u00a0=\u00a0q/y*\n\nEach of these modes of expressing the same equation corresponds to one\nmode of effecting a measurement.\n\nWhen the standard quantity is greater than that to be measured, we\noften adopt the first mode, and subdivide the unit until we get a\nmagnitude equal to that measured. The angles observed in surveying,\nin astronomy, or in goniometry are usually smaller than a whole\nrevolution, and the measuring circle is divided by the use of the\nscrew and microscope, until we obtain an angle undistinguishable from\nthat observed. The dimensions of minute objects are determined by\nsubdividing the inch or centimetre, the screw micrometer being the most\naccurate means of subdivision. Ordinary temperatures are estimated by\ndivision of the standard interval between the freezing and boiling\npoints of water, as marked on a thermometer tube.\n\nIn a still greater number of cases, perhaps, we multiply the standard\nunit until we get a magnitude equal to that to be measured. Ordinary\nmeasurement by a foot rule, a surveyor\u2019s chain, or the excessively\ncareful measurements of the base line of a trigonometrical survey by\nstandard bars, are sufficient instances of this procedure.\n\nIn the second case, where *p(y/x)\u00a0=\u00a0q*, we multiply or divide a\nmagnitude until we get what is equal to the unit, or to some magnitude\neasily comparable with it. As a general rule the quantities which we\ndesire to measure in physical science are too small rather than too\ngreat for easy determination, and the problem consists in multiplying\nthem without introducing error. Thus the expansion of a metallic bar\nwhen heated from 0\u00b0C to 100\u00b0 may be multiplied by a train of levers or\ncog wheels. In the common thermometer the expansion of the mercury,\nthough slight, is rendered very apparent, and easily measurable by the\nfineness of the tube, and many other cases might be quoted. There are\nsome phenomena, on the contrary, which are too great or rapid to come\nwithin the easy range of our senses, and our task is then the opposite\none of diminution. Galileo found it difficult to measure the velocity\nof a falling body, owing to the considerable velocity acquired in a\nsingle second. He adopted the elegant device, therefore, of lessening\nthe rapidity by letting the body roll down an inclined plane, which\nenables us to reduce the accelerating force in any required ratio.\nThe same purpose is effected in the well-known experiments performed\non Attwood\u2019s machine, and the measurement of gravity by the pendulum\nreally depends on the same principle applied in a far more advantageous\nmanner. Wheatstone invented a beautiful method of galvanometry for\nstrong currents, which consists in drawing off from the main current a\ncertain determinate portion, which is equated by the galvanometer to a\nstandard current. In short, he measures not the current itself but a\nknown fraction of it.\n\nIn many electrical and other experiments, we wish to measure the\nmovements of a needle or other body, which are not only very slight\nin themselves, but the manifestations of exceedingly small forces. We\ncannot even approach a delicately balanced needle without disturbing\nit. Under these circumstances the only mode of proceeding with\naccuracy, is to attach a very small mirror to the moving body, and\nemploy a ray of light reflected from the mirror as an index of its\nmovements. The ray may be considered quite incapable of affecting the\nbody, and yet by allowing the ray to pass to a sufficient distance, the\nmotions of the mirror may be increased to almost any extent. A ray of\nlight is in fact a perfectly weightless finger or index of indefinite\nlength, with the additional advantage that the angular deviation is\nby the law of reflection double that of the mirror. This method was\nintroduced by Gauss, and is now of great importance; but in Wollaston\u2019s\nreflecting goniometer a ray of light had previously been employed as an\nindex. Lavoisier and Laplace had also used a telescope in connection\nwith the pyrometer.\n\nIt is a great advantage in some instruments that they can be readily\nmade to manifest a phenomenon in a greater or less degree, by a very\nslight change in the construction. Thus either by enlarging the bulb\nor contracting the tube of the thermometer, we can make it give\nmore conspicuous indications of change of temperature. The ordinary\nbarometer, on the other hand, always gives the variations of pressure\non one scale. The torsion balance is remarkable for the extreme\ndelicacy which may be attained by increasing the length and lightness\nof the rod, and the length and thinness of the supporting thread.\nForces so minute as the attraction of gravitation between two balls, or\nthe magnetic and diamagnetic attraction of common liquids and gases,\nmay thus be made apparent, and even measured. The common chemical\nbalance, too, is capable theoretically of unlimited sensibility.\n\nThe third mode of measurement, which may be called the Method of\nRepetition, is of such great importance and interest that we must\nconsider it in a separate section. It consists in multiplying both\nmagnitudes to be compared until some multiple of the first is found\nto coincide very nearly with some multiple of the second. If the\nmultiplication can be effected to an unlimited extent, without the\nintroduction of countervailing errors, the accuracy with which the\nrequired ratio can be determined is unlimited, and we thus account for\nthe extraordinary precision with which intervals of time in astronomy\nare compared together.\n\nThe fourth mode of measurement, in which we equate submultiples of\ntwo magnitudes, is comparatively seldom employed, because it does not\nconduce to accuracy. In the photometer, perhaps, we may be said to use\nit; we compare the intensity of two sources of light, by placing them\nboth at such distances from a given surface, that the light falling\non the surface is tolerable to the eye, and equally intense from each\nsource. Since the intensity of light varies inversely as the square\nof the distance, the relative intensities of the luminous bodies are\nproportional to the squares of their distances. The equal intensity of\ntwo rays of similarly coloured light may be most accurately ascertained\nin the mode suggested by Arago, namely, by causing the rays to pass in\nopposite directions through two nearly flat lenses pressed together.\nThere is an exact equation between the intensities of the beams when\nNewton\u2019s rings disappear, the ring created by one ray being exactly the\ncomplement of that created by the other.\n\n\n*The Method of Repetition.*\n\nThe ratio of two quantities can be determined with unlimited accuracy,\nif we can multiply both the object of measurement and the standard unit\nwithout error, and then observe what multiple of the one coincides or\nnearly coincides with some multiple of the other. Although perfect\ncoincidence can never be really attained, the error thus arising\nmay be indefinitely reduced. For if the equation *py*\u00a0=\u00a0*qx* be\nuncertain to the amount *e*, so that *py*\u00a0=\u00a0*qx*\u00a0\u00b1\u00a0*e*, then we have\n*p*\u00a0=\u00a0*q(x/y)*\u00a0\u00b1\u00a0*e/y* , and as we are supposed to be able to make *y*\nas great as we like without increasing the error *e*, it follows that\nwe can make *e*\u00a0\u00f7\u00a0*y* as small as we like, and thus approximate within\nan inconsiderable quantity to the required ratio *x*\u00a0\u00f7\u00a0*y*.\n\nThis method of repetition is naturally employed whenever quantities\ncan be repeated, or repeat themselves without error of juxtaposition,\nwhich is especially the case with the motions of the earth and heavenly\nbodies. In determining the length of the sidereal day, we determine the\nratio between the earth\u2019s revolution round the sun, and its rotation on\nits own axis. We might ascertain the ratio by observing the successive\npassages of a star across the zenith, and comparing the interval by a\ngood clock with that between two passages of the sun, the difference\nbeing due to the angular movement of the earth round the sun. In such\nobservations we should have an error of a considerable part of a second\nat each observation, in addition to the irregularities of the clock.\nBut the revolutions of the earth repeat themselves day after day, and\nyear after year, without the slightest interval between the end of one\nperiod and the beginning of another. The operation of multiplication\nis perfectly performed for us by nature. If, then, we can find an\nobservation of the passage of a star across the meridian a hundred\nyears ago, that is of the interval of time between the passage of the\nsun and the star, the instrumental errors in measuring this interval by\na clock and telescope may be greater than in the present day, but will\nbe divided by about 36,524 days, and rendered excessively small. It is\nthus that astronomers have been able to ascertain the ratio of the mean\nsolar to the sidereal day to the 8th place of decimals (1\u00b700273791 to\n1), or to the hundred millionth part, probably the most accurate result\nof measurement in the whole range of science.\n\nThe antiquity of this mode of comparison is almost as great as that of\nastronomy itself. Hipparchus made the first clear application of it,\nwhen he compared his own observations with those of Aristarchus, made\n145 years previously, and thus ascertained the length of the year.\nThis calculation may in fact be regarded as the earliest attempt at\nan exact determination of the constants of nature. The method is the\nmain resource of astronomers; Tycho, for instance, detected the slow\ndiminution of the obliquity of the earth\u2019s axis, by the comparison of\nobservations at long intervals. Living astronomers use the method as\nmuch as earlier ones; but so superior in accuracy are all observations\ntaken during the last hundred years to all previous ones, that it is\noften found preferable to take a shorter interval, rather than incur\nthe risk of greater instrumental errors in the earlier observations.\n\nIt is obvious that many of the slower changes of the heavenly bodies\nmust require the lapse of large intervals of time to render their\namount perceptible. Hipparchus could not possibly have discovered the\nsmaller inequalities of the heavenly motions, because there were no\nprevious observations of sufficient age or exactness to exhibit them.\nAnd just as the observations of Hipparchus formed the starting-point\nfor subsequent comparisons, so a large part of the labour of present\nastronomers is directed to recording the present state of the heavens\nso exactly, that future generations of astronomers may detect changes,\nwhich cannot possibly become known in the present age.\n\nThe principle of repetition was very ingeniously employed in an\ninstrument first proposed by Mayer in 1767, and carried into practice\nin the Repeating Circle of Borda. The exact measurement of angles\nis indispensable, not only in astronomy but also in trigonometrical\nsurveys, and the highest skill in the mechanical execution of the\ngraduated circle and telescope will not prevent terminal errors of\nconsiderable amount. If instead of one telescope, the circle be\nprovided with two similar telescopes, these may be alternately directed\nto two distant points, say the marks in a trigonometrical survey, so\nthat the circle shall be turned through any multiple of the angle\nsubtended by those marks, before the amount of the angular revolution\nis read off upon the graduated circle. Theoretically speaking, all\nerror arising from imperfect graduation might thus be indefinitely\nreduced, being divided by the number of repetitions. In practice, the\nadvantage of the invention is not found to be very great, probably\nbecause a certain error is introduced at each observation in the\nchanging and fixing of the telescopes. It is moreover inapplicable to\nmoving objects like the heavenly bodies, so that its use is confined to\nimportant trigonometrical surveys.\n\nThe pendulum is the most perfect of all instruments, chiefly because\nit admits of almost endless repetition. Since the force of gravity\nnever ceases, one swing of the pendulum is no sooner ended than the\nother is begun, so that the juxtaposition of successive units is\nabsolutely perfect. Provided that the oscillations be equal, one\nthousand oscillations will occupy exactly one thousand times as great\nan interval of time as one oscillation. Not only is the subdivision of\ntime entirely dependent on this fact, but in the accurate measurement\nof gravity, and many other important determinations, it is of the\ngreatest service. In the deepest mine, we could not observe the\nrapidity of fall of a body for more than a quarter of a minute, and\nthe measurement of its velocity would be difficult, and subject to\nuncertain errors from resistance of air, &c. In the pendulum, we have a\nbody which can be kept rising and falling for many hours, in a medium\nentirely under our command or if desirable in a vacuum. Moreover, the\ncomparative force of gravity at different points, at the top and bottom\nof a mine for instance, can be determined with wonderful precision, by\ncomparing the oscillations of two exactly similar pendulums, with the\naid of electric clock signals.\n\nTo ascertain the comparative times of vibration of two pendulums, it\nis only requisite to swing them one in front of the other, to record\nby a clock the moment when they coincide in swing, so that one hides\nthe other, and then count the number of vibrations until they again\ncome to coincidence. If one pendulum makes *m* vibrations and the other\n*n*, we at once have our equation *pn*\u00a0=\u00a0*qm*; which gives the length\nof vibration of either pendulum in terms of the other. This method of\ncoincidence, embodying the principle of repetition in perfection, was\nemployed with wonderful skill by Sir George Airy, in his experiments on\nthe Density of the Earth at the Harton Colliery, the pendulums above\nand below being compared with clocks, which again were compared with\neach other by electric signals. So exceedingly accurate was this method\nof observation, as carried out by Sir George Airy, that he was able to\nmeasure a total difference in the vibrations at the top and bottom of\nthe shaft, amounting to only 2\u00b724 seconds in the twenty-four hours,\nwith an error of less than one hundredth part of a second, or one part\nin 8,640,000 of the whole day.[185]\n\n  [185] *Philosophical Transactions*, (1856) vol.\u00a0146, Part i. p.\u00a0297.\n\nThe principle of repetition has been elegantly applied in observing\nthe motion of waves in water. If the canal in which the experiments are\nmade be short, say twenty feet long, the waves will pass through it\nso rapidly that an observation of one length, as practised by Walker,\nwill be subject to much terminal error, even when the observer is very\nskilful. But it is a result of the undulatory theory that a wave is\nunaltered, and loses no time by complete reflection, so that it may be\nallowed to travel backwards and forwards in the same canal, and its\nmotion, say through sixty lengths, or 1200 feet, may be observed with\nthe same accuracy as in a canal 1200 feet long, with the advantage of\ngreater uniformity in the condition of the canal and water.[186] It\nis always desirable, if possible, to bring an experiment into a small\ncompass, so that it may be well under command, and yet we may often by\nrepetition enjoy at the same time the advantage of extensive trial.\n\n  [186] Airy, *On Tides and Waves*, Encyclop\u00e6dia Metropolitana, p.\u00a0345.\n  Scott Russell, *British Association Report*, 1837, p.\u00a0432.\n\nOne reason of the great accuracy of weighing with a good balance is\nthe fact, that weights placed in the same scale are naturally added\ntogether without the slightest error. There is no difficulty in the\nprecise juxtaposition of two grams, but the juxtaposition of two metre\nmeasures can only be effected with tolerable accuracy, by the use of\nmicroscopes and many precautions. Hence, the extreme trouble and cost\nattaching to the exact measurement of a base line for a survey, the\nrisk of error entering at every juxtaposition of the measuring bars,\nand indefatigable attention to all the requisite precautions being\nnecessary throughout the operation.\n\n\n*Measurements by Natural Coincidence.*\n\nIn certain cases a peculiar conjunction of circumstances enables us to\ndispense more or less with instrumental aids, and to obtain very exact\nnumerical results in the simplest manner. The mere fact, for instance,\nthat no human being has ever seen a different face of the moon from\nthat familiar to us, conclusively proves that the period of rotation\nof the moon on its own axis is equal to that of its revolution round\nthe earth. Not only have we the repetition of these movements during\n1000 or 2000 years at least, but we have observations made for us\nat very remote periods, free from instrumental error, no instrument\nbeing needed. We learn that the seventh satellite of Saturn is subject\nto a similar law, because its light undergoes a variation in each\nrevolution, owing to the existence of some dark tract of land; now\nthis failure of light always occurs while it is in the same position\nrelative to Saturn, clearly proving the equality of the axial and\nrevolutional periods, as Huygens perceived.[187] A like peculiarity in\nthe motions of Jupiter\u2019s fourth satellite was similarly detected by\nMaraldi in 1713.\n\n  [187] *Hugenii Cosmotheoros*, pp.\u00a0117, 118. Laplace\u2019s *Syst\u00e8me*,\n  translated, vol.\u00a0i. p.\u00a067.\n\nRemarkable conjunctions of the planets may sometimes allow us to\ncompare their periods of revolution, through great intervals of time,\nwith much accuracy. Laplace in explaining the long inequality in the\nmotions of Jupiter and Saturn, was assisted by a conjunction of these\nplanets, observed at Cairo, towards the close of the eleventh century.\nLaplace calculated that such a conjunction must have happened on the\n31st of October, A.D. 1087; and the discordance between the distances\nof the planets as recorded, and as assigned by theory, was less than\none-fifth part of the apparent diameter of the sun. This difference\nbeing less than the probable error of the early record, the theory was\nconfirmed as far as facts were available.[188]\n\n  [188] Grant\u2019s *History of Physical Astronomy*, p.\u00a0129.\n\nAncient astronomers often showed the highest ingenuity in turning\nany opportunities of measurement which occurred to good account.\nEratosthenes, as early as 250 B.C., happening to hear that the sun at\nSyene, in Upper Egypt, was visible at the summer solstice at the bottom\nof a well, proving that it was in the zenith, proposed to determine\nthe dimensions of the earth, by measuring the length of the shadow of\na rod at Alexandria on the same day of the year. He thus learnt in a\nrude manner the difference of latitude between Alexandria and Syene and\nfinding it to be about one fiftieth part of the whole circumference, he\nascertained the dimensions of the earth within about one sixth part\nof the truth. The use of wells in astronomical observation appears to\nhave been occasionally practised in comparatively recent times as by\nFlamsteed in 1679.[189] The Alexandrian astronomers employed the moon\nas an instrument of measurement in several sagacious modes. When the\nmoon is exactly half full, the moon, sun, and earth, are at the angles\nof a right-angled triangle. Aristarchus measured at such a time the\nmoon\u2019s elongation from the sun, which gave him the two other angles of\nthe triangle, and enabled him to judge of the comparative distances\nof the moon and sun from the earth. His result, though very rude, was\nfar more accurate than any notions previously entertained, and enabled\nhim to form some estimate of the comparative magnitudes of the bodies.\nEclipses of the moon were very useful to Hipparchus in ascertaining\nthe longitude of the stars, which are invisible when the sun is above\nthe horizon. For the moon when eclipsed must be 180\u00b0 distant from the\nsun; hence it is only requisite to measure the distance of a fixed star\nin longitude from the eclipsed moon to obtain with ease its angular\ndistance from the sun.\n\n  [189] Baily\u2019s *Account of Flamsteed*, p. lix.\n\nIn later times the eclipses of Jupiter have served to measure an angle;\nfor at the middle moment of the eclipse the satellite must be in the\nsame straight line with the planet and sun, so that we can learn from\nthe known laws of movement of the satellite the longitude of Jupiter\nas seen from the sun. If at the same time we measure the elongation or\napparent angular distance of Jupiter from the sun, as seen from the\nearth, we have all the angles of the triangle between Jupiter, the sun,\nand the earth, and can calculate the comparative magnitudes of the\nsides of the triangle by trigonometry.\n\nThe transits of Venus over the sun\u2019s face are other natural events\nwhich give most accurate measurements of the sun\u2019s parallax, or\napparent difference of position as seen from distant points of the\nearth\u2019s surface. The sun forms a kind of background on which the place\nof the planet is marked, and serves as a measuring instrument free\nfrom all the errors of construction which affect human instruments.\nThe rotation of the earth, too, by variously affecting the apparent\nvelocity of ingress or egress of Venus, as seen from different places,\ndiscloses the amount of the parallax. It has been sufficiently shown\nthat by rightly choosing the moments of observation, the planetary\nbodies may often be made to reveal their relative distance, to measure\ntheir own position, to record their own movements with a high degree\nof accuracy. With the improvement of astronomical instruments, such\nconjunctions become less necessary to the progress of the science,\nbut it will always remain advantageous to choose those moments for\nobservation when instrumental errors enter with the least effect.\n\nIn other sciences, exact quantitative laws can occasionally be obtained\nwithout instrumental measurement, as when we learn the exactly equal\nvelocity of sounds of different pitch, by observing that a peal of\nbells or a musical performance is heard harmoniously at any distance\nto which the sound penetrates; this could not be the case, as Newton\nremarked, if one sound overtook the other. One of the most important\nprinciples of the atomic theory, was proved by implication before the\nuse of the balance was introduced into chemistry. Wenzel observed,\nbefore 1777, that when two neutral substances decompose each other,\nthe resulting salts are also neutral. In mixing sodium sulphate and\nbarium nitrate, we obtain insoluble barium sulphate and neutral sodium\nnitrate. This result could not follow unless the nitric acid, requisite\nto saturate one atom of sodium, were exactly equal to that required\nby one atom of barium, so that an exchange could take place without\nleaving either acid or base in excess.\n\nAn important principle of mechanics may also be established by a simple\nacoustical observation. When a rod or tongue of metal fixed at one\nend is set in vibration, the pitch of the sound may be observed to\nbe exactly the same, whether the vibrations be small or great; hence\nthe oscillations are isochronous, or equally rapid, independently of\ntheir magnitude. On the ground of theory, it can be shown that such a\nresult only happens when the flexure is proportional to the deflecting\nforce. Thus the simple observation that the pitch of the sound of a\nharmonium, for instance, does not change with its loudness establishes\nan exact law of nature.[190]\n\n  [190] Jamin, *Cours de Physique*, vol.\u00a0i. p.\u00a0152.\n\nA closely similar instance is found in the proof that the intensity\nof light or heat rays varies inversely as the square of the distance\nincreases. For the apparent magnitude certainly varies according to\nthis law; hence, if the intensity of light varied according to any\nother law, the brightness of an object would be different at different\ndistances, which is not observed to be the case. Melloni applied the\nsame kind of reasoning, in a somewhat different form, to the radiation\nof heat-rays.\n\n\n*Modes of Indirect Measurement.*\n\nSome of the most conspicuously beautiful experiments in the whole range\nof science, have been devised for the purpose of indirectly measuring\nquantities, which in their extreme greatness or smallness surpass the\npowers of sense. All that we need to do, is to discover some other\nconveniently measurable phenomenon, which is related in a known ratio\nor according to a known law, however complicated, with that to be\nmeasured. Having once obtained experimental data, there is no further\ndifficulty beyond that of arithmetic or algebraic calculation.\n\nGold is reduced by the gold-beater to leaves so thin, that the most\npowerful microscope would not detect any measurable thickness. If we\nlaid several hundred leaves upon each other to multiply the thickness,\nwe should still have no more than 1/100th of an inch at the most to\nmeasure, and the errors arising in the superposition and measurement\nwould be considerable. But we can readily obtain an exact result\nthrough the connected amount of weight. Faraday weighed 2000 leaves of\ngold, each 3-3/8 inch square, and found them equal to 384 grains. From\nthe known specific gravity of gold it was easy to calculate that the\naverage thickness of the leaves was 1/282,000 of an inch.[191]\n\n  [191] Faraday, *Chemical Researches*, p.\u00a0393.\n\nWe must ascribe to Newton the honour of leading the way in methods of\nminute measurement. He did not call waves of light by their right name,\nand did not understand their nature; yet he measured their length,\nthough it did not exceed the 2,000,000th part of a metre or the one\nfifty-thousandth part of an inch. He pressed together two lenses of\nlarge but known radii. It was easy to calculate the interval between\nthe lenses at any point, by measuring the distance from the central\npoint of contact. Now, with homogeneous rays the successive rings of\nlight and darkness mark the points at which the interval between the\nlenses is equal to one half, or any multiple of half a vibration of\nthe light, so that the length of the vibration became known. In a\nsimilar manner many phenomena of interference of rays of light admit\nof the measurement of the wave lengths. Fringes of interference arise\nfrom rays of light which cross each other at a small angle, and an\nexcessively minute difference in the lengths of the waves makes a very\nperceptible difference in the position of the point at which two rays\nwill interfere and produce darkness.\n\nFizeau has recently employed Newton\u2019s rings to measure small amounts of\nmotion. By merely counting the number of rings of sodium monochromatic\nlight passing a certain point where two glass plates are in close\nproximity, he is able to ascertain with the greatest accuracy and ease\nthe change of distance between these glasses, produced, for instance,\nby the expansion of a metallic bar, connected with one of the glass\nplates.[192]\n\n  [192] *Proceedings of the Royal Society*, 30th November, 1866.\n\nNothing excites more admiration than the mode in which scientific\nobservers can occasionally measure quantities, which seem beyond\nthe bounds of human observation. We know the *average* depth of the\nPacific Ocean to be 14,190 feet, not by actual sounding, which would\nbe impracticable in sufficient detail, but by noticing the rate of\ntransmission of earthquake waves from the South American to the\nopposite coasts, the rate of movement being connected by theory with\nthe depth of the water.[193] In the same way the average depth of\nthe Atlantic Ocean is inferred to be no less than 22,157 feet, from\nthe velocity of the ordinary tidal waves. A tidal wave again gives\nbeautiful evidence of an effect of the law of gravity, which we could\nnever in any other way detect. Newton estimated that the moon\u2019s force\nin moving the ocean is only one part in 2,871,400 of the whole force of\ngravity, so that even the pendulum, used with the utmost skill, would\nfail to render it apparent. Yet, the immense extent of the ocean allows\nthe accumulation of the effect into a very palpable amount; and from\nthe comparative heights of the lunar and solar tides, Newton roughly\nestimated the comparative forces of the moon\u2019s and sun\u2019s gravity at the\nearth.[194]\n\n  [193] Herschel, *Physical Geography*, \u00a7 40.\n\n  [194] *Principia*, bk. iii. Prop. 37, *Corollaries*, 2 and 3. Motte\u2019s\n  translation, vol.\u00a0ii. p.\u00a0310.\n\nA few years ago it might have seemed impossible that we should ever\nmeasure the velocity with which a star approaches or recedes from the\nearth, since the apparent position of the star is thereby unaltered.\nBut the spectroscope now enables us to detect and even measure such\nmotions with considerable accuracy, by the alteration which it\ncauses in the apparent rapidity of vibration, and consequently in\nthe refrangibility of rays of light of definite colour. And while\nour estimates of the lateral movements of stars depend upon our very\nuncertain knowledge of their distances, the spectroscope gives the\nmotions of approach and recess irrespective of other motions excepting\nthat of the earth. It gives in short the motions of approach and recess\nof the stars relatively to the earth.[195]\n\n  [195] Roscoe\u2019s *Spectrum Analysis*, 1st ed. p.\u00a0296.\n\nThe rapidity of vibration for each musical tone, having been accurately\ndetermined by comparison with the Syren (p.\u00a010), we can use sounds as\nindirect indications of rapid vibrations. It is now known that the\ncontraction of a muscle arises from the periodical contractions of each\nseparate fibre, and from a faint sound or susurrus which accompanies\nthe action of a muscle, it is inferred that each contraction lasts for\nabout one 300th part of a second. Minute quantities of radiant heat are\nnow always measured indirectly by the electricity which they produce\nwhen falling upon a thermopile. The extreme delicacy of the method\nseems to be due to the power of multiplication at several points in the\napparatus. The number of elements or junctions of different metals in\nthe thermopile can be increased so that the tension of the electric\ncurrent derived from the same intensity of radiation is multiplied;\nthe effect of the current upon the magnetic needle can be multiplied\nwithin certain bounds, by passing the current many times round it in\na coil; the excursions of the needle can be increased by rendering it\nastatic and increasing the delicacy of its suspension; lastly, the\nangular divergence can be observed, with any required accuracy, by the\nuse of an attached mirror and distant scale viewed through a telescope\n(p.\u00a0287). Such is the delicacy of this method of measuring heat, that\nDr. Joule succeeded in making a thermopile which would indicate a\ndifference of 0\u00b0\u00b7000114 Cent.[196]\n\n  [196] *Philosophical Transactions* (1859), vol.\u00a0cxlix. p.\u00a094.\n\nA striking case of indirect measurement is furnished by the revolving\nmirror of Wheatstone and Foucault, whereby a minute interval of time\nis estimated in the form of an angular deviation. Wheatstone viewed an\nelectric spark in a mirror rotating so rapidly, that if the duration\nof the spark had been more than one 72,000th part of a second, the\npoint of light would have appeared elongated to an angular extent\nof one-half degree. In the spark, as drawn directly from a Leyden\njar, no elongation was apparent, so that the duration of the spark\nwas immeasurably small; but when the discharge took place through\na bad conductor, the elongation of the spark denoted a sensible\nduration.[197] In the hands of Foucault the rotating mirror gave a\nmeasure of the time occupied by light in passing through a few metres\nof space.\n\n  [197] Watts\u2019 *Dictionary of Chemistry*, vol.\u00a0ii. p.\u00a0393.\n\n\n*Comparative Use of Measuring Instruments.*\n\nIn almost every case a measuring instrument serves, and should serve\nonly as a means of comparison between two or more magnitudes. As a\ngeneral rule, we should not attempt to make the divisions of the\nmeasuring scale exact multiples or submultiples of the unit, but,\nregarding them as arbitrary marks, should determine their values by\ncomparison with the standard itself. The perpendicular wires in the\nfield of a transit telescope, are fixed at nearly equal but arbitrary\ndistances, and those distances are afterwards determined, as first\nsuggested by Malvasia, by watching the passage of star after star\nacross them, and noting the intervals of time by the clock. Owing\nto the perfectly regular motion of the earth, these time intervals\ngive exact determinations of the angular intervals. In the same way,\nthe angular value of each turn of the screw micrometer attached to a\ntelescope, can be easily and accurately ascertained.\n\nWhen a thermopile is used to observe radiant heat, it would be almost\nimpossible to calculate on *\u00e0 priori* grounds what is the value of\neach division of the galvanometer circle, and still more difficult\nto construct a galvanometer, so that each division should have a\ngiven value. But this is quite unnecessary, because by placing the\nthermopile before a body of known dimensions, at a known distance, with\na known temperature and radiating power, we measure a known amount\nof radiant heat, and inversely measure the value of the indications\nof the thermopile. In a similar way Dr. Joule ascertained the actual\ntemperature produced by the compression of bars of metal. For having\ninserted a small thermopile composed of a single junction of copper and\niron wire, and noted the deflections of the galvanometer, he had only\nto dip the bars into water of different temperatures, until he produced\na like deflection, in order to ascertain the temperature developed by\npressure.[198]\n\n  [198] *Philosophical Transactions* (1859), vol.\u00a0cxlix. p.\u00a0119, &c.\n\nIn some cases we are obliged to accept a very carefully constructed\ninstrument as a standard, as in the case of a standard barometer or\nthermometer. But it is then best to treat all inferior instruments\ncomparatively only, and determine the values of their scales by\ncomparison with the assumed standard.\n\n\n*Systematic Performance of Measurements.*\n\nWhen a large number of accurate measurements have to be effected, it\nis usually desirable to make a certain number of determinations with\nscrupulous care, and afterwards use them as points of reference for the\nremaining determinations. In the trigonometrical survey of a country,\nthe principal triangulation fixes the relative positions and distances\nof a few points with rigid accuracy. A minor triangulation refers every\nprominent hill or village to one of the principal points, and then the\ndetails are filled in by reference to the secondary points. The survey\nof the heavens is effected in a like manner. The ancient astronomers\ncompared the right ascensions of a few principal stars with the moon,\nand thus ascertained their positions with regard to the sun; the minor\nstars were afterwards referred to the principal stars. Tycho followed\nthe same method, except that he used the more slowly moving planet\nVenus instead of the moon. Flamsteed was in the habit of using about\nseven stars, favourably situated at points all round the heavens. In\nhis early observations the distances of the other stars from these\nstandard points were determined by the use of the quadrant.[199] Even\nsince the introduction of the transit telescope and the mural circle,\ntables of standard stars are formed at Greenwich, the positions being\ndetermined with all possible accuracy, so that they can be employed for\npurposes of reference by astronomers.\n\n  [199] Baily\u2019s *Account of Flamsteed*, pp.\u00a0378\u2013380.\n\nIn ascertaining the specific gravities of substances, all gases are\nreferred to atmospheric air at a given temperature and pressure;\nall liquids and solids are referred to water. We require to compare\nthe densities of water and air with great care, and the comparative\ndensities of any two substances whatever can then be ascertained.\n\nIn comparing a very great with a very small magnitude, it is\nusually desirable to break up the process into several steps, using\nintermediate terms of comparison. We should never think of measuring\nthe distance from London to Edinburgh by laying down measuring rods,\nthroughout the whole length. A base of several miles is selected on\nlevel ground, and compared on the one hand with the standard yard,\nand on the other with the distance of London and Edinburgh, or any\nother two points, by trigonometrical survey. Again, it would be\nexceedingly difficult to compare the light of a star with that of the\nsun, which would be about thirty thousand million times greater; but\nHerschel[200] effected the comparison by using the full moon as an\nintermediate unit. Wollaston ascertained that the sun gave 801,072\ntimes as much light as the full moon, and Herschel determined that the\nlight of the latter exceeded that of \u03b1 Centauri 27,408 times, so that\nwe find the ratio between the light of the sun and star to be that of\nabout 22,000,000,000 to 1.\n\n  [200] Herschel\u2019s *Astronomy*, \u00a7 817, 4th. ed. p.\u00a0553.\n\n\n*The Pendulum.*\n\nBy far the most perfect and beautiful of all instruments of measurement\nis the pendulum. Consisting merely of a heavy body suspended freely\nat an invariable distance from a fixed point, it is most simple in\nconstruction; yet all the highest problems of physical measurement\ndepend upon its careful use. Its excessive value arises from two\ncircumstances.\n\n(1) The method of repetition is eminently applicable to it, as already\ndescribed (p.\u00a0290).\n\n(2) Unlike other instruments, it connects together three different\nquantities, those of space, time, and force.\n\nIn most works on natural philosophy it is shown, that when the\noscillations of the pendulum are infinitely small, the square of the\ntime occupied by an oscillation is directly proportional to the length\nof the pendulum, and indirectly proportional to the force affecting it,\nof whatever kind. The whole theory of the pendulum is contained in the\nformula, first given by Huygens in his *Horologium Oscillatorium*.\n\n  Time of oscillation = 3\u00b714159\u00a0\u00d7 \u221a(length of pendulum/force).\n\nThe quantity 3\u00b714159 is the constant ratio of the circumference and\nradius of a circle, and is of course known with accuracy. Hence, any\ntwo of the three quantities concerned being given, the third may be\nfound; or any two being maintained invariable, the third will be\ninvariable. Thus a pendulum of invariable length suspended at the\nsame place, where the force of gravity may be considered constant,\nfurnishes a measure of time. The same invariable pendulum being made\nto vibrate at different points of the earth\u2019s surface, and the times\nof vibration being astronomically determined, the force of gravity\nbecomes accurately known. Finally, with a known force of gravity, and\ntime of vibration ascertained by reference to the stars, the length is\ndeterminate.\n\nAll astronomical observations depend upon the first manner of using the\npendulum, namely, in the astronomical clock. In the second employment\nit has been almost equally indispensable. The primary principle that\ngravity is equal in all matter was proved by Newton\u2019s and Gauss\u2019\npendulum experiments. The torsion pendulum of Michell, Cavendish, and\nBaily, depending upon exactly the same principles as the ordinary\npendulum, gave the density of the earth, one of the foremost natural\nconstants. Kater and Sabine, by pendulum observations in different\nparts of the earth, ascertained the variation of gravity, whence comes\na determination of the earth\u2019s ellipticity. The laws of electric\nand magnetic attraction have also been determined by the method\nof vibrations, which is in constant use in the measurement of the\nhorizontal force of terrestrial magnetism.\n\nWe must not confuse with the ordinary use of the pendulum its\napplication by Newton, to show the absence of internal friction against\nspace,[201] or to ascertain the laws of motion and elasticity.[202] In\nthese cases the extent of vibration is the quantity measured, and the\nprinciples of the instrument are different.\n\n  [201] *Principia*, bk. ii. Sect. 6. Prop. 31. Motte\u2019s Translation,\n  vol.\u00a0ii. p. 107.\n\n  [202] Ibid. bk. i. Law iii. Corollary 6. Motte\u2019s Translation, vol.\u00a0i.\n  p.\u00a033.\n\n\n*Attainable Accuracy of Measurement.*\n\nIt is a matter of some interest to compare the degrees of accuracy\nwhich can be attained in the measurement of different kinds of\nmagnitude. Few measurements of any kind are exact to more than six\nsignificant figures,[203] but it is seldom that such accuracy can be\nhoped for. Time is the magnitude which seems to be capable of the most\nexact estimation, owing to the properties of the pendulum, and the\nprinciple of repetition described in previous sections. As regards\nshort intervals of time, it has already been stated that Sir George\nAiry was able to estimate one part in 8,640,000, an exactness, as he\ntruly remarks, \u201calmost beyond conception.\u201d[204] The ratio between the\nmean solar and the sidereal day is known to be about one part in one\nhundred millions, or to the eighth place of decimals, (p.\u00a0289).\n\n  [203] Thomson and Tait\u2019s *Natural Philosophy*, vol.\u00a0i. p.\u00a0333.\n\n  [204] *Philosophical Transactions*, (1856), vol.\u00a0cxlvi. pp.\u00a0330, 331.\n\nDeterminations of weight seem to come next in exactness, owing to the\nfact that repetition without error is applicable to them. An ordinary\ngood balance should show about one part in 500,000 of the load. The\nfinest balance employed by M. Stas, turned with one part in 825,000 of\nthe load.[205] But balances have certainly been constructed to show\none part in a million,[206] and Ramsden is said to have constructed a\nbalance for the Royal Society, to indicate one part in seven millions,\nthough this is hardly credible. Professor Clerk Maxwell takes it for\ngranted that one part in five millions can be detected, but we ought to\ndiscriminate between what a balance can do when first constructed, and\nwhen in continuous use.\n\n  [205] *First Annual Report of the Mint*, p.\u00a0106.\n\n  [206] Jevons, in Watts\u2019 *Dictionary of Chemistry*, vol.\u00a0i. p.\u00a0483.\n\nDeterminations of length, unless performed with extraordinary care,\nare open to much error in the junction of the measuring bars. Even\nin measuring the base line of a trigonometrical survey, the accuracy\ngenerally attained is only that of about one part in 60,000, or an\ninch in the mile; but it is said that in four measurements of a base\nline carried out very recently at Cape Comorin, the greatest error was\n0\u00b7077 inch in 1\u00b768 mile, or one part in 1,382,400, an almost incredible\ndegree of accuracy. Sir J. Whitworth has shown that touch is even a\nmore delicate mode of measuring lengths than sight, and by means of\na splendidly executed screw, and a small cube of iron placed between\ntwo flat-ended iron bars, so as to be suspended when touching them, he\ncan detect a change of dimension in a bar, amounting to no more than\none-millionth of an inch.[207]\n\n  [207] British Association, Glasgow, 1856. *Address of the President\n  of the Mechanical Section*.\n\n\n\n\nCHAPTER XIV.\n\nUNITS AND STANDARDS OF MEASUREMENT.\n\n\nAs we have seen, instruments of measurement are only means of\ncomparison between one magnitude and another, and as a general rule we\nmust assume some one arbitrary magnitude, in terms of which all results\nof measurement are to be expressed. Mere ratios between any series of\nobjects will never tell us their absolute magnitudes; we must have at\nleast one ratio for each, and we must have one absolute magnitude.\nThe number of ratios *n* are expressible in *n* equations, which will\ncontain at least *n*\u00a0+\u00a01 quantities, so that if we employ them to make\nknown *n* magnitudes, we must have one magnitude known. Hence, whether\nwe are measuring time, space, density, mass, weight, energy, or any\nother physical quantity, we must refer to some concrete standard, some\nactual object, which if once lost and irrecoverable, all our measures\nlose their absolute meaning. This concrete standard is in all cases\narbitrary in point of theory, and its selection a question of practical\nconvenience.\n\nThere are two kinds of magnitude, indeed, which do not need to be\nexpressed in terms of arbitrary concrete units, since they pre-suppose\nthe existence of natural standard units. One case is that of abstract\nnumber itself, which needs no special unit, because any object which\nexists or is thought of as separate from other objects (p.\u00a0157)\nfurnishes us with a unit, and is the only standard required.\n\nAngular magnitude is the second case in which we have a natural unit\nof reference, namely the whole revolution or *perigon*, as it has\nbeen called by Mr. Sandeman.[208] It is a necessary result of the\nuniform properties of space, that all complete revolutions are equal\nto each other, so that we need not select any one revolution, but can\nalways refer anew to space itself. Whether we take the whole perigon,\nits half, or its quarter, is really immaterial; Euclid took the right\nangle, because the Greek geometers had never generalised their notions\nof angular magnitude sufficiently to treat angles of all magnitudes,\nor of unlimited *quantity of revolution*. Euclid defines a right angle\nas half that made by a line with its own continuation, which is of\ncourse equal to half a revolution, but which was not treated as an\nangle by him. In mathematical analysis a different fraction of the\nperigon is taken, namely, such a fraction that the arc or portion of\nthe circumference included within it is equal to the radius of the\ncircle. In this point of view angular magnitude is an abstract ratio,\nnamely, the ratio between the length of arc subtended and the length\nof the radius. The geometrical unit is then necessarily the angle\ncorresponding to the ratio unity. This angle is equal to about 57\u00b0,\n17\u2032, 44\u2033\u00b78, or decimally 57\u00b0\u00b7295779513...\u00a0.[209] It was called by De\nMorgan the *arcual unit*, but a more convenient name for common use\nwould be *radian*, as suggested by Professor Everett. Though this\nstandard angle is naturally employed in mathematical analysis, and any\nother unit would introduce great complexity, we must not look upon it\nas a distinct unit, since its amount is connected with that of the half\nperigon, by the natural constant 3\u00b714159... usually denoted by the\nletter \u03c0.\n\n  [208] *Pelicotetics, or the Science of Quantity; an Elementary\n  Treatise on Algebra, and its groundwork Arithmetic.* By Archibald\n  Sandeman, M.\u00a0A. Cambridge (Deighton, Bell, and Co.), 1868, p.\u00a0304.\n\n  [209] De Morgan\u2019s *Trigonometry and Double Algebra*, p.\u00a05.\n\nWhen we pass to other species of quantity, the choice of unit is found\nto be entirely arbitrary. There is absolutely no mode of defining a\nlength, but by selecting some physical object exhibiting that length\nbetween certain obvious points--as, for instance, the extremities of a\nbar, or marks made upon its surface.\n\n\n*Standard Unit of Time.*\n\nTime is the great independent variable of all change--that which itself\nflows on uninterruptedly, and brings the variety which we call motion\nand life. When we reflect upon its intimate nature, Time, like every\nother element of existence, proves to be an inscrutable mystery. We\ncan only say with St. Augustin, to one who asks us what is time, \u201cI\nknow when you do not ask me.\u201d The mind of man will ask what can never\nbe answered, but one result of a true and rigorous logical philosophy\nmust be to convince us that scientific explanation can only take place\nbetween phenomena which have something in common, and that when we get\ndown to primary notions, like those of time and space, the mind must\nmeet a point of mystery beyond which it cannot penetrate. A definition\nof time must not be looked for; if we say with Hobbes,[210] that it is\n\u201cthe phantasm of before and after in motion,\u201d or with Aristotle that it\nis \u201cthe number of motion according to former and latter,\u201d we obviously\ngain nothing, because the notion of time is involved in the expressions\n*before and after*, *former and latter*. Time is undoubtedly one of\nthose primary notions which can only be defined physically, or by\nobservation of phenomena which proceed in time.\n\n  [210] *English Works of Thos. Hobbes*, Edit. by Molesworth, vol.\u00a0i.\n  p.\u00a095.\n\nIf we have not advanced a step beyond Augustin\u2019s acute reflections on\nthis subject,[211] it is curious to observe the wonderful advances\nwhich have been made in the practical measurement of its efflux. In\nearlier centuries the rude sun-dial or the rising of a conspicuous star\ngave points of reference, while the flow of water from the clepsydra,\nthe burning of a candle, or, in the monastic ages, even the continuous\nchanting of psalms, were the means of roughly subdividing periods, and\nmarking the hours of the day and night.[212] The sun and stars still\nfurnish the standard of time, but means of accurate subdivision have\nbecome requisite, and this has been furnished by the pendulum and the\nchronograph. By the pendulum we can accurately divide the day into\nseconds of time. By the chronograph we can subdivide the second into\na hundred, a thousand, or even a million parts. Wheatstone measured\nthe duration of an electric spark, and found it to be no more than one\n115,200th part of a second, while more recently Captain Noble has been\nable to appreciate intervals of time not exceeding the millionth part\nof a second.\n\n  [211] *Confessions*, bk. xi. chapters 20\u201328.\n\n  [212] Sir G.\u00a0C. Lewis gives many curious particulars concerning the\n  measurement of time in his *Astronomy of the Ancients*, pp.\u00a0241, &c.\n\nWhen we come to inquire precisely what phenomenon it is that we thus\nso minutely measure, we meet insurmountable difficulties. Newton\ndistinguished time according as it was *absolute* or *apparent* time,\nin the following words:--\u201cAbsolute, true, and mathematical time,\nof itself and from its own nature, flows equably without regard to\nanything external, and by another name is called *duration*; relative,\napparent and common time, is some sensible and external measure of\nduration by the means of motion.\u201d[213] Though we are perhaps obliged to\nassume the existence of a uniformly increasing quantity which we call\ntime, yet we cannot feel or know abstract and absolute time. Duration\nmust be made manifest to us by the recurrence of some phenomenon. The\nsuccession of our own thoughts is no doubt the first and simplest\nmeasure of time, but a very rude one, because in some persons and\ncircumstances the thoughts evidently flow with much greater rapidity\nthan in other persons and circumstances. In the absence of all\nother phenomena, the interval between one thought and another would\nnecessarily become the unit of time, but the most cursory observations\nshow that there are changes in the outward world much better fitted by\ntheir constancy to measure time than the change of thoughts within us.\n\n  [213] *Principia*, bk. i. *Scholium to Definitions*. Translated by\n  Motte, vol.\u00a0i. p.\u00a09. See also p.\u00a011.\n\nThe earth, as I have already said, is the real clock of the astronomer,\nand is practically assumed as invariable in its movements. But on\nwhat ground is it so assumed? According to the first law of motion,\nevery body perseveres in its state of rest or of uniform motion in\na right line, unless it is compelled to change that state by forces\nimpressed thereon. Rotatory motion is subject to a like condition,\nnamely, that it perseveres uniformly unless disturbed by extrinsic\nforces. Now uniform motion means motion through equal spaces in equal\ntimes, so that if we have a body entirely free from all resistance\nor perturbation, and can measure equal spaces of its path, we have a\nperfect measure of time. But let it be remembered that this law has\nnever been absolutely proved by experience; for we cannot point to any\nbody, and say that it is wholly unresisted or undisturbed; and even if\nwe had such a body, we should need some independent standard of time\nto ascertain whether its motion was really uniform. As it is in moving\nbodies that we find the best standard of time, we cannot use them to\nprove the uniformity of their own movements, which would amount to a\n*petitio principii*. Our experience comes to this, that when we examine\nand compare the movements of bodies which seem to us nearly free from\ndisturbance, we find them giving nearly harmonious measures of time.\nIf any one body which seems to us to move uniformly is not doing so,\nbut is subject to fits and starts unknown to us, because we have no\nabsolute standard of time, then all other bodies must be subject to the\nsame arbitrary fits and starts, otherwise there would be discrepancy\ndisclosing the irregularities. Just as in comparing together a number\nof chronometers, we should soon detect bad ones by their going\nirregularly, as compared with the others, so in nature we detect\ndisturbed movement by its discrepancy from that of other bodies which\nwe believe to be undisturbed, and which agree nearly among themselves.\nBut inasmuch as the measure of motion involves time, and the measure\nof time involves motion, there must be ultimately an assumption. We\nmay define equal times, as times during which a moving body under the\ninfluence of no force describes equal spaces;[214] but all we can\nsay in support of this definition is, that it leads us into no known\ndifficulties, and that to the best of our experience one freely moving\nbody gives the same results as any other.\n\n  [214] Rankine, *Philosophical Magazine*, Feb. 1867, vol.\u00a0xxxiii.\n  p.\u00a091.\n\nWhen we inquire where the freely moving body is, no perfectly\nsatisfactory answer can be given. Practically the rotating globe is\nsufficiently accurate, and Thomson and Tait say: \u201cEqual times are\ntimes during which the earth turns through equal angles.\u201d[215] No long\ntime has passed since astronomers thought it impossible to detect any\ninequality in its movement. Poisson was supposed to have proved that a\nchange in the length of the sidereal day amounting to one ten-millionth\npart in 2,500 years was incompatible with an ancient eclipse recorded\nby the Chald\u00e6ans, and similar calculations were made by Laplace. But\nit is now known that these calculations were somewhat in error, and\nthat the dissipation of energy arising out of the friction of tidal\nwaves, and the radiation of the heat into space, has slightly decreased\nthe rapidity of the earth\u2019s rotatory motion. The sidereal day is now\nlonger by one part in 2,700,000, than it was in 720 B.C. Even before\nthis discovery, it was known that invariability of rotation depended\nupon the perfect maintenance of the earth\u2019s internal heat, which is\nrequisite in order that the earth\u2019s dimensions shall be unaltered. Now\nthe earth being superior in temperature to empty space, must cool more\nor less rapidly, so that it cannot furnish an absolute measure of time.\nSimilar objections could be raised to all other rotating bodies within\nour cognisance.\n\n  [215] *Treatise on Natural Philosophy*, vol.\u00a0i. p.\u00a0179.\n\nThe moon\u2019s motion round the earth, and the earth\u2019s motion round the\nsun, form the next best measure of time. They are subject, indeed,\nto disturbance from other planets, but it is believed that these\nperturbations must in the course of time run through their rhythmical\ncourses, leaving the mean distances unaffected, and consequently, by\nthe third Law of Kepler, the periodic times unchanged. But there is\nmore reason than not to believe that the earth encounters a slight\nresistance in passing through space, like that which is so apparent\nin Encke\u2019s comet. There may also be dissipation of energy in the\nelectrical relations of the earth to the sun, possibly identical with\nthat which is manifested in the retardation of comets.[216] It is\nprobably an untrue assumption then, that the earth\u2019s orbit remains\nquite invariable. It is just possible that some other body may be found\nin the course of time to furnish a better standard of time than the\nearth in its annual motion. The greatly superior mass of Jupiter and\nits satellites, and their greater distance from the sun, may render the\nelectrical dissipation of energy less considerable than in the case of\nthe earth. But the choice of the best measure will always be an open\none, and whatever moving body we choose may ultimately be shown to be\nsubject to disturbing forces.\n\n  [216] *Proceedings of the Manchester Philosophical Society*, 28th\n  Nov. 1871, vol.\u00a0xi. p.\u00a033.\n\nThe pendulum, although so admirable an instrument for subdivision of\ntime, fails as a standard; for though the same pendulum affected by the\nsame force of gravity performs equal vibrations in equal times, yet\nthe slightest change in the form or weight of the pendulum, the least\ncorrosion of any part, or the most minute displacement of the point of\nsuspension, falsifies the results, and there enter many other difficult\nquestions of temperature, friction, resistance, length of vibration, &c.\n\nThomson and Tait are of opinion[217] that the ultimate standard of\nchronometry must be founded on the physical properties of some body\nof more constant character than the earth; for instance, a carefully\narranged metallic spring, hermetically sealed in an exhausted glass\nvessel. But it is hard to see how we can be sure that the dimensions\nand elasticity of a piece of wrought metal will remain perfectly\nunchanged for the few millions of years contemplated by them. A nearly\nperfect gas, like hydrogen, is perhaps the only kind of substance in\nthe unchanged elasticity of which we could have confidence. Moreover,\nit is difficult to perceive how the undulations of such a spring could\nbe observed with the requisite accuracy. More recently Professor Clerk\nMaxwell has made the novel suggestion, discussed in a subsequent\nsection, that undulations of light *in vacuo* would form the most\nuniversal standard of reference, both as regards time and space.\nAccording to this system the unit of time would be the time occupied\nby one vibration of the particular kind of light whose wave length is\ntaken as the unit of length.\n\n  [217] *The Elements of Natural Philosophy*, part i. p.\u00a0119.\n\n\n*The Unit of Space and the Bar Standard.*\n\nNext in importance after the measurement of time is that of space.\nTime comes first in theory, because phenomena, our internal thoughts\nfor instance, may change in time without regard to space. As to the\nphenomena of outward nature, they tend more and more to resolve\nthemselves into motions of molecules, and motion cannot be conceived or\nmeasured without reference both to time and space.\n\nTurning now to space measurement, we find it almost equally difficult\nto fix and define once and for ever, a unit magnitude. There are\nthree different modes in which it has been proposed to attempt the\nperpetuation of a standard length.\n\n(1) By constructing an actual specimen of the standard yard or metre,\nin the form of a bar.\n\n(2) By assuming the globe itself to be the ultimate standard of\nmagnitude, the practical unit being a submultiple of some dimension of\nthe globe.\n\n(3) By adopting the length of the simple seconds pendulum, as a\nstandard of reference.\n\nAt first sight it might seem that there was no great difficulty in this\nmatter, and that any one of these methods might serve well enough;\nbut the more minutely we inquire into the details, the more hopeless\nappears to be the attempt to establish an invariable standard. We must\nin the first place point out a principle not of an obvious character,\nnamely, that *the standard length must be defined by one single\nobject*.[218] To make two bars of exactly the same length, or even two\nbars bearing a perfectly defined ratio to each other, is beyond the\npower of human art. If two copies of the standard metre be made and\ndeclared equally correct, future investigators will certainly discover\nsome discrepancy between them, proving of course that they cannot both\nbe the standard, and giving cause for dispute as to what magnitude\nshould then be taken as correct.\n\n  [218] See Harris\u2019 *Essay upon Money and Coins*, part. ii. [1758]\n  p.\u00a0127.\n\nIf one invariable bar could be constructed and maintained as the\nabsolute standard, no such inconvenience could arise. Each successive\ngeneration as it acquired higher powers of measurement, would detect\nerrors in the copies of the standard, but the standard itself would be\nunimpeached, and would, as it were, become by degrees more and more\naccurately known. Unfortunately to construct and preserve a metre or\nyard is also a task which is either impossible, or what comes nearly\nto the same thing, cannot be shown to be possible. Passing over the\npractical difficulty of defining the ends of the standard length\nwith complete accuracy, whether by dots or lines on the surface, or\nby the terminal points of the bar, we have no means of proving that\nsubstances remain of invariable dimensions. Just as we cannot tell\nwhether the rotation of the earth is uniform, except by comparing it\nwith other moving bodies, believed to be more uniform in motion, so\nwe cannot detect the change of length in a bar, except by comparing\nit with some other bar supposed to be invariable. But how are we to\nknow which is the invariable bar? It is certain that many rigid and\napparently invariable substances do change in dimensions. The bulb of\na thermometer certainly contracts by age, besides undergoing rapid\nchanges of dimensions when warmed or cooled through 100\u00b0\u00a0Cent. Can\nwe be sure that even the most solid metallic bars do not slightly\ncontract by age, or undergo variations in their structure by change\nof temperature. Fizeau was induced to try whether a quartz crystal,\nsubjected to several hundred alternations of temperature, would be\nmodified in its physical properties, and he was unable to detect any\nchange in the coefficient of expansion.[219] It does not follow,\nhowever, that, because no apparent change was discovered in a quartz\ncrystal, newly-constructed bars of metal would undergo no change.\n\n  [219] *Philosophical Magazine*, (1868), 4th Series, vol.\u00a0xxxvi. p.\u00a032.\n\nThe best principle, as it seems to me, upon which the perpetuation of\na standard of length can be rested, is that, if a variation of length\noccurs, it will in all probability be of different amount in different\nsubstances. If then a great number of standard metres were constructed\nof all kinds of different metals and alloys; hard rocks, such as\ngranite, serpentine, slate, quartz, limestone; artificial substances,\nsuch as porcelain, glass, &c., &c., careful comparison would show from\ntime to time the comparative variations of length of these different\nsubstances. The most variable substances would be the most divergent,\nand the standard would be furnished by the mean length of those which\nagreed most closely with each other just as uniform motion is that of\nthose bodies which agree most closely in indicating the efflux of time.\n\n\n*The Terrestrial Standard.*\n\nThe second method assumes that the globe itself is a body of invariable\ndimensions and the founders of the metrical system selected the\nten-millionth part of the distance from the equator to the pole as\nthe definition of the metre. The first imperfection in such a method\nis that the earth is certainly not invariable in size; for we know\nthat it is superior in temperature to surrounding space, and must be\nslowly cooling and contracting. There is much reason to believe that\nall earthquakes, volcanoes, mountain elevations, and changes of sea\nlevel are evidences of this contraction as asserted by Mr. Mallet.[220]\nBut such is the vast bulk of the earth and the duration of its past\nexistence, that this contraction is perhaps less rapid in proportion\nthan that of any bar or other material standard which we can construct.\n\n  [220] *Proceedings of the Royal Society*, 20th June, 1872, vol.\u00a0xx.\n  p.\u00a0438.\n\nThe second and chief difficulty of this method arises from the vast\nsize of the earth, which prevents us from making any comparison with\nthe ultimate standard, except by a trigonometrical survey of a most\nelaborate and costly kind. The French physicists, who first proposed\nthe method, attempted to obviate this inconvenience by carrying out\nthe survey once for all, and then constructing a standard metre, which\nshould be exactly the one ten millionth part of the distance from the\npole to the equator. But since all measuring operations are merely\napproximate, it was impossible that this operation could be perfectly\nachieved. Accordingly, it was shown in 1838 that the supposed French\nmetre was erroneous to the considerable extent of one part in 5527. It\nthen became necessary either to alter the length of the assumed metre,\nor to abandon its supposed relation to the earth\u2019s dimensions. The\nFrench Government and the International Metrical Commission have for\nobvious reasons decided in favour of the latter course, and have thus\nreverted to the first method of defining the metre by a given bar. As\nfrom time to time the ratio between this assumed standard metre and the\nquadrant of the earth becomes more accurately known, we have better\nmeans of restoring that metre by reference to the globe if required.\nBut until lost, destroyed, or for some clear reason discredited,\nthe bar metre and not the globe is the standard. Thomson and Tait\nremark that any of the more accurate measurements of the English\ntrigonometrical survey might in like manner be employed to restore our\nstandard yard, in terms of which the results are recorded.\n\n\n*The Pendulum Standard.*\n\nThe third method of defining a standard length, by reference to the\nseconds pendulum, was first proposed by Huyghens, and was at one time\nadopted by the English Government. From the principle of the pendulum\n(p.\u00a0302) it clearly appears that if the time of oscillation and the\nforce actuating the pendulum be the same, the length of the pendulum\nmust be the same. We do not get rid of theoretical difficulties, for\nwe must assume the attraction of gravity at some point of the earth\u2019s\nsurface, say London, to be unchanged from time to time, and the\nsidereal day to be invariable, neither assumption being absolutely\ncorrect so far as we can judge. The pendulum, in short, is only an\nindirect means of making one physical quantity of space depend upon two\nother physical quantities of time and force.\n\nThe practical difficulties are, however, of a far more serious\ncharacter than the theoretical ones. The length of a pendulum is not\nthe ordinary length of the instrument, which might be greatly varied\nwithout affecting the duration of a vibration, but the distance from\nthe centre of suspension to the centre of oscillation. There are no\ndirect means of determining this latter centre, which depends upon\nthe average momentum of all the particles of the pendulum as regards\nthe centre of suspension. Huyghens discovered that the centres of\nsuspension and oscillation are interchangeable, and Kater pointed\nout that if a pendulum vibrates with exactly the same rapidity when\nsuspended from two different points, the distance between these points\nis the true length of the equivalent simple pendulum.[221] But the\npractical difficulties in employing Kater\u2019s reversible pendulum are\nconsiderable, and questions regarding the disturbance of the air, the\nforce of gravity, or even the interference of electrical attractions\nhave to be entertained. It has been shown that all the experiments made\nunder the authority of Government for determining the ratio between\nthe standard yard and the seconds pendulum, were vitiated by an error\nin the corrections for the resisting, adherent, or buoyant power of\nthe air in which the pendulums were swung. Even if such corrections\nwere rendered unnecessary by operating in a vacuum, other difficult\nquestions remain.[222] Gauss\u2019 mode of comparing the vibrations of a\nwire pendulum when suspended at two different lengths is open to equal\nor greater practical difficulties. Thus it is found that the pendulum\nstandard cannot compete in accuracy and certainty with the simple bar\nstandard, and the method would only be useful as an accessory mode of\nrestoring the bar standard if at any time again destroyed.\n\n  [221] Kater\u2019s *Treatise on Mechanics*, Cabinet Cyclop\u00e6dia, p.\u00a0154.\n\n  [222] Grant\u2019s *History of Physical Astronomy*, p.\u00a0156.\n\n\n*Unit of Density.*\n\nBefore we can measure the phenomena of nature, we require a third\nindependent unit, which shall enable us to define the quantity of\nmatter occupying any given space. All the changes of nature, as we\nshall see, are probably so many manifestations of energy; but energy\nrequires some substratum or material machinery of molecules, in and by\nwhich it may be manifested. Observation shows that, as regards force,\nthere may be two modes of variation of matter. As Newton says in the\nfirst definition of the Principia, \u201cthe quantity of matter is the\nmeasure of the same, arising from its density and bulk conjunctly.\u201d\nThus the force required to set a body in motion varies both according\nto the bulk of the matter, and also according to its quality. Two cubic\ninches of iron of uniform quality, will require twice as much force\nas one cubic inch to produce a certain velocity in a given time; but\none cubic inch of gold will require more force than one cubic inch of\niron. There is then some new measurable quality in matter apart from\nits bulk, which we may call *density*, and which is, strictly speaking,\nindicated by its capacity to resist and absorb the action of force.\nFor the unit of density we may assume that of any substance which is\nuniform in quality, and can readily be referred to from time to time.\nPure water at any definite temperature, for instance that of snow\nmelting under inappreciable pressure, furnishes an invariable standard\nof density, and by comparing equal bulks of various substances with\na like bulk of ice-cold water, as regards the velocity produced in a\nunit of time by the same force, we should ascertain the densities of\nthose substances as expressed in that of water. Practically the force\nof gravity is used to measure density; for a beautiful experiment with\nthe pendulum, performed by Newton and repeated by Gauss, shows that all\nkinds of matter gravitate equally. Two portions of matter then which\nare in equilibrium in the balance, may be assumed to possess equal\ninertia, and their densities will therefore be inversely as their cubic\ndimensions.\n\n\n*Unit of Mass.*\n\nMultiplying the number of units of density of a portion of matter,\nby the number of units of space occupied by it, we arrive at the\nquantity of matter, or, as it is usually called, the *unit of mass*, as\nindicated by the inertia and gravity it possesses. To proceed in the\nmost simple manner, the unit of mass ought to be that of a cubic unit\nof matter of the standard density; but the founders of the metrical\nsystem took as their unit of mass, the cubic centimetre of water, at\nthe temperature of maximum density (about 4\u00b0\u00a0Cent.). They called this\nunit of mass the *gramme*, and constructed standard specimens of the\nkilogram, which might be readily referred to by all who required to\nemploy accurate weights. Unfortunately the determination of the bulk\nof a given weight of water at a certain temperature is an operation\ninvolving many difficulties, and it cannot be performed in the present\nday with a greater exactness than that of about one part in 5000, the\nresults of careful observers being sometimes found to differ as much as\none part in 1000.[223]\n\n  [223] Clerk Maxwell\u2019s *Theory of Heat*, p.\u00a079.\n\nWeights, on the other hand, can be compared with each other to at least\none part in a million. Hence if different specimens of the kilogram be\nprepared by direct weighing against water, they will not agree closely\nwith each other; the two principal standard kilograms agree neither\nwith each other, nor with their definition. According to Professor\nMiller the so-called Kilogramme des Archives weighs 15432\u00b734874 grains,\nwhile the kilogram deposited at the Ministry of the Interior in Paris,\nas the standard for commercial purposes, weighs 15432\u00b7344 grains. Since\na standard weight constructed of platinum, or platinum and iridium, can\nbe preserved free from any appreciable alteration, and since it can be\nvery accurately compared with other weights, we shall ultimately attain\nthe greatest exactness in our measurements of mass, by assuming some\nsingle kilogram as a *provisional standard*, leaving the determination\nof its actual mass in units of space and density for future\ninvestigation. This is what is practically done at the present day,\nand thus a unit of mass takes the place of the unit of density, both\nin the French and English systems. The English pound is defined by a\ncertain lump of platinum, preserved at Westminster, and is an arbitrary\nmass, chosen merely that it may agree as nearly as possible with old\nEnglish pounds. The gallon, the old English unit of cubic measurement,\nis defined by the condition that it shall contain exactly ten pounds\nweight of water at 62\u00b0\u00a0Fahr.; and although it is stated that it has the\ncapacity of about 277\u00b7274 cubic inches, this ratio between the cubic\nand linear systems of measurement is not legally enacted, but left open\nto investigation. While the French metric system as originally designed\nwas theoretically perfect, it does not differ practically in this point\nfrom the English system.\n\n\n*Natural System of Standards.*\n\nQuite recently Professor Clerk Maxwell has suggested that the\nvibrations of light and the atoms of matter might conceivably be\nemployed as the ultimate standards of length, time, and mass. We\nshould thus arrive at a *natural system of standards*, which,\nthough possessing no present practical importance, has considerable\ntheoretical interest. \u201cIn the present state of science,\u201d he says, \u201cthe\nmost universal standard of length which we could assume would be the\nwave-length in vacuum of a particular kind of light, emitted by some\nwidely diffused substance such as sodium, which has well-defined lines\nin its spectrum. Such a standard would be independent of any changes in\nthe dimensions of the earth, and should be adopted by those who expect\ntheir writings to be more permanent than that body.\u201d[224] In the same\nway we should get a universal standard unit of time, independent of\nall questions about the motion of material bodies, by taking as the\nunit the periodic time of vibration of that particular kind of light\nwhose wave-length is the unit of length. It would follow that with\nthese units of length and time the unit of velocity would coincide with\nthe velocity of light in empty space. As regards the unit of mass,\nProfessor Maxwell, humorously as I should think, remarks that if we\nexpect soon to be able to determine the mass of a single molecule of\nsome standard substance, we may wait for this determination before\nfixing a universal standard of mass.\n\n  [224] *Treatise on Electricity and Magnetism*, vol.\u00a0i. p.\u00a03.\n\nIn a theoretical point of view there can be no reasonable doubt that\nvibrations of light are, as far as we can tell, the most fixed in\nmagnitude of all phenomena. There is as usual no certainty in the\nmatter, for the properties of the basis of light may vary to some\nextent in different parts of space. But no differences could ever\nbe established in the velocity of light in different parts of the\nsolar system, and the spectra of the stars show that the times of\nvibration there do not differ perceptibly from those in this part\nof the universe. Thus all presumption is in favour of the absolute\nconstancy of the vibrations of light--absolute, that is, so far as\nregards any means of investigation we are likely to possess. Nearly\nthe same considerations apply to the atomic weight as the standard of\nmass. It is impossible to prove that all atoms of the same substance\nare of equal mass, and some physicists think that they differ, so that\nthe fixity of combining proportions may be due only to the approximate\nconstancy of the mean of countless millions of discrepant weights. But\nin any case the detection of difference is probably beyond our powers.\nIn a theoretical point of view, then, the magnitudes suggested by\nProfessor Maxwell seem to be the most fixed ones of which we have any\nknowledge, so that they necessarily become the natural units.\n\nIn a practical point of view, as Professor Maxwell would be the first\nto point out, they are of little or no value, because in the present\nstate of science we cannot measure a vibration or weigh an atom with\nany approach to the accuracy which is attainable in the comparison\nof standard metres and kilograms. The velocity of light is not known\nprobably within a thousandth part, and as we progress in the knowledge\nof light, so we shall progress in the accurate fixation of other\nstandards. All that can be said then, is that it is very desirable\nto determine the wave-lengths and periods of the principal lines of\nthe solar spectrum, and the absolute atomic weights of the elements,\nwith all attainable accuracy, in terms of our existing standards. The\nnumbers thus obtained would admit of the reproduction of our standards\nin some future age of the world to a corresponding degree of accuracy,\nwere there need of such reference; but so far as we can see at present,\nthere is no considerable probability that this mode of reproduction\nwould ever be the best mode.\n\n\n*Subsidiary Units.*\n\nHaving once established the standard units of time, space, and density\nor mass, we might employ them for the expression of all quantities\nof such nature. But it is often convenient in particular branches of\nscience to use multiples or submultiples of the original units, for the\nexpression of quantities in a simple manner. We use the mile rather\nthan the yard when treating of the magnitude of the globe, and the\nmean distance of the earth and sun is not too large a unit when we\nhave to describe the distances of the stars. On the other hand, when\nwe are occupied with microscopic objects, the inch, the line or the\nmillimetre, become the most convenient terms of expression.\n\nIt is allowable for a scientific man to introduce a new unit in any\nbranch of knowledge, provided that it assists precise expression,\nand is carefully brought into relation with the primary units. Thus\nProfessor A.\u00a0W. Williamson has proposed as a convenient unit of\nvolume in chemical science, an absolute volume equal to about 11\u00b72\nlitres representing the bulk of one gram of hydrogen gas at standard\ntemperature and pressure, or the *equivalent* weight of any other\ngas, such as 16 grams of oxygen, 14 grams of nitrogen, &c.; in short,\nthe bulk of that quantity of any one of those gases which weighs as\nmany grams as there are units in the number expressing its atomic\nweight.[225] Hofmann has proposed a new unit of weight for chemists,\ncalled a *crith*, to be defined by the weight of one litre of hydrogen\ngas at 0\u00b0\u00a0C. and 0\u00b0\u00b776\u00a0mm., weighing about 0\u00b70896 gram.[226] Both of\nthese units must be regarded as purely subordinate units, ultimately\ndefined by reference to the primary units, and not involving any new\nassumption.\n\n  [225] *Chemistry for Students*, by A.\u00a0W. Williamson. Clarendon Press\n  Series, 2nd ed. Preface p. vi.\n\n  [226] *Introduction to Chemistry*, p.\u00a0131.\n\n\n*Derived Units.*\n\nThe standard units of time, space, and mass having been once fixed,\nmany kinds of magnitude are naturally measured by units derived from\nthem. From the metre, the unit of linear magnitude follows in the most\nobvious manner the centiare or square metre, the unit of superficial\nmagnitude, and the litre that is the cube of the tenth part of a metre,\nthe unit of capacity or volume. Velocity of motion is expressed by the\nratio of the space passed over, when the motion is uniform, to the time\noccupied; hence the unit of velocity is that of a body which passes\nover a unit of space in a unit of time. In physical science the unit of\nvelocity might be taken as one metre per second. Momentum is measured\nby the mass moving, regard being paid both to the amount of matter and\nthe velocity at which it is moving. Hence the unit of momentum will be\nthat of a unit volume of matter of the unit density moving with the\nunit velocity, or in the French system, a cubic centimetre of water of\nthe maximum density moving one metre per second.\n\nAn accelerating force is measured by the ratio of the momentum\ngenerated to the time occupied, the force being supposed to act\nuniformly. The unit of force will therefore be that which generates\na unit of momentum in a unit of time, or which causes, in the French\nsystem, one cubic centimetre of water at maximum density to acquire in\none second a velocity of one metre per second. The force of gravity is\nthe most familiar kind of force, and as, when acting unimpeded upon any\nsubstance, it produces in a second a velocity of 9\u00b780868\u00a0.\u00a0. metres per\nsecond in Paris, it follows that the absolute unit of force is about\nthe tenth part of the force of gravity. If we employ British weights\nand measures, the absolute unit of force is represented by the gravity\nof about half an ounce, since the force of gravity of any portion of\nmatter acting upon that matter during one second, produces a final\nvelocity of 32\u00b71889 feet per second or about 32 units of velocity.\nAlthough from its perpetual action and approximate uniformity we find\nin gravity the most convenient force for reference, and thus habitually\nemploy it to estimate quantities of matter, we must remember that it\nis only one of many instances of force. Strictly speaking, we should\nexpress weight in terms of force, but practically we express other\nforces in terms of weight.\n\nWe still require the unit of energy, a more complex notion. The\nmomentum of a body expresses the quantity of motion which belongs or\nwould belong to the aggregate of the particles; but when we consider\nhow this motion is related to the action of a force producing or\nremoving it, we find that the effect of a force is proportional to the\nmass multiplied by the square of the velocity and it is convenient to\ntake half this product as the expression required. But it is shown in\nbooks upon dynamics that it will be exactly the same thing if we define\nenergy by a force acting through a space. The natural unit of energy\nwill then be that which overcomes a unit of force acting through a unit\nof space; when we lift one kilogram through one metre, against gravity,\nwe therefore accomplish 9\u00b780868\u00a0.\u00a0. units of work, that is, we turn so\nmany units of potential energy existing in the muscles, into potential\nenergy of gravitation. In lifting one pound through one foot there is\nin like manner a conversion of 32\u00b71889 units of energy. Accordingly the\nunit of energy will be in the English system, that required to lift\none pound through about the thirty-second part of a foot; in terms of\nmetric units, it will be that required to lift a kilogram through about\none tenth part of a metre.\n\nEvery person is at liberty to measure and record quantities in terms of\nany unit which he likes. He may use the yard for linear measurement and\nthe litre for cubic measurement, only there will then be a complicated\nrelation between his different results. The system of derived units\nwhich we have been briefly considering, is that which gives the most\nsimple and natural relations between quantitative expressions of\ndifferent kinds, and therefore conduces to ease of comprehension and\nsaving of laborious calculation.\n\nIt would evidently be a source of great convenience if scientific men\ncould agree upon some single system of units, original and derived, in\nterms of which all quantities could be expressed. Statements would thus\nbe rendered easily comparable, a large part of scientific literature\nwould be made intelligible to all, and the saving of mental labour\nwould be immense. It seems to be generally allowed, too, that the\nmetric system of weights and measures presents the best basis for the\nultimate system; it is thoroughly established in Western Europe; it is\nlegalised in England; it is already commonly employed by scientific\nmen; it is in itself the most simple and scientific of systems. There\nis every reason then why the metric system should be accepted at least\nin its main features.\n\n\n*Provisional Units.*\n\nUltimately, as we can hardly doubt, all phenomena will be recognised\nas so many manifestations of energy; and, being expressed in terms of\nthe unit of energy, will be referable to the primary units of space,\ntime, and density. To effect this reduction, however, in any particular\ncase, we must not only be able to compare different quantities of\nthe phenomenon, but to trace the whole series of steps by which it\nis connected with the primary notions. We can readily observe that\nthe intensity of one source of light is greater than that of another;\nand, knowing that the intensity of light decreases as the square of\nthe distance increases, we can easily determine their comparative\nbrilliance. Hence we can express the intensity of light falling upon\nany surface, if we have a unit in which to make the expression. Light\nis undoubtedly one form of energy, and the unit ought therefore to be\nthe unit of energy. But at present it is quite impossible to say how\nmuch energy there is in any particular amount of light. The question\nthen arises,--Are we to defer the measurement of light until we can\nassign its relation to other forms of energy? If we answer Yes, it\nis equivalent to saying that the science of light must stand still\nperhaps for a generation; and not only this science but many others.\nThe true course evidently is to select, as the provisional unit of\nlight, some light of convenient intensity, which can be reproduced from\ntime to time in the same intensity, and which is defined by physical\ncircumstances. All the phenomena of light may be experimentally\ninvestigated relatively to this unit, for instance that obtained after\nmuch labour by Bunsen and Roscoe.[227] In after years it will become a\nmatter of inquiry what is the energy exerted in such unit of light; but\nit may be long before the relation is exactly determined.\n\n  [227] *Philosophical Transactions* (1859), vol.\u00a0cxlix. p.\u00a0884, &c.\n\nA provisional unit, then, means one which is assumed and physically\ndefined in a safe and reproducible manner, in order that particular\nquantities may be compared *inter se* more accurately than they can\nyet be referred to the primary units. In reality the great majority\nof our measurements are expressed in terms of such provisionally\nindependent units, and even the unit of mass, as we have seen, ought to\nbe considered as provisional.\n\nThe unit of heat ought to be simply the unit of energy, already\ndescribed. But a weight can be measured to the one-millionth part, and\ntemperature to less than the thousandth part of a degree Fahrenheit,\nand to less therefore than the five-hundred thousandth part of the\nabsolute temperature, whereas the mechanical equivalent of heat\nis probably not known to the thousandth part. Hence the need of a\nprovisional unit of heat, which is often taken as that requisite to\nraise one gram of water through one degree Centigrade, that is from\n0\u00b0 to 1\u00b0. This quantity of heat is capable of approximate expression\nin terms of time, space, and mass; for by the natural constant,\ndetermined by Dr. Joule, and called the mechanical equivalent of heat,\nwe know that the assumed unit of heat is equal to the energy of 423\u00b755\ngram-metres, or that energy which will raise the mass of 423\u00b755 grams\nthrough one metre against 9\u00b78... absolute units of force. Heat may also\nbe expressed in terms of the quantity of ice at 0\u00b0\u00a0Cent., which it is\ncapable of converting into water under inappreciable pressure.\n\n\n*Theory of Dimensions.*\n\nIn order to understand the relations between the quantities dealt with\nin physical science, it is necessary to pay attention to the Theory of\nDimensions, first clearly stated by Joseph Fourier,[228] but in later\nyears developed by several physicists. This theory investigates the\nmanner in which each derived unit depends upon or involves one or more\nof the fundamental units. The number of units in a rectangular area\nis found by multiplying together the numbers of units in the sides;\nthus the unit of length enters twice into the unit of area, which is\ntherefore said to have two dimensions with respect to length. Denoting\nlength by *L*, we may say that the dimensions of area are *L*\u00a0\u00d7\u00a0*L* or\n*L*^{2}. It is obvious in the same way that the dimensions of volume or\nbulk will be *L*^{3}.\n\n  [228] *Th\u00e9orie Analytique de la Chaleur*, Paris; 1822, \u00a7\u00a7 157\u2013162.\n\nThe number of units of mass in a body is found by multiplying the\nnumber of units of volume, by those of density. Hence mass is of\nthree dimensions as regards length, and one as regards density.\nCalling density *D*, the dimensions of mass are *L*^{3}*D*. As already\nexplained, however, it is usual to substitute an arbitrary provisional\nunit of mass, symbolised by *M*; according to the view here taken we\nmay say that the dimensions of *M* are *L*^{3}*D*.\n\nIntroducing time, denoted by *T*, it is easy to see that the dimensions\nof velocity will be *L/T* or *LT*^{-1}, because the number of units\nin the velocity of a body is found by *dividing* the units of length\npassed over by the units of time occupied in passing. The acceleration\nof a body is measured by the increase of velocity in relation to the\ntime, that is, we must divide the units of velocity gained by the units\nof time occupied in gaining it; hence its dimensions will be *LT*^{-2}.\nMomentum is the product of mass and velocity, so that its dimensions\nare *MLT*^{-1}. The effect of a force is measured by the acceleration\nproduced in a unit of mass in a unit of time; hence the dimensions of\nforce are *MLT*^{-2}. Work done is proportional to the force acting and\nto the space through which it acts; so that it has the dimensions of\nforce with that of length added, giving *ML*^{2}*T*^{-2}.\n\nIt should be particularly noticed that angular magnitude has no\ndimensions at all, being measured by the ratio of the arc to the radius\n(p.\u00a0305). Thus we have the dimensions *LL*^{-1} or *L*^{0}. This\nagrees with the statement previously made, that no arbitrary unit of\nangular magnitude is needed. Similarly, all pure numbers expressing\nratios only, such as sines and other trigonometrical functions,\nlogarithms, exponents, &c., are devoid of dimensions. They are absolute\nnumbers necessarily expressed in terms of unity itself, and are quite\nunaffected by the selection of the arbitrary physical units. Angular\nmagnitude, however, enters into other quantities, such as angular\nvelocity, which has the dimensions 1/*T* or *T*^{-1}, the units of\nangle being divided by the units of time occupied. The dimensions of\nangular acceleration are denoted by *T*^{-2}.\n\nThe quantities treated in the theories of heat and electricity\nare numerous and complicated as regards their dimensions. Thermal\ncapacity has the dimensions *ML*^{-3}, thermal conductivity,\n*ML*^{-1}*T*^{-1}. In Magnetism the dimensions of the strength\nof pole are *M*^{1/2}*L*^{3/2}*T*^{-1}, the dimensions of\nfield-intensity are *M*^{1/2}*L*^{-1/2}*T*^{-1}, and the intensity\nof magnetisation has the same dimensions. In the science of\nelectricity physicists have to deal with numerous kinds of quantity,\nand their dimensions are different too in the electro-static\nand the electro-magnetic systems. Thus electro-motive force has\nthe dimensions *M*^{1/2}*L*^{1/2}*T*^{-1}, in the former, and\n*M*^{1/2}*L*^{3/2}*T*^{-2} in the latter system. Capacity simply\ndepends upon length in electro-statics, but upon *L*^{-1}*T*^{2} in\nelectro-magnetics. It is worthy of particular notice that electrical\nquantities have simple dimensions when expressed in terms of density\ninstead of mass. The instances now given are sufficient to show the\ndifficulty of conceiving and following out the relations of the\nquantities treated in physical science without a systematic method of\ncalculating and exhibiting their dimensions. It is only in quite recent\nyears that clear ideas about these quantities have been attained. Half\na century ago probably no one but Fourier could have explained what\nhe meant by temperature or capacity for heat. The notion of measuring\nelectricity had hardly been entertained.\n\nBesides affording us a clear view of the complex relations of physical\nquantities, this theory is specially useful in two ways. Firstly, it\naffords a test of the correctness of mathematical reasoning. According\nto the *Principle of Homogeneity*, all the quantities *added* together,\nand equated in any equation, must have the same dimensions. Hence if,\non estimating the dimensions of the terms in any equation, they be not\nhomogeneous, some blunder must have been committed. It is impossible\nto add a force to a velocity, or a mass to a momentum. Even if the\nnumerical values of the two members of a non-homogeneous equation were\nequal, this would be accidental, and any alteration in the physical\nunits would produce inequality and disclose the falsity of the law\nexpressed in the equation.\n\nSecondly, the theory of units enables us readily and infallibly to\ndeduce the change in the numerical expression of any physical quantity,\nproduced by a change in the fundamental units. It is of course obvious\nthat in order to represent the same absolute quantity, a number must\nvary inversely as the magnitude of the units which are numbered. The\nyard expressed in feet is 3; taking the inch as the unit instead of\nthe foot it becomes 36. Every quantity into which the dimension length\nenters positively must be altered in like manner. Changing the unit\nfrom the foot to the inch, numerical expressions of volume must be\nmultiplied by 12\u00a0\u00d7\u00a012\u00a0\u00d7\u00a012. When a dimension enters negatively the\nopposite rule will hold. If for the minute we substitute the second\nas unit of time, then we must divide all numbers expressing angular\nvelocities by 60, and numbers expressing angular acceleration by\n60\u00a0\u00d7\u00a060. The rule is that a numerical expression varies inversely as\nthe magnitude of the unit as regards each whole dimension entering\npositively, and it varies directly as the magnitude of the unit for\neach whole dimension entering negatively. In the case of fractional\nexponents, the proper root of the ratio of change has to be taken.\n\nThe study of this subject may be continued in Professor J.\u00a0D. Everett\u2019s\n\u201cIllustrations of the Centimetre-gramme-second System of Units,\u201d\npublished by Taylor and Francis, 1875; in Professor Maxwell\u2019s \u201cTheory\nof Heat;\u201d or Professor Fleeming Jenkin\u2019s \u201cText Book of Electricity.\u201d\n\n\n*Natural Constants.*\n\nHaving acquired accurate measuring instruments, and decided upon the\nunits in which the results shall be expressed, there remains the\nquestion, What use shall be made of our powers of measurement? Our\nprincipal object must be to discover general quantitative laws of\nnature; but a very large amount of preliminary labour is employed in\nthe accurate determination of the dimensions of existing objects, and\nthe numerical relations between diverse forces and phenomena. Step\nby step every part of the material universe is surveyed and brought\ninto known relations with other parts. Each manifestation of energy is\ncorrelated with each other kind of manifestation. Professor Tyndall has\ndescribed the care with which such operations are conducted.[229]\n\n  [229] Tyndall\u2019s *Sound*, 1st ed. p.\u00a026.\n\n\u201cThose who are unacquainted with the details of scientific\ninvestigation, have no idea of the amount of labour expended on\nthe determination of those numbers on which important calculations\nor inferences depend. They have no idea of the patience shown by a\nBerzelius in determining atomic weights; by a Regnault in determining\ncoefficients of expansion; or by a Joule in determining the mechanical\nequivalent of heat. There is a morality brought to bear upon such\nmatters which, in point of severity, is probably without a parallel in\nany other domain of intellectual action.\u201d\n\nEvery new natural constant which is recorded brings many fresh\ninferences within our power. For if *n* be the number of such constants\nknown, then 1/2\u00a0(*n*^{2}--*n*) is the number of ratios which are within\nour powers of calculation, and this increases with the square of *n*.\nWe thus gradually piece together a map of nature, in which the lines of\ninference from one phenomenon to another rapidly grow in complexity,\nand the powers of scientific prediction are correspondingly augmented.\n\nBabbage[230] proposed the formation of a collection of the constant\nnumbers of nature, a work which has at last been taken in hand by the\nSmithsonian Institution.[231] It is true that a complete collection of\nsuch numbers would be almost co-extensive with scientific literature,\nsince almost all the numbers occurring in works on chemistry,\nmineralogy, physics, astronomy, &c., would have to be included.\nStill a handy volume giving all the more important numbers and their\nlogarithms, referred when requisite to the different units in common\nuse, would be very useful. A small collection of constant numbers will\nbe found at the end of Babbage\u2019s, Hutton\u2019s, and many other tables of\nlogarithms, and a somewhat larger collection is given in Templeton\u2019s\n*Millwright and Engineer\u2019s Pocket Companion*.\n\n  [230] British Association, Cambridge, 1833. Report, pp.\u00a0484\u2013490.\n\n  [231] *Smithsonian Miscellaneous Collections*, vol.\u00a0xii., the\n  Constants of Nature, part. i. Specific gravities compiled by F.\u00a0W.\n  Clarke, 8vo. Washington, 1873.\n\nOur present object will be to classify these constant numbers roughly,\naccording to their comparative generality and importance, under the\nfollowing heads:--\n\n  (1) Mathematical constants.\n  (2) Physical constants.\n  (3) Astronomical constants.\n  (4) Terrestrial numbers.\n  (5) Organic numbers.\n  (6) Social numbers.\n\n\n*Mathematical Constants.*\n\nAt the head of the list of natural constants must come those which\nexpress the necessary relations of numbers to each other. The ordinary\nMultiplication Table is the most familiar and the most important of\nsuch series of constants, and is, theoretically speaking, infinite in\nextent. Next we must place the Arithmetical Triangle, the significance\nof which has already been pointed out (p.\u00a0182). Tables of logarithms\nalso contain vast series of natural constants, arising out of the\nrelations of pure numbers. At the base of all logarithmic theory is\nthe mysterious natural constant commonly denoted by *e*, or \u03b5, being\nequal to the infinite series 1\u00a0+\u00a01/1\u00a0+\u00a01/1.2\u00a0+\u00a01/1.2.3\u00a0+\u00a01/1.2.3.4\n+...., and thus consisting of the sum of the ratios between the numbers\nof permutations and combinations of 0, 1, 2, 3, 4, &c. things. Tables\nof prime numbers and of the factors of composite numbers must not be\nforgotten.\n\nAnother vast and in fact infinite series of numerical constants\ncontains those connected with the measurement of angles, and embodied\nin trigonometrical tables, whether as natural or logarithmic sines,\ncosines, and tangents. It should never be forgotten that though these\nnumbers find their chief employment in connection with trigonometry,\nor the measurement of the sides of a right-angled triangle, yet the\nnumbers themselves arise out of numerical relations bearing no special\nrelation to space. Foremost among trigonometrical constants is the\nwell known number \u03c0, usually employed as expressing the ratio of the\ncircumference and the diameter of a circle; from \u03c0 follows the value of\nthe arcual or natural unit of angular value as expressed in ordinary\ndegrees (p.\u00a0306).\n\nAmong other mathematical constants not uncommonly used may be mentioned\ntables of factorials (p.\u00a0179), tables of Bernoulli\u2019s numbers, tables of\nthe error function,[232] which latter are indispensable not only in the\ntheory of probability but also in several other branches of science.\n\n  [232] J.\u00a0W. L. Glaisher, *Philosophical Magazine*, 4th Series,\n  vol.\u00a0xlii. p. 421.\n\nIt should be clearly understood that the mathematical constants and\ntables of reference already in our possession, although very extensive,\nare only an infinitely small part of what might be formed. With the\nprogress of science the tabulation of new functions will be continually\ndemanded, and it is worthy of consideration whether public money\nshould not be available to reward the severe, long continued, and\ngenerally thankless labour which must be gone through in calculating\ntables. Such labours are a benefit to the whole human race as long as\nit shall exist, though there are few who can appreciate the extent\nof this benefit. A most interesting and excellent description of\nmany mathematical tables will be found in De Morgan\u2019s article on\n*Tables*, in the *English Cyclop\u00e6dia*, Division of Arts and Sciences,\nvol.\u00a0vii. p.\u00a0976. An almost exhaustive critical catalogue of extant\ntables is being published by a Committee of the British Association,\ntwo portions, drawn up chiefly by Mr. J.\u00a0W.\u00a0L. Glaisher and Professor\nCayley, having appeared in the Reports of the Association for 1873 and\n1875.\n\n\n*Physical Constants.*\n\nThe second class of constants contains those which refer to the\nactual constitution of matter. For the most part they depend upon\nthe peculiarities of the chemical substance in question, but we may\nbegin with those which are of the most general character. In a first\nsub-class we may place the velocity of light or heat undulations, the\nnumbers expressing the relation between the lengths of the undulations,\nand the rapidity of the undulations, these numbers depending only on\nthe properties of the ethereal medium, and being probably the same in\nall parts of the universe. The theory of heat gives rise to several\nnumbers of the highest importance, especially Joule\u2019s mechanical\nequivalent of heat, the absolute zero of temperature, the mean\ntemperature of empty space, &c.\n\nTaking into account the diverse properties of the elements we must\nhave tables of the atomic weights, the specific heats, the specific\ngravities, the refractive powers, not only of the elements, but their\nalmost infinitely numerous compounds. The properties of hardness,\nelasticity, viscosity, expansion by heat, conducting powers for heat\nand electricity, must also be determined in immense detail. There are,\nhowever, certain of these numbers which stand out prominently because\nthey serve as intermediate units or terms of comparison. Such are, for\ninstance, the absolute coefficients of expansion of air, water and\nmercury, the temperature of the maximum density of water, the latent\nheats of water and steam, the boiling-point of water under standard\npressure, the melting and boiling-points of mercury, and so forth.\n\n\n*Astronomical Constants.*\n\nThe third great class consists of numbers possessing far less\ngenerality because they refer not to the properties of matter, but to\nthe special forms and distances in which matter has been disposed in\nthe part of the universe open to our examination. We have, first of\nall, to define the magnitude and form of the earth, its mean density,\nthe constant of aberration of light expressing the relation between\nthe earth\u2019s mean velocity in space and the velocity of light. From\nthe earth, as our observatory, we then proceed to lay down the mean\ndistances of the sun, and of the planets from the same centre; all the\nelements of the planetary orbits, the magnitudes, densities, masses,\nperiods of axial rotation of the several planets are by degrees\ndetermined with growing accuracy. The same labours must be gone through\nfor the satellites. Catalogues of comets with the elements of their\norbits, as far as ascertainable, must not be omitted.\n\nFrom the earth\u2019s orbit as a new base of observations, we next proceed\nto survey the heavens and lay down the apparent positions, magnitudes,\nmotions, distances, periods of variation, &c. of the stars. All\ncatalogues of stars from those of Hipparchus and Tycho, are full of\nnumbers expressing rudely the conformation of the visible universe.\nBut there is obviously no limit to the labours of astronomers; not\nonly are millions of distant stars awaiting their first measurements,\nbut those already registered require endless scrutiny as regards\ntheir movements in the three dimensions of space, their periods of\nrevolution, their changes of brilliance and colour. It is obvious that\nthough astronomical numbers are conventionally called *constant*, they\nare probably in all cases subject to more or less rapid variation.\n\n\n*Terrestrial Numbers.*\n\nOur knowledge of the globe we inhabit involves many numerical\ndeterminations, which have little or no connection with astronomical\ntheory. The extreme heights of the principal mountains, the mean\nelevations of continents, the mean or extreme depths of the oceans,\nthe specific gravities of rocks, the temperature of mines, the host of\nnumbers expressing the meteorological or magnetic conditions of every\npart of the surface, must fall into this class. Many such numbers\nare not to be called constant, being subject to periodic or secular\nchanges, but they are hardly more variable in fact than some which in\nastronomical science are set down as constant. In many cases quantities\nwhich seem most variable may go through rhythmical changes resulting\nin a nearly uniform average, and it is only in the long progress of\nphysical investigation that we can hope to discriminate successfully\nbetween those elemental numbers which are fixed and those which vary.\nIn the latter case the law of variation becomes the constant relation\nwhich is the object of our search.\n\n*Organic Numbers.*\n\nThe forms and properties of brute nature having been sufficiently\ndefined by the previous classes of numbers, the organic world, both\nvegetable and animal, remains outstanding, and offers a higher series\nof phenomena for our investigation. All exact knowledge relating to\nthe forms and sizes of living things, their numbers, the quantities\nof various compounds which they consume, contain, or excrete, their\nmuscular or nervous energy, &c. must be placed apart in a class by\nthemselves. All such numbers are doubtless more or less subject to\nvariation, and but in a minor degree capable of exact determination.\nMan, so far as he is an animal, and as regards his physical form, must\nalso be treated in this class.\n\n\n*Social Numbers.*\n\nLittle allusion need be made in this work to the fact that man in\nhis economic, sanitary, intellectual, \u00e6sthetic, or moral relations\nmay become the subject of sciences, the highest and most useful of\nall sciences. Every one who is engaged in statistical inquiry must\nacknowledge the possibility of natural laws governing such statistical\nfacts. Hence we must allot a distinct place to numerical information\nrelating to the numbers, ages, physical and sanitary condition,\nmortality, &c., of different peoples, in short, to vital statistics.\nEconomic statistics, comprehending the quantities of commodities\nproduced, existing, exchanged and consumed, constitute another\nextensive body of science. In the progress of time exact investigation\nmay possibly subdue regions of phenomena which at present defy all\nscientific treatment. That scientific method can ever exhaust the\nphenomena of the human mind is incredible.\n\n\n\n\nCHAPTER XV.\n\nANALYSIS OF QUANTITATIVE PHENOMENA.\n\n\nIn the two preceding chapters we have been engaged in considering how\na phenomenon may be accurately measured and expressed. So delicate\nand complex an operation is a measurement which pretends to any\nconsiderable degree of exactness, that no small part of the skill\nand patience of physicists is usually spent upon this work. Much\nof this difficulty arises from the fact that it is scarcely ever\npossible to measure a single effect at a time. The ultimate object\nmust be to discover the mathematical equation or law connecting a\nquantitative cause with its quantitative effect; this purpose usually\ninvolves, as we shall see, the varying of one condition at a time,\nthe other conditions being maintained constant. The labours of the\nexperimentalist would be comparatively light if he could carry out\nthis rule of varying one circumstance at a time. He would then obtain\na series of corresponding values of the variable quantities concerned,\nfrom which he might by proper hypothetical treatment obtain the\nrequired law of connection. But in reality it is seldom possible to\ncarry out this direction except in an approximate manner. Before then\nwe proceed to the consideration of the actual process of quantitative\ninduction, it is necessary to review the several devices by which a\ncomplicated series of effects can be disentangled. Every phenomenon\nmeasured will usually be the sum, difference, or it may be the product\nor quotient, of two or more different effects, and these must be in\nsome way analysed and separately measured before we possess the\nmaterials for inductive treatment.\n\n\n*Illustrations of the Complication of Effects.*\n\nIt is easy to bring forward a multitude of instances to show that a\nphenomenon is seldom to be observed simple and alone. A more or less\nelaborate process of analysis is almost always necessary. Thus if an\nexperimentalist wishes to observe and measure the expansion of a liquid\nby heat, he places it in a thermometer tube and registers the rise of\nthe column of liquid in the narrow tube. But he cannot heat the liquid\nwithout also heating the glass, so that the change observed is really\nthe difference between the expansions of the liquid and the glass. More\nminute investigation will show the necessity perhaps of allowing for\nfurther minute effects, namely the compression of the liquid and the\nexpansion of the bulb due to the increased pressure of the column as it\nbecomes lengthened.\n\nIn a great many cases an observed effect will be apparently at least\nthe simple sum of two separate and independent effects. The heat\nevolved in the combustion of oil is partly due to the carbon and partly\nto the hydrogen. A measurement of the heat yielded by the two jointly,\ncannot inform us how much proceeds from the one and how much from the\nother. If by some separate determination we can ascertain how much the\nhydrogen yields, then by mere subtraction we learn what is due to the\ncarbon; and *vice vers\u00e2*. The heat conveyed by a liquid, may be partly\nconveyed by true conduction, partly by convection. The light dispersed\nin the interior of a liquid consists both of what is reflected by\nfloating particles and what is due to true fluorescence;[233] and we\nmust find some mode of determining one portion before we can learn the\nother. The apparent motion of the spots on the sun, is the algebraic\nsum of the sun\u2019s axial rotation, and of the proper motion of the spots\nupon the sun\u2019s surface; hence the difficulty of ascertaining by direct\nobservations the period of the sun\u2019s rotation.\n\n  [233] Stokes, *Philosophical Transactions* (1852), vol.\u00a0cxlii. p.\u00a0529.\n\nWe cannot obtain the weight of a portion of liquid in a chemical\nbalance without weighing it with the containing vessel. Hence to have\nthe real weight of the liquid operated upon in an experiment, we must\nmake a separate weighing of the vessel, with or without the adhering\nfilm of liquid according to circumstances. This is likewise the mode\nin which a cart and its load are weighed together, the *tare* of the\ncart previously ascertained being deducted. The variation in the height\nof the barometer is a joint effect, partly due to the real variation\nof the atmospheric pressure, partly to the expansion of the mercurial\ncolumn by heat. The effects may be discriminated, if, instead of one\nbarometer tube we have two tubes containing mercury placed closely side\nby side, so as to have the same temperature. If one of them be closed\nat the bottom so as to be unaffected by the atmospheric pressure, it\nwill show the changes due to temperature only, and, by subtracting\nthese changes from those shown in the other tube, employed as a\nbarometer, we get the real oscillations of atmospheric pressure. But\nthis correction, as it is called, of the barometric reading, is better\neffected by calculation from the readings of an ordinary thermometer.\n\nIn other cases a quantitative effect will be the difference of\ntwo causes acting in opposite directions. Sir John Herschel\ninvented an instrument like a large thermometer, which he called\nthe Actinometer,[234] and Pouillet constructed a somewhat similar\ninstrument called the Pyrheliometer, for ascertaining the heating power\nof the sun\u2019s rays. In both instruments the heat of the sun was absorbed\nby a reservoir containing water, and the rise of temperature of the\nwater was exactly observed, either by its own expansion, or by the\nreadings of a delicate thermometer immersed in it. But in exposing the\nactinometer to the sun, we do not obtain the full effect of the heat\nabsorbed, because the receiving surface is at the same time radiating\nheat into empty space. The observed increment of temperature is in\nshort the difference between what is received from the sun and lost\nby radiation. The latter quantity is capable of ready determination;\nwe have only to shade the instrument from the direct rays of the sun,\nleaving it exposed to the sky, and we can observe how much it cools\nin a certain time. The total effect of the sun\u2019s rays will obviously\nbe the apparent effect *plus* the cooling effect in an equal time. By\nalternate exposure in sun and shade during equal intervals the desired\nresult may be obtained with considerable accuracy.[235]\n\n  [234] *Admiralty Manual of Scientific Enquiry*, 2nd ed. p.\u00a0299.\n\n  [235] Pouillet, *Taylor\u2019s Scientific Memoirs*, vol.\u00a0iv. p.\u00a045.\n\nTwo quantitative effects were beautifully distinguished in an\nexperiment of John Canton, devised in 1761 for the purpose of\ndemonstrating the compressibility of water. He constructed a\nthermometer with a large bulb full of water and a short capillary\ntube, the part of which above the water was freed from air. Under\nthese circumstances the water was relieved from the pressure of the\natmosphere, but the glass bulb in bearing that pressure was somewhat\ncontracted. He next placed the instrument under the receiver of an\nair-pump, and on exhausting the air, the water sank in the tube. Having\nthus obtained a measure of the effect of atmospheric pressure on the\nbulb, he opened the top of the thermometer tube and admitted the air.\nThe level of the water now sank still more, partly from the pressure\non the bulb being now compensated, and partly from the compression of\nthe water by the atmospheric pressure. It is obvious that the amount of\nthe latter effect was approximately the difference of the two observed\ndepressions.\n\nNot uncommonly the actual phenomenon which we wish to measure is\nconsiderably less than various disturbing effects which enter into\nthe question. Thus the compressibility of mercury is considerably\nless than the expansion of the vessels in which it is measured under\npressure, so that the attention of the experimentalist has chiefly\nto be concentrated on the change of magnitude of the vessels. Many\nastronomical phenomena, such as the parallax or the proper motions of\nthe fixed stars, are far less than the errors caused by instrumental\nimperfections, or motions arising from precession, nutation, and\naberration. We need not be surprised that astronomers have from time to\ntime mistaken one phenomenon for another, as when Flamsteed imagined\nthat he had discovered the parallax of the Pole star.[236]\n\n  [236] Baily\u2019s *Account of the Rev. John Flamsteed*, p.\u00a058.\n\n\n*Methods of Eliminating Error.*\n\nIn any particular experiment it is the object of the experimentalist to\nmeasure a single effect only, and he endeavours to obtain that effect\nfree from interfering effects. If this cannot be, as it seldom or\nnever can really be, he makes the effect as considerable as possible\ncompared with the other effects, which he reduces to a minimum, and\ntreats as noxious errors. Those quantities, which are called *errors*\nin one case, may really be most important and interesting phenomena in\nanother investigation. When we speak of eliminating error we really\nmean disentangling the complicated phenomena of nature. The physicist\nrightly wishes to treat one thing at a time, but as this object can\nseldom be rigorously carried into practice, he has to seek some mode of\ncounteracting the irrelevant and interfering causes.\n\nThe general principle is that a single observation can render known\nonly a single quantity. Hence, if several different quantitative\neffects are known to enter into any investigation, we must have at\nleast as many distinct observations as there are quantities to be\ndetermined. Every complete experiment will therefore consist in general\nof several operations. Guided if possible by previous knowledge of the\ncauses in action, we must arrange the determinations, so that by a\nsimple mathematical process we may distinguish the separate quantities.\nThere appear to be five principal methods by which we may accomplish\nthis object; these methods are specified below and illustrated in the\nsucceeding sections.\n\n(1) *The Method of Avoidance.* The physicist may seek for some special\nmode of experiment or opportunity of observation, in which the error is\nnon-existent or inappreciable.\n\n(2) *The Differential Method.* He may find opportunities of observation\nwhen all interfering phenomena remain constant, and only the subject\nof observation is at one time present and another time absent; the\ndifference between two observations then gives its amount.\n\n(3) *The Method of Correction.* He may endeavour to estimate the amount\nof the interfering effect by the best available mode, and then make a\ncorresponding correction in the results of observation.\n\n(4) *The Method of Compensation.* He may invent some mode of\nneutralising the interfering cause by balancing against it an exactly\nequal and opposite cause of unknown amount.\n\n(5) *The Method of Reversal.* He may so conduct the experiment that\nthe interfering cause may act in opposite directions, in alternate\nobservations, the mean result being free from interference.\n\n\nI. *Method of Avoidance of Error.*\n\nAstronomers seek opportunities of observation when errors will\nbe as small as possible. In spite of elaborate observations and\nlong-continued theoretical investigation, it is not practicable to\nassign any satisfactory law to the refractive power of the atmosphere.\nAlthough the apparent change of place of a heavenly body produced by\nrefraction may be more or less accurately calculated yet the error\ndepends upon the temperature and pressure of the atmosphere, and, when\na ray is highly inclined to the perpendicular, the uncertainty in the\nrefraction becomes very considerable. Hence astronomers always make\ntheir observations, if possible, when the object is at the highest\npoint of its daily course, *i.e.* on the meridian. In some kinds of\ninvestigation, as, for instance, in the determination of the latitude\nof an observatory, the astronomer is at liberty to select one or\nmore stars out of the countless number visible. There is an evident\nadvantage in such a case, in selecting a star which passes close to\nthe zenith, so that it may be observed almost entirely free from\natmospheric refraction, as was done by Hooke.\n\nAstronomers endeavour to render their clocks as accurate as\npossible, by removing the source of variation. The pendulum is\nperfectly isochronous so long as its length remains invariable, and\nthe vibrations are exactly of equal length. They render it nearly\ninvariable in length, that is in the distance between the centres of\nsuspension and oscillation, by a compensatory arrangement for the\nchange of temperature. But as this compensation may not be perfectly\naccomplished, some astronomers place their chief controlling clock in\na cellar, or other apartment, where the changes of temperature may\nbe as slight as possible. At the Paris Observatory a clock has been\nplaced in the caves beneath the building, where there is no appreciable\ndifference between the summer and winter temperature.\n\nTo avoid the effect of unequal oscillations Huyghens made his beautiful\ninvestigations, which resulted in the discovery that a pendulum, of\nwhich the centre of oscillation moved upon a cycloidal path, would\nbe perfectly isochronous, whatever the variation in the length of\noscillations. But though a pendulum may be easily rendered in some\ndegree cycloidal by the use of a steel suspension spring, it is found\nthat the mechanical arrangements requisite to produce a truly cycloidal\nmotion introduce more error than they remove. Hence astronomers seek\nto reduce the error to the smallest amount by maintaining their clock\npendulums in uniform movement; in fact, while a clock is in good order\nand has the same weights, there need be little change in the length\nof oscillation. When a pendulum cannot be made to swing uniformly,\nas in experiments upon the force of gravity, it becomes requisite to\nresort to the third method, and a correction is introduced, calculated\non theoretical grounds from the amount of the observed change in the\nlength of vibration.\n\nIt has been mentioned that the apparent expansion of a liquid by heat,\nwhen contained in a thermometer tube or other vessel, is the difference\nbetween the real expansion of the liquid and that of the containing\nvessel. The effects can be accurately distinguished provided that we\ncan learn the real expansion by heat of any one convenient liquid;\nfor by observing the apparent expansion of the same liquid in any\nrequired vessel we can by difference learn the amount of expansion of\nthe vessel due to any given change of temperature. When we once know\nthe change of dimensions of the vessel, we can of course determine the\nabsolute expansion of any other liquid tested in it. Thus it became an\nall-important object in scientific research to measure with accuracy\nthe absolute dilatation by heat of some one liquid, and mercury owing\nto several circumstances was by far the most suitable. Dulong and\nPetit devised a beautiful mode of effecting this by simply avoiding\naltogether the effect of the change of size of the vessel. Two upright\ntubes full of mercury were connected by a fine tube at the bottom, and\nwere maintained at two different temperatures. As mercury was free to\nflow from one tube to the other by the connecting tube, the two columns\nnecessarily exerted equal pressures by the principles of hydrostatics.\nHence it was only necessary to measure very accurately by a\ncathetometer the difference of level of the surfaces of the two columns\nof mercury, to learn the difference of length of columns of equal\nhydrostatic pressure, which at once gives the difference of density of\nthe mercury, and the dilatation by heat. The changes of dimension in\nthe containing tubes became a matter of entire indifference, and the\nlength of a column of mercury at different temperatures was measured\nas easily as if it had formed a solid bar. The experiment was carried\nout by Regnault with many improvements of detail, and the absolute\ndilatation of mercury, at temperatures between 0\u00b0\u00a0Cent. and 350\u00b0, was\ndetermined almost as accurately as was needful.[237]\n\n  [237] Jamin, *Cours de Physique*, vol.\u00a0ii. pp.\u00a015\u201328.\n\nThe presence of a large and uncertain amount of error may render a\nmethod of experiment valueless. Foucault devised a beautiful experiment\nwith the pendulum for demonstrating popularly the rotation of the\nearth, but it could be of no use for measuring the rotation exactly. It\nis impossible to make the pendulum swing in a perfect plane, and the\nslightest lateral motion gives it an elliptic path with a progressive\nmotion of the axis of the ellipse, which disguises and often entirely\noverpowers that due to the rotation of the earth.[238]\n\n  [238] *Philosophical Magazine*, 1851, 4th Series, vol.\u00a0ii. *passim*.\n\nFaraday\u2019s laborious experiments on the relation of gravity and\nelectricity were much obstructed by the fact that it is impossible\nto move a large weight of metal without generating currents of\nelectricity, either by friction or induction. To distinguish the\nelectricity, if any, directly due to the action of gravity from the\ngreater quantities indirectly produced was a problem of excessive\ndifficulty. Baily in his experiments on the density of the earth was\naware of the existence of inexplicable disturbances which have since\nbeen referred with much probability to the action of electricity.[239]\nThe skill and ingenuity of the experimentalist are often exhausted\nin trying to devise a form of apparatus in which such causes of error\nshall be reduced to a minimum.\n\n  [239] Hearn, *Philosophical Transactions*, 1847, vol.\u00a0cxxxvii.\n  pp.\u00a0217\u2013221.\n\nIn some rudimentary experiments we wish merely to establish the\nexistence of a quantitative effect without precisely measuring its\namount; if there exist causes of error of which we can neither\nrender the amount known or inappreciable, the best way is to make\nthem all negative so that the quantitative effects will be less than\nthe truth rather than greater. Grove, for instance, in proving that\nthe magnetisation or demagnetisation of a piece of iron raises its\ntemperature, took care to maintain the electro-magnet by which the iron\nwas magnetised at a lower temperature than the iron, so that it would\ncool rather than warm the iron by radiation or conduction.[240]\n\n  [240] *The Correlation of Physical Forces*, 3rd ed. p.\u00a0159.\n\nRumford\u2019s celebrated experiment to prove that heat was generated out\nof mechanical force in the boring of a cannon was subject to the\ndifficulty that heat might be brought to the cannon by conduction\nfrom neighbouring bodies. It was an ingenious device of Davy to\nproduce friction by a piece of clock-work resting upon a block of\nice in an exhausted receiver; as the machine rose in temperature\nabove 32\u00b0, it was certain that no heat was received by conduction\nfrom the support.[241] In many other experiments ice may be employed\nto prevent the access of heat by conduction, and this device, first\nput in practice by Murray,[242] is beautifully employed in Bunsen\u2019s\ncalorimeter.\n\n  [241] *Collected Works of Sir H. Davy*, vol.\u00a0ii. pp.\u00a012\u201314. *Elements\n  of Chemical Philosophy*, p.\u00a094.\n\n  [242] *Nicholson\u2019s Journal*, vol.\u00a0i. p.\u00a0241; quoted in *Treatise on\n  Heat*, Useful Knowledge Society, p.\u00a024.\n\nTo observe the true temperature of the air, though apparently so easy,\nis really a very difficult matter, because the thermometer is sure to\nbe affected either by the sun\u2019s rays, the radiation from neighbouring\nobjects, or the escape of heat into space. These sources of error are\ntoo fluctuating to allow of correction, so that the only accurate\nmode of procedure is that devised by Dr. Joule, of surrounding the\nthermometer with a copper cylinder ingeniously adjusted to the\ntemperature of the air, as described by him, so that the effect of\nradiation shall be nullified.[243]\n\n  [243] Clerk Maxwell, *Theory of Heat*, p.\u00a0228. *Proceedings of the\n  Manchester Philosophical Society*, Nov. 26, 1867, vol.\u00a0vii. p.\u00a035.\n\nWhen the avoidance of error is not practicable, it will yet be\ndesirable to reduce the absolute amount of the interfering error as\nmuch as possible before employing the succeeding methods to correct\nthe result. As a general rule we can determine a quantity with less\ninaccuracy as it is smaller, so that if the error itself be small\nthe error in determining that error will be of a still lower order\nof magnitude. But in some cases the absolute amount of an error is\nof no consequence, as in the index error of a divided circle, or the\ndifference between a chronometer and astronomical time. Even the rate\nat which a clock gains or loses is a matter of little importance\nprovided it remain constant, so that a sure calculation of its amount\ncan be made.\n\n\n2. *Differential Method.*\n\nWhen we cannot avoid the existence of error, we can often resort\nwith success to the second mode by measuring phenomena under such\ncircumstances that the error shall remain very nearly the same in all\nthe observations, and neutralise itself as regards the purposes in\nview. This mode is available whenever we want a difference between\nquantities and not the absolute quantity of either. The determination\nof the parallax of the fixed stars is exceedingly difficult, because\nthe amount of parallax is far less than most of the corrections for\natmospheric refraction, nutation, aberration, precession, instrumental\nirregularities, &c., and can with difficulty be detected among these\nphenomena of various magnitude. But, as Galileo long ago suggested,\nall such difficulties would be avoided by the differential observation\nof stars, which, though apparently close together, are really far\nseparated on the line of sight. Two such stars in close apparent\nproximity will be subject to almost exactly equal errors, so that all\nwe need do is to observe the apparent change of place of the nearer\nstar as referred to the more distant one. A good telescope furnished\nwith an accurate micrometer is alone needed for the application of\nthe method. Huyghens appears to have been the first observer who\nactually tried to employ the method practically, but it was not until\n1835 that the improvement of telescopes and micrometers enabled\nStruve to detect in this way the parallax of the star \u03b1 Lyr\u00e6. It is\none of the many advantages of the observation of transits of Venus\nfor the determination of the solar parallax that the refraction of\nthe atmosphere affects in an exactly equal degree the planet and\nthe portion of the sun\u2019s face over which it is passing. Thus the\nobservations are strictly of a differential nature.\n\nBy the process of substitutive weighing it is possible to ascertain\nthe equality or inequality of two weights with almost perfect freedom\nfrom error. If two weights A and B be placed in the scales of the\nbest balance we cannot be sure that the equilibrium of the beam\nindicates exact equality, because the arms of the beam may be unequal\nor unbalanced. But if we take B out and put another weight C in, and\nequilibrium still exists, it is apparent that the same causes of\nerroneous weighing exist in both cases, supposing that the balance has\nnot been disarranged; B then must be exactly equal to C, since it has\nexactly the same effect under the same circumstances. In like manner it\nis a general rule that, if by any uniform mechanical process we get a\ncopy of an object, it is unlikely that this copy will be precisely the\nsame as the original in magnitude and form, but two copies will equally\ndiverge from the original, and will therefore almost exactly resemble\neach other.\n\nLeslie\u2019s Differential Thermometer[244] was well adapted to the\nexperiments for which it was invented. Having two equal bulbs any\nalteration in the temperature of the air will act equally by conduction\non each and produce no change in the indications of the instrument.\nOnly that radiant heat which is purposely thrown upon one of the bulbs\nwill produce any effect. This thermometer in short carries out the\nprinciple of the differential method in a mechanical manner.\n\n  [244] Leslie, *Inquiry into the Nature of Heat*, p.\u00a010.\n\n\n3. *Method of Correction.*\n\nWhenever the result of an experiment is affected by an interfering\ncause to a calculable amount, it is sufficient to add or subtract this\namount. We are said to correct observations when we thus eliminate\nwhat is due to extraneous causes, although of course we are only\nseparating the correct effects of several agents. The variation in the\nheight of the barometer is partly due to the change of temperature,\nbut since the coefficient of absolute dilatation of mercury has been\nexactly determined, as already described (p.\u00a0341), we have only to make\ncalculations of a simple character, or, what is better still, tabulate\na series of such calculations for general use, and the correction for\ntemperature can be made with all desired accuracy. The height of the\nmercury in the barometer is also affected by capillary attraction,\nwhich depresses it by a constant amount depending mainly on the\ndiameter of the tube. The requisite corrections can be estimated with\naccuracy sufficient for most purposes, more especially as we can check\nthe correctness of the reading of a barometer by comparison with a\nstandard barometer, and introduce if need be an index error including\nboth the error in the affixing of the scale and the effect due to\ncapillarity. But in constructing the standard barometer itself we must\ntake greater precautions; the capillary depression depends somewhat\nupon the quality of the glass, the absence of air, and the perfect\ncleanliness of the mercury, so that we cannot assign the exact amount\nof the effect. Hence a standard barometer is constructed with a wide\ntube, sometimes even an inch in diameter, so that the capillary effect\nmay be rendered almost zero.[245] Gay-Lussac made barometers in the\nform of a uniform siphon tube, so that the capillary forces acting at\nthe upper and lower surfaces should balance and destroy each other;\nbut the method fails in practice because the lower surface, being\nopen to the air, becomes sullied and subject to a different force of\ncapillarity.\n\n  [245] Jevons, Watts\u2019 *Dictionary of Chemistry*, vol.\u00a0i. pp.\u00a0513\u2013515.\n\nIn mechanical experiments friction is an interfering condition, and\ndrains away a portion of the energy intended to be operated upon in a\ndefinite manner. We should of course reduce the friction in the first\nplace to the lowest possible amount, but as it cannot be altogether\nprevented, and is not calculable with certainty from any general\nlaws, we must determine it separately for each apparatus by suitable\nexperiments. Thus Smeaton, in his admirable but almost forgotten\nresearches concerning water-wheels, eliminated friction in the most\nsimple manner by determining by trial what weight, acting by a cord\nand roller upon his model water-wheel, would make it turn without\nwater as rapidly as the water made it turn. In short, he ascertained\nwhat weight concurring with the water would exactly compensate for the\nfriction.[246] In Dr. Joule\u2019s experiments to determine the mechanical\nequivalent of heat by the condensation of air, a considerable amount\nof heat was produced by friction of the condensing pump, and a small\nportion by stirring the water employed to absorb the heat. This heat of\nfriction was measured by simply repeating the experiment in an exactly\nsimilar manner except that no condensation was effected, and observing\nthe change of temperature then produced.[247]\n\n  [246] *Philosophical Transactions*, vol.\u00a0li. p.\u00a0100.\n\n  [247] *Philosophical Magazine*, 3rd Series, vol.\u00a0xxvi. p.\u00a0372.\n\nWe may describe as *test experiments* any in which we perform\noperations not intended to give the quantity of the principal\nphenomenon, but some quantity which would otherwise remain as an\nerror in the result. Thus in astronomical observations almost every\ninstrumental error may be avoided by increasing the number of\nobservations and distributing them in such a manner as to produce in\nthe final mean as much error in one way as in the other. But there\nis one source of error, first discovered by Maskelyne, which cannot\nbe thus avoided, because it affects all observations in the same\ndirection and to the same average amount, namely the Personal Error of\nthe observer or the inclination to record the passage of a star across\nthe wires of the telescope a little too soon or a little too late.\nThis personal error was first carefully described in the *Edinburgh\nJournal of Science*, vol.\u00a0i. p. 178. The difference between the\njudgment of observers at the Greenwich Observatory usually varies from\n1/100 to 1/3 of a second, and remains pretty constant for the same\nobservers.[248] One practised observer in Sir George Airy\u2019s pendulum\nexperiments recorded all his time observations half a second too early\non the average as compared with the chief observer.[249] In some\nobservers it has amounted to seven or eight-tenths of a second.[250]\nDe Morgan appears to have entertained the opinion that this source of\nerror was essentially incapable of elimination or correction.[251]\nBut it seems clear, as I suggested without knowing what had been\ndone,[252] that this personal error might be determined absolutely with\nany desirable degree of accuracy by test experiments, consisting in\nmaking an artificial star move at a considerable distance and recording\nby electricity the exact moment of its passage over the wire. This\nmethod has in fact been successfully employed in Leyden, Paris, and\nNeuchatel.[253] More recently, observers were trained for the Transit\nof Venus Expeditions by means of a mechanical model representing the\nmotion of Venus over the sun, this model being placed at a little\ndistance and viewed through a telescope, so that differences in the\njudgments of different observers would become apparent. It seems likely\nthat tests of this nature might be employed with advantage in other\ncases.\n\n  [248] *Greenwich Observations for* 1866, p. xlix.\n\n  [249] *Philosophical Transactions*, 1856, p.\u00a0309.\n\n  [250] Penny *Cyclop\u00e6dia*, art. *Transit*, vol.\u00a0xxv. pp.\u00a0129, 130.\n\n  [251] Ibid. art. *Observation*, p.\u00a0390.\n\n  [252] *Nature*, vol.\u00a0i. p.\u00a085.\n\n  [253] *Nature*, vol.\u00a0i. p 337. See references to the Memoirs\n  describing the method.\n\nNewton employed the pendulum for making experiments on the impact of\nballs. Two balls were hung in contact, and one of them, being drawn\naside through a measured arc, was then allowed to strike the other,\nthe arcs of vibration giving sufficient data for calculating the\ndistribution of energy at the moment of impact. The resistance of the\nair was an interfering cause which he estimated very simply by causing\none of the balls to make several complete vibrations without impact\nand then marking the reduction in the lengths of the arcs, a proper\nfraction of which reduction was added to each of the other arcs of\nvibration when impact took place.[254]\n\n  [254] *Principia*, Book I. Law III. Corollary VI. Scholium. Motte\u2019s\n  translation, vol.\u00a0i. p.\u00a033.\n\nThe exact definition of the standard of length is one of the most\nimportant, as it is one of the most difficult questions in physical\nscience, and the different practice of different nations introduces\nneedless confusion. Were all standards constructed so as to give\nthe true length at a fixed uniform temperature, for instance the\nfreezing-point, then any two standards could be compared without the\ninterference of temperature by bringing them both to exactly the same\nfixed temperature. Unfortunately the French metre was defined by a\nbar of platinum at 0\u00b0C, while our yard was defined by a bronze bar at\n62\u00b0F. It is quite impossible, then, to make a comparison of the yard\nand metre without the introduction of a correction, either for the\nexpansion of platinum or bronze, or both. Bars of metal differ too so\nmuch in their rates of expansion according to their molecular condition\nthat it is dangerous to infer from one bar to another.\n\nWhen we come to use instruments with great accuracy there are many\nminute sources of error which must be guarded against. If a thermometer\nhas been graduated when perpendicular, it will read somewhat\ndifferently when laid flat, as the pressure of a column of mercury is\nremoved from the bulb. The reading may also be somewhat altered if it\nhas recently been raised to a higher temperature than usual, if it be\nplaced under a vacuous receiver, or if the tube be unequally heated\nas compared with the bulb. For these minute causes of error we may\nhave to introduce troublesome corrections, unless we adopt the simple\nprecaution of using the thermometer in circumstances of position, &c.,\nexactly similar to those in which it was graduated. There is no end to\nthe number of minute corrections which may ultimately be required. A\nlarge number of experiments on gases, standard weights and measures,\n&c., depend upon the height of the barometer; but when experiments\nin different parts of the world are compared together we ought as a\nfurther refinement to take into account the varying force of gravity,\nwhich even between London and Paris makes a difference of \u00b7008 inch of\nmercury.\n\nThe measurement of quantities of heat is a matter of great difficulty,\nbecause there is no known substance impervious to heat, and the problem\nis therefore as difficult as to measure liquids in porous vessels.\nTo determine the latent heat of steam we must condense a certain\namount of the steam in a known weight of water, and then observe the\nrise of temperature of the water. But while we are carrying out the\nexperiment, part of the heat will escape by radiation and conduction\nfrom the condensing vessel or calorimeter. We may indeed reduce the\nloss of heat by using vessels with double sides and bright surfaces,\nsurrounded with swans-down wool or other non-conducting materials; and\nwe may also avoid raising the temperature of the water much above that\nof the surrounding air. Yet we cannot by any such means render the\nloss of heat inconsiderable. Rumford ingeniously proposed to reduce\nthe loss to zero by commencing the experiment when the temperature\nof the calorimeter is as much below that of the air as it is at the\nend of the experiment above it. Thus the vessel will first gain and\nthen lose by radiation and conduction, and these opposite errors will\napproximately balance each other. But Regnault has shown that the loss\nand gain do not proceed by exactly the same laws, so that in very\naccurate investigations Rumford\u2019s method is not sufficient. There\nremains the method of correction which was beautifully carried out by\nRegnault in his determination of the latent heat of steam. He employed\ntwo calorimeters, made in exactly the same way and alternately used to\ncondense a certain amount of steam, so that while one was measuring\nthe latent heat, the other calorimeter was engaged in determining\nthe corrections to be applied, whether on account of radiation and\nconduction from the vessel or on account of heat reaching the vessel by\nmeans of the connecting pipes.[255]\n\n  [255] Graham\u2019s *Chemical Reports and Memoirs*, Cavendish Society, pp.\n  247, 268, &c.\n\n\n4. *Method of Compensation.*\n\nThere are many cases in which a cause of error cannot conveniently be\nrendered null, and is yet beyond the reach of the third method, that\nof calculating the requisite correction from independent observations.\nThe magnitude of an error may be subject to continual variations, on\naccount of change of weather, or other fickle circumstances beyond\nour control. It may either be impracticable to observe the variation\nof those circumstances in sufficient detail, or, if observed, the\ncalculation of the amount of error may be subject to doubt. In these\ncases, and only in these cases, it will be desirable to invent some\nartificial mode of counterpoising the variable error against an equal\nerror subject to exactly the same variation.\n\nWe cannot weigh an object with great accuracy unless we make a\ncorrection for the weight of the air displaced by the object, and add\nthis to the apparent weight. In very accurate investigations relating\nto standard weights, it is usual to note the barometer and thermometer\nat the time of making a weighing, and, from the measured bulks of\nthe objects compared, to calculate the weight of air displaced; the\nthird method in fact is adopted. To make these calculations in the\nfrequent weighings requisite in chemical analysis would be exceedingly\nlaborious, hence the correction is usually neglected. But when the\nchemist wishes to weigh gas contained in a large glass globe for the\npurpose of determining its specific gravity, the correction becomes of\nmuch importance. Hence chemists avoid at once the error, and the labour\nof correcting it, by attaching to the opposite scale of the balance a\ndummy sealed glass globe of equal capacity to that containing the gas\nto be weighed, noting only the difference of weight when the operating\nglobe is full and empty. The correction, being the same for both\nglobes, may be entirely neglected.[256]\n\n  [256] Regnault\u2019s *Cours El\u00e9mentaire de Chimie*, 1851, vol i. p.\u00a0141.\n\nA device of nearly the same kind is employed in the construction of\ngalvanometers which measure the force of an electric current by the\ndeflection of a suspended magnetic needle. The resistance of the needle\nis partly due to the directive influence of the earth\u2019s magnetism, and\npartly to the torsion of the thread. But the former force may often be\ninconveniently great as well as troublesome to determine for different\ninclinations. Hence it is customary to connect together two equally\nmagnetised needles, with their poles pointing in opposite directions,\none needle being within and another without the coil of wire. As\nregards the earth\u2019s magnetism, the needles are now *astatic* or\nindifferent, the tendency of one needle towards the pole being balanced\nby that of the other.\n\nAn elegant instance of the elimination of a disturbing force by\ncompensation is found in Faraday\u2019s researches upon the magnetism of\ngases. To observe the magnetic attraction or repulsion of a gas seems\nimpossible unless we enclose the gas in an envelope, probably best made\nof glass. But any such envelope is sure to be more or less affected\nby the magnet, so that it becomes difficult to distinguish between\nthree forces which enter into the problem, namely, the magnetism of\nthe gas in question, that of the envelope, and that of the surrounding\natmospheric air. Faraday avoided all difficulties by employing two\nequal and similar glass tubes connected together, and so suspended from\nthe arm of a torsion balance that the tubes were in similar parts of\nthe magnetic field. One tube being filled with nitrogen and the other\nwith oxygen, it was found that the oxygen seemed to be attracted and\nthe nitrogen repelled. The suspending thread of the balance was then\nturned until the force of torsion restored the tubes to their original\nplaces, where the magnetism of the tubes as well as that of the\nsurrounding air, being the same and in the opposite directions upon the\ntwo tubes, could not produce any interference. The force required to\nrestore the tubes was measured by the amount of torsion of the thread,\nand it indicated correctly the difference between the attractive powers\nof oxygen and nitrogen. The oxygen was then withdrawn from one of the\ntubes, and a second experiment made, so as to compare a vacuum with\nnitrogen. No force was now required to maintain the tubes in their\nplaces, so that nitrogen was found to be, approximately speaking,\nindifferent to the magnet, that is, neither magnetic nor diamagnetic,\nwhile oxygen was proved to be positively magnetic.[257] It required\nthe highest experimental skill on the part of Faraday and Tyndall, to\ndistinguish between what is apparent and real in magnetic attraction\nand repulsion.\n\n  [257] Tyndall\u2019s *Faraday*, pp.\u00a0114, 115.\n\nExperience alone can finally decide when a compensating arrangement\nis conducive to accuracy. As a general rule mechanical compensation\nis the last resource, and in the more accurate observations it is\nlikely to introduce more uncertainty than it removes. A multitude\nof instruments involving mechanical compensation have been devised,\nbut they are usually of an unscientific character,[258] because the\nerrors compensated can be more accurately determined and allowed for.\nBut there are exceptions to this rule, and it seems to be proved that\nin the delicate and tiresome operation of measuring a base line,\ninvariable bars, compensated for expansion by heat, give the most\naccurate results. This arises from the fact that it is very difficult\nto determine accurately the temperature of the measuring bars under\nvarying conditions of weather and manipulation.[259] Again, the last\nrefinement in the measurement of time at Greenwich Observatory depends\nupon mechanical compensation. Sir George Airy, observing that the\nstandard clock increased its losing rate 0\u00b730 second for an increase of\none inch in atmospheric pressure, placed a magnet moved by a barometer\nin such a position below the pendulum, as almost entirely to neutralise\nthis cause of irregularity. The thorough remedy, however, would be to\nremove the cause of error altogether by placing the clock in a vacuous\ncase.\n\n  [258] See, for instance, the Compensated Sympiesometer,\n  *Philosophical Magazine*, 4th Series, vol.\u00a0xxxix. p.\u00a0371.\n\n  [259] Grant, *History of Physical Astronomy*, pp.\u00a0146, 147.\n\nWe thus see that the choice of one or other mode of eliminating an\nerror depends entirely upon circumstances and the object in view; but\nwe may safely lay down the following conclusions. First of all, seek\nto avoid the source of error altogether if it can be conveniently\ndone; if not, make the experiment so that the error may be as small,\nbut more especially as constant, as possible. If the means are at hand\nfor determining its amount by calculation from other experiments and\nprinciples of science, allow the error to exist and make a correction\nin the result. If this cannot be accurately done or involves too\nmuch labour for the purposes in view, then throw in a counteracting\nerror which shall as nearly as possible be of equal amount in all\ncircumstances with that to be eliminated. There yet remains, however,\none important method, that of Reversal, which will form an appropriate\ntransition to the succeeding chapters on the Method of Mean Results and\nthe Law of Error.\n\n\n5. *Method of Reversal.*\n\nThe fifth method of eliminating error is most potent and satisfactory\nwhen it can be applied, but it requires that we shall be able to\nreverse the apparatus and mode of procedure, so as to make the\ninterfering cause act alternately in opposite directions. If we can\nget two experimental results, one of which is as much too great as\nthe other is too small, the error is equal to half the difference,\nand the true result is the mean of the two apparent results. It is an\nunavoidable defect of the chemical balance, for instance, that the\npoints of suspension of the pans cannot be fixed at exactly equal\ndistances from the centre of suspension of the beam. Hence two weights\nwhich seem to balance each other will never be quite equal in reality.\nThe difference is detected by reversing the weights, and it may be\nestimated by adding small weights to the deficient side to restore\nequilibrium, and then taking as the true weight the geometric mean\nof the two apparent weights of the same object. If the difference is\nsmall, the arithmetic mean, that is half the sum, may be substituted\nfor the geometric mean, from which it will not appreciably differ.\n\nThis method of reversal is most extensively employed in practical\nastronomy. The apparent elevation of a heavenly body is observed by a\ntelescope moving upon a divided circle, upon which the inclination of\nthe telescope is read off. Now this reading will be erroneous if the\ncircle and the telescope have not accurately the same centre. But if we\nread off at the same time both ends of the telescope, the one reading\nwill be about as much too small as the other is too great, and the\nmean will be nearly free from error. In practice the observation is\ndifferently conducted, but the principle is the same; the telescope is\nfixed to the circle, which moves with it, and the angle through which\nit moves is read off at three, six, or more points, disposed at equal\nintervals round the circle. The older astronomers, down even to the\ntime of Flamsteed, were accustomed to use portions only of a divided\ncircle, generally quadrants, and R\u00f6mer made a vast improvement when he\nintroduced the complete circle.\n\nThe transit circle, employed to determine the meridian passage of\nheavenly bodies, is so constructed that the telescope and the axis\nbearing it, in fact the whole moving part of the instrument, can be\ntaken out of the bearing sockets and turned over, so that what was\nformerly the western pivot becomes the eastern one, and *vice vers\u00e2*.\nIt is impossible that the instrument could have been so perfectly\nconstructed, mounted, and adjusted that the telescope should point\nexactly to the meridian, but the effect of the reversal is that it will\npoint as much to the west in one position as it does to the east in the\nother, and the mean result of observations in the two positions must be\nfree from such cause of error.\n\nThe accuracy with which the inclination of the compass needle can be\ndetermined depends almost entirely on the method of reversal. The\ndip needle consists of a bar of magnetised steel, suspended somewhat\nlike the beam of a delicate balance on a slender axis passing through\nthe centre of gravity of the bar, so that it is at liberty to rest\nin that exact degree of inclination in the magnetic meridian which\nthe magnetism of the earth induces. The inclination is read off\nupon a vertical divided circle, but to avoid error arising from the\ncentring of the needle and circle, both ends are read, and the mean\nof the results is taken. The whole instrument is now turned carefully\nround through 180\u00b0, which causes the needle to assume a new position\nrelatively to the circle and gives two new readings, in which any error\ndue to the wrong position of the zero of the division will be reversed.\nAs the axis of the needle may not be exactly horizontal, it is now\nreversed in the same manner as the transit instrument, the end of the\naxis which formerly pointed east being made to point west, and a new\nset of four readings is taken.\n\nFinally, error may arise from the axis not passing accurately through\nthe centre of gravity of the bar, and this error can only be detected\nand eliminated on changing the magnetic poles of the bar by the\napplication of a strong magnet. The error is thus made to act in\nopposite directions. To ensure all possible accuracy each reversal\nought to be combined with each other reversal, so that the needle will\nbe observed in eight different positions by sixteen readings, the mean\nof the whole of which will give the required inclination free from all\neliminable errors.[260]\n\n  [260] Quetelet, *Sur la Physique du Globe*, p.\u00a0174. Jamin, *Cours de\n  Physique*, vol.\u00a0i. p.\u00a0504.\n\nThere are certain cases in which a disturbing cause can with ease\nbe made to act in opposite directions, in alternate observations,\nso that the mean of the results will be free from disturbance. Thus\nin direct experiments upon the velocity of sound in passing through\nthe air between stations two or three miles apart, the wind is a\ncause of error. It will be well, in the first place, to choose a\ntime for the experiment when the air is very nearly at rest, and the\ndisturbance slight, but if at the same moment signal sounds be made\nat each station and observed at the other, two sounds will be passing\nin opposite directions through the same body of air and the wind will\naccelerate one sound almost exactly as it retards the other. Again, in\ntrigonometrical surveys the apparent height of a point will be affected\nby atmospheric refraction and the curvature of the earth. But if in\nthe case of two points the apparent elevation of each as seen from\nthe other be observed, the corrections will be the same in amount,\nbut reversed in direction, and the mean between the two apparent\ndifferences of altitude will give the true difference of level.\n\nIn the next two chapters we really pursue the Method of Reversal into\nmore complicated applications.\n\n\n\n\nCHAPTER XVI.\n\nTHE METHOD OF MEANS.\n\n\nAll results of the measurement of continuous quantity can be only\napproximately true. Were this assertion doubted, it could readily be\nproved by direct experience. If any person, using an instrument of\nthe greatest precision, makes and registers successive observations\nin an unbiassed manner, it will almost invariably be found that the\nresults differ from each other. When we operate with sufficient care\nwe cannot perform so simple an experiment as weighing an object in a\ngood balance without getting discrepant numbers. Only the rough and\ncareless experimenter will think that his observations agree, but\nin reality he will be found to overlook the differences. The most\nelaborate researches, such as those undertaken in connection with\nstandard weights and measures, always render it apparent that complete\ncoincidence is out of the question, and that the more accurate our\nmodes of observation are rendered, the more numerous are the sources\nof minute error which become apparent. We may look upon the existence\nof error in all measurements as the normal state of things. It is\nabsolutely impossible to eliminate separately the multitude of small\ndisturbing influences, except by balancing them off against each other.\nEven in drawing a mean it is to be expected that we shall come near\nthe truth rather than exactly to it. In the measurement of continuous\nquantity, absolute coincidence, if it seems to occur, must be only\napparent, and is no indication of precision. It is one of the most\nembarrassing things we can meet when experimental results agree\ntoo closely. Such coincidences should raise our suspicion that the\napparatus in use is in some way restricted in its operation, so as\nnot really to give the true result at all, or that the actual results\nhave not been faithfully recorded by the assistant in charge of the\napparatus.\n\nIf then we cannot get twice over exactly the same result, the question\narises, How can we ever attain the truth or select the result which may\nbe supposed to approach most nearly to it? The quantity of a certain\nphenomenon is expressed in several numbers which differ from each\nother; no more than one of them at the most can be true, and it is more\nprobable that they are all false. It may be suggested, perhaps, that\nthe observer should select the one observation which he judged to be\nthe best made, and there will often doubtless be a feeling that one\nor more results were satisfactory, and the others less trustworthy.\nThis seems to have been the course adopted by the early astronomers.\nFlamsteed, when he had made several observations of a star, probably\nchose in an arbitrary manner that which seemed to him nearest to the\ntruth.[261]\n\n  [261] Baily\u2019s *Account of Flamsteed*, p.\u00a0376.\n\nWhen Horrocks selected for his estimate of the sun\u2019s semi-diameter a\nmean between the results of Kepler and Tycho, he professed not to do\nit from any regard to the idle adage, \u201cMedio tutissimus ibis,\u201d but\nbecause he thought it from his own observations to be correct.[262] But\nthis method will not apply at all when the observer has made a number\nof measurements which are equally good in his opinion, and it is quite\napparent that in using an instrument or apparatus of considerable\ncomplication the observer will not necessarily be able to judge whether\nslight causes have affected its operation or not.\n\n  [262] *The Transit of Venus across the Sun*, by Horrocks, London,\n  1859, p. 146.\n\nIn this question, as indeed throughout inductive logic, we deal only\nwith probabilities. There is no infallible mode of arriving at the\nabsolute truth, which lies beyond the reach of human intellect, and\ncan only be the distant object of our long-continued and painful\napproximations. Nevertheless there is a mode pointed out alike by\ncommon sense and the highest mathematical reasoning, which is more\nlikely than any other, as a general rule, to bring us near the truth.\nThe \u1f04\u03c1\u03b9\u03c3\u03c4\u03bf\u03bd \u03bc\u03ad\u03c4\u03c1\u03bf\u03bd, or the *aurea mediocritas*, was highly esteemed\nin the ancient philosophy of Greece and Rome; but it is not probable\nthat any of the ancients should have been able clearly to analyse and\nexpress the reasons why they advocated the *mean* as the safest course.\nBut in the last two centuries this apparently simple question of the\nmean has been found to afford a field for the exercise of the utmost\nmathematical skill. Roger Cotes, the editor of the *Principia*, appears\nto have had some insight into the value of the mean; but profound\nmathematicians such as De Moivre, Daniel Bernoulli, Laplace, Lagrange,\nGauss, Quetelet, De Morgan, Airy, Leslie Ellis, Boole, Glaisher, and\nothers, have hardly exhausted the subject.\n\n\n*Several uses of the Mean Result.*\n\nThe elimination of errors of unknown sources, is almost always\naccomplished by the simple arithmetical process of taking the *mean*,\nor, as it is often called, the *average* of several discrepant numbers.\nTo take an average is to add the several quantities together, and\ndivide by the number of quantities thus added, which gives a quotient\nlying among, or in the *middle* of, the several quantities. Before\nhowever inquiring fully into the grounds of this procedure, it is\nessential to observe that this one arithmetical process is really\napplied in at least three different cases, for different purposes, and\nupon different principles, and we must take great care not to confuse\none application of the process with another. A *mean result*, then, may\nhave any one of the following significations.\n\n(1) It may give a merely representative number, expressing the general\nmagnitude of a series of quantities, and serving as a convenient mode\nof comparing them with other series of quantities. Such a number is\nproperly called *The fictitious mean* or *The average result*.\n\n(2) It may give a result approximately free from disturbing quantities,\nwhich are known to affect some results in one direction, and other\nresults equally in the opposite direction. We may say that in this case\nwe get a *Precise mean result*.\n\n(3) It may give a result more or less free from unknown and uncertain\nerrors; this we may call the *Probable mean result*.\n\nOf these three uses of the mean the first is entirely different in\nnature from the two last, since it does not yield an approximation\nto any natural quantity, but furnishes us with an arithmetic result\ncomparing the aggregate of certain quantities with their number. The\nthird use of the mean rests entirely upon the theory of probability,\nand will be more fully considered in a later part of this chapter. The\nsecond use is closely connected, or even identical with, the Method of\nReversal already described, but it will be desirable to enter somewhat\nfully into all the three employments of the same arithmetical process.\n\n\n*The Mean and the Average.*\n\nMuch confusion exists in the popular, or even the scientific employment\nof the terms *mean* and *average*, and they are commonly taken as\nsynonymous. It is necessary to ascertain carefully what significations\nwe ought to attach to them. The English word *mean* is equivalent to\n*medium*, being derived, perhaps through the French *moyen*, from the\nLatin *medius*, which again is undoubtedly kindred with the Greek\n\u03bc\u03b5\u03c3\u03bf\u03c2. Etymologists believe, too, that this Greek word is connected\nwith the preposition \u03bc\u03b5\u03c4\u03b1, the German *mitte*, and the true English\n*mid* or *middle*; so that after all the *mean* is a technical term\nidentical in its root with the more popular equivalent *middle*.\n\nIf we inquire what is the mean in a mathematical point of view, the\ntrue answer is that there are several or many kinds of means. The old\narithmeticians recognised ten kinds, which are stated by Boethius, and\nan eleventh was added by Jordanus.[263]\n\n  [263] De Morgan, Supplement to the *Penny Cyclop\u00e6dia*, art. *Old\n  Appellations of Numbers*.\n\nThe *arithmetic mean* is the one by far the most commonly denoted by\nthe term, and that which we may understand it to signify in the absence\nof any qualification. It is the sum of a series of quantities divided\nby their number, and may be represented by the formula 1/2(*a*\u00a0+\u00a0*b*).\nBut there is also the *geometric mean*, which is the square root of the\nproduct, \u221a(*a*\u00a0\u00d7\u00a0*b*), or that quantity the logarithm of which\nis the arithmetic mean of the logarithms of the quantities. There is\nalso the *harmonic mean*, which is the reciprocal of the arithmetic\nmean of the reciprocals of the quantities. Thus if *a* and *b* be the\nquantities, as before, their reciprocals are 1/*a* and 1/*b*, the\nmean of which is 1/2\u00a0(1/*a*\u00a0+\u00a01/*b*), and the reciprocal again is\n(2*ab*)/(*a*\u00a0+\u00a0*b*), which is the harmonic mean. Other kinds of means\nmight no doubt be invented for particular purposes, and we might apply\nthe term, as De Morgan pointed out,[264] to any quantity a function of\nwhich is equal to a function of two or more other quantities, and is\nsuch that the interchange of these latter quantities among themselves\nwill make no alteration in the value of the function. Symbolically, if\n\u03a6(*y*, *y*, *y*\u00a0....)\u00a0= \u03a6(*x*_{1}, *x*_{2}, *x*_{3}\u00a0....), then *y* is\na kind of mean of the quantities, *x*_{1}, *x*_{2}, &c.\n\n  [264] *Penny Cyclop\u00e6dia*, art. *Mean*.\n\nThe geometric mean is necessarily adopted in certain cases. When we\nestimate the work done against a force which varies inversely as\nthe square of the distance from a fixed point, the mean force is\nthe geometric mean between the forces at the beginning and end of\nthe path. When in an imperfect balance, we reverse the weights to\neliminate error, the true weight will be the geometric mean of the two\napparent weights. In almost all the calculations of statistics and\ncommerce the geometric mean ought, strictly speaking, to be used. If a\ncommodity rises in price 100 per cent. and another remains unaltered,\nthe mean rise of a price is not 50 per cent. because the ratio\n150\u00a0:\u00a0200 is not the same as 100\u00a0:\u00a0150. The mean ratio is as unity to\n\u221a(1\u00b700\u00a0\u00d7\u00a02\u00b700) or 1 to 1\u00b741. The difference between the three\nkinds of means in such a case[265] is very considerable; while the\nrise of price estimated by the Arithmetic mean would be 50 per cent.\nit would be only 41 and 33 per cent. respectively according to the\nGeometric and Harmonic means.\n\n  [265] Jevons, *Journal of the Statistical Society*, June 1865,\n  vol.\u00a0xxviii, p. 296.\n\nIn all calculations concerning the average rate of progress of a\ncommunity, or any of its operations, the geometric mean should be\nemployed. For if a quantity increases 100 per cent. in 100 years, it\nwould not on the average increase 10 per cent. in each ten years, as\nthe 10 per cent. would at the end of each decade be calculated upon\nlarger and larger quantities, and give at the end of 100 years much\nmore than 100 per cent., in fact as much as 159 per cent. The true mean\nrate in each decade would be ^{10}\u221a2 or about 1\u00b707, that is, the\nincrease would be about 7 per cent. in each ten years. But when the\nquantities differ very little, the arithmetic and geometric means are\napproximately the same. Thus the arithmetic mean of 1\u00b7000 and 1\u00b7001\nis 1\u00b70005, and the geometric mean is about 1\u00b70004998, the difference\nbeing of an order inappreciable in almost all scientific and practical\nmatters. Even in the comparison of standard weights by Gauss\u2019 method\nof reversal, the arithmetic mean may usually be substituted for the\ngeometric mean which is the true result.\n\nRegarding the mean in the absence of express qualification to the\ncontrary as the common arithmetic mean, we must still distinguish\nbetween its two uses where it gives with more or less accuracy and\nprobability a really existing quantity, and where it acts as a mere\nrepresentative of other quantities. If I make many experiments to\ndetermine the atomic weight of an element, there is a certain number\nwhich I wish to approximate to, and the mean of my separate results\nwill, in the absence of any reasons to the contrary, be the most\nprobable approximate result. When we determine the mean density of\nthe earth, it is not because any part of the earth is of that exact\ndensity; there may be no part exactly corresponding to the mean\ndensity, and as the crust of the earth has only about half the mean\ndensity, the internal matter of the globe must of course be above the\nmean. Even the density of a homogeneous substance like carbon or gold\nmust be regarded as a mean between the real density of its atoms, and\nthe zero density of the intervening vacuous space.\n\nThe very different signification of the word \u201cmean\u201d in these two uses\nwas fully explained by Quetelet,[266] and the importance of the\ndistinction was pointed out by Sir John Herschel in reviewing his\nwork.[267] It is much to be desired that scientific men would mark the\ndifference by using the word *mean* only in the former sense when it\ndenotes approximation to a definite existing quantity; and *average*,\nwhen the mean is only a fictitious quantity, used for convenience\nof thought and expression. The etymology of this word \u201caverage\u201d is\nsomewhat obscure; but according to De Morgan[268] it comes from\n*averia*, \u201chavings or possessions,\u201d especially applied to farm stock.\nBy the accidents of language *averagium* came to mean the labour of\nfarm horses to which the lord was entitled, and it probably acquired in\nthis manner the notion of distributing a whole into parts, a sense in\nwhich it was early applied to maritime averages or contributions of the\nother owners of cargo to those whose goods have been thrown overboard\nor used for the safety of the vessel.\n\n  [266] *Letters on the Theory of Probabilities*, transl. by Downes,\n  Part ii.\n\n  [267] Herschel\u2019s *Essays*, &c. pp.\u00a0404, 405.\n\n  [268] *On the Theory of Errors of Observations, Cambridge\n  Philosophical Transactions*, vol.\u00a0x. Part ii. 416.\n\n\n*On the Average or Fictitious Mean.*\n\nAlthough the average when employed in its proper sense of a fictitious\nmean, represents no really existing quantity, it is yet of the\nhighest scientific importance, as enabling us to conceive in a single\nresult a multitude of details. It enables us to make a hypothetical\nsimplification of a problem, and avoid complexity without committing\nerror. The weight of a body is the sum of the weights of infinitely\nsmall particles, each acting at a different place, so that a mechanical\nproblem resolves itself, strictly speaking, into an infinite number\nof distinct problems. We owe to Archimedes the first introduction of\nthe beautiful idea that one point may be discovered in a gravitating\nbody such that the weight of all the particles may be regarded as\nconcentrated in that point, and yet the behaviour of the whole body\nwill be exactly represented by the behaviour of this heavy point. This\nCentre of Gravity may be within the body, as in the case of a sphere,\nor it may be in empty space, as in the case of a ring. Any two bodies,\nwhether connected or separate, may be conceived as having a centre of\ngravity, that of the sun and earth lying within the sun and only 267\nmiles from its centre.\n\nAlthough we most commonly use the notion of a centre or average point\nwith regard to gravity, the same notion is applicable to other cases.\nTerrestrial gravity is a case of approximately parallel forces, and\nthe centre of gravity is but a special case of the more general Centre\nof Parallel Forces. Wherever a number of forces of whatever amount\nact in parallel lines, it is possible to discover a point at which\nthe algebraic sum of the forces may be imagined to act with exactly\nthe same effect. Water in a cistern presses against the side with a\npressure varying according to the depth, but always in a direction\nperpendicular to the side. We may then conceive the whole pressure\nas exerted on one point, which will be one-third from the bottom of\nthe cistern, and may be called the Centre of Pressure. The Centre\nof Oscillation of a pendulum, discovered by Huyghens, is that point\nat which the whole weight of the pendulum may be considered as\nconcentrated, without altering the time of oscillation (p.\u00a0315). When\none body strikes another the Centre of Percussion is that point in\nthe striking body at which all its mass might be concentrated without\naltering the effect of the stroke. In position the Centre of Percussion\ndoes not differ from the Centre of Oscillation. Mathematicians have\nalso described the Centre of Gyration, the Centre of Conversion, the\nCentre of Friction, &c.\n\nWe ought carefully to distinguish between those cases in which an\n*invariable* centre can be assigned, and those in which it cannot.\nIn perfect strictness, there is no such thing as a true invariable\ncentre of gravity. As a general rule a body is capable of possessing an\ninvariable centre only for perfectly parallel forces, and gravity never\ndoes act in absolutely parallel lines. Thus, as usual, we find that our\nconceptions are only hypothetically correct, and only approximately\napplicable to real circumstances. There are indeed certain geometrical\nforms called *Centrobaric*,[269] such that a body of that shape would\nattract another exactly as if the mass were concentrated at the centre\nof gravity, whether the forces act in a parallel manner or not.\nNewton showed that uniform spheres of matter have this property,\nand this truth proved of the greatest importance in simplifying his\ncalculations. But it is after all a purely hypothetical truth, because\nwe can nowhere meet with, nor can we construct, a perfectly spherical\nand homogeneous body. The slightest irregularity or protrusion from\nthe surface will destroy the rigorous correctness of the assumption.\nThe spheroid, on the other hand, has no invariable centre at which its\nmass may always be regarded as concentrated. The point from which its\nresultant attraction acts will move about according to the distance\nand position of the other attracting body, and it will only coincide\nwith the centre as regards an infinitely distant body whose attractive\nforces may be considered as acting in parallel lines.\n\n  [269] Thomson and Tait, *Treatise on Natural Philosophy*, vol.\u00a0i.\n  p.\u00a0394.\n\nPhysicists speak familiarly of the poles of a magnet, and the term\nmay be used with convenience. But, if we attach any definite meaning\nto the word, the poles are not the ends of the magnet, nor any fixed\npoints within, but the variable points from which the resultants of\nall the forces exerted by the particles in the bar upon exterior\nmagnetic particles may be considered as acting. The poles are, in\nshort, Centres of Magnetic Forces; but as those forces are never\nreally parallel, these centres will vary in position according to\nthe relative place of the object attracted. Only when we regard the\nmagnet as attracting a very distant, or, strictly speaking, infinitely\ndistant particle, do its centres become fixed points, situated in short\nmagnets approximately at one-sixth of the whole length from each end of\nthe bar. We have in the above instances of centres or poles of force\nsufficient examples of the mode in which the Fictitious Mean or Average\nis employed in physical science.\n\n\n*The Precise Mean Result.*\n\nWe now turn to that mode of employing the mean result which is\nanalogous to the method of reversal, but which is brought into practice\nin a most extensive manner throughout many branches of physical\nscience. We find the simplest possible case in the determination of\nthe latitude of a place by observations of the Pole-star. Tycho Brahe\nsuggested that if the elevation of any circumpolar star were observed\nat its higher and lower passages across the meridian, half the sum of\nthe elevations would be the latitude of the place, which is equal to\nthe height of the pole. Such a star is as much above the pole at its\nhighest passage, as it is below at its lowest, so that the mean must\nnecessarily give the height of the pole itself free from doubt, except\nas regards incidental errors. The Pole-star is usually selected for the\npurpose of such observations because it describes the smallest circle,\nand is thus on the whole least affected by atmospheric refraction.\n\nWhenever several causes are in action, each of which at one time\nincreases and at another time decreases the joint effect by equal\nquantities, we may apply this method and disentangle the effects. Thus\nthe solar and lunar tides roll on in almost complete independence of\neach other. When the moon is new or full the solar tide coincides, or\nnearly so, with that caused by the moon, and the joint effect is the\nsum of the separate effects. When the moon is in quadrature, or half\nfull, the two tides are acting in opposition, one raising and the other\ndepressing the water, so that we observe only the difference of the\neffects. We have in fact--\n\n  Spring tide = lunar tide + solar tide;\n  Neap tide   = lunar tide - solar tide.\n\nWe have only then to add together the heights of the maximum spring\ntide and the minimum neap tide, and half the sum is the true height of\nthe lunar tide. Half the difference of the spring and neap tides on the\nother hand gives the solar tide.\n\nEffects of very small amount may be detected with great approach to\ncertainty among much greater fluctuations, provided that we have\na series of observations sufficiently numerous and long continued\nto enable us to balance all the larger effects against each other.\nFor this purpose the observations should be continued over at least\none complete cycle, in which the effects run through all their\nvariations, and return exactly to the same relative positions as at the\ncommencement. If casual or irregular disturbing causes exist, we should\nprobably require many such cycles of results to render their effect\ninappreciable. We obtain the desired result by taking the mean of all\nthe observations in which a cause acts positively, and the mean of all\nin which it acts negatively. Half the difference of these means will\ngive the effect of the cause in question, provided that no other effect\nhappens to vary in the same period or nearly so.\n\nSince the moon causes a movement of the ocean, it is evident that\nits attraction must have some effect upon the atmosphere. The laws\nof atmospheric tides were investigated by Laplace, but as it would\nbe impracticable by theory to calculate their amounts we can only\ndetermine them by observation, as Laplace predicted that they would\none day be determined.[270] But the oscillations of the barometer thus\ncaused are far smaller than the oscillations due to several other\ncauses. Storms, hurricanes, or changes of weather produce movements\nof the barometer sometimes as much as a thousand times as great as\nthe tides in question. There are also regular daily, yearly, or other\nfluctuations, all greater than the desired quantity. To detect and\nmeasure the atmospheric tide it was desirable that observations should\nbe made in a place as free as possible from irregular disturbances.\nOn this account several long series of observations were made at St.\nHelena, where the barometer is far more regular in its movements than\nin a continental climate. The effect of the moon\u2019s attraction was\nthen detected by taking the mean of all the readings when the moon\nwas on the meridian and the similar mean when she was on the horizon.\nThe difference of these means was found to be only \u00b700365, yet it was\npossible to discover even the variation of this tide according as the\nmoon was nearer to or further from the earth, though this difference\nwas only \u00b700056 inch.[271] It is quite evident that such minute effects\ncould never be discovered in a purely empirical manner. Having no\ninformation but the series of observations before us, we could have\nno clue as to the mode of grouping them which would give so small a\ndifference. In applying this method of means in an extensive manner\nwe must generally then have *\u00e0 priori* knowledge as to the periods at\nwhich a cause will act in one direction or the other.\n\n  [270] *Essai Philosophique sur les Probabilit\u00e9s*, pp.\u00a049, 50.\n\n  [271] Grant, *History of Physical Astronomy*, p.\u00a0163.\n\nWe are sometimes able to eliminate fluctuations and take a mean result\nby purely mechanical arrangements. The daily variations of temperature,\nfor instance, become imperceptible one or two feet below the surface\nof the earth, so that a thermometer placed with its bulb at that\ndepth gives very nearly the true daily mean temperature. At a depth\nof twenty feet even the yearly fluctuations are nearly effaced, and\nthe thermometer stands a little above the true mean temperature of the\nlocality. In registering the rise and fall of the tide by a tide-gauge,\nit is desirable to avoid the oscillations arising from surface waves,\nwhich is very readily accomplished by placing the float in a cistern\ncommunicating by a small hole with the sea. Only a general rise or\nfall of the level is then perceptible, just as in the marine barometer\nthe narrow tube prevents any casual fluctuations and allows only a\ncontinued change of pressure to manifest itself.\n\n\n*Determination of the Zero point.*\n\nIn many important observations the chief difficulty consists in\ndefining exactly the zero point from which we are to measure. We can\npoint a telescope with great precision to a star and can measure to\na second of arc the angle through which the telescope is raised or\nlowered; but all this precision will be useless unless we know exactly\nthe centre point of the heavens from which we measure, or, what comes\nto the same thing, the horizontal line 90\u00b0 distant from it. Since the\ntrue horizon has reference to the figure of the earth at the place of\nobservation, we can only determine it by the direction of gravity,\nas marked either by the plumb-line or the surface of a liquid. The\nquestion resolves itself then into the most accurate mode of observing\nthe direction of gravity, and as the plumb-line has long been found\nhopelessly inaccurate, astronomers generally employ the surface of\nmercury in repose as the criterion of horizontality. They ingeniously\nobserve the direction of the surface by making a star the index. From\nthe laws of reflection it follows that the angle between the direct\nray from a star and that reflected from a surface of mercury will\nbe exactly double the angle between the surface and the direct ray\nfrom the star. Hence the horizontal or zero point is the mean between\nthe apparent place of any star or other very distant object and its\nreflection in mercury.\n\nA plumb-line is perpendicular, or a liquid surface is horizontal only\nin an approximate sense; for any irregularity of the surface of the\nearth, a mountain, or even a house must cause some deviation by its\nattracting power. To detect such deviation might seem very difficult,\nbecause every other plumb-line or liquid surface would be equally\naffected by gravity. Nevertheless it can be detected; for if we place\none plumb-line to the north of a mountain, and another to the south,\nthey will be about equally deflected in opposite directions, and if\nby observations of the same star we can measure the angle between the\nplumb-lines, half the inclination will be the deviation of either,\nafter allowance has been made for the inclination due to the difference\nof latitude of the two places of observation. By this mode of\nobservation applied to the mountain Schiehallion the deviation of the\nplumb-line was accurately measured by Maskelyne, and thus a comparison\ninstituted between the attractive forces of the mountain and the whole\nglobe, which led to a probable estimate of the earth\u2019s density.\n\nIn some cases it is actually better to determine the zero point by the\naverage of equally diverging quantities than by direct observation. In\ndelicate weighings by a chemical balance it is requisite to ascertain\nexactly the point at which the beam comes to rest, and when standard\nweights are being compared the position of the beam is ascertained\nby a carefully divided scale viewed through a microscope. But when\nthe beam is just coming to rest, friction, small impediments or other\naccidental causes may readily obstruct it, because it is near the\npoint at which the force of stability becomes infinitely small. Hence\nit is found better to let the beam vibrate and observe the terminal\npoints of the vibrations. The mean between two extreme points will\nnearly indicate the position of rest. Friction and the resistance of\nair tend to reduce the vibrations, so that this mean will be erroneous\nby half the amount of this effect during a half vibration. But by\ntaking several observations we may determine this retardation and\nallow for it. Thus if *a*, *b*, *c* be the readings of the terminal\npoints of three excursions of the beam from the zero of the scale,\nthen 1/2(*a*\u00a0+\u00a0*b*) will be about as much erroneous in one direction\nas 1/2(*b*\u00a0+\u00a0*c*) in the other, so that the mean of these two means,\nor 1/4(*a*\u00a0+\u00a02*b*\u00a0+\u00a0*c*), will be exceedingly near to the point of\nrest.[272] A still closer approximation may be made by taking four\nreadings and reducing them by the formula 1/6(*a*\u00a0+\u00a02*b*\u00a0+\u00a02*c*\u00a0+\u00a0*d*).\n\n  [272] Gauss, Taylor\u2019s *Scientific Memoirs*, vol.\u00a0ii. p.\u00a043, &c.\n\nThe accuracy of Baily\u2019s experiments, directed to determine the\ndensity of the earth, entirely depended upon this mode of observing\noscillations. The balls whose gravitation was measured were so\ndelicately suspended by a torsion balance that they never came to\nrest. The extreme points of the oscillations were observed both when\nthe heavy leaden attracting ball was on one side and on the other. The\ndifference of the mean points when the leaden ball was on the right\nhand and that when it was on the left hand gave double the amount of\nthe deflection.\n\nA beautiful instance of avoiding the use of a zero point is found in\nMr. E.\u00a0J. Stone\u2019s observations on the radiant heat of the fixed stars.\nThe difficulty of these observations arose from the comparatively great\namounts of heat which were sent into the telescope from the atmosphere,\nand which were sufficient to disguise almost entirely the feeble heat\nrays of a star. But Mr. Stone fixed at the focus of his telescope a\ndouble thermo-electric pile of which the two parts were reversed in\norder. Now any disturbance of temperature which acted uniformly upon\nboth piles produced no effect upon the galvanometer needle, and when\nthe rays of the star were made to fall alternately upon one pile and\nthe other, the total amount of the deflection represented double the\nheating power of the star. Thus Mr. Stone was able to detect with\nmuch certainty a heating effect of the star Arcturus, which even when\nconcentrated by the telescope amounted only to 0\u00b0\u00b702 Fahr., and which\nrepresents a heating effect of the direct ray of only about 0\u00b0\u00b700000137\nFahr., equivalent to the heat which would be received from a three-inch\ncubic vessel full of boiling water at the distance of 400 yards.[273]\nIt is probable that Mr. Stone\u2019s arrangement of the pile might be\nusefully employed in other delicate thermometric experiments subject to\nconsiderable disturbing influences.\n\n  [273] *Proceedings of the Royal Society*, vol.\u00a0xviii. p.\u00a0159 (Jan.\n  13, 1870). *Philosophical Magazine* (4th Series), vol.\u00a0xxxix. p.\u00a0376.\n\n\n*Determination of Maximum Points.*\n\nWe employ the method of means in a certain number of observations\ndirected to determine the moment at which a phenomenon reaches its\nhighest point in quantity. In noting the place of a fixed star at a\ngiven time there is no difficulty in ascertaining the point to be\nobserved, for a star in a good telescope presents an exceedingly small\ndisc. In observing a nebulous body which from a bright centre fades\ngradually away on all sides, it will not be possible to select with\ncertainty the middle point. In many such cases the best method is not\nto select arbitrarily the supposed middle point, but points of equal\nbrightness on either side, and then take the mean of the observations\nof these two points for the centre. As a general rule, a variable\nquantity in reaching its maximum increases at a less and less rate,\nand after passing the highest point begins to decrease by insensible\ndegrees. The maximum may indeed be defined as that point at which\nthe increase or decrease is null. Hence it will usually be the most\nindefinite point, and if we can accurately measure the phenomenon we\nshall best determine the place of the maximum by determining points on\neither side at which the ordinates are equal. There is moreover this\nadvantage in the method that several points may be determined with the\ncorresponding ones on the other side, and the mean of the whole taken\nas the true place of the maximum. But this method entirely depends upon\nthe existence of symmetry in the curve, so that of two equal ordinates\none shall be as far on one side of the maximum as the other is on the\nother side. The method fails when other laws of variation prevail.\n\nIn tidal observations great difficulty is encountered in fixing\nthe moment of high water, because the rate at which the water is\nthen rising or falling, is almost imperceptible. Whewell proposed,\ntherefore, to note the time at which the water passes a fixed point\nsomewhat below the maximum both in rising and falling, and take\nthe mean time as that of high water. But this mode of proceeding\nunfortunately does not give a correct result, because the tide follows\ndifferent laws in rising and in falling. There is a difficulty again in\nselecting the highest spring tide, another object of much importance in\ntidology. Laplace discovered that the tide of the second day preceding\nthe conjunction of the sun and moon is nearly equal to that of the\nfifth day following; and, believing that the increase and decrease of\nthe tides proceeded in a nearly symmetrical manner, he decided that the\nhighest tide would occur about thirty-six hours after the conjunction,\nthat is half-way between the second day before and the fifth day\nafter.[274]\n\n  [274] Airy *On Tides and Waves*, Encycl. Metrop. pp.\u00a0364*-366*.\n\nThis method is also employed in determining the time of passage of the\nmiddle or densest point of a stream of meteors. The earth takes two\nor three days in passing completely through the November stream; but\nastronomers need for their calculations to have some definite point\nfixed within a few minutes if possible. When near to the middle they\nobserve the numbers of meteors which come within the sphere of vision\nin each half hour, or quarter hour, and then, assuming that the law of\nvariation is symmetrical, they select a moment for the passage of the\ncentre equidistant between times of equal frequency.\n\nThe eclipses of Jupiter\u2019s satellites are not only of great interest as\nregards the motions of the satellites themselves, but were, and perhaps\nstill are, of use in determining longitudes, because they are events\noccurring at fixed moments of absolute time, and visible in all parts\nof the planetary system at the same time, allowance being made for the\ninterval occupied by the light in travelling. But, as is explained by\nHerschel,[275] the moment of the event is wanting in definiteness,\npartly because the long cone of Jupiter\u2019s shadow is surrounded by\na penumbra, and partly because the satellite has itself a sensible\ndisc, and takes time in entering the shadow. Different observers using\ndifferent telescopes would usually select different moments for that\nof the eclipse. But the increase of light in the emersion will proceed\naccording to a law the reverse of that observed in the immersion,\nso that if an observer notes the time of both events with the same\ntelescope, he will be as much too soon in one observation as he is too\nlate in the other, and the mean moment of the two observations will\nrepresent with considerable accuracy the time when the satellite is in\nthe middle of the shadow. Error of judgment of the observer is thus\neliminated, provided that he takes care to act at the emersion as he\ndid at the immersion.\n\n  [275] *Outlines of Astronomy*, 4th edition, \u00a7 538.\n\n\n\n\nCHAPTER XVII.\n\nTHE LAW OF ERROR.\n\n\nTo bring error itself under law might seem beyond human power. He who\nerrs surely diverges from law, and it might be deemed hopeless out of\nerror to draw truth. One of the most remarkable achievements of the\nhuman intellect is the establishment of a general theory which not\nonly enables us among discrepant results to approximate to the truth,\nbut to assign the degree of probability which fairly attaches to this\nconclusion. It would be a mistake indeed to suppose that this law is\nnecessarily the best guide under all circumstances. Every measuring\ninstrument and every form of experiment may have its own special law\nof error; there may in one instrument be a tendency in one direction\nand in another in the opposite direction. Every process has its\npeculiar liabilities to disturbance, and we are never relieved from the\nnecessity of providing against special difficulties. The general Law of\nError is the best guide only when we have exhausted all other means of\napproximation, and still find discrepancies, which are due to unknown\ncauses. We must treat such residual differences in some way or other,\nsince they will occur in all accurate experiments, and as their origin\nis assumed to be unknown, there is no reason why we should treat them\ndifferently in different cases. Accordingly the ultimate Law of Error\nmust be a uniform and general one.\n\nIt is perfectly recognised by mathematicians that in each case a\nspecial Law of Error may exist, and should be discovered if possible.\n\u201cNothing can be more unlikely than that the errors committed in all\nclasses of observations should follow the same law,\u201d[276] and the\nspecial Laws of Error which will apply to certain instruments, as for\ninstance the repeating circle, have been investigated by Bravais.[277]\nHe concludes that every distinct cause of error gives rise to a curve\nof possibility of errors, which may have any form,--a curve which\nwe may either be able or unable to discover, and which in the first\ncase may be determined by *\u00e0 priori* considerations on the peculiar\nnature of this cause, or which may be determined *\u00e0 posteriori* by\nobservation. Whenever it is practicable and worth the labour, we ought\nto investigate these special conditions of error; nevertheless, when\nthere are a great number of different sources of minute error, the\ngeneral resultant will always tend to obey that general law which we\nare about to consider.\n\n  [276] *Philosophical Magazine*, 3rd Series, vol.\u00a0xxxvii. p.\u00a0324.\n\n  [277] *Letters on the Theory of Probabilities*, by Quetelet,\n  translated by O.\u00a0G. Downes, Notes to Letter XXVI. pp.\u00a0286\u2013295.\n\n\n*Establishment of the Law of Error.*\n\nMathematicians agree far better as to the form of the Law of Error\nthan they do as to the manner in which it can be deduced and proved.\nThey agree that among a number of discrepant results of observation,\nthat mean quantity is probably the best approximation to the truth\nwhich makes the sum of the squares of the errors as small as possible.\nBut there are three principal ways in which this law has been arrived\nat respectively by Gauss, by Laplace and Quetelet, and by Sir John\nHerschel. Gauss proceeds much upon assumption; Herschel rests upon\ngeometrical considerations; while Laplace and Quetelet regard the Law\nof Error as a development of the doctrine of combinations. A number\nof other mathematicians, such as Adrain of New Brunswick, Bessel,\nIvory, Donkin, Leslie Ellis, Tait, and Crofton have either attempted\nindependent proofs or have modified or commented on those here to be\ndescribed. For full accounts of the literature of the subject the\nreader should refer either to Mr. Todhunter\u2019s *History of the Theory of\nProbability* or to the able memoir of Mr. J.\u00a0W.\u00a0L. Glaisher.[278]\n\n  [278] *On the Law of Facility of Errors of Observations, and on the\n  Method of Least Squares*, Memoirs of the Royal Astronomical Society,\n  vol.\u00a0xxxix. p.\u00a075.\n\nAccording to Gauss the Law of Error expresses the comparative\nprobability of errors of various magnitude, and partly from experience,\npartly from *\u00e0 priori* considerations, we may readily lay down certain\nconditions to which the law will certainly conform. It may fairly\nbe assumed as a first principle to guide us in the selection of the\nlaw, that large errors will be far less frequent and probable than\nsmall ones. We know that very large errors are almost impossible, so\nthat the probability must rapidly decrease as the amount of the error\nincreases. A second principle is that positive and negative errors\nshall be equally probable, which may certainly be assumed, because we\nare supposed to be devoid of any knowledge as to the causes of the\nresidual errors. It follows that the probability of the error must be a\nfunction of an even power of the magnitude, that is of the square, or\nthe fourth power, or the sixth power, otherwise the probability of the\nsame amount of error would vary according as the error was positive or\nnegative. The even powers *x*^{2}, *x*^{4}, *x*^{6}, &c., are always\nintrinsically positive, whether *x* be positive or negative. There is\nno *\u00e0 priori* reason why one rather than another of these even powers\nshould be selected. Gauss himself allows that the fourth or sixth power\nwould fulfil the conditions as well as the second;[279] but in the\nabsence of any theoretical reasons we should prefer the second power,\nbecause it leads to formul\u00e6 of great comparative simplicity. Did the\nLaw of Error necessitate the use of the higher powers of the error, the\ncomplexity of the necessary calculations would much reduce the utility\nof the theory.\n\n  [279] *M\u00e9thode des Moindres Carr\u00e9s. M\u00e9moires sur la Combinaison\n  des Observations, par Ch. Fr. Gauss. Traduit en Fran\u00e7ais par J.\n  Bertrand*, Paris, 1855, pp.\u00a06, 133, &c.\n\nBy mathematical reasoning which it would be undesirable to attempt\nto follow in this book, it is shown that under these conditions, the\nfacility of occurrence, or in other, words, the probability of error\nis expressed by a function of the general form \u03b5^{\u2013*h*^{2}\u00a0*x*^{2}}, in\nwhich *x* represents the variable amount of errors. From this law, to\nbe more fully described in the following sections, it at once follows\nthat the most probable result of any observations is that which makes\nthe sum of the squares of the consequent errors the least possible.\nLet *a*, *b*, *c*, &c., be the results of observation, and *x* the\nquantity selected as the most probable, that is the most free from\nunknown errors: then we must determine *x* so that (*a*\u00a0-\u00a0*x*)^{2}\u00a0+\n(*b*\u00a0-\u00a0*x*)^{2}\u00a0+ (*c*\u00a0-\u00a0*x*)^{2}\u00a0+\u00a0... shall be the least possible\nquantity. Thus we arrive at the celebrated *Method of Least Squares*,\nas it is usually called, which appears to have been first distinctly\nput in practice by Gauss in 1795, while Legendre first published in\n1806 an account of the process in his work, entitled, *Nouvelles\nM\u00e9thodes pour la D\u00e9termination des Orbites des Com\u00e8tes*. It is worthy\nof notice, however, that Roger Cotes had long previously recommended a\nmethod of equivalent nature in his tract, \u201cEstimatio Erroris in Mixta\nMathesi.\u201d[280]\n\n  [280] De Morgan, *Penny Cyclop\u00e6dia*, art. *Least Squares*.\n\n\n*Herschel\u2019s Geometrical Proof.*\n\nA second way of arriving at the Law of Error was proposed by Herschel,\nand although only applicable to geometrical cases, it is remarkable as\nshowing that from whatever point of view we regard the subject, the\nsame principle will be detected. After assuming that some general law\nmust exist, and that it is subject to the principles of probability,\nhe supposes that a ball is dropped from a high point with the\nintention that it shall strike a given mark on a horizontal plane. In\nthe absence of any known causes of deviation it will either strike\nthat mark, or, as is infinitely more probable, diverge from it by an\namount which we must regard as error of unknown origin. Now, to quote\nthe words of Herschel,[281] \u201cthe probability of that error is the\nunknown function of its square, *i.e.* of the sum of the squares of\nits deviations in any two rectangular directions. Now, the probability\nof any deviation depending solely on its magnitude, and not on its\ndirection, it follows that the probability of each of these rectangular\ndeviations must be the same function of *its* square. And since the\nobserved oblique deviation is equivalent to the two rectangular ones,\nsupposed concurrent, and which are essentially independent of one\nanother, and is, therefore, a compound event of which they are the\nsimple independent constituents, therefore its probability will be the\nproduct of their separate probabilities. Thus the form of our unknown\nfunction comes to be determined from this condition, viz., that the\nproduct of such functions of two independent elements is equal to the\nsame function of their sum. But it is shown in every work on algebra\nthat this property is the peculiar characteristic of, and belongs only\nto, the exponential or antilogarithmic function. This, then, is the\nfunction of the square of the error, which expresses the probability\nof committing that error. That probability decreases, therefore, in\ngeometrical progression, as the square of the error increases in\narithmetical.\u201d\n\n  [281] *Edinburgh Review*, July 1850, vol.\u00a0xcii. p.\u00a017. Reprinted\n  *Essays*, p. 399. This method of demonstration is discussed by Boole,\n  *Transactions of Royal Society of Edinburgh*, vol.\u00a0xxi. pp.\u00a0627\u2013630.\n\n\n*Laplace\u2019s and Quetelet\u2019s Proof of the Law.*\n\nHowever much presumption the modes of determining the Law of Error,\nalready described, may give in favour of the law usually adopted, it is\ndifficult to feel that the arguments are satisfactory. The law adopted\nis chosen rather on the grounds of convenience and plausibility, than\nbecause it can be seen to be the necessary law. We can however approach\nthe subject from an entirely different point of view, and yet get to\nthe same result.\n\nLet us assume that a particular observation is subject to four chances\nof error, each of which will increase the result one inch if it occurs.\nEach of these errors is to be regarded as an event independent of the\nrest and we can therefore assign, by the theory of probability, the\ncomparative probability and frequency of each conjunction of errors.\nFrom the Arithmetical Triangle (pp.\u00a0182\u2013188) we learn that no error\nat all can happen only in one way; an error of one inch can happen\nin 4 ways; and the ways of happening of errors of 2, 3 and 4 inches\nrespectively, will be 6, 4 and 1 in number.\n\nWe may infer that the error of two inches is the most likely to occur,\nand will occur in the long run in six cases out of sixteen. Errors\nof one and three inches will be equally likely, but will occur less\nfrequently; while no error at all, or one of four inches will be a\ncomparatively rare occurrence. If we now suppose the errors to act\nas often in one direction as the other, the effect will be to alter\nthe average error by the amount of two inches, and we shall have the\nfollowing results:--\n\n  Negative error of 2 inches     1 way.\n  Negative error of 1 inch       4 ways.\n  No error at all                6 ways.\n  Positive error of 1 inch       4 ways.\n  Positive error of 2 inches     1 way.\n\nWe may now imagine the number of causes of error increased and the\namount of each error decreased, and the arithmetical triangle will\ngive us the frequency of the resulting errors. Thus if there be five\npositive causes of error and five negative causes, the following\ntable shows the numbers of errors of various amount which will be the\nresult:--\n\n  +----------------------+-------------------+---+-------------------+\n  | Direction of Error.  |  Positive Error.  |   |  Negative Error.  |\n  +----------------------+-------------------+---+-------------------+\n  |   Amount of Error.   |5,  4,  3,   2,   1| 0 | 1,   2,  3,  4,  5|\n  +----------------------+-------------------+---+-------------------+\n  |Number of such Errors.|1, 10, 45, 120, 210|252|210, 120, 45, 10, 1|\n  +----------------------+-------------------+---+-------------------+\n\nIt is plain that from such numbers I can ascertain the probability\nof any particular amount of error under the conditions supposed. The\nprobability of a positive error of exactly one inch is 210/1024, in\nwhich fraction the numerator is the number of combinations giving\none inch positive error, and the denominator the whole number of\npossible errors of all magnitudes. I can also, by adding together the\nappropriate numbers get the probability of an error not exceeding a\ncertain amount. Thus the probability of an error of three inches or\nless, positive or negative, is a fraction whose numerator is the sum of\n45\u00a0+\u00a0120\u00a0+\u00a0210\u00a0+\u00a0252\u00a0+\u00a0210\u00a0+\u00a0120\u00a0+\u00a045, and the denominator, as before,\ngiving the result 1002/1024. We may see at once that, according to\nthese principles, the probability of small errors is far greater than\nof large ones: the odds are 1002 to 22, or more than 45 to 1, that the\nerror will not exceed three inches; and the odds are 1022 to 2 against\nthe occurrence of the greatest possible error of five inches.\n\nIf any case should arise in which the observer knows the number and\nmagnitude of the chief errors which may occur, he ought certainly to\ncalculate from the Arithmetical Triangle the special Law of Error\nwhich would apply. But the general law, of which we are in search,\nis to be used in the dark, when we have no knowledge whatever of the\nsources of error. To assume any special number of causes of error\nis then an arbitrary proceeding, and mathematicians have chosen the\nleast arbitrary course of imagining the existence of an infinite\nnumber of infinitely small errors, just as, in the inverse method of\nprobabilities, an infinite number of infinitely improbable hypotheses\nwere submitted to calculation (p.\u00a0255).\n\nThe reasons in favour of this choice are of several different kinds.\n\n1. It cannot be denied that there may exist infinitely numerous causes\nof error in any act of observation.\n\n2. The law resulting from the hypothesis of a moderate number of causes\nof error, does not appreciably differ from that given by the hypothesis\nof an infinite number of causes of error.\n\n3. We gain by the hypothesis of infinity a general law capable of ready\ncalculation, and applicable by uniform rules to all problems.\n\n4. This law, when tested by comparison with extensive series of\nobservations, is strikingly verified, as will be shown in a later\nsection.\n\nWhen we imagine the existence of any large number of causes of\nerror, for instance one hundred, the numbers of combinations become\nimpracticably large, as may be seen to be the case from a glance at\nthe Arithmetical Triangle, which proceeds only up to the seventeenth\nline. Quetelet, by suitable abbreviating processes, calculated out\na table of probability of errors on the hypothesis of one thousand\ndistinct causes;[282] but mathematicians have generally proceeded on\nthe hypothesis of infinity, and then, by the devices of analysis,\nhave substituted a general law of easy treatment. In mathematical\nworks upon the subject, it is shown that the standard Law of Error is\nexpressed in the formula\n\n  *y* = *Y*\u03b5^{-*cx*^{2}},\n\n  [282] *Letters on the Theory of Probabilities*, Letter XV. and\n  Appendix, note pp.\u00a0256\u2013266.\n\nin which *x* is the amount of the error, *Y* the maximum ordinate\nof the curve of error, and *c* a number constant for each series of\nobservations, and expressing the amount of the tendency to error,\nvarying between one series of observations and another. The letter \u03b5\nis the mathematical constant, the sum of ratios between the numbers of\npermutations and combinations, previously referred to (p.\u00a0330).\n\n[Illustration]\n\nTo show the close correspondence of this general law with the special\nlaw which might be derived from the supposition of a moderate number\nof causes of error, I have in the accompanying figure drawn a curved\nline representing accurately the variation of *y* when *x* in the above\nformula is taken equal 0, 1/2, 1, 3/2, 2, &c., positive or negative,\nthe arbitrary quantities *Y* and *c* being each assumed equal to unity,\nin order to simplify the calculations. In the same figure are inserted\neleven dots, whose heights above the base line are proportional to\nthe numbers in the eleventh line of the Arithmetical Triangle, thus\nrepresenting the comparative probabilities of errors of various amounts\narising from ten equal causes of error. The correspondence of the\ngeneral and the special Law of Error is almost as close as can be\nexhibited in the figure, and the assumption of a greater number of\nequal causes of error would render the correspondence far more close.\n\nIt may be explained that the ordinates NM, *nm*, *n\u2032m\u2032*, represent\nvalues of *y* in the equation expressing the Law of Error. The\noccurrence of any one definite amount of error is infinitely\nimprobable, because an infinite number of such ordinates might be\ndrawn. But the probability of an error occurring between certain limits\nis finite, and is represented by a portion of the *area* of the curve.\nThus the probability that an error, positive or negative, not exceeding\nunity will occur, is represented by the area M*mnn\u2032m\u2032*, in short, by\nthe area standing upon the line *nn\u2032*. Since every observation must\neither have some definite error or none at all, it follows that the\nwhole area of the curve should be considered as the unit expressing\ncertainty, and the probability of an error falling between particular\nlimits will then be expressed by the ratio which the area of the curve\nbetween those limits bears to the whole area of the curve.\n\nThe mere fact that the Law of Error allows of the possible existence of\nerrors of every assignable amount shows that it is only approximately\ntrue. We may fairly say that in measuring a mile it would be impossible\nto commit an error of a hundred miles, and the length of life\nwould never allow of our committing an error of one million miles.\nNevertheless the general Law of Error would assign a probability for\nan error of that amount or more, but so small a probability as to be\nutterly inconsiderable and almost inconceivable. All that can, or in\nfact need, be said in defence of the law is, that it may be made to\nrepresent the errors in any special case to a very close approximation,\nand that the probability of large and practically impossible errors, as\ngiven by the law, will be so small as to be entirely inconsiderable.\nAnd as we are dealing with error itself, and our results pretend to\nnothing more than approximation and probability, an indefinitely small\nerror in our process of approximation is of no importance whatever.\n\n\n*Logical Origin of the Law of Error.*\n\nIt is worthy of notice that this Law of Error, abstruse though the\nsubject may seem, is really founded upon the simplest principles.\nIt arises entirely out of the difference between permutations and\ncombinations, a subject upon which I may seem to have dwelt with\nunnecessary prolixity in previous pages (pp.\u00a0170, 189). The order in\nwhich we add quantities together does not affect the amount of the sum,\nso that if there be three positive and five negative causes of error\nin operation, it does not matter in which order they are considered as\nacting. They may be intermixed in any arrangement, and yet the result\nwill be the same. The reader should not fail to notice how laws or\nprinciples which appeared to be absurdly simple and evident when first\nnoticed, reappear in the most complicated and mysterious processes of\nscientific method. The fundamental Laws of Identity and Difference gave\nrise to the Logical Alphabet which, after abstracting the character of\nthe differences, led to the Arithmetical Triangle. The Law of Error is\ndefined by an infinitely high line of that triangle, and the law proves\nthat the mean is the most probable result, and that divergencies from\nthe mean become much less probable as they increase in amount. Now\nthe comparative greatness of the numbers towards the middle of each\nline of the Arithmetical Triangle is entirely due to the indifference\nof order in space or time, which was first prominently pointed out\nas a condition of logical relations, and the symbols indicating them\n(pp.\u00a032\u201335), and which was afterwards shown to attach equally to\nnumerical symbols, the derivatives of logical terms (p.\u00a0160).\n\n\n*Verification of the Law of Error.*\n\nThe theory of error which we have been considering rests entirely\nupon an assumption, namely that when known sources of disturbances\nare allowed for, there yet remain an indefinite, possibly an infinite\nnumber of other minute sources of error, which will as often produce\nexcess as deficiency. Granting this assumption, the Law of Error must\nbe as it is usually taken to be, and there is no more need to verify\nit empirically than to test the truth of one of Euclid\u2019s propositions\nmechanically. Nevertheless, it is an interesting occupation to verify\neven the propositions of geometry, and it is still more instructive to\ntry whether a large number of observations will justify our assumption\nof the Law of Error.\n\nEncke has given an excellent instance of the correspondence of theory\nwith experience, in the case of observations of the differences of\nRight Ascension of the sun and two stars, namely \u03b1 Aquil\u00e6 and \u03b1 Canis\nminoris. The observations were 470 in number, and were made by Bradley\nand reduced by Bessel, who found the probable error of the final result\nto be only about one-fourth part of a second (0\u00b72637). He then compared\nthe numbers of errors of each magnitude from 0\u00b71 second upwards, as\nactually given by the observations, with what should occur according to\nthe Law of Error.\n\nThe results were as follow:--[283]\n\n  +-------------------------+--------------------------+\n  |                         | Number of errors of each |\n  | Magnitude of the errors |  magnitude according to  |\n  |  in parts of a second.  +-------------+------------+\n  |                         | Observation.|   Theory.  |\n  +-------------------------+-------------+------------+\n  |       0\u00b70  to 0\u00b71       |      94     |     95     |\n  |        \u00b71  \"   \u00b72       |      88     |     89     |\n  |        \u00b72  \"   \u00b73       |      78     |     78     |\n  |        \u00b73  \"   \u00b74       |      58     |     64     |\n  |        \u00b74  \"   \u00b75       |      51     |     50     |\n  |        \u00b75  \"   \u00b76       |      36     |     36     |\n  |        \u00b76  \"   \u00b77       |      26     |     24     |\n  |        \u00b77  \"   \u00b78       |      14     |     15     |\n  |        \u00b78  \"   \u00b79       |      10     |      9     |\n  |        \u00b79  \"  1\u00b70       |       7     |      5     |\n  |        above  1\u00b70       |       8     |      5     |\n  +-------------------------+-------------+------------+\n\n  [283] Encke, *On the Method of Least Squares*, Taylor\u2019s *Scientific\n  Memoirs*, vol.\u00a0ii. pp.\u00a0338, 339.\n\nThe reader will remark that the correspondence is very close, except\nas regards larger errors, which are excessive in practice. It is one\nobjection, indeed, to the theory of error, that, being expressed in\na continuous mathematical function, it contemplates the existence of\nerrors of every magnitude, such as could not practically occur; yet\nin this case the theory seems to under-estimate the number of large\nerrors.\n\nAnother comparison of the law with observation was made by Quetelet,\nwho investigated the errors of 487 determinations in time of the Right\nAscension of the Pole-Star made at Greenwich during the four years\n1836\u201339. These observations, although carefully corrected for all known\ncauses of error, as well as for nutation, precession, &c., are yet of\ncourse found to differ, and being classified as regards intervals of\none-half second of time, and then proportionately increased in number,\nso that their sum may be one thousand, give the following results as\ncompared with what Quetelet\u2019s theory would lead us to expect:--[284]\n\n  +------------+--------------------+------------+--------------------+\n  |Magnitude of|  Number of Errors  |Magnitude of|  Number of Errors  |\n  |   error    +------------+-------+   error    +------------+-------+\n  | in tenths  |    by      |  by   | in tenths  |    by      |  by   |\n  |of a second.|Observation.|Theory.|of a second.|Observation.|Theory.|\n  +------------+------------+-------+------------+------------+-------+\n  |     0\u00b70    |    168     |  163  |     --     |      --    |   --  |\n  |    +0\u00b75    |    148     |  147  |    -0\u00b75    |     150    |  152  |\n  |    +1\u00b70    |    129     |  112  |    -1\u00b70    |     126    |  121  |\n  |    +1\u00b75    |     78     |   72  |    -1\u00b75    |      74    |   82  |\n  |    +2\u00b70    |     33     |   40  |    -2\u00b70    |      43    |   46  |\n  |    +2\u00b75    |     10     |   19  |    -2\u00b75    |      25    |   22  |\n  |    +3\u00b70    |      2     |   10  |    -3\u00b70    |      12    |   10  |\n  |     --     |     --     |   --  |    -3\u00b75    |       2    |    4  |\n  +------------+------------+-------+------------+------------+-------+\n\n  [284] Quetelet, *Letters on the Theory of Probabilities*, translated\n  by Downes, Letter XIX. p.\u00a088. See also Galton\u2019s *Hereditary Genius*,\n  p. 379.\n\nIn this instance also the correspondence is satisfactory, but the\ndivergence between theory and fact is in the opposite direction to\nthat discovered in the former comparison, the larger errors being less\nfrequent than theory would indicate. It will be noticed that Quetelet\u2019s\ntheoretical results are not symmetrical.\n\n\n*The Probable Mean Result.*\n\nOne immediate result of the Law of Error, as thus stated, is that the\nmean result is the most probable one; and when there is only a single\nvariable this mean is found by the familiar arithmetical process. An\nunfortunate error has crept into several works which allude to this\nsubject. Mill, in treating of the \u201cElimination of Chance,\u201d remarks in\na note[285] that \u201cthe mean is spoken of as if it were exactly the\nsame thing as the average. But the mean, for purposes of inductive\ninquiry, is not the average, or arithmetical mean, though in a familiar\nillustration of the theory the difference may be disregarded.\u201d He goes\non to say that, according to mathematical principles, the most probable\nresult is that for which the sums of the squares of the deviations is\nthe least possible. It seems probable that Mill and other writers were\nmisled by Whewell, who says[286] that \u201cThe method of least squares is\nin fact a method of means, but with some peculiar characters.... The\nmethod proceeds upon this supposition: that all errors are not equally\nprobable, but that small errors are more probable than large ones.\u201d He\nadds that this method \u201cremoves much that is arbitrary in the method\nof means.\u201d It is strange to find a mathematician like Whewell making\nsuch remarks, when there is no doubt whatever that the Method of Means\nis only an application of the Method of Least Squares. They are, in\nfact, the same method, except that the latter method may be applied to\ncases where two or more quantities have to be determined at the same\ntime. Lubbock and Drinkwater say,[287] \u201cIf only one quantity has to\nbe determined, this method evidently resolves itself into taking the\nmean of all the values given by observation.\u201d Encke says,[288] that the\nexpression for the probability of an error \u201cnot only contains in itself\nthe principle of the arithmetical mean, but depends so immediately upon\nit, that for all those magnitudes for which the arithmetical mean holds\ngood in the simple cases in which it is principally applied, no other\nlaw of probability can be assumed than that which is expressed by this\nformula.\u201d\n\n  [285] *System of Logic*, bk. iii. chap. 17, \u00a7 3. 5th ed. vol.\u00a0ii.\n  p.\u00a056.\n\n  [286] *Philosophy of the Inductive Sciences*, 2nd ed. vol.\u00a0ii.\n  pp.\u00a0408, 409.\n\n  [287] *Essay on Probability*, Useful Knowledge Society, 1833, p.\u00a041.\n\n  [288] Taylor\u2019s *Scientific Memoirs*, vol.\u00a0ii. p.\u00a0333.\n\n\n*The Probable Error of Results.*\n\nWhen we draw a conclusion from the numerical results of observations we\nought not to consider it sufficient, in cases of importance, to content\nourselves with finding the simple mean and treating it as true. We\nought also to ascertain what is the degree of confidence we may place\nin this mean, and our confidence should be measured by the degree of\nconcurrence of the observations from which it is derived. In some cases\nthe mean may be approximately certain and accurate. In other cases it\nmay really be worth little or nothing. The Law of Error enables us to\ngive exact expression to the degree of confidence proper in any case;\nfor it shows how to calculate the probability of a divergence of any\namount from the mean, and we can thence ascertain the probability that\nthe mean in question is within a certain distance from the true number.\nThe *probable error* is taken by mathematicians to mean the limits\nwithin which it is as likely as not that the truth will fall. Thus\nif 5\u00b745 be the mean of all the determinations of the density of the\nearth, and \u00b720 be approximately the probable error, the meaning is that\nthe probability of the real density of the earth falling between 5\u00b725\nand 5\u00b765 is 1/2. Any other limits might have been selected at will.\nWe might calculate the limits within which it was one hundred or one\nthousand to one that the truth would fall; but there is a convention to\ntake the even odds one to one, as the quantity of probability of which\nthe limits are to be estimated.\n\nMany books on probability give rules for making the calculations,\nbut as, in the progress of science, persons ought to become more\nfamiliar with these processes, I propose to repeat the rules here and\nillustrate their use. The calculations, when made in accordance with\nthe directions, involve none but arithmetic or logarithmic operations.\n\nThe following are the rules for treating a mean result, so as\nthoroughly to ascertain its trustworthiness.\n\n1. Draw the mean of all the observed results.\n\n2. Find the excess or defect, that is, the error of each result from\nthe mean.\n\n3. Square each of these reputed errors.\n\n4. Add together all these squares of the errors, which are of course\nall positive.\n\n5. Divide by one less than the number of observations. This gives the\n*square of the mean error*.\n\n6. Take the square root of the last result; it is the *mean error of a\nsingle observation*.\n\n7. Divide now by the square root of the number of observations, and we\nget the *mean error of the mean result*.\n\n8. Lastly, multiply by the natural constant 0\u00b76745 (or approximately\nby 0\u00b7674, or even by 2/3), and we arrive at the *probable error of the\nmean result*.\n\nSuppose, for instance, that five measurements of the height of a\nhill, by the barometer or otherwise, have given the numbers of feet\nas 293, 301, 306, 307, 313; we want to know the probable error of the\nmean, namely 304. Now the differences between this mean and the above\nnumbers, *paying no regard to direction*, are 11, 3, 2, 3, 9; their\nsquares are 121, 9, 4, 9, 81, and the sum of the squares of the errors\nconsequently 224. The number of observations being 5, we divide by\n1 less, or 4, getting 56. This is the square of the mean error, and\ntaking its square root we have 7\u00b748 (say 7-1/2), the mean error of\na single observation. Dividing by 2\u00b7236, the square root of 5, the\nnumber of observations, we find the mean error of the *mean* result to\nbe 3\u00b735, or say 3-1/3, and lastly, multiplying by \u00b76745, we arrive at\nthe *probable error of the mean result*, which is found to be 2\u00b7259,\nor say 2-1/4. The meaning of this is that the probability is one half,\nor the odds are even that the true height of the mountain lies between\n301-3/4 and 306-1/4 feet. We have thus an exact measure of the degree\nof credibility of our mean result, which mean indicates the most likely\npoint for the truth to fall upon.\n\nThe reader should observe that as the object in these calculations\nis only to gain a notion of the degree of confidence with which we\nview the mean, there is no real use in carrying the calculations to\nany great degree of precision; and whenever the neglect of decimal\nfractions, or even the slight alteration of a number, will much\nabbreviate the computations, it may be fearlessly done, except in\ncases of high importance and precision. Brodie has shown how the law\nof error may be usefully applied in chemical investigations, and some\nillustrations of its employment may be found in his paper.[289]\n\n  [289] *Philosophical Transactions*, 1873, p.\u00a083.\n\nThe experiments of Benzenberg to detect the revolution of the earth, by\nthe deviation of a ball from the perpendicular line in falling down a\ndeep pit, have been cited by Encke[290] as an interesting illustration\nof the Law of Error. The mean deviation was 5\u00b7086 lines, and its\nprobable error was calculated by Encke to be not more than \u00b7950 line,\nthat is, the odds were even that the true result lay between 4\u00b7136 and\n6\u00b7036. As the deviation, according to astronomical theory, should be\n4\u00b76 lines, which lies well within the limits, we may consider that the\nexperiments are consistent with the Copernican system of the universe.\n\n  [290] Taylor\u2019s *Scientific Memoirs*, vol.\u00a0ii. pp.\u00a0330, 347, &c.\n\nIt will of course be understood that the probable error has regard only\nto those causes of errors which in the long run act as much in one\ndirection as another; it takes no account of constant errors. The true\nresult accordingly will often fall far beyond the limits of probable\nerror, owing to some considerable constant error or errors, of the\nexistence of which we are unaware.\n\n\n*Rejection of the Mean Result.*\n\nWe ought always to bear in mind that the mean of any series of\nobservations is the best, that is, the most probable approximation\nto the truth, only in the absence of knowledge to the contrary. The\nselection of the mean rests entirely upon the probability that unknown\ncauses of error will in the long run fall as often in one direction as\nthe opposite, so that in drawing the mean they will balance each other.\nIf we have any reason to suppose that there exists a tendency to error\nin one direction rather than the other, then to choose the mean would\nbe to ignore that tendency. We may certainly approximate to the length\nof the circumference of a circle, by taking the mean of the perimeters\nof inscribed and circumscribed polygons of an equal and large number\nof sides. The length of the circular line undoubtedly lies between the\nlengths of the two perimeters, but it does not follow that the mean is\nthe best approximation. It may in fact be shown that the circumference\nof the circle is *very nearly* equal to the perimeter of the inscribed\npolygon, together with one-third part of the difference between the\ninscribed and circumscribed polygons of the same number of sides.\nHaving this knowledge, we ought of course to act upon it, instead of\ntrusting to probability.\n\nWe may often perceive that a series of measurements tends towards an\nextreme limit rather than towards a mean. In endeavouring to obtain\na correct estimate of the apparent diameter of the brightest fixed\nstars, we find a continuous diminution in estimates as the powers of\nobservation increased. Kepler assigned to Sirius an apparent diameter\nof 240 seconds; Tycho Brahe made it 126; Gassendi 10 seconds; Galileo,\nHevelius, and J. Cassini, 5 or 6 seconds. Halley, Michell, and\nsubsequently Sir W. Herschel came to the conclusion that the brightest\nstars in the heavens could not have real discs of a second, and were\nprobably much less in diameter. It would of course be absurd to take\nthe mean of quantities which differ more than 240 times; and as the\ntendency has always been to smaller estimates, there is a considerable\npresumption in favour of the smallest.[291]\n\n  [291] Quetelet, *Letters*, &c. p.\u00a0116.\n\nIn many experiments and measurements we know that there is a\npreponderating tendency to error in one direction. The readings of a\nthermometer tend to rise as the age of the instrument increases, and\nno drawing of means will correct this result. Barometers, on the other\nhand, are likely to read too low instead of too high, owing to the\nimperfection of the vacuum and the action of capillary attraction. If\nthe mercury be perfectly pure and no appreciable error be due to the\nmeasuring apparatus, the best barometer will be that which gives the\nhighest result. In determining the specific gravity of a solid body the\nchief danger of error arises from bubbles of air adhering to the body,\nwhich would tend to make the specific gravity too small. Much attention\nmust always be given to one-sided errors of this kind, since the\nmultiplication of experiments does not remove the error. In such cases\none very careful experiment is better than any number of careless ones.\n\nWhen we have reasonable grounds for supposing that certain experimental\nresults are liable to grave errors, we should exclude them in drawing\na mean. If we want to find the most probable approximation to the\nvelocity of sound in air, it would be absurd to go back to the old\nexperiments which made the velocity from 1200 to 1474 feet per second;\nfor we know that the old observers did not guard against errors arising\nfrom wind and other causes. Old chemical experiments are valueless as\nregards quantitative results. The old chemists found the atmosphere\nin different places to differ in composition nearly ten per cent.,\nwhereas modern accurate experimenters find very slight variations.\nAny method of measurement which we know to avoid a source of error is\nfar to be preferred to others which trust to probabilities for the\nelimination of the error. As Flamsteed says,[292] \u201cOne good instrument\nis of as much worth as a hundred indifferent ones.\u201d But an instrument\nis good or bad only in a comparative sense, and no instrument gives\ninvariable and truthful results. Hence we must always ultimately fall\nback upon probabilities for the selection of the final mean, when other\nprecautions are exhausted.\n\n  [292] Baily, *Account of Flamsteed*, p.\u00a056.\n\nLegendre, the discoverer of the method of Least Squares,\nrecommended that observations differing very much from the results\nof his method should be rejected. The subject has been carefully\ninvestigated by Professor Pierce, who has proposed a criterion\nfor the rejection of doubtful observations based on the following\nprinciple:[293]\u2032\u201c--observations should be rejected when the probability\nof the system of errors obtained by retaining them is less than that\nof the system of errors obtained by their rejection multiplied by the\nprobability of making so many and no more abnormal observations.\u201d\nProfessor Pierce\u2019s investigation is given nearly in his own words in\nProfessor W. Chauvenet\u2019s \u201cManual of Spherical and Practical Astronomy,\u201d\nwhich contains a full and excellent discussion of the methods of\ntreating numerical observations.[294]\n\n  [293] Gould\u2019s *Astronomical Journal*, Cambridge, Mass., vol.\u00a0ii.\n  p.\u00a0161.\n\n  [294] Philadelphia (London, Tr\u00fcbner) 1863. Appendix, vol.\u00a0ii. p.\u00a0558.\n\nVery difficult questions sometimes arise when one or more results of\na method of experiment diverge widely from the mean of the rest. Are\nwe or are we not to exclude them in adopting the supposed true mean\nresult of the method? The drawing of a mean result rests, as I have\nfrequently explained, upon the assumption that every error acting in\none direction will probably be balanced by other errors acting in an\nopposite direction. If then we know or can possibly discover any causes\nof error not agreeing with this assumption, we shall be justified in\nexcluding results which seem to be affected by this cause.\n\nIn reducing large series of astronomical observations, it is not\nuncommon to meet with numbers differing from others by a whole degree\nor half a degree, or some considerable integral quantity. These are\nerrors which could hardly arise in the act of observation or in\ninstrumental irregularity; but they might readily be accounted for\nby misreading of figures or mistaking of division marks. It would be\nabsurd to trust to chance that such mistakes would balance each other\nin the long run, and it is therefore better to correct arbitrarily\nthe supposed mistake, or better still, if new observations can be\nmade, to strike out the divergent numbers altogether. When results\ncome sometimes too great or too small in a regular manner, we should\nsuspect that some part of the instrument slips through a definite\nspace, or that a definite cause of error enters at times, and not at\nothers. We should then make it a point of prime importance to discover\nthe exact nature and amount of such an error, and either prevent its\noccurrence for the future or else introduce a corresponding correction.\nIn many researches the whole difficulty will consist in this detection\nand avoidance of sources of error. Professor Roscoe found that the\npresence of phosphorus caused serious and almost unavoidable errors in\nthe determination of the atomic weight of vanadium.[295] Herschel, in\nreducing his observations of double stars at the Cape of Good Hope,\nwas perplexed by an unaccountable difference of the angles of position\nas measured by the seven-feet equatorial and the twenty-feet reflector\ntelescopes, and after a careful investigation was obliged to be\ncontented with introducing a correction experimentally determined.[296]\n\n  [295] Bakerian Lecture, *Philosophical Transactions* (1868),\n  vol.\u00a0clviii. p. 6.\n\n  [296] *Results of Observations at the Cape of Good Hope*, p.\u00a0283.\n\nWhen observations are sufficiently numerous it seems desirable to\nproject the apparent errors into a curve, and then to observe whether\nthis curve exhibits the symmetrical and characteristic form of the\ncurve of error. If so, it may be inferred that the errors arise from\nmany minute independent sources, and probably compensate each other\nin the mean result. Any considerable irregularity will indicate the\nexistence of one-sided or large causes of error, which should be made\nthe subject of investigation.\n\nEven the most patient and exhaustive investigations will sometimes\nfail to disclose any reason why some results diverge from others.\nThe question again recurs--Are we arbitrarily to exclude them? The\nanswer should be in the negative as a general rule. The mere fact\nof divergence ought not to be taken as conclusive against a result,\nand the exertion of arbitrary choice would open the way to the fatal\ninfluence of bias, and what is commonly known as the \u201ccooking\u201d of\nfigures. It would amount to judging fact by theory instead of theory\nby fact. The apparently divergent number may prove in time to be the\ntrue one. It may be an exception of that valuable kind which upsets our\nfalse theories, a real exception, exploding apparent coincidences, and\nopening a way to a new view of the subject. To establish this position\nfor the divergent fact will require additional research; but in the\nmeantime we should give it some weight in our mean conclusions, and\nshould bear in mind the discrepancy as one demanding attention. To\nneglect a divergent result is to neglect the possible clue to a great\ndiscovery.\n\n\n*Method of Least Squares.*\n\nWhen two or more unknown quantities are so involved that they cannot\nbe separately determined by the Simple Method of Means, we can yet\nobtain their most probable values by the Method of Least Squares,\nwithout more difficulty than arises from the length of the arithmetical\ncomputations. If the result of each observation gives an equation\nbetween two unknown quantities of the form\n\n  *ax* + *by* = *c*\n\nthen, if the observations were free from error, we should need only two\nobservations giving two equations; but for the attainment of greater\naccuracy, we may take many observations, and reduce the equations\nso as to give only a pair with mean coefficients. This reduction is\neffected by (1.), multiplying the coefficients of each equation by the\nfirst coefficient, and adding together all the similar coefficients\nthus resulting for the coefficients of a new equation; and (2.), by\nrepeating this process, and multiplying the coefficients of each\nequation by the coefficient of the second term. Meaning by (sum of\n*a*^{2}) the sum of all quantities of the same kind, and having the\nsame place in the equations as *a*^{2}, we may briefly describe the two\nresulting mean equations as follows:--\n\n  (sum of *a*^{2})\u00a0.\u00a0*x*\u00a0+\u00a0(sum of *ab*)\u00a0.\u00a0*y*\u00a0=\u00a0(sum of *ac*),\n  (sum of *ab*)\u00a0.\u00a0*x*\u00a0+\u00a0(sum of *b*^{2})\u00a0.\u00a0*y*\u00a0=\u00a0(sum of *bc*).\n\nWhen there are three or more unknown quantities the process is exactly\nthe same in nature, and we get additional mean equations by multiplying\nby the third, fourth, &c., coefficients. As the numbers are in any\ncase approximate, it is usually unnecessary to make the computations\nwith accuracy, and places of decimals may be freely cut off to save\narithmetical work. The mean equations having been computed, their\nsolution by the ordinary methods of algebra gives the most probable\nvalues of the unknown quantities.\n\n\n*Works upon the Theory of Probability.*\n\nRegarding the Theory of Probability and the Law of Error as most\nimportant subjects of study for any one who desires to obtain a\ncomplete comprehension of scientific method as actually applied in\nphysical investigations, I will briefly indicate the works in one or\nother of which the reader will best pursue the study.\n\nThe best popular, and at the same time profound English work on the\nsubject is De Morgan\u2019s \u201cEssay on Probabilities and on their Application\nto Life Contingencies and Insurance Offices,\u201d published in the *Cabinet\nCyclop\u00e6dia*, and to be obtained (in print) from Messrs. Longman. Mr.\nVenn\u2019s work on *The Logic of Chance* can now be procured in a greatly\nenlarged second edition;[297] it contains a most interesting and able\ndiscussion of the metaphysical basis of probability and of related\nquestions concerning causation, belief, design, testimony, &c.; but I\ncannot always agree with Mr. Venn\u2019s opinions. No mathematical knowledge\nbeyond that of common arithmetic is required in reading these works.\nQuetelet\u2019s *Letters* form a good introduction to the subject, and the\nmathematical notes are of value. Sir George Airy\u2019s brief treatise *On\nthe Algebraical and Numerical Theory of Errors of Observations and\nthe Combination of Observations*, contains a complete explanation of\nthe Law of Error and its practical applications. De Morgan\u2019s treatise\n\u201cOn the Theory of Probabilities\u201d in the *Encyclop\u00e6dia Metropolitana*,\npresents an abstract of the more abstruse investigations of\nLaplace, together with a multitude of profound and original remarks\nconcerning the theory generally. In Lubbock and Drinkwater\u2019s work on\n*Probability*, in the Library of Useful Knowledge, we have a concise\nbut good statement of a number of important problems. The Rev.\nW.\u00a0A. Whitworth has given, in a work entitled *Choice and Chance*,\na number of good illustrations of calculations both in combinations\nand probabilities. In Mr. Todhunter\u2019s admirable History we have an\nexhaustive critical account of almost all writings upon the subject\nof probability down to the culmination of the theory in Laplace\u2019s\nworks. The Memoir of Mr. J.\u00a0W.\u00a0L. Glaisher has already been mentioned\n(p.\u00a0375). In spite of the existence of these and some other good\nEnglish works, there seems to be a want of an easy and yet pretty\ncomplete mathematical introduction to the study of the theory.\n\n  [297] *The Logic of Chance*, an Essay on the Foundations and Province\n  of the Theory of Probability, with especial reference to its\n  Logical Bearings and its Application to Moral and Social Science.\n  (Macmillan), 1876.\n\nAmong French works the Trait\u00e9 *\u00c9l\u00e9mentaire du Calcul des Probabilit\u00e9s*,\nby S.\u00a0E. Lacroix, of which several editions have been published, and\nwhich is not difficult to obtain, forms probably the best elementary\ntreatise. Poisson\u2019s *Recherches sur la Probabilit\u00e9 des Jugements*\n(Paris 1837), commence with an admirable investigation of the grounds\nand methods of the theory. While Laplace\u2019s great *Th\u00e9orie Analytique\ndes Probabilit\u00e9s* is of course the \u201cPrincipia\u201d of the subject; his\n*Essai Philosophique sur les Probabilit\u00e9s* is a popular discourse, and\nis one of the most profound and interesting essays ever published. It\nshould be familiar to every student of logical method, and has lost\nlittle or none of its importance by lapse of time.\n\n\n*Detection of Constant Errors.*\n\nThe Method of Means is absolutely incapable of eliminating any error\nwhich is always the same, or which always lies in one direction. We\nsometimes require to be roused from a false feeling of security, and\nto be urged to take suitable precautions against such occult errors.\n\u201cIt is to the observer,\u201d says Gauss,[298] \u201cthat belongs the task of\ncarefully removing the causes of constant errors,\u201d and this is quite\ntrue when the error is absolutely constant. When we have made a number\nof determinations with a certain apparatus or method of measurement,\nthere is a great advantage in altering the arrangement, or even\ndevising some entirely different method of getting estimates of the\nsame quantity. The reason obviously consists in the improbability that\nthe same error will affect two or more different methods of experiment.\nIf a discrepancy is found to exist, we shall at least be aware of the\nexistence of error, and can take measures for finding in which way it\nlies. If we can try a considerable number of methods, the probability\nbecomes great that errors constant in one method will be balanced or\nnearly so by errors of an opposite effect in the others. Suppose that\nthere be three different methods each affected by an error of equal\namount. The probability that this error will in all fall in the same\ndirection is only 1/4; and with four methods similarly 1/8. If each\nmethod be affected, as is always the case, by several independent\nsources of error, the probability becomes much greater that in the mean\nresult of all the methods some of the errors will partially compensate\nthe others. In this case as in all others, when human vigilance has\nexhausted itself, we must trust the theory of probability.\n\n  [298] Gauss, translated by Bertrand, p.\u00a025.\n\nIn the determination of a zero point, of the magnitude of the\nfundamental standards of time and space, in the personal equation of\nan astronomical observer, we have instances of fixed errors; but as a\ngeneral rule a change of procedure is likely to reverse the character\nof the error, and many instances may be given of the value of this\nprecaution. If we measure over and over again the same angular\nmagnitude by the same divided circle, maintained in exactly the same\nposition, it is evident that the same mark in the circle will be the\ncriterion in each case, and any error in the position of that mark will\nequally affect all our results. But if in each measurement we use a\ndifferent part of the circle, a new mark will come into use, and as the\nerror of each mark cannot be in the same direction, the average result\nwill be nearly free from errors of division. It will be better still to\nuse more than one divided circle.\n\nEven when we have no perception of the points at which error is\nlikely to enter, we may with advantage vary the construction of our\napparatus in the hope that we shall accidentally detect some latent\ncause of error. Baily\u2019s purpose in repeating the experiments of\nMichell and Cavendish on the density of the earth was not merely to\nfollow the same course and verify the previous numbers, but to try\nwhether variations in the size and substance of the attracting balls,\nthe mode of suspension, the temperature of the surrounding air, &c.,\nwould yield different results. He performed no less than 62 distinct\nseries, comprising 2153 experiments, and he carefully classified and\ndiscussed the results so as to disclose the utmost differences. Again,\nin experimenting upon the resistance of the air to the motion of a\npendulum, Baily employed no less than 80 pendulums of various forms\nand materials, in order to ascertain exactly upon what conditions\nthe resistance depends. Regnault, in his exact researches upon the\ndilatation of gases, made arbitrary changes in the magnitude of parts\nof his apparatus. He thinks that if, in spite of such modification,\nthe results are unchanged, the errors are probably of inconsiderable\namount;[299] but in reality it is always possible, and usually likely,\nthat we overlook sources of error which a future generation will\ndetect. Thus the pendulum experiments of Baily and Sabine were directed\nto ascertain the nature and amount of a correction for air resistance,\nwhich had been entirely misunderstood in the experiments by means of\nthe seconds pendulum, upon which was founded the definition of the\nstandard yard, in the Act of 5th George IV. c. 74. It has already been\nmentioned that a considerable error was discovered in the determination\nof the standard metre as the ten-millionth part of the distance from\nthe pole to the equator (p.\u00a0314).\n\n  [299] Jamin, *Cours de Physique*, vol.\u00a0ii. p.\u00a060.\n\nWe shall return in Chapter XXV. to the further consideration of the\nmethods by which we may as far as possible secure ourselves against\npermanent and undetected sources of error. In the meantime, having\ncompleted the consideration of the special methods requisite for\ntreating quantitative phenomena, we must pursue our principal subject,\nand endeavour to trace out the course by which the physicist, from\nobservation and experiment, collects the materials of knowledge, and\nthen proceeds by hypothesis and inverse calculation to induce from them\nthe laws of nature.\n\n\n\n\nBook IV.\n\nINDUCTIVE INVESTIGATION.\n\n\n\n\nCHAPTER XVIII.\n\nOBSERVATION.\n\n\nALL knowledge proceeds originally from experience. Using the name in a\nwide sense, we may say that experience comprehends all that we *feel*,\nexternally or internally--the aggregate of the impressions which we\nreceive through the various apertures of perception--the aggregate\nconsequently of what is in the mind, except so far as some portions\nof knowledge may be the reasoned equivalents of other portions. As\nthe word experience expresses, we *go through* much in life, and the\nimpressions gathered intentionally or unintentionally afford the\nmaterials from which the active powers of the mind evolve science.\n\nNo small part of the experience actually employed in science is\nacquired without any distinct purpose. We cannot use the eyes without\ngathering some facts which may prove useful. A great science has in\nmany cases risen from an accidental observation. Erasmus Bartholinus\nthus first discovered double refraction in Iceland spar; Galvani\nnoticed the twitching of a frog\u2019s leg; Oken was struck by the form of\na vertebra; Malus accidentally examined light reflected from distant\nwindows with a double refracting substance; and Sir John Herschel\u2019s\nattention was drawn to the peculiar appearance of a solution of quinine\nsulphate. In earlier times there must have been some one who first\nnoticed the strange behaviour of a loadstone, or the unaccountable\nmotions produced by amber. As a general rule we shall not know in what\ndirection to look for a great body of phenomena widely different from\nthose familiar to us. Chance then must give us the starting point; but\none accidental observation well used may lead us to make thousands of\nobservations in an intentional and organised manner, and thus a science\nmay be gradually worked out from the smallest opening.\n\n\n*Distinction of Observation and Experiment.*\n\nIt is usual to say that the two sources of experience are Observation\nand Experiment. When we merely note and record the phenomena which\noccur around us in the ordinary course of nature we are said *to\nobserve*. When we change the course of nature by the intervention\nof our muscular powers, and thus produce unusual combinations and\nconditions of phenomena, we are said *to experiment*. Herschel justly\nremarked[300] that we might properly call these two modes of experience\n*passive and active observation*. In both cases we must certainly\nemploy our senses to observe, and an experiment differs from a mere\nobservation in the fact that we more or less influence the character\nof the events which we observe. Experiment is thus observation *plus*\nalteration of conditions.\n\n  [300] *Preliminary Discourse on the Study of Natural Philosophy*,\n  p.\u00a077.\n\nIt may readily be seen that we pass upwards by insensible gradations\nfrom pure observation to determinate experiment. When the earliest\nastronomers simply noticed the ordinary motions of the sun, moon, and\nplanets upon the face of the starry heavens, they were pure observers.\nBut astronomers now select precise times and places for important\nobservations of stellar parallax, or the transits of planets. They make\nthe earth\u2019s orbit the basis of a well arranged *natural experiment*,\nas it were, and take well considered advantage of motions which they\ncannot control. Meteorology might seem to be a science of pure\nobservation, because we cannot possibly govern the changes of weather\nwhich we record. Nevertheless we may ascend mountains or rise in\nballoons, like Gay-Lussac and Glaisher, and may thus so vary the points\nof observation as to render our procedure experimental. We are wholly\nunable either to produce or prevent earth-currents of electricity,\nbut when we construct long lines of telegraph, we gather such strong\ncurrents during periods of disturbance as to render them capable of\neasy observation.\n\nThe best arranged systems of observation, however, would fail to give\nus a large part of the facts which we now possess. Many processes\ncontinually going on in nature are so slow and gentle as to escape\nour powers of observation. Lavoisier remarked that the decomposition\nof water must have been constantly proceeding in nature, although its\npossibility was unknown till his time.[301] No substance is wholly\ndestitute of magnetic or diamagnetic powers; but it required all the\nexperimental skill of Faraday to prove that iron and a few other\nmetals had no monopoly of these powers. Accidental observation long\nago impressed upon men\u2019s minds the phenomena of lightning, and the\nattractive properties of amber. Experiment only could have shown that\nphenomena so diverse in magnitude and character were manifestations of\nthe same agent. To observe with accuracy and convenience we must have\nagents under our control, so as to raise or lower their intensity,\nto stop or set them in action at will. Just as Smeaton found it\nrequisite to create an artificial and governable supply of wind for\nhis investigation of windmills, so we must have governable supplies of\nlight, heat, electricity, muscular force, or whatever other agents we\nare examining.\n\n  [301] Lavoisier\u2019s *Elements of Chemistry*, translated by Kerr, 3rd\n  ed. p. 148.\n\nIt is hardly needful to point out too that on the earth\u2019s surface we\nlive under nearly constant conditions of gravity, temperature, and\natmospheric pressure, so that if we are to extend our inferences to\nother parts of the universe where conditions are widely different, we\nmust be prepared to imitate those conditions on a small scale here.\nWe must have intensely high and low temperatures; we must vary the\ndensity of gases from approximate vacuum upwards; we must subject\nliquids and solids to pressures or strains of almost unlimited amount.\n\n\n*Mental Conditions of Correct Observation.*\n\nEvery observation must in a certain sense be true, for the observing\nand recording of an event is in itself an event. But before we proceed\nto deal with the supposed meaning of the record, and draw inferences\nconcerning the course of nature, we must take care to ascertain that\nthe character and feelings of the observer are not to a great extent\nthe phenomena recorded. The mind of man, as Francis Bacon said, is\nlike an uneven mirror, and does not reflect the events of nature\nwithout distortion. We need hardly take notice of intentionally false\nobservations, nor of mistakes arising from defective memory, deficient\nlight, and so forth. Even where the utmost fidelity and care are used\nin observing and recording, tendencies to error exist, and fallacious\nopinions arise in consequence.\n\nIt is difficult to find persons who can with perfect fairness register\nfacts for and against their own peculiar views. Among uncultivated\nobservers the tendency to remark favourable and forget unfavourable\nevents is so great, that no reliance can be placed upon their supposed\nobservations. Thus arises the enduring fallacy that the changes of the\nweather coincide in some way with the changes of the moon, although\nexact and impartial registers give no countenance to the fact. The\nwhole race of prophets and quacks live on the overwhelming effect of\none success, compared with hundreds of failures which are unmentioned\nand forgotten. As Bacon says, \u201cMen mark when they hit, and never mark\nwhen they miss.\u201d And we should do well to bear in mind the ancient\nstory, quoted by Bacon, of one who in Pagan times was shown a temple\nwith a picture of all the persons who had been saved from shipwreck,\nafter paying their vows. When asked whether he did not now acknowledge\nthe power of the gods, \u201cAy,\u201d he answered; \u201cbut where are they painted\nthat were drowned after their vows?\u201d\n\nIf indeed we could estimate the amount of *bias* existing in any\nparticular observations, it might be treated like one of the forces\nof the problem, and the true course of external nature might still be\nrendered apparent. But the feelings of an observer are usually too\nindeterminate, so that when there is reason to suspect considerable\nbias, rejection is the only safe course. As regards facts casually\nregistered in past times, the capacity and impartiality of the observer\nare so little known that we should spare no pains to replace these\nstatements by a new appeal to nature. An indiscriminate medley of\ntruth and absurdity, such as Francis Bacon collected in his *Natural\nHistory*, is wholly unsuited to the purposes of science. But of course\nwhen records relate to past events like eclipses, conjunctions,\nmeteoric phenomena, earthquakes, volcanic eruptions, changes of sea\nmargins, the existence of now extinct animals, the migrations of\ntribes, remarkable customs, &c., we must make use of statements however\nunsatisfactory, and must endeavour to verify them by the comparison of\nindependent records or traditions.\n\nWhen extensive series of observations have to be made, as in\nastronomical, meteorological, or magnetical observatories,\ntrigonometrical surveys, and extensive chemical or physical researches,\nit is an advantage that the numerical work should be executed by\nassistants who are not interested in, and are perhaps unaware of, the\nexpected results. The record is thus rendered perfectly impartial.\nIt may even be desirable that those who perform the purely routine\nwork of measurement and computation should be unacquainted with the\nprinciples of the subject. The great table of logarithms of the\nFrench Revolutionary Government was worked out by a staff of sixty or\neighty computers, most of whom were acquainted only with the rules of\narithmetic, and worked under the direction of skilled mathematicians;\nyet their calculations were usually found more correct than those of\npersons more deeply versed in mathematics.[302] In the Indian Ordnance\nSurvey the actual measurers were selected so that they should not have\nsufficient skill to falsify their results without detection.\n\n  [302] Babbage, *Economy of Manufactures*, p.\u00a0194.\n\nBoth passive observation and experimentation must, however, be\ngenerally conducted by persons who know for what they are to look. It\nis only when excited and guided by the hope of verifying a theory that\nthe observer will notice many of the most important points; and, where\nthe work is not of a routine character, no assistant can supersede the\nmind-directed observations of the philosopher. Thus the successful\ninvestigator must combine diverse qualities; he must have clear notions\nof the result he expects and confidence in the truth of his theories,\nand yet he must have that candour and flexibility of mind which enable\nhim to accept unfavourable results and abandon mistaken views.\n\n\n*Instrumental and Sensual Conditions of Observation.*\n\nIn every observation one or more of the senses must be employed, and\nwe should ever bear in mind that the extent of our knowledge may be\nlimited by the power of the sense concerned. What we learn of the world\nonly forms the lower limit of what is to be learned, and, for all that\nwe can tell, the processes of nature may infinitely surpass in variety\nand complexity those which are capable of coming within our means of\nobservation. In some cases inference from observed phenomena may make\nus indirectly aware of what cannot be directly felt, but we can never\nbe sure that we thus acquire any appreciable fraction of the knowledge\nthat might be acquired.\n\nIt is a strange reflection that space may be filled with dark wandering\nstars, whose existence could not have yet become in any way known to\nus. The planets have already cooled so far as to be no longer luminous,\nand it may well be that other stellar bodies of various size have\nfallen into the same condition. From the consideration, indeed, of\nvariable and extinguished stars, Laplace inferred that there probably\nexist opaque bodies as great and perhaps as numerous as those we\nsee.[303] Some of these dark stars might ultimately become known to\nus, either by reflecting light, or more probably by their gravitating\neffects upon luminous stars. Thus if one member of a double star\nwere dark, we could readily detect its existence, and even estimate\nits size, position, and motions, by observing those of its visible\ncompanion. It was a favourite notion of Huyghens that there may exist\nstars and vast universes so distant that their light has never yet\nhad time to reach our eyes; and we must also bear in mind that light\nmay possibly suffer slow extinction in space, so that there is more\nthan one way in which an absolute limit to the powers of telescopic\ndiscovery may exist.\n\n  [303] *System of the World*, translated by Harte, vol.\u00a0ii. p.\u00a0335.\n\nThere are natural limits again to the power of our senses in detecting\nundulations of various kinds. It is commonly said that vibrations of\nmore than 38,000 strokes per second are not audible as sound; and\nas some ears actually do hear sounds of much higher pitch, even two\noctaves higher than what other ears can detect, it is exceedingly\nprobable that there are incessant vibrations which we cannot call sound\nbecause they are never heard. Insects may communicate by such acute\nsounds, constituting a language inaudible to us; and the remarkable\nagreement apparent among bodies of ants or bees might thus perhaps be\nexplained. Nay, as Fontenelle long ago suggested in his scientific\nromance, there may exist unlimited numbers of senses or modes of\nperception which we can never feel, though Darwin\u2019s theory would render\nit probable that any useful means of knowledge in an ancestor would\nbe developed and improved in the descendants. We might doubtless have\nbeen endowed with a sense capable of feeling electric phenomena with\nacuteness, so that the positive or negative state of charge of a body\ncould be at once estimated. The absence of such a sense is probably due\nto its comparative uselessness.\n\nHeat undulations are subject to the same considerations. It is now\napparent that what we call light is the affection of the eye by certain\nvibrations, the less rapid of which are invisible and constitute the\ndark rays of radiant heat, in detecting which we must substitute\nthe thermometer or the thermopile for the eye. At the other end of\nthe spectrum, again, the ultra-violet rays are invisible, and only\nindirectly brought to our knowledge in the phenomena of fluorescence or\nphoto-chemical action. There is no reason to believe that at either end\nof the spectrum an absolute limit has yet been reached.\n\nJust as our knowledge of the stellar universe is limited by the\npower of the telescope and other conditions, so our knowledge of the\nminute world has its limit in the powers and optical conditions of\nthe microscope. There was a time when it would have been a reasonable\ninduction that vegetables are motionless, and animals alone endowed\nwith power of locomotion. We are astonished to discover by the\nmicroscope that minute plants are if anything more active than\nminute animals. We even find that mineral substances seem to lose\ntheir inactive character and dance about with incessant motion when\nreduced to sufficiently minute particles, at least when suspended\nin a non-conducting medium.[304] Microscopists will meet a natural\nlimit to observation when the minuteness of the objects examined\nbecomes comparable to the length of light undulations, and the extreme\ndifficulty already encountered in determining the forms of minute marks\non Diatoms appears to be due to this cause. According to Helmholtz the\nsmallest distance which can be accurately defined depends upon the\ninterference of light passing through the centres of the bright spaces.\nWith a theoretically perfect microscope and a dry lense the smallest\nvisible object would not be less than one 80,000th part of an inch in\nred light.\n\n  [304] This curious phenomenon, which I propose to call *pedesis*, or\n  the *pedetic movement*, from \u03c0\u03b7\u03b4\u03cc\u03c9, to jump, is carefully described\n  in my paper published in the *Quarterly Journal of Science* for\n  April, 1878, vol.\u00a0viii. (N.S.) p. 167. See also *Proceedings of the\n  Literary and Philosophical Society of Manchester*, 25th January,\n  1870, vol.\u00a0ix. p.\u00a078, *Nature*, 22nd August, 1878, vol.\u00a0xviii.\n  p.\u00a0440, or the *Quarterly Journal of Science*, vol.\u00a0viii. (N.S.)\n  p.\u00a0514.\n\nOf the errors likely to arise in estimating quantities by the senses I\nhave already spoken, but there are some cases in which we actually see\nthings differently from what they are. A jet of water appears to be a\ncontinuous thread, when it is really a wonderfully organised succession\nof small and large drops, oscillating in form. The drops fall so\nrapidly that their impressions upon the eye run into each other, and in\norder to see the separate drops we require some device for giving an\ninstantaneous view.\n\nOne insuperable limit to our powers of observation arises from the\nimpossibility of following and identifying the ultimate atoms of\nmatter. One atom of oxygen is probably undistinguishable from another\natom; only by keeping a certain volume of oxygen safely inclosed in a\nbottle can we assure ourselves of its identity; allow it to mix with\nother oxygen, and we lose all power of identification. Accordingly\nwe seem to have no means of directly proving that every gas is in a\nconstant state of diffusion of every part into every part. We can only\ninfer this to be the case from observing the behaviour of distinct\ngases which we can distinguish in their course, and by reasoning on the\ngrounds of molecular theory.[305]\n\n  [305] Maxwell, *Theory of Heat*, p.\u00a0301.\n\n\n*External Conditions of Correct Observation.*\n\nBefore we proceed to draw inferences from any series of recorded facts,\nwe must take care to ascertain perfectly, if possible, the external\nconditions under which the facts are brought to our notice. Not only\nmay the observing mind be prejudiced and the senses defective, but\nthere may be circumstances which cause one kind of event to come more\nfrequently to our notice than another. The comparative numbers of\nobjects of different kinds existing may in any degree differ from the\nnumbers which come to our notice. This difference must if possible be\ntaken into account before we make any inferences.\n\nThere long appeared to be a strong presumption that all comets moved\nin elliptic orbits, because no comet had been proved to move in any\nother kind of path. The theory of gravitation admitted of the existence\nof comets moving in hyperbolic orbits, and the question arose whether\nthey were really non-existent or were only beyond the bounds of easy\nobservation. From reasonable suppositions Laplace calculated that\nthe probability was at least 6000 to 1 against a comet which comes\nwithin the planetary system sufficiently to be visible at the earth\u2019s\nsurface, presenting an orbit which could be discriminated from a very\nelongated ellipse or parabola in the part of its orbit within the reach\nof our telescopes.[306] In short, the chances are very much in favour\nof our seeing elliptic rather than hyperbolic comets. Laplace\u2019s views\nhave been confirmed by the discovery of six hyperbolic comets, which\nappeared in the years 1729, 1771, 1774, 1818, 1840, and 1843,[307] and\nas only about 800 comets altogether have been recorded, the proportion\nof hyperbolic ones is quite as large as should be expected.\n\n  [306] Laplace, *Essai Philosophique*, p.\u00a059. Todhunter\u2019s *History*,\n  pp. 491\u2013494.\n\n  [307] Chambers\u2019 *Astronomy*, 1st ed. p.\u00a0203.\n\nWhen we attempt to estimate the numbers of objects which may have\nexisted, we must make large allowances for the limited sphere of our\nobservations. Probably not more than 4000 or 5000 comets have been seen\nin historical times, but making allowance for the absence of observers\nin the southern hemisphere, and for the small probability that we see\nany considerable fraction of those which are in the neighbourhood of\nour system, we must accept Kepler\u2019s opinion, that there are more comets\nin the regions of space than fishes in the depths of the ocean. When\nlike calculations are made concerning the numbers of meteors visible to\nus, it is astonishing to find that the number of meteors entering the\nearth\u2019s atmosphere in every twenty-four hours is probably not less than\n400,000,000, of which 13,000 exist in every portion of space equal to\nthat filled by the earth.\n\nSerious fallacies may arise from overlooking the inevitable conditions\nunder which the records of past events are brought to our notice.\nThus it is only the durable objects manufactured by former races of\nmen, such as flint implements, which can have come to our notice as a\ngeneral rule. The comparative abundance of iron and bronze articles\nused by an ancient nation must not be supposed to be coincident with\ntheir comparative abundance in our museums, because bronze is far the\nmore durable. There is a prevailing fallacy that our ancestors built\nmore strongly than we do, arising from the fact that the more fragile\nstructures have long since crumbled away. We have few or no relics of\nthe habitations of the poorer classes among the Greeks or Romans, or in\nfact of any past race; for the temples, tombs, public buildings, and\nmansions of the wealthier classes alone endure. There is an immense\nexpanse of past events necessarily lost to us for ever, and we must\ngenerally look upon records or relics as exceptional in their character.\n\nThe same considerations apply to geological relics. We could not\ngenerally expect that animals would be preserved unless as regards the\nbones, shells, strong integuments, or other hard and durable parts. All\nthe infusoria and animals devoid of mineral framework have probably\nperished entirely, distilled perhaps into oils. It has been pointed\nout that the peculiar character of some extinct floras may be due to\nthe unequal preservation of different families of plants. By various\naccidents, however, we gain glimpses of a world that is usually lost\nto us--as by insects embedded in amber, the great mammoth preserved in\nice, mummies, casts in solid material like that of the Roman soldier at\nPompeii, and so forth.\n\nWe should also remember, that just as there may be conjunctions of the\nheavenly bodies that can have happened only once or twice in the period\nof history, so remarkable terrestrial conjunctions may take place.\nGreat storms, earthquakes, volcanic eruptions, landslips, floods,\nirruptions of the sea, may, or rather must, have occurred, events of\nsuch unusual magnitude and such extreme rarity that we can neither\nexpect to witness them nor readily to comprehend their effects. It is\na great advantage of the study of probabilities, as Laplace himself\nremarked, to make us mistrust the extent of our knowledge, and pay\nproper regard to the probability that events would come within the\nsphere of our observations.\n\n\n*Apparent Sequence of Events.*\n\nDe Morgan has excellently pointed out[308] that there are no less than\nfour modes in which one event may seem to follow or be connected with\nanother, without being really so. These involve mental, sensual, and\nexternal causes of error, and I will briefly state and illustrate them.\n\n  [308] *Essay on Probabilities*, Cabinet Cyclop\u00e6dia, p.\u00a0121.\n\nInstead of A causing B, it may be *our perception of A that causes B*.\nThus it is that prophecies, presentiments, and the devices of sorcery\nand witchcraft often work their own ends. A man dies on the day which\nhe has always regarded as his last, from his own fears of the day. An\nincantation effects its purpose, because care is taken to frighten the\nintended victim, by letting him know his fate. In all such cases the\nmental condition is the cause of apparent coincidence.\n\nIn a second class of cases, *the event A may make our perception of\nB follow, which would otherwise happen without being perceived*.\nThus it was believed to be the result of investigation that more\ncomets appeared in hot than cold summers. No account was taken of\nthe fact that hot summers would be comparatively cloudless, and\nafford better opportunities for the discovery of comets. Here the\ndisturbing condition is of a purely external character. Certain ancient\nphilosophers held that the moon\u2019s rays were cold-producing, mistaking\nthe cold caused by radiation into space for an effect of the moon,\nwhich is more likely to be visible at a time when the absence of clouds\npermits radiation to proceed.\n\nIn a third class of cases, *our perception of A may make our perception\nof B follow*. The event B may be constantly happening, but our\nattention may not be drawn to it except by our observing A. This case\nseems to be illustrated by the fallacy of the moon\u2019s influence on\nclouds. The origin of this fallacy is somewhat complicated. In the\nfirst place, when the sky is densely clouded the moon would not be\nvisible at all; it would be necessary for us to see the full moon in\norder that our attention should be strongly drawn to the fact, and this\nwould happen most often on those nights when the sky is cloudless. Mr.\nW. Ellis,[309] moreover, has ingeniously pointed out that there is a\ngeneral tendency for clouds to disperse at the commencement of night,\nwhich is the time when the full moon rises. Thus the change of the sky\nand the rise of the full moon are likely to attract attention mutually,\nand the coincidence in time suggests the relation of cause and effect.\nMr. Ellis proves from the results of observations at the Greenwich\nObservatory that the moon possesses no appreciable power of the kind\nsupposed, and yet it is remarkable that so sound an observer as Sir\nJohn Herschel was convinced of the connection. In his \u201cResults of\nObservations at the Cape of Good Hope,\u201d[310] he mentions many evenings\nwhen a full moon occurred with a peculiarly clear sky.\n\n  [309] *Philosophical Magazine*, 4th Series (1867), vol.\u00a0xxxiv. p.\u00a064.\n\n  [310] See *Notes to Measures of Double Stars*, 1204, 1336, 1477,\n  1686, 1786, 1816, 1835, 1929, 2081, 2186, pp.\u00a0265, &c. See also\n  Herschel\u2019s *Familiar Lectures on Scientific Subjects*, p.\u00a0147, and\n  *Outlines of Astronomy*, 7th ed. p.\u00a0285.\n\nThere is yet a fourth class of cases, in which *B is really the\nantecedent event, but our perception of A, which is a consequence\nof B, may be necessary to bring about our perception of B*. There\ncan be no doubt, for instance, that upward and downward currents are\ncontinually circulating in the lowest stratum of the atmosphere during\nthe day-time; but owing to the transparency of the atmosphere we have\nno evidence of their existence until we perceive cumulous clouds, which\nare the consequence of such currents. In like manner an interfiltration\nof bodies of air in the higher parts of the atmosphere is probably in\nnearly constant progress, but unless threads of cirrous cloud indicate\nthese motions we remain ignorant of their occurrence.[311] The highest\nstrata of the atmosphere are wholly imperceptible to us, except when\nrendered luminous by auroral currents of electricity, or by the passage\nof meteoric stones. Most of the visible phenomena of comets probably\narise from some substance which, existing previously invisible, becomes\ncondensed or electrified suddenly into a visible form. Sir John\nHerschel attempted to explain the production of comet tails in this\nmanner by evaporation and condensation.[312]\n\n  [311] Jevons, *On the Cirrous Form of Cloud*, Philosophical Magazine,\n  July, 1857, 4th Series, vol.\u00a0xiv. p.\u00a022.\n\n  [312] *Astronomy*, 4th ed. p.\u00a0358.\n\n\n*Negative Arguments from Non-observation.*\n\nFrom what has been suggested in preceding sections, it will plainly\nappear that the non-observation of a phenomenon is not generally to\nbe taken as proving its non-occurrence. As there are sounds which we\ncannot hear, rays of heat which we cannot feel, multitudes of worlds\nwhich we cannot see, and myriads of minute organisms of which not the\nmost powerful microscope can give us a view, we must as a general rule\ninterpret our experience in an affirmative sense only. Accordingly\nwhen inferences have been drawn from the non-occurrence of particular\nfacts or objects, more extended and careful examination has often\nproved their falsity. Not many years since it was quite a well credited\nconclusion in geology that no remains of man were found in connection\nwith those of extinct animals, or in any deposit not actually at\npresent in course of formation. Even Babbage accepted this conclusion\nas strongly confirmatory of the Mosaic accounts.[313] While the opinion\nwas yet universally held, flint implements had been found disproving\nsuch a conclusion, and overwhelming evidence of man\u2019s long-continued\nexistence has since been forthcoming. At the end of the last century,\nwhen Herschel had searched the heavens with his powerful telescopes,\nthere seemed little probability that planets yet remained unseen\nwithin the orbit of Jupiter. But on the first day of this century such\nan opinion was overturned by the discovery of Ceres, and more than a\nhundred other small planets have since been added to the lists of the\nplanetary system.\n\n  [313] Babbage, *Ninth Bridgewater Treatise*, p.\u00a067.\n\nThe discovery of the Eozo\u00f6n Canadense in strata of much greater age\nthan any previously known to contain organic remains, has given a shock\nto groundless opinions concerning the origin of organic forms; and\nthe oceanic dredging expeditions under Dr. Carpenter and Sir Wyville\nThomson have modified some opinions of geologists by disclosing the\ncontinued existence of forms long supposed to be extinct. These and\nmany other cases which might be quoted show the extremely unsafe\ncharacter of negative inductions.\n\nBut it must not be supposed that negative arguments are of no force and\nvalue. The earth\u2019s surface has been sufficiently searched to render it\nhighly improbable that any terrestrial animals of the size of a camel\nremain to be discovered. It is believed that no new large animal has\nbeen encountered in the last eighteen or twenty centuries,[314] and\nthe probability that if existent they would have been seen, increases\nthe probability that they do not exist. We may with somewhat less\nconfidence discredit the existence of any large unrecognised fish, or\nsea animals, such as the alleged sea-serpent. But, as we descend to\nforms of smaller size negative evidence loses weight from the less\nprobability of our seeing smaller objects. Even the strong induction in\nfavour of the four-fold division of the animal kingdom into Vertebrata,\nAnnulosa, Mollusca, and C\u0153lenterata, may break down by the discovery\nof intermediate or anomalous forms. As civilisation spreads over the\nsurface of the earth, and unexplored tracts are gradually diminished,\nnegative conclusions will increase in force; but we have much to learn\nyet concerning the depths of the ocean, almost wholly unexamined as\nthey are, and covering three-fourths of the earth\u2019s surface.\n\n  [314] Cuvier, *Essay on the Theory of the Earth*, translation, p.\u00a061,\n  &c.\n\nIn geology there are many statements to which considerable probability\nattaches on account of the large extent of the investigations already\nmade, as, for instance, that true coal is found only in rocks of a\nparticular geological epoch; that gold occurs in secondary and tertiary\nstrata only in exceedingly small quantities,[315] probably derived\nfrom the disintegration of earlier rocks. In natural history negative\nconclusions are exceedingly treacherous and unsatisfactory. The utmost\npatience will not enable a microscopist or the observer of any living\nthing to watch the behaviour of the organism under all circumstances\ncontinuously for a great length of time. There is always a chance\ntherefore that the critical act or change may take place when the\nobserver\u2019s eyes are withdrawn. This certainly happens in some cases;\nfor though the fertilisation of orchids by agency of insects is proved\nas well as any fact in natural history, Mr. Darwin has never been\nable by the closest watching to detect an insect in the performance\nof the operation. Mr. Darwin has himself adopted one conclusion on\nnegative evidence, namely, that the *Orchis pyramidalis* and certain\nother orchidaceous flowers secrete no nectar. But his caution and\nunwearying patience in verifying the conclusion give an impressive\nlesson to the observer. For twenty-three consecutive days, as he tells\nus, he examined flowers in all states of the weather, at all hours, in\nvarious localities. As the secretion in other flowers sometimes takes\nplace rapidly and might happen at early dawn, that inconvenient hour\nof observation was specially adopted. Flowers of different ages were\nsubjected to irritating vapours, to moisture, and to every condition\nlikely to bring on the secretion; and only after invariable failure of\nthis exhaustive inquiry was the barrenness of the nectaries assumed to\nbe proved.[316]\n\n  [315] Murchison\u2019s *Siluria*, 1st ed. p.\u00a0432.\n\n  [316] Darwin\u2019s *Fertilisation of Orchids*, p.\u00a048.\n\nIn order that a negative argument founded on the non-observation of\nan object shall have any considerable force, it must be shown to be\nprobable that the object if existent would have been observed, and it\nis this probability which defines the value of the negative conclusion.\nThe failure of astronomers to see the planet Vulcan, supposed by some\nto exist within Mercury\u2019s orbit, is no sufficient disproof of its\nexistence. Similarly it would be very difficult, or even impossible, to\ndisprove the existence of a second satellite of small size revolving\nround the earth. But if any person make a particular assertion,\nassigning place and time, then observation will either prove or\ndisprove the alleged fact. If it is true that when a French observer\nprofessed to have seen a planet on the sun\u2019s face, an observer in\nBrazil was carefully scrutinising the sun and failed to see it, we have\na negative proof. False facts in science, it has been well said, are\nmore mischievous than false theories. A false theory is open to every\nperson\u2019s criticism, and is ever liable to be judged by its accordance\nwith facts. But a false or grossly erroneous assertion of a fact\noften stands in the way of science for a long time, because it may be\nextremely difficult or even impossible to prove the falsity of what has\nbeen once recorded.\n\nIn other sciences the force of a negative argument will often depend\nupon the number of possible alternatives which may exist. It was long\nbelieved that the quality of a musical sound as distinguished from\nits pitch, must depend upon the form of the undulation, because no\nother cause of it had ever been suggested or was apparently possible.\nThe truth of the conclusion was proved by Helmholtz, who applied a\nmicroscope to luminous points attached to the strings of various\ninstruments, and thus actually observed the different modes of\nundulation. In mathematics negative inductive arguments have seldom\nmuch force, because the possible forms of expression, or the possible\ncombinations of lines and circles in geometry, are quite unlimited in\nnumber. An enormous number of attempts were made to trisect the angle\nby the ordinary methods of Euclid\u2019s geometry, but their invariable\nfailure did not establish the impossibility of the task. This was shown\nin a totally different manner, by proving that the problem involves an\nirreducible cubic equation to which there could be no corresponding\nplane geometrical solution.[317] This is a case of *reductio ad\nabsurdum*, a form of argument of a totally different character.\nSimilarly no number of failures to obtain a general solution of\nequations of the fifth degree would establish the impossibility of the\ntask, but in an indirect mode, equivalent to a *reductio ad absurdum*,\nthe impossibility is considered to be proved.[318]\n\n  [317] Peacock, *Algebre*, vol.\u00a0ii. p.\u00a0344.\n\n  [318] Ibid, p.\u00a0359. Serret, *Alg\u00e8bre Sup\u00e9rieure*, 2nd ed. p.\u00a0304.\n\n\n\n\nCHAPTER XIX.\n\nEXPERIMENT.\n\n\nWe may now consider the great advantages which we enjoy in examining\nthe combinations of phenomena when things are within our reach and\ncapable of being experimented on. We are said *to experiment* when we\nbring substances together under various conditions of temperature,\npressure, electric disturbance, chemical action, &c., and then record\nthe changes observed. Our object in inductive investigation is to\nascertain exactly the group of circumstances or conditions which being\npresent, a certain other group of phenomena will follow. If we denote\nby A the antecedent group, and by X subsequent phenomena, our object\nwill usually be to discover a law of the form A\u00a0=\u00a0AX, the meaning of\nwhich is that where A is X will happen.\n\nThe circumstances which might be enumerated as present in the simplest\nexperiment are very numerous, in fact almost infinite. Rub two sticks\ntogether and consider what would be an exhaustive statement of the\nconditions. There are the form, hardness, organic structure, and all\nthe chemical qualities of the wood; the pressure and velocity of the\nrubbing; the temperature, pressure, and all the chemical qualities of\nthe surrounding air; the proximity of the earth with its attractive\nand electric powers; the temperature and other properties of the\npersons producing motion; the radiation from the sun, and to and from\nthe sky; the electric excitement possibly existing in any overhanging\ncloud; even the positions of the heavenly bodies must be mentioned.\nOn *\u00e0 priori* grounds it is unsafe to assume that any one of these\ncircumstances is without effect, and it is only by experience that we\ncan single out those precise conditions from which the observed heat of\nfriction proceeds.\n\nThe great method of experiment consists in removing, one at a time,\neach of those conditions which may be imagined to have an influence\non the result. Our object in the experiment of rubbing sticks is to\ndiscover the exact circumstances under which heat appears. Now the\npresence of air may be requisite; therefore prepare a vacuum, and\nrub the sticks in every respect as before, except that it is done\n*in vacuo*. If heat still appears we may say that air is not, in\nthe presence of the other circumstances, a requisite condition. The\nconduction of heat from neighbouring bodies may be a condition. Prevent\nthis by making all the surrounding bodies ice cold, which is what Davy\naimed at in rubbing two pieces of ice together. If heat still appears\nwe have eliminated another condition, and so we may go on until it\nbecomes apparent that the expenditure of energy in the friction of two\nbodies is the sole condition of the production of heat.\n\nThe great difficulty of experiment arises from the fact that we must\nnot assume the conditions to be independent. Previous to experiment we\nhave no right to say that the rubbing of two sticks will produce heat\nin the same way when air is absent as before. We may have heat produced\nin one way when air is present, and in another when air is absent.\nThe inquiry branches out into two lines, and we ought to try in both\ncases whether cutting off a supply of heat by conduction prevents its\nevolution in friction. The same branching out of the inquiry occurs\nwith regard to every circumstance which enters into the experiment.\n\nRegarding only four circumstances, say A, B, C, D, we ought to test\nnot only the combinations ABCD, ABC*d*, AB*c*D, A*b*CD, *a*BCD, but\nwe ought really to go through the whole of the combinations given in\nthe fifth column of the Logical Alphabet. The effect of the absence\nof each condition should be tried both in the presence and absence of\nevery other condition, and every selection of those conditions. Perfect\nand exhaustive experimentation would, in short, consist in examining\nnatural phenomena in all their possible combinations and registering\nall relations between conditions and results which are found capable\nof existence. It would thus resemble the exclusion of contradictory\ncombinations carried out in the Indirect Method of Inference, except\nthat the exclusion of combinations is grounded not on prior logical\npremises, but on *\u00e0 posteriori* results of actual trial.\n\nThe reader will perceive, however, that such exhaustive investigation\nis practically impossible, because the number of requisite experiments\nwould be immensely great. Four antecedents only would require sixteen\nexperiments; twelve antecedents would require 4096, and the number\nincreases as the powers of two. The result is that the experimenter\nhas to fall back upon his own tact and experience in selecting those\nexperiments which are most likely to yield him significant facts. It\nis at this point that logical rules and forms begin to fail in giving\naid. The logical rule is--Try all possible combinations; but this being\nimpracticable, the experimentalist necessarily abandons strict logical\nmethod, and trusts to his own insight. Analogy, as we shall see, gives\nsome assistance, and attention should be concentrated on those kinds\nof conditions which have been found important in like cases. But we\nare now entirely in the region of probability, and the experimenter,\nwhile he is confidently pursuing what he thinks the right clue, may\nbe overlooking the one condition of importance. It is an impressive\nlesson, for instance, that Newton pursued all his exquisite researches\non the spectrum unsuspicious of the fact that if he reduced the hole in\nthe shutter to a narrow slit, all the mysteries of the bright and dark\nlines were within his grasp, provided of course that his prisms were\nsufficiently good to define the rays. In like manner we know not what\nslight alteration in the most familiar experiments may not open the way\nto realms of new discovery.\n\nPractical difficulties, also, encumber the progress of the physicist.\nIt is often impossible to alter one condition without altering others\nat the same time; and thus we may not get the pure effect of the\ncondition in question. Some conditions may be absolutely incapable of\nalteration; others may be with great difficulty, or only in a certain\ndegree, removable. A very treacherous source of error is the existence\nof unknown conditions, which of course we cannot remove except by\naccident. These difficulties we will shortly consider in succession.\n\nIt is beautiful to observe how the alteration of a single circumstance\nsometimes conclusively explains a phenomenon. An instance is found in\nFaraday\u2019s investigation of the behaviour of Lycopodium spores scattered\non a vibrating plate. It was observed that these minute spores\ncollected together at the points of greatest motion, whereas sand and\nall heavy particles collected at the nodes, where the motion was least.\nIt happily occurred to Faraday to try the experiment in the exhausted\nreceiver of an air-pump, and it was then found that the light powder\nbehaved exactly like heavy powder. A conclusive proof was thus obtained\nthat the presence of air was the condition of importance, doubtless\nbecause it was thrown into eddies by the motion of the plate, and\ncarried the Lycopodium to the points of greatest agitation. Sand was\ntoo heavy to be carried by the air.\n\n\n*Exclusion of Indifferent Circumstances.*\n\nFrom what has been already said it will be apparent that the detection\nand exclusion of indifferent circumstances is a work of importance,\nbecause it allows the concentration of attention upon circumstances\nwhich contain the principal condition. Many beautiful instances may be\ngiven where all the most obvious antecedents have been shown to have no\npart in the production of a phenomenon. A person might suppose that the\npeculiar colours of mother-of-pearl were due to the chemical qualities\nof the substance. Much trouble might have been spent in following out\nthat notion by comparing the chemical qualities of various iridescent\nsubstances. But Brewster accidentally took an impression from a piece\nof mother-of-pearl in a cement of resin and bees\u2019-wax, and finding\nthe colours repeated upon the surface of the wax, he proceeded to\ntake other impressions in balsam, fusible metal, lead, gum arabic,\nisinglass, &c., and always found the iridescent colours the same. He\nthus proved that the chemical nature of the substance is a matter of\nindifference, and that the form of the surface is the real condition\nof such colours.[319] Nearly the same may be said of the colours\nexhibited by thin plates and films. The rings and lines of colour will\nbe nearly the same in character whatever may be the nature of the\nsubstance; nay, a void space, such as a crack in glass, would produce\nthem even though the air were withdrawn by an air-pump. The conditions\nare simply the existence of two reflecting surfaces separated by a very\nsmall space, though it should be added that the refractive index of the\nintervening substance has some influence on the exact nature of the\ncolour produced.\n\n  [319] *Treatise on Optics*, by Brewster, Cab. Cyclo. p.\u00a0117.\n\nWhen a ray of light passes close to the edge of an opaque body, a\nportion of the light appears to be bent towards it, and produces\ncoloured fringes within the shadow of the body. Newton attributed\nthis inflexion of light to the attraction of the opaque body for the\nsupposed particles of light, although he was aware that the nature\nof the surrounding medium, whether air or other pellucid substance,\nexercised no apparent influence on the phenomena. Gravesande proved,\nhowever, that the character of the fringes is exactly the same, whether\nthe body be dense or rare, compound or elementary. A wire produces\nexactly the same fringes as a hair of the same thickness. Even the\nform of the obstructing edge was subsequently shown to be a matter of\nindifference by Fresnel, and the interference spectrum, or the spectrum\nseen when light passes through a fine grating, is absolutely the same\nwhatever be the form or chemical nature of the bars making the grating.\nThus it appears that the stoppage of a portion of a beam of light is\nthe sole necessary condition for the diffraction or inflexion of light,\nand the phenomenon is shown to bear no analogy the refraction of light,\nin which the form and nature of the substance are all important.\n\nIt is interesting to observe how carefully Newton, in his researches\non the spectrum, ascertained the indifference of many circumstances by\nactual trial. He says:[320] \u201cNow the different magnitude of the hole\nin the window-shut, and different thickness of the prism where the\nrays passed through it, and different inclinations of the prism to the\nhorizon, made no sensible changes in the length of the image. Neither\ndid the different matter of the prisms make any: for in a vessel made\nof polished plates of glass cemented together in the shape of a prism,\nand filled with water, there is the like success of the experiment\naccording to the quantity of the refraction.\u201d But in the latter\nstatement, as I shall afterwards remark (p.\u00a0432), Newton assumed an\nindifference which does not exist, and fell into an unfortunate mistake.\n\n  [320] *Opticks*, 3rd. ed. p.\u00a025.\n\nIn the science of sound it is shown that the pitch of a sound depends\nsolely upon the number of impulses in a second, and the material\nexciting those impulses is a matter of indifference. Whatever fluid,\nair or water, gas or liquid, be forced into the Siren, the sound\nproduced is the same; and the material of which an organ-pipe is\nconstructed does not at all affect the pitch of its sound. In the\nscience of statical electricity it is an important principle that\nthe nature of the interior of a conducting body is a matter of no\nimportance. The electrical charge is confined to the conducting\nsurface, and the interior remains in a neutral state. A hollow copper\nsphere takes exactly the same charge as a solid sphere of the same\nmetal.\n\nSome of Faraday\u2019s most elegant and successful researches were devoted\nto the exclusion of conditions which previous experimenters had\nthought essential for the production of electrical phenomena. Davy\nasserted that no known fluids, except such as contain water, could be\nmade the medium of connexion between the poles of a battery; and some\nchemists believed that water was an essential agent in electro-chemical\ndecomposition. Faraday gave abundant experiments to show that other\nfluids allowed of electrolysis, and he attributed the erroneous opinion\nto the very general use of water as a solvent, and its presence in most\nnatural bodies.[321] It was, in fact, upon the weakest kind of negative\nevidence that the opinion had been founded.\n\nMany experimenters attributed peculiar powers to the poles of a\nbattery, likening them to magnets, which, by their attractive powers,\ntear apart the elements of a substance. By a beautiful series of\nexperiments,[322] Faraday proved conclusively that, on the contrary,\nthe substance of the poles is of no importance, being merely the path\nthrough which the electric force reaches the liquid acted upon. Poles\nof water, charcoal, and many diverse substances, even air itself,\nproduced similar results; if the chemical nature of the pole entered at\nall into the question, it was as a disturbing agent.\n\n  [321] *Experimental Researches in Electricity*, vol.\u00a0i. pp.\u00a0133, 134.\n\n  [322] Ibid. vol i. pp.\u00a0127, 162, &c.\n\nIt is an essential part of the theory of gravitation that the proximity\nof other attracting particles is without effect upon the attraction\nexisting between any two molecules. Two pound weights weigh as much\ntogether as they do separately. Every pair of molecules in the world\nhave, as it were, a private communication, apart from their relations\nto all other molecules. Another undoubted result of experience pointed\nout by Newton[323] is that the weight of a body does not in the least\ndepend upon its form or texture. It may be added that the temperature,\nelectric condition, pressure, state of motion, chemical qualities,\nand all other circumstances concerning matter, except its mass, are\nindifferent as regards its gravitating power.\n\n  [323] *Principia*, bk. iii. Prop. vi. Corollary i.\n\nAs natural science progresses, physicists gain a kind of insight\nand tact in judging what qualities of a substance are likely to be\nconcerned in any class of phenomena. The physical astronomer treats\nmatter in one point of view, the chemist in another, and the students\nof physical optics, sound, mechanics, electricity, &c., make a fair\ndivision of the qualities among them. But errors will arise if too\nmuch confidence be placed in this independence of various kinds of\nphenomena, so that it is desirable from time to time, especially\nwhen any unexplained discrepancies come into notice, to question the\nindifference which is assumed to exist, and to test its real existence\nby appropriate experiments.\n\n\n*Simplification of Experiments.*\n\nOne of the most requisite precautions in experimentation is to\nvary only one circumstance at a time, and to maintain all other\ncircumstances rigidly unchanged. There are two distinct reasons for\nthis rule, the first and most obvious being that if we vary two\nconditions at a time, and find some effect, we cannot tell whether\nthe effect is due to one or the other condition, or to both jointly.\nA second reason is that if no effect ensues we cannot safely conclude\nthat either of them is indifferent; for the one may have neutralised\nthe effect of the other. In our symbolic logic AB \ua58c A*b* was shown to\nbe identical with A (p.\u00a097), so that B denotes a circumstance which is\nindifferently present or absent. But if B always goes together with\nanother antecedent C, we cannot show the same independence, for ABC \ua58c\nA*bc* is not identical with A and none of our logical processes enables\nus to reduce it to A.\n\nIf we want to prove that oxygen is necessary to life, we must not\nput a rabbit into a vessel from which the oxygen has been exhausted\nby a burning candle. We should then have not only an absence of\noxygen, but an addition of carbonic acid, which may have been the\ndestructive agent. For a similar reason Lavoisier avoided the use of\natmospheric air in experiments on combustion, because air was not a\nsimple substance, and the presence of nitrogen might impede or even\nalter the effect of oxygen. As Lavoisier remarks,[324] \u201cIn performing\nexperiments, it is a necessary principle, which ought never to be\ndeviated from, that they be simplified as much as possible, and that\nevery circumstance capable of rendering their results complicated be\ncarefully removed.\u201d It has also been well said by Cuvier[325] that\nthe method of physical inquiry consists in isolating bodies, reducing\nthem to their utmost simplicity, and bringing each of their properties\nseparately into action, either mentally or by experiment.\n\n  [324] Lavoisier\u2019s *Chemistry*, translated by Kerr, p.\u00a0103.\n\n  [325] Cuvier\u2019s *Animal Kingdom*, introduction, pp.\u00a01, 2.\n\nThe electro-magnet has been of the utmost service in the investigation\nof the magnetic properties of matter, by allowing of the production\nor removal of a most powerful magnetic force without disturbing any\nof the other arrangements of the experiment. Many of Faraday\u2019s most\nvaluable experiments would have been impossible had it been necessary\nto introduce a heavy permanent magnet, which could not be suddenly\nmoved without shaking the whole apparatus, disturbing the air,\nproducing currents by changes of temperature, &c. The electro-magnet\nis perfectly under control, and its influence can be brought into\naction, reversed, or stopped by merely touching a button. Thus Faraday\nwas enabled to prove the rotation of the plane of circularly polarised\nlight by the fact that certain light ceased to be visible when the\nelectric current of the magnet was cut off, and re-appeared when the\ncurrent was made. \u201cThese phenomena,\u201d he says, \u201ccould be reversed at\npleasure, and at any instant of time, and upon any occasion, showing a\nperfect dependence of cause and effect.\u201d[326]\n\n  [326] *Experimental Researches in Electricity*, vol.\u00a0iii. p.\u00a04.\n\nIt was Newton\u2019s omission to obtain the solar spectrum under the\nsimplest conditions which prevented him from discovering the dark\nlines. Using a broad beam of light which had passed through a round\nhole or a triangular slit, he obtained a brilliant spectrum, but one\nin which many different coloured rays overlapped each other. In the\nrecent history of the science of the spectrum, one main difficulty has\nconsisted in the mixture of the lines of several different substances,\nwhich are usually to be found in the light of any flame or spark. It\nis seldom possible to obtain the light of any element in a perfectly\nsimple manner. Angstr\u00f6m greatly advanced this branch of science by\nexamining the light of the electric spark when formed between poles of\nvarious metals, and in the presence of various gases. By varying the\npole alone, or the gaseous medium alone, he was able to discriminate\ncorrectly between the lines due to the metal and those due to the\nsurrounding gas.[327]\n\n  [327] *Philosophical Magazine*, 4th Series, vol.\u00a0ix. p.\u00a0327.\n\n\n*Failure in the Simplification of Experiments.*\n\nIn some cases it seems to be impossible to carry out the rule of\nvarying one circumstance at a time. When we attempt to obtain two\ninstances or two forms of experiment in which a single circumstance\nshall be present in one case and absent in another, it may be found\nthat this single circumstance entails others. Benjamin Franklin\u2019s\nexperiment concerning the comparative absorbing powers of different\ncolours is well known. \u201cI took,\u201d he says, \u201ca number of little square\npieces of broadcloth from a tailor\u2019s pattern card, of various colours.\nThey were black, deep blue, lighter blue, green, purple, red, yellow,\nwhite, and other colours and shades of colour. I laid them all out upon\nthe snow on a bright sunshiny morning. In a few hours the black, being\nmost warmed by the sun, was sunk so low as to be below the stroke of\nthe sun\u2019s rays; the dark blue was almost as low; the lighter blue not\nquite so much as the dark; the other colours less as they were lighter.\nThe white remained on the surface of the snow, not having entered it at\nall.\u201d This is a very elegant and apparently simple experiment; but when\nLeslie had completed his series of researches upon the nature of heat,\nhe came to the conclusion that the colour of a surface has very little\neffect upon the radiating power, the mechanical nature of the surface\nappearing to be more influential. He remarks[328] that \u201cthe question\nis incapable of being positively resolved, since no substance can be\nmade to assume different colours without at the same time changing its\ninternal structure.\u201d Recent investigation has shown that the subject\nis one of considerable complication, because the absorptive power of a\nsurface may be different according to the character of the rays which\nfall upon it; but there can be no doubt as to the acuteness with which\nLeslie points out the difficulty. In Well\u2019s investigations concerning\nthe nature of dew, we have, again, very complicated conditions. If we\nexpose plates of various material, such as rough iron, glass, polished\nmetal, to the midnight sky, they will be dewed in various degrees; but\nsince these plates differ both in the nature of the surface and the\nconducting power of the material, it would not be plain whether one\nor both circumstances were of importance. We avoid this difficulty by\nexposing the same material polished or varnished, so as to present\ndifferent conditions of surface;[329] and again by exposing different\nsubstances with the same kind of surface.\n\n  [328] *Inquiry into the Nature of Heat*, p.\u00a095.\n\n  [329] Herschel, *Preliminary Discourse*, p.\u00a0161.\n\nWhen we are quite unable to isolate circumstances we must resort to\nthe procedure described by Mill under the name of the Joint Method\nof Agreement and Difference. We must collect as many instances as\npossible in which a given circumstance produces a given result, and\nas many as possible in which the absence of the circumstance is\nfollowed by the absence of the result. To adduce his example, we\ncannot experiment upon the cause of double refraction in Iceland spar,\nbecause we cannot alter its crystalline condition without altering\nit altogether, nor can we find substances exactly like calc spar in\nevery circumstance except one. We resort therefore to the method\nof comparing together all known substances which have the property\nof doubly-refracting light, and we find that they agree in being\ncrystalline.[330] This indeed is nothing but an ordinary process of\nperfect or probable induction, already partially described, and to\nbe further discussed under Classification. It may be added that the\nsubject does admit of perfect experimental treatment, since glass,\nwhen compressed in one direction, becomes capable of doubly-refracting\nlight, and as there is probably no alteration in the glass but change\nof elasticity, we learn that the power of double refraction is probably\ndue to a difference of elasticity in different directions.\n\n  [330] *System of Logic*, bk. iii. chap. viii. \u00a7 4, 5th ed. vol.\u00a0i.\n  p.\u00a0433.\n\n\n*Removal of Usual Conditions.*\n\nOne of the great objects of experiment is to enable us to judge\nof the behaviour of substances under conditions widely different\nfrom those which prevail upon the surface of the earth. We live in\nan atmosphere which does not vary beyond certain narrow limits in\ntemperature or pressure. Many of the powers of nature, such as gravity,\nwhich constantly act upon us, are of almost fixed amount. Now it\nwill afterwards be shown that we cannot apply a quantitative law to\ncircumstances much differing from those in which it was observed. In\nthe other planets, the sun, the stars, or remote parts of the Universe,\nthe conditions of existence must often be widely different from what\nwe commonly experience here. Hence our knowledge of nature must remain\nrestricted and hypothetical, unless we can subject substances to\nunusual conditions by suitable experiments.\n\nThe electric arc is an invaluable means of exposing metals or other\nconducting substances to the highest known temperature. By its aid we\nlearn not only that all the metals can be vaporised, but that they\nall give off distinctive rays of light. At the other extremity of the\nscale, the intensely powerful freezing mixture devised by Faraday,\nconsisting of solid carbonic acid and ether mixed *in vacuo*, enables\nus to observe the nature of substances at temperatures immensely below\nany we meet with naturally on the earth\u2019s surface.\n\nWe can hardly realise now the importance of the invention of the\nair-pump, previous to which invention it was exceedingly difficult\nto experiment except under the ordinary pressure of the atmosphere.\nThe Torricellian vacuum had been employed by the philosophers of the\nAccademia del Cimento to show the behaviour of water, smoke, sound,\nmagnets, electric substances, &c., *in vacuo*, but their experiments\nwere often unsuccessful from the difficulty of excluding air.[331]\n\n  [331] *Essayes of Natural Experiments made in the Accademia del\n  Cimento.* Englished by Richard Waller, 1684, p.\u00a040, &c.\n\nAmong the most constant circumstances under which we live is the force\nof gravity, which does not vary, except by a slight fraction of its\namount, in any part of the earth\u2019s crust or atmosphere to which we\ncan attain. This force is sufficient to overbear and disguise various\nactions, for instance, the mutual gravitation of small bodies. It\nwas an interesting experiment of Plateau to neutralise the action of\ngravity by placing substances in liquids of exactly the same specific\ngravity. Thus a quantity of oil poured into the middle of a suitable\nmixture of alcohol and water assumes a spherical shape; on being made\nto rotate it becomes spheroidal, and then successively separates into a\nring and a group of spherules. Thus we have an illustration of the mode\nin which the planetary system may have been produced,[332] though the\nextreme difference of scale prevents our arguing with confidence from\nthe experiment to the conditions of the nebular theory.\n\n  [332] Plateau, *Taylor\u2019s Scientific Memoirs*, vol.\u00a0iv. pp.\u00a016\u201343.\n\nIt is possible that the so-called elements are elementary only to us,\nbecause we are restricted to temperatures at which they are fixed.\nLavoisier carefully defined an element as a substance which cannot be\ndecomposed *by any known means*; but it seems almost certain that some\nseries of elements, for instance Iodine, Bromine, and Chlorine, are\nreally compounds of a simpler substance. We must look to the production\nof intensely high temperatures, yet quite beyond our means, for the\ndecomposition of these so-called elements. Possibly in this age and\npart of the universe the dissipation of energy has so far proceeded\nthat there are no sources of heat sufficiently intense to effect the\ndecomposition.\n\n\n*Interference of Unsuspected Conditions.*\n\nIt may happen that we are not aware of all the conditions under which\nour researches are made. Some substance may be present or some power\nmay be in action, which escapes the most vigilant examination. Not\nbeing aware of its existence, we are unable to take proper measures\nto exclude it, and thus determine the share which it has in the\nresults of our experiments. There can be no doubt that the alchemists\nwere misled and encouraged in their vain attempts by the unsuspected\npresence of traces of gold and silver in the substances they proposed\nto transmute. Lead, as drawn from the smelting furnace, almost always\ncontains some silver, and gold is associated with many other metals.\nThus small quantities of noble metal would often appear as the result\nof experiment and raise delusive hopes.\n\nIn more than one case the unsuspected presence of common salt in the\nair has caused great trouble. In the early experiments on electrolysis\nit was found that when water was decomposed, an acid and an alkali\nwere produced at the poles, together with oxygen and hydrogen. In\nthe absence of any other explanation, some chemists rushed to the\nconclusion that electricity must have the power of *generating*\nacids and alkalies, and one chemist thought he had discovered a new\nsubstance called *electric acid*. But Davy proceeded to a systematic\ninvestigation of the circumstances, by varying the conditions. Changing\nthe glass vessel for one of agate or gold, he found that far less\nalkali was produced; excluding impurities by the use of carefully\ndistilled water, he found that the quantities of acid and alkali were\nstill further diminished; and having thus obtained a clue to the cause,\nhe completed the exclusion of impurities by avoiding contact with his\nfingers, and by placing the apparatus under an exhausted receiver,\nno acid or alkali being then detected. It would be difficult to meet\nwith a more elegant case of the detection of a condition previously\nunsuspected.[333]\n\n  [333] *Philosophical Transactions* [1826], vol.\u00a0cxvi. pp.\u00a0388, 389.\n  Works of Sir Humphry Davy, vol.\u00a0v. pp.\u00a01\u201312.\n\nIt is remarkable that the presence of common salt in the air, proved\nto exist by Davy, nevertheless continued a stumbling-block in the\nscience of spectrum analysis, and probably prevented men, such as\nBrewster, Herschel, and Talbot, from anticipating by thirty years\nthe discoveries of Bunsen and Kirchhoff. As I pointed out,[334] the\nutility of the spectrum was known in the middle of the last century\nto Thomas Melvill, a talented Scotch physicist, who died at the early\nage of 27 years.[335] But Melvill was struck in his examination of\ncoloured flames by the extraordinary predominance of homogeneous yellow\nlight, which was due to some circumstance escaping his attention.\nWollaston and Fraunhofer were equally struck by the prominence of the\nyellow line in the spectrum of nearly every kind of light. Talbot\nexpressly recommended the use of the prism for detecting the presence\nof substances by what we now call spectrum analysis, but he found that\nall substances, however different the light they yielded in other\nrespects, were identical as regards the production of yellow light.\nTalbot knew that the salts of soda gave this coloured light, but in\nspite of Davy\u2019s previous difficulties with salt in electrolysis, it\ndid not occur to him to assert that where the light is, there sodium\nmust be. He suggested water as the most likely source of the yellow\nlight, because of its frequent presence; but even substances which were\napparently devoid of water gave the same yellow light.[336] Brewster\nand Herschel both experimented upon flames almost at the same time as\nTalbot, and Herschel unequivocally enounced the principle of spectrum\nanalysis.[337] Nevertheless Brewster, after numerous experiments\nattended with great trouble and disappointment, found that yellow light\nmight be obtained from the combustion of almost any substance. It\nwas not until 1856 that Swan discovered that an almost infinitesimal\nquantity of sodium chloride, say a millionth part of a grain, was\nsufficient to tinge a flame of a bright yellow colour. The universal\ndiffusion of the salts of sodium, joined to this unique light-producing\npower, was thus shown to be the unsuspected condition which had\ndestroyed the confidence of all previous experimenters in the use of\nthe prism. Some references concerning the history of this curious point\nare given below.[338]\n\n  [334] *National Review*, July, 1861, p.\u00a013.\n\n  [335] His published works are contained in *The Edinburgh Physical\n  and Literary Essays*, vol.\u00a0ii. p.\u00a034; *Philosophical Transactions*\n  [1753], vol.\u00a0xlviii. p.\u00a0261; see also Morgan\u2019s Papers in\n  *Philosophical Transactions* [1785], vol.\u00a0lxxv. p.\u00a0190.\n\n  [336] *Edinburgh Journal of Science*, vol.\u00a0v. p.\u00a079.\n\n  [337] *Encyclop\u00e6dia Metropolitana*, art. *Light*, \u00a7 524; Herschel\u2019s\n  *Familiar Lectures*, p.\u00a0266.\n\n  [338] Talbot, *Philosophical Magazine*, 3rd Series, vol.\u00a0ix. p.\u00a01\n  (1836); Brewster, *Transactions of the Royal Society of Edinburgh*\n  [1823], vol.\u00a0ix. pp.\u00a0433, 455; Swan, ibid. [1856] vol.\u00a0xxi. p.\u00a0411;\n  *Philosophical Magazine*, 4th Series, vol.\u00a0xx. p.\u00a0173 [Sept. 1860];\n  Roscoe, *Spectrum Analysis*, Lecture III.\n\nIn the science of radiant heat, early inquirers were led to the\nconclusion that radiation proceeded only from the surface of a solid,\nor from a very small depth below it. But they happened to experiment\nupon surfaces covered by coats of varnish, which is highly athermanous\nor opaque to heat. Had they properly varied the character of the\nsurface, using a highly diathermanous substance like rock salt, they\nwould have obtained very different results.[339]\n\nOne of the most extraordinary instances of an erroneous opinion due\nto overlooking interfering agents is that concerning the increase of\nrainfall near to the earth\u2019s surface. More than a century ago it was\nobserved that rain-gauges placed upon church steeples, house tops, and\nother elevated places, gave considerably less rain than if they were\non the ground, and it has been recently shown that the variation is\nmost rapid in the close neighbourhood of the ground.[340] All kinds\nof theories have been started to explain this phenomenon; but I have\nshown[341] that it is simply due to the interference of wind, which\ndeflects more or less rain from all the gauges which are exposed to it.\n\n  [339] Balfour Stewart, *Elementary Treatise on Heat*, p.\u00a0192.\n\n  [340] British Association, Liverpool, 1870. *Report on Rainfall*,\n  p.\u00a0176.\n\n  [341] *Philosophical Magazine.*, Dec. 1861. 4th Series, vol.\u00a0xxii.\n  p.\u00a0421.\n\nThe great magnetic power of iron renders it a source of disturbance in\nmagnetic experiments. In building a magnetic observatory great care\nmust therefore be taken that no iron is employed in the construction,\nand that no masses of iron are near at hand. In some cases magnetic\nobservations have been seriously disturbed by the existence of masses\nof iron ore in the neighbourhood. In Faraday\u2019s experiments upon feebly\nmagnetic or diamagnetic substances he took the greatest precautions\nagainst the presence of disturbing substances in the copper wire, wax,\npaper, and other articles used in suspending the test objects. It was\nhis custom to try the effect of the magnet upon the apparatus in the\nabsence of the object of experiment, and without this preliminary trial\nno confidence could be placed in the results.[342] Tyndall has also\nemployed the same mode for testing the freedom of electro-magnetic\ncoils from iron, and was thus enabled to obtain them devoid of any\ncause of disturbance.[343] It is worthy of notice that in the very\ninfancy of the science of magnetism, the acute experimentalist Gilbert\ncorrectly accounted for the opinion existing in his day that magnets\nwould attract silver, by pointing out that the silver contained iron.\n\n  [342] *Experimental Researches in Electricity*, vol.\u00a0iii. p.\u00a084, &c.\n\n  [343] *Lectures on Heat*, p.\u00a021.\n\nEven when we are not aware by previous experience of the probable\npresence of a special disturbing agent, we ought not to assume the\nabsence of unsuspected interference. If an experiment is of really\nhigh importance, so that any considerable branch of science rests\nupon it, we ought to try it again and again, in as varied conditions\nas possible. We should intentionally disturb the apparatus in various\nways, so as if possible to hit by accident upon any weak point.\nEspecially when our results are more regular than we have fair grounds\nfor anticipating, ought we to suspect some peculiarity in the apparatus\nwhich causes it to measure some other phenomenon than that in question,\njust as Foucault\u2019s pendulum almost always indicates the movement of the\naxes of its own elliptic path instead of the rotation of the globe.\n\nIt was in this cautious spirit that Baily acted in his experiments on\nthe density of the earth. The accuracy of his results depended upon the\nelimination of all disturbing influences, so that the oscillation of\nhis torsion balance should measure gravity alone. Hence he varied the\napparatus in many ways, changing the small balls subject to attraction,\nchanging the connecting rod, and the means of suspension. He observed\nthe effect of disturbances, such as the presence of visitors, the\noccurrence of violent storms, &c., and as no real alteration was\nproduced in the results, he confidently attributed them to gravity.[344]\n\n  [344] Baily, *Memoirs of the Royal Astronomical Society*, vol.\u00a0xiv.\n  pp. 29, 30.\n\nNewton would probably have discovered the mode of constructing\nachromatic lenses, but for the unsuspected effect of some sugar of\nlead which he is supposed to have dissolved in the water of a prism.\nHe tried, by means of a glass prism combined with a water prism, to\nproduce dispersion of light without refraction, and if he had succeeded\nthere would have been an obvious mode of producing refraction without\ndispersion. His failure is attributed to his adding lead acetate to\nthe water for the purpose of increasing its refractive power, the lead\nhaving a high dispersive power which frustrated his purpose.[345]\nJudging from Newton\u2019s remarks, in the *Philosophical Transactions*,\nit would appear as if he had not, without many unsuccessful trials,\ndespaired of the construction of achromatic glasses.[346]\n\n  [345] Grant, *History of Physical Astronomy*, p.\u00a0531.\n\n  [346] *Philosophical Transactions*, abridged by Lowthorp, 4th\n  edition, vol.\u00a0i. p.\u00a0202.\n\nThe Academicians of Cimento, in their early and ingenious experiments\nupon the vacuum, were often misled by the mechanical imperfections\nof their apparatus. They concluded that the air had nothing to do\nwith the production of sounds, evidently because their vacuum was not\nsufficiently perfect. Otto von Guericke fell into a like mistake in the\nuse of his newly-constructed air-pump, doubtless from the unsuspected\npresence of air sufficiently dense to convey the sound of the bell.\n\nIt is hardly requisite to point out that the doctrine of spontaneous\ngeneration is due to the unsuspected presence of germs, even after the\nmost careful efforts to exclude them, and in the case of many diseases,\nboth of animals and plants, germs which we have no means as yet of\ndetecting are doubtless the active cause. It has long been a subject\nof dispute, again, whether the plants which spring from newly turned\nland grow from seeds long buried in that land, or from seeds brought\nby the wind. Argument is unphilosophical when direct trial can readily\nbe applied; for by turning up some old ground, and covering a portion\nof it with a glass case, the conveyance of seeds by the wind can be\nentirely prevented, and if the same plants appear within and without\nthe case, it will become clear that the seeds are in the earth. By\ngross oversight some experimenters have thought before now that crops\nof rye had sprung up where oats had been sown.\n\n\n*Blind or Test Experiments.*\n\nEvery conclusive experiment necessarily consists in the comparison of\nresults between two different combinations of circumstances. To give a\nfair probability that A is the cause of X, we must maintain invariable\nall surrounding objects and conditions, and we must then show that\nwhere A is X is, and where A is not X is not. This cannot really be\naccomplished in a single trial. If, for instance, a chemist places\na certain suspected substance in Marsh\u2019s test apparatus, and finds\nthat it gives a small deposit of metallic arsenic, he cannot be sure\nthat the arsenic really proceeds from the suspected substance; the\nimpurity of the zinc or sulphuric acid may have been the cause of its\nappearance. It is therefore the practice of chemists to make what they\ncall a *blind experiment*, that is to try whether arsenic appears in\nthe absence of the suspected substance. The same precaution ought to be\ntaken in all important analytical operations. Indeed, it is not merely\na precaution, it is an essential part of any experiment. If the blind\ntrial be not made, the chemist merely assumes that he knows what would\nhappen. Whenever we assert that because A and X are found together A\nis the cause of X, we assume that if A were absent X would be absent.\nBut wherever it is possible, we ought not to take this as a mere\nassumption, or even as a matter of inference. Experience is ultimately\nthe basis of all our inferences, but if we can bring immediate\nexperience to bear upon the point in question we should not trust to\nanything more remote and liable to error. When Faraday examined the\nmagnetic properties of the bearing apparatus, in the absence of the\nsubstance to be experimented on, he really made a blind experiment\n(p.\u00a0431).\n\nWe ought, also, to test the accuracy of a method of experiment whenever\nwe can, by introducing known amounts of the substance or force to be\ndetected. A new analytical process for the quantitative estimation of\nan element should be tested by performing it upon a mixture compounded\nso as to contain a known quantity of that element. The accuracy of the\ngold assay process greatly depends upon the precaution of assaying\nalloys of gold of exactly known composition.[347] Gabriel Plattes\u2019\nworks give evidence of much scientific spirit, and when discussing the\nsupposed merits of the divining rod for the discovery of subterranean\ntreasure, he sensibly suggests that the rod should be tried in places\nwhere veins of metal are known to exist.[348]\n\n  [347] Jevons in Watts\u2019 *Dictionary of Chemistry*, vol.\u00a0ii. pp.\u00a0936,\n  937.\n\n  [348] *Discovery of Subterraneal Treasure.* London, 1639, p.\u00a048.\n\n\n*Negative Results of Experiment.*\n\nWhen we pay proper regard to the imperfection of all measuring\ninstruments and the possible minuteness of effects, we shall see\nmuch reason for interpreting with caution the negative results of\nexperiments. We may fail to discover the existence of an expected\neffect, not because that effect is really non-existent, but because it\nis of a magnitude inappreciable to our senses, or confounded with other\neffects of much greater amount. As there is no limit on *\u00e0 priori*\ngrounds to the smallness of a phenomenon, we can never, by a single\nexperiment, prove the non-existence of a supposed effect. We are always\nat liberty to assume that a certain amount of effect might have been\ndetected by greater delicacy of measurement. We cannot safely affirm\nthat the moon has no atmosphere at all. We may doubtless show that the\natmosphere, if present, is less dense than the air in the so-called\nvacuum of an air-pump, as did Du Sejour. It is equally impossible to\nprove that gravity occupies *no time* in transmission. Laplace indeed\nascertained that the velocity of propagation of the influence was at\nleast fifty million times greater than that of light;[349] but it does\nnot really follow that it is instantaneous; and were there any means\nof detecting the action of one star upon another exceedingly distant\nstar, we might possibly find an appreciable interval occupied in the\ntransmission of the gravitating impulse. Newton could not demonstrate\nthe absence of all resistance to matter moving through empty space; but\nhe ascertained by an experiment with the pendulum (p.\u00a0443), that if\nsuch resistance existed, it was in amount less than one five-thousandth\npart of the external resistance of the air.[350]\n\n  [349] Laplace, *System of the World*, translated by Harte, vol.\u00a0ii.\n  p.\u00a0322.\n\n  [350] *Principia*, bk. ii. sect. 6, Prop. xxxi. Motte\u2019s translation,\n  vol.\u00a0ii. p. 108.\n\nA curious instance of false negative inference is furnished by\nexperiments on light. Euler rejected the corpuscular theory on the\nground that particles of matter moving with the immense velocity of\nlight would possess momentum, of which there was no evidence. Bennet\nhad attempted to detect the momentum of light by concentrating the rays\nof the sun upon a delicately balanced body. Observing no result, it\nwas considered to be proved that light had no momentum. Mr. Crookes,\nhowever, having suspended thin vanes, blacked on one side, in a nearly\nvacuous globe, found that they move under the influence of light. It\nis now allowed that this effect can be explained in accordance with\nthe undulatory theory of light, and the molecular theory of gases. It\ncomes to this--that Bennet failed to detect an effect which he might\nhave detected with a better method of experimenting; but if he had\nfound it, the phenomenon would have confirmed, not the corpuscular\ntheory of light, as was expected, but the rival undulatory theory. The\nconclusion drawn from Bennet\u2019s experiment was falsely drawn, but it was\nnevertheless true in matter.\n\nMany incidents in the history of science tend to show that phenomena,\nwhich one generation has failed to discover, may become accurately\nknown to a succeeding generation. The compressibility of water which\nthe Academicians of Florence could not detect, because at a low\npressure the effect was too small to perceive, and at a high pressure\nthe water oozed through their silver vessel,[351] has now become the\nsubject of exact measurement and precise calculation. Independently of\nNewton, Hooke entertained very remarkable notions concerning the nature\nof gravitation. In this and other subjects he showed, indeed, a genius\nfor experimental investigation which would have placed him in the first\nrank in any other age than that of Newton. He correctly conceived that\nthe force of gravity would decrease as we recede from the centre of\nthe earth, and he boldly attempted to prove it by experiment. Having\nexactly counterpoised two weights in the scales of a balance, or rather\none weight against another weight and a long piece of fine cord, he\nremoved his balance to the top of the dome of St. Paul\u2019s, and tried\nwhether the balance remained in equilibrium after one weight was\nallowed to hang down to a depth of 240 feet. No difference could be\nperceived when the weights were at the same and at different levels,\nbut Hooke rightly held that the failure arose from the insufficient\nelevation. He says, \u201cYet I am apt to think some difference might be\ndiscovered in greater heights.\u201d[352] The radius of the earth being\nabout 20,922,000 feet, we can now readily calculate from the law of\ngravity that a height of 240 would not make a greater difference than\none part in 40,000 of the weight. Such a difference would doubtless\nbe inappreciable in the balances of that day, though it could readily\nbe detected by balances now frequently constructed. Again, the\nmutual gravitation of bodies at the earth\u2019s surface is so small that\nNewton appears to have made no attempt to demonstrate its existence\nexperimentally, merely remarking that it was too small to fall under\nthe observation of our senses.[353] It has since been successfully\ndetected and measured by Cavendish, Baily, and others.\n\n  [351] *Essayes of Natural Experiments*, &c. p.\u00a0117.\n\n  [352] Hooke\u2019s *Posthumous Works*, p.\u00a0182.\n\n  [353] *Principia*, bk. iii. Prop. vii. Corollary 1.\n\nThe smallness of the quantities which we can sometimes observe is\nastonishing. A balance will weigh to one millionth part of the load.\nWhitworth can measure to the millionth part of an inch. A rise of\ntemperature of the 8800th part of a degree centigrade has been\ndetected by Dr. Joule. The spectroscope has revealed the presence of\nthe 10,000,000th part of a gram. It is said that the eye can observe\nthe colour produced in a drop of water by the 50,000,000th part of a\ngram of fuschine, and about the same quantity of cyanine. By the sense\nof smell we can probably feel still smaller quantities of odorous\nmatter.[354] We must nevertheless remember that quantitative effects of\nfar less amount than these must exist, and we should state our negative\nresults with corresponding caution. We can only disprove the existence\nof a quantitative phenomenon by showing deductively from the laws of\nnature, that if present it would amount to a perceptible quantity. As\nin the case of other negative arguments (p.\u00a0414), we must demonstrate\nthat the effect would appear, where it is by experiment found not to\nappear.\n\n  [354] Keill\u2019s *Introduction to Natural Philosophy*, 3rd ed., London,\n  1733, pp.\u00a048\u201354.\n\n\n*Limits of Experiment.*\n\nIt will be obvious that there are many operations of nature which we\nare quite incapable of imitating in our experiments. Our object is to\nstudy the conditions under which a certain effect is produced; but\none of those conditions may involve a great length of time. There\nare instances on record of experiments extending over five or ten\nyears, and even over a large part of a lifetime; but such intervals\nof time are almost nothing to the time during which nature may have\nbeen at work. The contents of a mineral vein in Cornwall may have been\nundergoing gradual change for a hundred million years. All metamorphic\nrocks have doubtless endured high temperature and enormous, pressure\nfor inconceivable periods of time, so that chemical geology is\ngenerally beyond the scope of experiment.\n\nArguments have been brought against Darwin\u2019s theory, founded upon the\nabsence of any clear instance of the production of a new species.\nDuring an historical interval of perhaps four thousand years, no\nanimal, it is said, has been so much domesticated as to become\ndifferent in species. It might as well be argued that no geological\nchanges are taking place, because no new mountain has risen in Great\nBritain within the memory of man. Our actual experience of geological\nchanges is like a point in the infinite progression of time. When we\nknow that rain water falling on limestone will carry away a minute\nportion of the rock in solution, we do not hesitate to multiply that\nquantity by millions, and infer that in course of time a mountain may\nbe dissolved away. We have actual experience concerning the rise of\nland in some parts of the globe and its fall in others to the extent\nof some feet. Do we hesitate to infer what may thus be done in course\nof geological ages? As Gabriel Plattes long ago remarked, \u201cThe sea\nnever resting, but perpetually winning land in one place and losing in\nanother, doth show what may be done in length of time by a continual\noperation, not subject unto ceasing or intermission.\u201d[355] The action\nof physical circumstances upon the forms and characters of animals by\nnatural selection is subject to exactly the same remarks. As regards\nanimals living in a state of nature, the change of circumstances which\ncan be ascertained to have occurred is so slight, that we could not\nexpect to observe any change in those animals whatever. Nature has made\nno experiment at all for us within historical times. Man, however, by\ntaming and domesticating dogs, horses, oxen, pigeons, &c., has made\nconsiderable change in their circumstances, and we find considerable\nchange also in their forms and characters. Supposing the state of\ndomestication to continue unchanged, these new forms would continue\npermanent so far as we know, and in this sense they are permanent. Thus\nthe arguments against Darwin\u2019s theory, founded on the non-observation\nof natural changes within the historical period, are of the weakest\ncharacter, being purely negative.\n\n  [355] *Discovery of Subterraneal Treasure*, 1639, p.\u00a052.\n\n\n\n\nCHAPTER XX.\n\nMETHOD OF VARIATIONS.\n\n\nExperiments may be of two kinds, experiments of simple fact, and\nexperiments of quantity. In the first class of experiments we combine\ncertain conditions, and wish to ascertain whether or not a certain\neffect of any quantity exists. Hooke wished to ascertain whether or not\nthere was any difference in the force of gravity at the top and bottom\nof St. Paul\u2019s Cathedral. The chemist continually performs analyses for\nthe purpose of ascertaining whether or not a given element exists in a\nparticular mineral or mixture; all such experiments and analyses are\nqualitative rather than quantitative, because though the result may be\nmore or less, the particular amount of the result is not the object of\nthe inquiry.\n\nSo soon, however, as a result is known to be discoverable, the\nscientific man ought to proceed to the quantitative inquiry, how great\na result follows from a certain amount of the conditions which are\nsupposed to constitute the cause? The possible numbers of experiments\nare now infinitely great, for every variation in a quantitative\ncondition will usually produce a variation in the amount of the effect.\nThe method of variation which thus arises is no narrow or special\nmethod, but it is the general application of experiment to phenomena\ncapable of continuous variation. As Mr. Fowler has well remarked,[356]\nthe observation of variations is really an integration of a supposed\ninfinite number of applications of the so-called method of difference,\nthat is of experiment in its perfect form.\n\n  [356] *Elements of Inductive Logic*, 1st edit. p.\u00a0175.\n\nIn induction we aim at establishing a general law, and if we deal\nwith quantities that law must really be expressed more or less\nobviously in the form of an equation, or equations. We treat as\nbefore of conditions, and of what happens under those conditions. But\nthe conditions will now vary, not in quality, but quantity, and the\neffect will also vary in quantity, so that the result of quantitative\ninduction is always to arrive at some mathematical expression involving\nthe quantity of each condition, and expressing the quantity of the\nresult. In other words, we wish to know what function the effect is\nof its conditions. We shall find that it is one thing to obtain the\nnumerical results, and quite another thing to detect the law obeyed\nby those results, the latter being an operation of an inverse and\ntentative character.\n\n\n*The Variable and the Variant.*\n\nAlmost every series of quantitative experiments is directed to obtain\nthe relation between the different values of one quantity which is\nvaried at will, and another quantity which is caused thereby to vary.\nWe may conveniently distinguish these as respectively the *variable*\nand the *variant*. When we are examining the effect of heat in\nexpanding bodies, heat, or one of its dimensions, temperature, is the\nvariable, length the variant. If we compress a body to observe how much\nit is thereby heated, pressure, or it may be the dimensions of the\nbody, forms the variable, heat the variant. In the thermo-electric pile\nwe make heat the variable and measure electricity as the variant. That\none of the two measured quantities which is an antecedent condition of\nthe other will be the variable.\n\nIt is always convenient to have the variable entirely under our\ncommand. Experiments may indeed be made with accuracy, provided we\ncan exactly measure the variable at the moment when the quantity of\nthe effect is determined. But if we have to trust to the action of\nsome capricious force, there may be great difficulty in making exact\nmeasurements, and those results may not be disposed over the whole\nrange of quantity in a convenient manner. It is one prime object of the\nexperimenter, therefore, to obtain a regular and governable supply\nof the force which he is investigating. To determine correctly the\nefficiency of windmills, when the natural winds were constantly varying\nin force, would be exceedingly difficult. Smeaton, therefore, in his\nexperiments on the subject, created a uniform wind of the required\nforce by moving his models against the air on the extremity of a\nrevolving arm.[357] The velocity of the wind could thus be rendered\ngreater or less, it could be maintained uniform for any length of\ntime, and its amount could be exactly ascertained. In determining the\nlaws of the chemical action of light it would be out of the question\nto employ the rays of the sun, which vary in intensity with the\nclearness of the atmosphere, and with every passing cloud. One great\ndifficulty in photometry and the investigation of the chemical action\nof light consists in obtaining a uniform and governable source of light\nrays.[358]\n\n  [357] *Philosophical Transactions*, vol.\u00a0li. p.\u00a0138; abridgment,\n  vol.\u00a0xi. p. 355.\n\n  [358] See Bunsen and Roscoe\u2019s researches, in *Philosophical\n  Transactions* (1859), vol.\u00a0cxlix. p.\u00a0880, &c., where they describe a\n  constant flame of carbon monoxide gas.\n\nFizeau\u2019s method of measuring the velocity of light enabled him\nto appreciate the time occupied by light in travelling through a\ndistance of eight or nine thousand metres. But the revolving mirror\nof Wheatstone subsequently enabled Foucault and Fizeau to measure the\nvelocity in a space of four metres. In this latter method there was\nthe advantage that various media could be substituted for air, and the\ntemperature, density, and other conditions of the experiment could be\naccurately governed and measured.\n\n\n*Measurement of the Variable.*\n\nThere is little use in obtaining exact measurements of an effect unless\nwe can also exactly measure its conditions.\n\nIt is absurd to measure the electrical resistance of a piece of metal,\nits elasticity, tenacity, density, or other physical qualities, if\nthese vary, not only with the minute impurities of the metal, but also\nwith its physical condition. If the same bar changes its properties\nby being heated and cooled, and we cannot exactly define the state\nin which it is at any moment, our care in measuring will be wasted,\nbecause it can lead to no law. It is of little use to determine very\nexactly the electric conductibility of carbon, which as graphite or gas\ncarbon conducts like a metal, as diamond is almost a non-conductor,\nand in several other forms possesses variable and intermediate\npowers of conduction. It will be of use only for immediate practical\napplications. Before measuring these we ought to have something to\nmeasure of which the conditions are capable of exact definition, and\nto which at a future time we can recur. Similarly the accuracy of our\nmeasurement need not much surpass the accuracy with which we can define\nthe conditions of the object treated.\n\nThe speed of electricity in passing through a conductor mainly depends\nupon the inductive capacity of the surrounding substances, and, except\nfor technical or special purposes, there is little use in measuring\nvelocities which in some cases are one hundred times as great as in\nother cases. But the maximum speed of electric conduction is probably\na constant quantity of great scientific importance, and according\nto Prof. Clerk Maxwell\u2019s determination in 1868 is 174,800 miles per\nsecond, or little less than that of light. The true boiling point of\nwater is a point on which practical thermometry depends, and it is\nhighly important to determine that point in relation to the absolute\nthermometric scale. But when water free from air and impurity is\nheated there seems to be no definite limit to the temperature it may\nreach, a temperature of 180\u00b0 Cent. having been actually observed.\nSuch temperatures, therefore, do not require accurate measurement.\nAll meteorological measurements depending on the accidental condition\nof the sky are of far less importance than physical measurements in\nwhich such accidental conditions do not intervene. Many profound\ninvestigations depend upon our knowledge of the radiant energy\ncontinually poured upon the earth by the sun; but this must be measured\nwhen the sky is perfectly clear, and the absorption of the atmosphere\nat its minimum. The slightest interference of cloud destroys the value\nof such a measurement, except for meteorological purposes, which are of\nvastly less generality and importance. It is seldom useful, again, to\nmeasure the height of a snow-covered mountain within a foot, when the\nthickness of the snow alone may cause it to vary 25 feet or more, when\nin short the height itself is indefinite to that extent.[359]\n\n  [359] Humboldt\u2019s *Cosmos* (Bohn), vol.\u00a0i. p.\u00a07.\n\n\n*Maintenance of Similar Conditions.*\n\nOur ultimate object in induction must be to obtain the complete\nrelation between the conditions and the effect, but this relation\nwill generally be so complex that we can only attack it in detail.\nWe must, as far as possible, confine the variation to one condition\nat a time, and establish a separate relation between each condition\nand the effect. This is at any rate the first step in approximating\nto the complete law, and it will be a subsequent question how far the\nsimultaneous variation of several conditions modifies their separate\nactions. In many experiments, indeed, it is only one condition which\nwe wish to study, and the others are interfering forces which we would\navoid if possible. One of the conditions of the motion of a pendulum is\nthe resistance of the air, or other medium in which it swings; but when\nNewton was desirous of proving the equal gravitation of all substances,\nhe had no interest in the air. His object was to observe a single force\nonly, and so it is in a great many other experiments. Accordingly,\none of the most important precautions in investigation consists in\nmaintaining all conditions constant except that which is to be studied.\nAs that admirable experimental philosopher, Gilbert, expressed it,[360]\n\u201cThere is always need of similar preparation, of similar figure, and\nof equal magnitude, for in dissimilar and unequal circumstances the\nexperiment is doubtful.\u201d\n\n  [360] Gilbert, *De Magnete*, p.\u00a0109.\n\nIn Newton\u2019s decisive experiment similar conditions were provided\nfor, with the simplicity which characterises the highest art. The\npendulums of which the oscillations were compared consisted of equal\nboxes of wood, hanging by equal threads, and filled with different\nsubstances, so that the total weights should be equal and the centres\nof oscillation at the same distance from the points of suspension.\nHence the resistance of the air became approximately a matter of\nindifference; for the outward size and shape of the pendulums being\nthe same, the absolute force of resistance would be the same, so long\nas the pendulums vibrated with equal velocity; and the weights being\nequal the resistance would diminish the velocity equally. Hence if any\ninequality were observed in the vibrations of the two pendulums, it\nmust arise from the only circumstance which was different, namely the\nchemical nature of the matter within the boxes. No inequality being\nobserved, the chemical nature of substances can have no appreciable\ninfluence upon the force of gravitation.[361]\n\n  [361] *Principia*, bk. iii. Prop. vi.\n\nA beautiful experiment was devised by Dr. Joule for the purpose of\nshowing that the gain or loss of heat by a gas is connected, not\nwith the mere change of its volume and density, but with the energy\nreceived or given out by the gas. Two strong vessels, connected by\na tube and stopcock, were placed in water after the air had been\nexhausted from one vessel and condensed in the other to the extent\nof twenty atmospheres. The whole apparatus having been brought to a\nuniform temperature by agitating the water, and the temperature having\nbeen exactly observed, the stopcock was opened, so that the air at\nonce expanded and filled the two vessels uniformly. The temperature\nof the water being again noted was found to be almost unchanged. The\nexperiment was then repeated in an exactly similar manner, except that\nthe strong vessels were placed in separate portions of the water. Now\ncold was produced in the vessel from which the air rushed, and an\nalmost exactly equal quantity of heat appeared in that to which it was\nconducted. Thus Dr. Joule clearly proved that rarefaction produces\nas much heat as cold, and that only when there is disappearance of\nmechanical energy will there be production of heat.[362] What we have\nto notice, however, is not so much the result of the experiment, as the\nsimple manner in which a single change in the apparatus, the separation\nof the portions of water surrounding the air vessels, is made to give\nindications of the utmost significance.\n\n  [362] *Philosophical Magazine*, 3rd Series, vol.\u00a0xxvi. p.\u00a0375.\n\n\n*Collective Experiments.*\n\nThere is an interesting class of experiments which enable us to observe\na number of quantitative results in one act. Generally speaking, each\nexperiment yields us but one number, and before we can approach the\nreal processes of reasoning we must laboriously repeat measurement\nafter measurement, until we can lay out a curve of the variation of\none quantity as depending on another. We can sometimes abbreviate\nthis labour, by making a quantity vary in different parts of the same\napparatus through every required amount. In observing the height to\nwhich water rises by the capillary attraction of a glass vessel, we may\ntake a series of glass tubes of different bore, and measure the height\nthrough which it rises in each. But if we take two glass plates, and\nplace them vertically in water, so as to be in contact at one vertical\nside, and slightly separated at the other side, the interval between\nthe plates varies through every intermediate width, and the water rises\nto a corresponding height, producing at its upper surface a hyperbolic\ncurve.\n\nThe absorption of light in passing through a coloured liquid may be\nbeautifully shown by enclosing the liquid in a wedge-shaped glass, so\nthat we have at a single glance an infinite variety of thicknesses in\nview. As Newton himself remarked, a red liquid viewed in this manner is\nfound to have a pale yellow colour at the thinnest part, and it passes\nthrough orange into red, which gradually becomes of a deeper and darker\ntint.[363] The effect may be noticed in a conical wine-glass. The\nprismatic analysis of light from such a wedge-shaped vessel discloses\nthe reason, by exhibiting the progressive absorption of different rays\nof the spectrum as investigated by Dr. J.\u00a0H. Gladstone.[364]\n\n  [363] *Opticks*, 3rd edit. p.\u00a0159.\n\n  [364] Watts, *Dictionary of Chemistry*, vol.\u00a0iii. p.\u00a0637.\n\nA moving body may sometimes be made to mark out its own course, like\na shooting star which leaves a tail behind it. Thus an inclined jet\nof water exhibits in the clearest manner the parabolic path of a\nprojectile. In Wheatstone\u2019s Kaleidophone the curves produced by the\ncombination of vibrations of different ratios are shown by placing\nbright reflective buttons on the tops of wires of various forms. The\nmotions are performed so quickly that the eye receives the impression\nof the path as a complete whole, just as a burning stick whirled round\nproduces a continuous circle. The laws of electric induction are\nbeautifully shown when iron filings are brought under the influence of\na magnet, and fall into curves corresponding to what Faraday called\nthe Lines of Magnetic Force. When Faraday tried to define what he\nmeant by his lines of force, he was obliged to refer to the filings.\n\u201cBy magnetic curves,\u201d he says,[365] \u201cI mean lines of magnetic forces\nwhich would be depicted by iron filings.\u201d Robison had previously\nproduced similar curves by the action of frictional electricity, and\nfrom a mathematical investigation of the forms of such curves we may\ninfer that magnetic and electric attractions obey the general law of\nemanation, that of the inverse square of the distance. In the electric\nbrush we have a similar exhibition of the laws of electric attraction.\n\n  [365] *Faraday\u2019s Life*, by Bence Jones, vol.\u00a0ii. p.\u00a05.\n\nThere are several branches of science in which collective experiments\nhave been used with great advantage. Lichtenberg\u2019s electric figures,\nproduced by scattering electrified powder on an electrified resin cake,\nso as to show the condition of the latter, suggested to Chladni the\nnotion of discovering the state of vibration of plates by strewing sand\nupon them. The sand collects at the points where the motion is least,\nand we gain at a glance a comprehension of the undulations of the\nplate. To this method of experiment we owe the beautiful observations\nof Savart. The exquisite coloured figures exhibited by plates of\ncrystal, when examined by polarised light, afford a more complicated\nexample of the same kind of investigation. They led Brewster and\nFresnel to an explanation of the properties of the optic axes of\ncrystals. The unequal conduction of heat in crystalline substances has\nalso been shown in a similar manner, by spreading a thin layer of wax\nover the plate of crystal, and applying heat to a single point. The\nwax then melts in a circular or elliptic area according as the rate of\nconduction is uniform or not. Nor should we forget that Newton\u2019s rings\nwere an early and most important instance of investigations of the\nsame kind, showing the effects of interference of light undulations\nof all magnitudes at a single view. Herschel gave to all such\nopportunities of observing directly the results of a general law, the\nname of *Collective Instances*,[366] and I propose to adopt the name\n*Collective Experiments*.\n\n  [366] *Preliminary Discourse*, &c., p.\u00a0185.\n\nSuch experiments will in many subjects only give the first hint of\nthe nature of the law in question, but will not admit of any exact\nmeasurements. The parabolic form of a jet of water may well have\nsuggested to Galileo his views concerning the path of a projectile;\nbut it would not serve now for the exact investigation of the laws of\ngravity. It is unlikely that capillary attraction could be exactly\nmeasured by the use of inclined plates of glass, and tubes would\nprobably be better for precise investigation. As a general rule, these\ncollective experiments would be most useful for popular illustration.\nBut when the curves are of a precise and permanent character, as in\nthe coloured figures produced by crystalline plates, they may admit of\nexact measurement. Newton\u2019s rings and diffraction fringes allow of very\naccurate measurements.\n\nUnder collective experiments we may perhaps place those in which we\nrender visible the motions of gas or liquid by diffusing some opaque\nsubstance in it. The behaviour of a body of air may often be studied\nin a beautiful way by the use of smoke, as in the production of smoke\nrings and jets. In the case of liquids lycopodium powder is sometimes\nemployed. To detect the mixture of currents or strata of liquid, I\nemployed very dilute solutions of common salt and silver nitrate,\nwhich produce a visible cloud wherever they come into contact.[367]\nAtmospheric clouds often reveal to us the movements of great volumes of\nair which would otherwise be quite unapparent.\n\n  [367] *Philosophical Magazine*, July, 1857, 4th Series, vol.\u00a0xiv.\n  p.\u00a024.\n\n\n*Periodic Variations.*\n\nA large class of investigations is concerned with Periodic Variations.\nWe may define a periodic phenomenon as one which, with the uniform\nchange of the variable, returns time after time to the same value.\nIf we strike a pendulum it presently returns to the point from which\nwe disturbed it, and while time, the variable, progresses uniformly,\nit goes on making excursions and returning, until stopped by the\ndissipation of its energy. If one body in space approaches by gravity\ntowards another, they will revolve round each other in elliptic\norbits, and return for an indefinite number of times to the same\nrelative positions. On the other hand a single body projected into\nempty space, free from the action of any extraneous force, would go\non moving for ever in a straight line, according to the first law of\nmotion. In the latter case the variation is called *secular*, because\nit proceeds during ages in a similar manner, and suffers no \u03c0\u03b5\u03c1\u03af\u03bf\u03b4\u03bf\u03c2\nor going round. It may be doubted whether there really is any motion\nin the universe which is not periodic. Mr. Herbert Spencer long since\nadopted the doctrine that all motion is ultimately rhythmical,[368] and\nabundance of evidence may be adduced in favour of his view.\n\n  [368] *First Principles*, 3rd edit. chap. x. p.\u00a0253.\n\nThe so-called secular acceleration of the moon\u2019s motion is certainly\nperiodic, and as, so far as we can tell, no body is beyond the\nattractive power of other bodies, rectilinear motion becomes purely\nhypothetical, or at least infinitely improbable. All the motions of all\nthe stars must tend to become periodic. Though certain disturbances\nin the planetary system seem to be uniformly progressive, Laplace is\nconsidered to have proved that they really have their limits, so that\nafter an immense time, all the planetary bodies might return to the\nsame places, and the stability of the system be established. Such a\ntheory of periodic stability is really hypothetical, and does not\ntake into account phenomena resulting in the dissipation of energy,\nwhich may be a really secular process. For our present purposes we\nneed not attempt to form an opinion on such questions. Any change\nwhich does not present the appearance of a periodic character will be\nempirically regarded as a secular change, so that there will be plenty\nof non-periodic variations.\n\nThe variations which we produce experimentally will often be\nnon-periodic. When we communicate heat to a gas it increases in\nbulk or pressure, and as far as we can go the higher the temperature\nthe higher the pressure. Our experiments are of course restricted\nin temperature both above and below, but there is every reason to\nbelieve that the bulk being the same, the pressure would never return\nto the same point at any two different temperatures. We may of course\nrepeatedly raise and lower the temperature at regular or irregular\nintervals entirely at our will, and the pressure of the gas will vary\nin like manner and exactly at the same intervals, but such an arbitrary\nseries of changes would not constitute Periodic Variation. It would\nconstitute a succession of distinct experiments, which would place\nbeyond reasonable doubt the connexion of cause and effect.\n\nWhenever a phenomenon recurs at equal or nearly equal intervals, there\nis, according to the theory of probability, considerable evidence\nof connexion, because if the recurrences were entirely casual it is\nunlikely that they would happen at equal intervals. The fact that a\nbrilliant comet had appeared in the years 1301, 1378, 1456, 1531, 1607,\nand 1682 gave considerable presumption in favour of the identity of the\nbody, apart from similarity of the orbit. There is nothing which so\nfascinates the attention of men as the recurrence time after time of\nsome unusual event. Things and appearances which remain ever the same,\nlike mountains and valleys, fail to excite the curiosity of a primitive\npeople. It has been remarked by Laplace that even in his day the rising\nof Venus in its brightest phase never failed to excite surprise and\ninterest. So there is little doubt that the first germ of science\narose in the attention given by Eastern people to the changes of the\nmoon and the motions of the planets. Perhaps the earliest astronomical\ndiscovery consisted in proving the identity of the morning and evening\nstars, on the grounds of their similarity of aspect and invariable\nalternation.[369] Periodical changes of a somewhat complicated kind\nmust have been understood by the Chaldeans, because they were aware\nof the cycle of 6585 days or 19 years which brings round the new and\nfull moon upon the same days, hours, and even minutes of the year.\nThe earliest efforts of scientific prophecy were founded upon this\nknowledge, and if at present we cannot help wondering at the precise\nanticipations of the nautical almanack, we may imagine the wonder\nexcited by such predictions in early times.\n\n  [369] Laplace, *System of the World*, vol.\u00a0i. pp.\u00a050, 54, &c.\n\n\n*Combined Periodic Changes.*\n\nWe shall seldom find a body subject to a single periodic variation,\nand free from other disturbances. We may expect the periodic variation\nitself to undergo variation, which may possibly be secular, but is more\nlikely to prove periodic; nor is there any limit to the complication of\nperiods beyond periods, or periods within periods, which may ultimately\nbe disclosed. In studying a phenomenon of rhythmical character we have\na succession of questions to ask. Is the periodic variation uniform? If\nnot, is the change uniform? If not, is the change itself periodic? Is\nthat new period uniform, or subject to any other change, or not? and so\non *ad infinitum*.\n\nIn some cases there may be many distinct causes of periodic variations,\nand according to the principle of the superposition of small effects,\nto be afterwards considered, these periodic effects will be simply\nadded together, or at least approximately so, and the joint result may\npresent a very complicated subject of investigation. The tides of the\nocean consist of a series of superimposed undulations. Not only are\nthere the ordinary semi-diurnal tides caused by sun and moon, but a\nseries of minor tides, such as the lunar diurnal, the solar diurnal,\nthe lunar monthly, the lunar fortnightly, the solar annual and solar\nsemi-annual are gradually being disentangled by the labours of Sir W.\nThomson, Professor Haughton and others.\n\nVariable stars present interesting periodic phenomena; while some\nstars, \u03b4 Cephei for instance, are subject to very regular variations,\nothers, like Mira Ceti, are less constant in the degrees of brilliancy\nwhich they attain or the rapidity of the changes, possibly on account\nof some longer periodic variation.[370] The star \u03b2 Lyr\u00e6 presents a\ndouble maximum and minimum in each of its periods of nearly 13 days,\nand since the discovery of this variation the period in a period has\nprobably been on the increase. \u201cAt first the variability was more\nrapid, then it became gradually slower; and this decrease in the length\nof time reached its limit between the years 1840 and 1844. During that\ntime its period was nearly invariable; at present it is again decidedly\non the decrease.\u201d[371] The tracing out of such complicated variations\npresents an unlimited field for interesting investigation. The number\nof such variable stars already known is considerable, and there is no\nreason to suppose that any appreciable fraction of the whole number has\nyet been detected.\n\n  [370] Herschel\u2019s *Outlines of Astronomy*, 4th edit. pp.\u00a0555\u2013557.\n\n  [371] Humboldt\u2019s *Cosmos* (Bohn), vol.\u00a0iii. p.\u00a0229.\n\n\n*Principle of Forced Vibrations.*\n\nInvestigations of the connection of periodic causes and effects\nrest upon a principle, which has been demonstrated by Sir John\nHerschel for some special cases, and clearly explained by him in\nseveral of his works.[372] The principle may be formally stated in\nthe following manner: \u201cIf one part of any system connected together\neither by material ties, or by the mutual attractions of its members,\nbe continually maintained by any cause, whether inherent in the\nconstitution of the system or external to it, in a state of regular\nperiodic motion, that motion will be propagated throughout the whole\nsystem, and will give rise, in every member of it, and in every part\nof each member, to periodic movements executed in equal periods, with\nthat to which they owe their origin, though not necessarily synchronous\nwith them in their maxima and minima.\u201d The meaning of the proposition\nis that the effect of a periodic cause will be periodic, and will recur\nat intervals equal to those of the cause. Accordingly when we find two\nphenomena which do proceed, time after time, through changes of the\nsame period, there is much probability that they are connected. In\nthis manner, doubtless, Pliny correctly inferred that the cause of the\ntides lies in the sun and the moon, the intervals between successive\nhigh tides being equal to the intervals between the moon\u2019s passage\nacross the meridian. Kepler and Descartes too admitted the connection\nprevious to Newton\u2019s demonstration of its precise nature. When Bradley\ndiscovered the apparent motion of the stars arising from the aberration\nof light, he was soon able to attribute it to the earth\u2019s annual\nmotion, because it went through its phases in a year.\n\n  [372] *Encyclop\u00e6dia Metropolitana*, art. *Sound*, \u00a7 323; *Outlines\n  of Astronomy*, 4th edit., \u00a7 650. pp.\u00a0410, 487\u201388; *Meteorology,\n  Encyclop\u00e6dia Britannica*, Reprint, p.\u00a0197.\n\nThe most beautiful instance of induction concerning periodic changes\nwhich can be cited, is the discovery of an eleven-year period in\nvarious meteorological phenomena. It would be difficult to mention any\ntwo things apparently more disconnected than the spots upon the sun and\nauroras. As long ago as 1826, Schwabe commenced a regular series of\nobservations of the spots upon the sun, which has been continued to the\npresent time, and he was able to show that at intervals of about eleven\nyears the spots increased much in size and number. Hardly was this\ndiscovery made known, when Lamont pointed out a nearly equal period of\nvariation in the declination of the magnetic needle. Magnetic storms or\nsudden disturbances of the needle were next shown to take place most\nfrequently at the times when sun-spots were prevalent, and as auroras\nare generally coincident with magnetic storms, these phenomena were\nbrought into the cycle. It has since been shown by Professor Piazzi\nSmyth and Mr. E.\u00a0J. Stone, that the temperature of the earth\u2019s surface\nas indicated by sunken thermometers gives some evidence of a like\nperiod. The existence of a periodic cause having once been established,\nit is quite to be expected, according to the principle of forced\nvibrations, that its influence will be detected in all meteorological\nphenomena.\n\n\n*Integrated Variations.*\n\nIn considering the various modes in which one effect may depend upon\nanother, we must set in a distinct class those which arise from the\naccumulated effects of a constantly acting cause. When water runs out\nof a cistern, the velocity of motion depends, according to Torricelli\u2019s\ntheorem, on the height of the surface of the water above the vent;\nbut the amount of water which leaves the cistern in a given time\ndepends upon the aggregate result of that velocity, and is only to\nbe ascertained by the mathematical process of integration. When one\ngravitating body falls towards another, the force of gravity varies\naccording to the inverse square of the distance; to obtain the velocity\nproduced we must integrate or sum the effects of that law; and to\nobtain the space passed over by the body in a given time, we must\nintegrate again.\n\nIn periodic variations the same distinction must be drawn. The heating\npower of the sun\u2019s rays at any place on the earth varies every day with\nthe height attained, and is greatest about noon; but the temperature of\nthe air will not be greatest at the same time. This temperature is an\nintegrated effect of the sun\u2019s heating power, and as long as the sun\nis able to give more heat to the air than the air loses in other ways,\nthe temperature continues to rise, so that the maximum is deferred\nuntil about 3\u00a0P.M. Similarly the hottest day of the year falls, on an\naverage, about one month later than the summer solstice, and all the\nseasons lag about a month behind the motions of the sun. In the case\nof the tides, too, the effect of the moon\u2019s attractive power is never\ngreatest when the power is greatest; the effect always lags more or\nless behind the cause. Yet the intervals between successive tides are\nequal, in the absence of disturbance, to the intervals between the\npassages of the moon across the meridian. Thus the principle of forced\nvibrations holds true.\n\nIn periodic phenomena, however, curious results sometimes follow from\nthe integration of effects. If we strike a pendulum, and then repeat\nthe stroke time after time at the same part of the vibration, all the\nstrokes concur in adding to the momentum, and we can thus increase the\nextent and violence of the vibrations to any degree. We can stop the\npendulum again by strokes applied when it is moving in the opposite\ndirection, and the effects being added together will soon bring it to\nrest. Now if we alter the intervals of the strokes so that each two\nsuccessive strokes act in opposite manners they will neutralise each\nother, and the energy expended will be turned into heat or sound at the\npoint of percussion. Similar effects occur in all cases of rhythmical\nmotion. If a musical note is sounded in a room containing a piano,\nthe string corresponding to it will be thrown into vibration, because\nevery successive stroke of the air-waves upon the string finds it in\nlike position as regards the vibration, and thus adds to its energy of\nmotion. But the other strings being incapable of vibrating with the\nsame rapidity are struck at various points of their vibrations, and one\nstroke will soon be opposed by one contrary in effect. All phenomena of\n*resonance* arise from this coincidence in time of undulation. The air\nin a pipe closed at one end, and about 12 inches in length, is capable\nof vibrating 512 times in a second. If, then, the note C is sounded in\nfront of the open end of the pipe, every successive vibration of the\nair is treasured up as it were in the motion of the air. In a pipe of\ndifferent length the pulses of air would strike each other, and the\nmechanical energy being transmuted into heat would become no longer\nperceptible as sound.\n\nAccumulated vibrations sometimes become so intense as to lead to\nunexpected results. A glass vessel if touched with a violin bow at\na suitable point may be fractured with the violence of vibration. A\nsuspension bridge may be broken down if a company of soldiers walk\nacross it in steps the intervals of which agree with the vibrations\nof the bridge itself. But if they break the step or march in either\nquicker or slower pace, they may have no perceptible effect upon the\nbridge. In fact if the impulses communicated to any vibrating body are\nsynchronous with its vibrations, the energy of those vibrations will be\nunlimited, and may fracture any body.\n\nLet us now consider what will happen if the strokes be not exactly at\nthe same intervals as the vibrations of the body, but, say, a little\nslower. Then a succession of strokes will meet the body in nearly but\nnot quite the same position, and their efforts will be accumulated.\nAfterwards the strokes will begin to fall when the body is in the\nopposite phase. Imagine that one pendulum moving from one extreme\npoint to another in a second, should be struck by another pendulum\nwhich makes 61 beats in a minute; then, if the pendulums commence\ntogether, they will at the end of 30-1/2 beats be moving in opposite\ndirections. Hence whatever energy was communicated in the first half\nminute will be neutralised by the opposite effect of that given in the\nsecond half. The effect of the strokes of the second pendulum will\ntherefore be alternately to increase and decrease the vibrations of\nthe first, so that a new kind of vibration will be produced running\nthrough its phases in 61 seconds. An effect of this kind was actually\nobserved by Ellicott, a member of the Royal Society, in the case of two\nclocks.[373] He found that through the wood-work by which the clocks\nwere connected a slight impulse was transmitted, and each pendulum\nalternately lost and gained momentum. Each clock, in fact, tended to\nstop the other at regular intervals, and in the intermediate times to\nbe stopped by the other.\n\n  [373] *Philosophical Transactions*, (1739), vol.\u00a0xli. p.\u00a0126.\n\nMany disturbances in the planetary system depend upon the same\nprinciple; for if one planet happens always to pull another in the\nsame direction in similar parts of their orbits, the effects, however\nslight, will be accumulated, and a disturbance of large ultimate\namount and of long period will be produced. The long inequality in\nthe motions of Jupiter and Saturn is thus due to the fact that five\ntimes the mean motion of Saturn is very nearly equal to twice the mean\nmotion of Jupiter, causing a coincidence in their relative positions\nand disturbing powers. The rolling of ships depends mainly upon the\nquestion whether the period of vibration of the ship corresponds or\nnot with the intervals at which the waves strike her. Much which seems\nat first sight unaccountable in the behaviour of vessels is thus\nexplained, and the loss of the *Captain* is a sad case in point.\n\n\n\n\nCHAPTER XXI.\n\nTHEORY OF APPROXIMATION.\n\n\nIn order that we may gain a true understanding of the kind, degree, and\nvalue of the knowledge which we acquire by experimental investigation,\nit is requisite that we should be fully conscious of its approximate\ncharacter. We must learn to distinguish between what we can know\nand cannot know--between the questions which admit of solution, and\nthose which only seem to be solved. Many persons may be misled by the\nexpression *exact science*, and may think that the knowledge acquired\nby scientific methods admits of our reaching absolutely true laws,\nexact to the last degree. There is even a prevailing impression that\nwhen once mathematical formul\u00e6 have been successfully applied to a\nbranch of science, this portion of knowledge assumes a new nature, and\nadmits of reasoning of a higher character than those sciences which are\nstill unmathematical.\n\nThe very satisfactory degree of accuracy attained in the science of\nastronomy gives a certain plausibility to erroneous notions of this\nkind. Some persons no doubt consider it to be *proved* that planets\nmove in ellipses, in such a manner that all Kepler\u2019s laws hold exactly\ntrue; but there is a double error in any such notions. In the first\nplace, Kepler\u2019s laws are *not proved*, if by proof we mean certain\ndemonstration of their exact truth. In the next place, even assuming\nKepler\u2019s laws to be exactly true in a theoretical point of view, the\nplanets never move according to those laws. Even if we could observe\nthe motions of a planet, of a perfect globular form, free from all\nperturbing or retarding forces, we could never prove that it moved\nin a perfect ellipse. To prove the elliptical form we should have to\nmeasure infinitely small angles, and infinitely small fractions of a\nsecond; we should have to perform impossibilities. All we can do is to\nshow that the motion of an unperturbed planet approaches *very nearly*\nto the form of an ellipse, and more nearly the more accurately our\nobservations are made. But if we go on to assert that the path *is* an\nellipse we pass beyond our data, and make an assumption which cannot be\nverified by observation.\n\nBut, secondly, as a matter of fact no planet does move in a perfect\nellipse, or manifest the truth of Kepler\u2019s laws exactly. The law of\ngravity prevents its own results from being clearly exhibited, because\nthe mutual perturbations of the planets distort the elliptical paths.\nThose laws, again, hold exactly true only of infinitely small bodies,\nand when two great globes, like the sun and Jupiter, attract each\nother, the law must be modified. The periodic time is then shortened\nin the ratio of the square root of the number expressing the sun\u2019s\nmass, to that of the sum of the numbers expressing the masses of the\nsun and planet, as was shown by Newton.[374] Even at the present day\ndiscrepancies exist between the observed dimensions of the planetary\norbits and their theoretical magnitudes, after making allowance for\nall disturbing causes.[375] Nothing is more certain in scientific\nmethod than that approximate coincidence alone can be expected. In\nthe measurement of continuous quantity perfect correspondence must\nbe accidental, and should give rise to suspicion rather than to\nsatisfaction.\n\n  [374] *Principia*, bk. iii. Prop. 15.\n\n  [375] Lockyer\u2019s *Lessons in Elementary Astronomy*, p.\u00a0301.\n\nOne remarkable result of the approximate character of our observations\nis that we could never prove the existence of perfectly circular or\nparabolic movement, even if it existed. The circle is a singular case\nof the ellipse, for which the eccentricity is zero; it is infinitely\nimprobable that any planet, even if undisturbed by other bodies, would\nhave a circle for its orbit; but if the orbit were a circle we could\nnever prove the entire absence of eccentricity. All that we could\ndo would be to declare the divergence from the circular form to be\ninappreciable. Delambre was unable to detect the slightest ellipticity\nin the orbit of Jupiter\u2019s first satellite, but he could only infer\nthat the orbit was *nearly* circular. The parabola is the singular\nlimit between the ellipse and the hyperbola. As there are elliptic and\nhyperbolic comets, so we might conceive the existence of a parabolic\ncomet. Indeed if an undisturbed comet fell towards the sun from an\ninfinite distance it would move in a parabola; but we could never prove\nthat it so moved.\n\n\n*Substitution of Simple Hypotheses.*\n\nIn truth men never can solve problems fulfilling the complex\ncircumstances of nature. All laws and explanations are in a certain\nsense hypothetical, and apply exactly to nothing which we can know\nto exist. In place of the actual objects which we see and feel, the\nmathematician substitutes imaginary objects, only partially resembling\nthose represented, but so devised that the discrepancies are not of\nan amount to alter seriously the character of the solution. When we\nprobe the matter to the bottom physical astronomy is as hypothetical as\nEuclid\u2019s elements. There may exist in nature perfect straight lines,\ntriangles, circles, and other regular geometrical figures; to our\nscience it is a matter of indifference whether they do or do not exist,\nbecause in any case they must be beyond our powers of perception. If\nwe submitted a perfect circle to the most rigorous scrutiny, it is\nimpossible that we should discover whether it were perfect or not.\nNevertheless in geometry we argue concerning perfect curves, and\nrectilinear figures, and the conclusions apply to existing objects so\nfar as we can assure ourselves that they agree with the hypothetical\nconditions of our reasoning. This is in reality all that we can do in\nthe most perfect of the sciences.\n\nDoubtless in astronomy we meet with the nearest approximation to\nactual conditions. The law of gravity is not a complex one in itself,\nand we believe it with much probability to be exactly true; but we\ncannot calculate out in any real case its accurate results. The law\nasserts that every particle of matter in the universe attracts every\nother particle, with a force depending on the masses of the particles\nand their distances. We cannot know the force acting on any particle\nunless we know the masses and distances and positions of all other\nparticles in the universe. The physical astronomer has made a sweeping\nassumption, namely, that all the millions of existing systems exert no\nperturbing effects on our planetary system, that is to say, no effects\nin the least appreciable. The problem at once becomes hypothetical,\nbecause there is little doubt that gravitation between our sun and\nplanets and other systems does exist. Even when they consider the\nrelations of our planetary bodies *inter se*, all their processes are\nonly approximate. In the first place they assume that each of the\nplanets is a perfect ellipsoid, with a smooth surface and a homogeneous\ninterior. That this assumption is untrue every mountain and valley,\nevery sea, every mine affords conclusive evidence. If astronomers are\nto make their calculations perfect, they must not only take account\nof the Himalayas and the Andes, but must calculate separately the\nattraction of every hill, nay, of every ant-hill. So far are they\nfrom having considered any local inequality of the surface, that they\nhave not yet decided upon the general form of the earth; it is still\na matter of speculation whether or not the earth is an ellipsoid with\nthree unequal axes. If, as is probable, the globe is irregularly\ncompressed in some directions, the calculations of astronomers will\nhave to be repeated and refined, in order that they may approximate\nto the attractive power of such a body. If we cannot accurately learn\nthe form of our own earth, how can we expect to ascertain that of\nthe moon, the sun, and other planets, in some of which probably are\nirregularities of greater proportional amount?\n\nIn a further way the science of physical astronomy is merely\napproximate and hypothetical. Given homogeneous ellipsoids acting upon\neach other according to the law of gravity, the best mathematicians\nhave never and perhaps never will determine exactly the resulting\nmovements. Even when three bodies simultaneously attract each other the\ncomplication of effects is so great that only approximate calculations\ncan be made. Astronomers have not even attempted the general problem\nof the simultaneous attractions of four, five, six, or more bodies;\nthey resolve the general problem into so many different problems of\nthree bodies. The principle upon which the calculations of physical\nastronomy proceed, is to neglect every quantity which does not seem\nlikely to lead to an effect appreciable in observation, and the\nquantities rejected are far more numerous and complex than the few\nlarger terms which are retained. All then is merely approximate.\n\nConcerning other branches of physical science the same statements are\neven more evidently true. We speak and calculate about inflexible\nbars, inextensible lines, heavy points, homogeneous substances,\nuniform spheres, perfect fluids and gases, and we deduce a great\nnumber of beautiful theorems; but all is hypothetical. There is no\nsuch thing as an inflexible bar, an inextensible line, nor any one\nof the other perfect objects of mechanical science; they are to be\nclassed with those mythical existences, the straight line, triangle,\ncircle, &c., about which Euclid so freely reasoned. Take the simplest\noperation considered in statics--the use of a crowbar in raising a\nheavy stone, and we shall find, as Thomson and Tait have pointed\nout, that we neglect far more than we observe.[376] If we suppose\nthe bar to be quite rigid, the fulcrum and stone perfectly hard, and\nthe points of contact real points, we may give the true relation of\nthe forces. But in reality the bar must bend, and the extension and\ncompression of different parts involve us in difficulties. Even if\nthe bar be homogeneous in all its parts, there is no mathematical\ntheory capable of determining with accuracy all that goes on; if, as\nis infinitely more probable, the bar is not homogeneous, the complete\nsolution will be immensely more complicated, but hardly more hopeless.\nNo sooner had we determined the change of form according to simple\nmechanical principles, than we should discover the interference of\nthermodynamic principles. Compression produces heat and extension\ncold, and thus the conditions of the problem are modified throughout.\nIn attempting a fourth approximation we should have to allow for the\nconduction of heat from one part of the bar to another. All these\neffects are utterly inappreciable in a practical point of view, if\nthe bar be a good stout one; but in a theoretical point of view they\nentirely prevent our saying that we have solved a natural problem. The\nfaculties of the human mind, even when aided by the wonderful powers\nof abbreviation conferred by analytical methods, are utterly unable to\ncope with the complications of any real problem. And had we exhausted\nall the known phenomena of a mechanical problem, how can we tell that\nhidden phenomena, as yet undetected, do not intervene in the commonest\nactions? It is plain that no phenomenon comes within the sphere of\nour senses unless it possesses a momentum capable of irritating the\nappropriate nerves. There may then be worlds of phenomena too slight to\nrise within the scope of our consciousness.\n\n  [376] *Treatise on Natural Philosophy*, vol.\u00a0i. pp.\u00a0337, &c.\n\nAll the instruments with which we perform our measurements are faulty.\nWe assume that a plumb-line gives a vertical line; but this is never\ntrue in an absolute sense, owing to the attraction of mountains\nand other inequalities in the surface of the earth. In an accurate\ntrigonometrical survey, the divergencies of the plumb-line must be\napproximately determined and allowed for. We assume a surface of\nmercury to be a perfect plane, but even in the breadth of 5 inches\nthere is a calculable divergence from a true plane of about one\nten-millionth part of an inch; and this surface further diverges from\ntrue horizontality as the plumb-line does from true verticality.\nThat most perfect instrument, the pendulum, is not theoretically\nperfect, except for infinitely small arcs of vibration, and the\ndelicate experiments performed with the torsion balance proceed on the\nassumption that the force of torsion of a wire is proportional to the\nangle of torsion, which again is only true for infinitely small angles.\n\nSuch is the purely approximate character of all our operations that it\nis not uncommon to find the theoretically worse method giving truer\nresults than the theoretically perfect method. The common pendulum\nwhich is not isochronous is better for practical purposes than the\ncycloidal pendulum, which is isochronous in theory but subject to\nmechanical difficulties. The spherical form is not the correct form for\na speculum or lense, but it differs so slightly from the true form, and\nis so much more easily produced mechanically, that it is generally\nbest to rest content with the spherical surface. Even in a six-feet\nmirror the difference between the parabola and the sphere is only about\none ten-thousandth part of an inch, a thickness which would be taken\noff in a few rubs of the polisher. Watts\u2019 ingenious parallel motion was\nintended to produce rectilinear movement of the piston-rod. In reality\nthe motion was always curvilinear, but for his purposes a certain part\nof the curve approximated sufficiently to a straight line.\n\n\n*Approximation to Exact Laws.*\n\nThough we can not prove numerical laws with perfect accuracy, it would\nbe a great mistake to suppose that there is any inexactness in the laws\nof nature. We may even discover a law which we believe to represent\nthe action of forces with perfect exactness. The mind may seem to pass\nin advance of its data, and choose out certain numerical results as\nabsolutely true. We can never really pass beyond our data, and so far\nas assumption enters in, so far want of certainty will attach to our\nconclusions; nevertheless we may sometimes rightly prefer a probable\nassumption of a precise law to numerical results, which are at the best\nonly approximate. We must accordingly draw a strong distinction between\nthe laws of nature which we believe to be accurately stated in our\nformulas, and those to which our statements only make an approximation,\nso that at a future time the law will be differently stated.\n\nThe law of gravitation is expressed in the form F\u00a0=\u00a0Mm/D^{2},\nmeaning that gravity is proportional directly to the product of\nthe gravitating masses, and indirectly to the square of their\ndistance. The latent heat of steam is expressed by the equation\nlog\u00a0F\u00a0=\u00a0*a*\u00a0+\u00a0*b*\u03b1^{t}\u00a0+\u00a0*c*\u03b2^{t}, in which are five quantities *a*,\n*b*, *c*, \u03b1, \u03b2, to be determined by experiment. There is every reason\nto believe that in the progress of science the law of gravity will\nremain entirely unaltered, and the only effect of further inquiry will\nbe to render it a more and more probable expression of the absolute\ntruth. The law of the latent heat of steam on the other hand, will be\nmodified by every new series of experiments, and it may not improbably\nbe shown that the assumed law can never be made to agree exactly with\nthe results of experiment.\n\nPhilosophers have not always supposed that the law of gravity was\nexactly true. Newton, though he had the highest confidence in its\ntruth, admitted that there were motions in the planetary system which\nhe could not reconcile with the law. Euler and Clairaut who were, with\nD\u2019Alembert, the first to apply the full powers of mathematical analysis\nto the theory of gravitation as explaining the perturbations of the\nplanets, did not think the law sufficiently established to attribute\nall discrepancies to the errors of calculation and observation. They\ndid not feel certain that the force of gravity exactly obeyed the\nwell-known rule. The law might involve other powers of the distance. It\nmight be expressed in the form\n\n  F\u00a0=\u00a0...\u00a0+\u00a0*a*/D\u00a0+\u00a0*b*/D^{2}\u00a0+ *c*/D^{3}\u00a0+\u00a0...\n\nand the coefficients *a* and *c* might be so small that those terms\nwould become apparent only in very accurate comparisons with fact.\nAttempts have been made to account for difficulties, by attributing\nvalue to such neglected terms. Gauss at one time thought the even more\nfundamental principle of gravity, that the force is dependent only on\nmass and distance, might not be exactly true, and he undertook accurate\npendulum experiments to test this opinion. Only as repeated doubts have\ntime after time been resolved in favour of the law of Newton, has it\nbeen assumed as precisely correct. But this belief does not rest on\nexperiment or observation only. The calculations of physical astronomy,\nhowever accurate, could never show that the other terms of the above\nexpression were absolutely devoid of value. It could only be shown that\nthey had such slight value as never to become apparent.\n\nThere are, however, other reasons why the law is probably complete\nand true as commonly stated. Whatever influence spreads from a point,\nand expands uniformly through space, will doubtless vary inversely\nin intensity as the square of the distance, because the area over\nwhich it is spread increases as the square of the radius. This part\nof the law of gravity may be considered as due to the properties of\nspace, and there is a perfect analogy in this respect between gravity\nand all other *emanating* forces, as was pointed out by Keill.[377]\nThus the undulations of light, heat, and sound, and the attractions\nof electricity and magnetism obey the very same law so far as we\ncan ascertain. If the molecules of a gas or the particles of matter\nconstituting odour were to start from a point and spread uniformly,\ntheir distances would increase and their density decrease according to\nthe same principle.\n\n  [377] *An Introduction to Natural Philosophy*, 3rd edit. 1733, p.\u00a05.\n\nOther laws of nature stand in a similar position. Dalton\u2019s laws of\ndefinite combining proportions never have been, and never can be,\nexactly proved; but chemists having shown, to a considerable degree of\napproximation, that the elements combine together as if each element\nhad atoms of an invariable mass, assume that this is exactly true. They\ngo even further. Prout pointed out in 1815 that the equivalent weights\nof the elements appeared to be simple numbers; and the researches of\nDumas, Pelouze, Marignac, Erdmann, Stas, and others have gradually\nrendered it likely that the atomic weights of hydrogen, carbon, oxygen,\nnitrogen, chlorine, and silver, are in the ratios of the numbers 1,\n12, 16, 14, 35\u00b75, and 108. Chemists then step beyond their data; they\nthrow aside their actual experimental numbers, and assume that the true\nratios are not those exactly indicated by any weighings, but the simple\nratios of these numbers. They boldly assume that the discrepancies are\ndue to experimental errors, and they are justified by the fact that\nthe more elaborate and skilful the researches on the subject, the more\nnearly their assumption is verified. Potassium is the only element\nwhose atomic weight has been determined with great care, but which has\nnot shown an approach to a simple ratio with the other elements. This\nexception may be due to some unsuspected cause of error.[378] A similar\nassumption is made in the law of definite combining volumes of gases,\nand Brodie has clearly pointed out the line of argument by which the\nchemist, observing that the discrepancies between the law and fact are\nwithin the limits of experimental error, assumes that they are due to\nerror.[379]\n\n  [378] Watts, *Dictionary of Chemistry*, vol.\u00a0i. p.\u00a0455.\n\n  [379] *Philosophical Transactions*, (1866) vol.\u00a0clvi. p.\u00a0809.\n\nFaraday, in one of his researches, expressly makes an assumption\nof the same kind. Having shown, with some degree of experimental\nprecision, that there exists a simple proportion between quantities of\nelectrical energy and the quantities of chemical substances which it\ncan decompose, so that for every atom dissolved in the battery cell an\natom ought theoretically, that is without regard to dissipation of some\nof the energy, to be decomposed in the electrolytic cell, he does not\nstop at his numerical results. \u201cI have not hesitated,\u201d he says,[380]\n\u201cto apply the more strict results of chemical analysis to correct the\nnumbers obtained as electrolytic results. This, it is evident, may be\ndone in a great number of cases, without using too much liberty towards\nthe due severity of scientific research.\u201d\n\n  [380] *Experimental Researches in Electricity*, vol.\u00a0i. p.\u00a0246.\n\nThe law of the conservation of energy, one of the widest of all\nphysical generalisations, rests upon the same footing. The most that\nwe can do by experiment is to show that the energy entering into any\nexperimental combination is almost equal to what comes out of it,\nand more nearly so the more accurately we perform the measurements.\nAbsolute equality is always a matter of assumption. We cannot even\nprove the indestructibility of matter; for were an exceedingly minute\nfraction of existing matter to vanish in any experiment, say one part\nin ten millions, we could never detect the loss.\n\n\n*Successive Approximations to Natural Conditions.*\n\nWhen we examine the history of scientific problems, we find that one\nman or one generation is usually able to make but a single step at\na time. A problem is solved for the first time by making some bold\nhypothetical simplification, upon which the next investigator makes\nhypothetical modifications approaching more nearly to the truth. Errors\nare successively pointed out in previous solutions, until at last there\nmight seem little more to be desired. Careful examination, however,\nwill show that a series of minor inaccuracies remain to be corrected\nand explained, were our powers of reasoning sufficiently great, and the\npurpose adequate in importance.\n\nNewton\u2019s successful solution of the problem of the planetary movements\nentirely depended at first upon a great simplification. The law of\ngravity only applies directly to two infinitely small particles, so\nthat when we deal with vast globes like the earth, Jupiter, and the\nsun, we have an immense aggregate of separate attractions to deal with,\nand the law of the aggregate need not coincide with the law of the\nelementary particles. But Newton, by a great effort of mathematical\nreasoning, was able to show that two homogeneous spheres of matter act\nas if the whole of their masses were concentrated at the centres; in\nshort, that such spheres are centrobaric bodies (p.\u00a0364). He was then\nable with comparative ease to calculate the motions of the planets on\nthe hypothesis of their being spheres, and to show that the results\nroughly agreed with observation. Newton, indeed, was one of the few men\nwho could make two great steps at once. He did not rest contented with\nthe spherical hypothesis; having reason to believe that the earth was\nreally a spheroid with a protuberance around the equator, he proceeded\nto a second approximation, and proved that the attraction of the\nprotuberant matter upon the moon accounted for the precession of the\nequinoxes, and led to various complicated effects. But, (p.\u00a0459), even\nthe spheroidal hypothesis is far from the truth. It takes no account\nof the irregularities of surface, the great protuberance of land in\nCentral Asia and South America, and the deficiency in the bed of the\nAtlantic.\n\nTo determine the law according to which a projectile, such as a cannon\nball, moves through the atmosphere is a problem very imperfectly\nsolved at the present day, but in which many successive advances have\nbeen made. So little was known concerning the subject three or four\ncenturies ago that a cannon ball was supposed to move at first in a\nstraight line, and after a time to be deflected into a curve. Tartaglia\nventured to maintain that the path was curved throughout, as by the\nprinciple of continuity it should be; but the ingenuity of Galileo\nwas required to prove this opinion, and to show that the curve was\napproximately a parabola. It is only, however, under forced hypotheses\nthat we can assert the path of a projectile to be truly a parabola: the\npath must be through a perfect vacuum, where there is no resisting\nmedium of any kind; the force of gravity must be uniform and act in\nparallel lines; or else the moving body must be either a mere point, or\na perfect centrobaric body, that is a body possessing a definite centre\nof gravity. These conditions cannot be really fulfilled in practice.\nThe next great step in the problem was made by Newton and Huyghens, the\nlatter of whom asserted that the atmosphere would offer a resistance\nproportional to the velocity of the moving body, and concluded that\nthe path would have in consequence a logarithmic character. Newton\ninvestigated in a general manner the subject of resisting media, and\ncame to the conclusion that the resistance is more nearly proportional\nto the square of the velocity. The subject then fell into the hands\nof Daniel Bernoulli, who pointed out the enormous resistance of the\nair in cases of rapid movement, and calculated that a cannon ball, if\nfired vertically in a vacuum, would rise eight times as high as in the\natmosphere. In recent times an immense amount both of theoretical and\nexperimental investigation has been spent upon the subject, since it is\none of importance in the art of war. Successive approximations to the\ntrue law have been made, but nothing like a complete and final solution\nhas been achieved or even hoped for.[381]\n\n  [381] Hutton\u2019s *Mathematical Dictionary*, vol.\u00a0ii. pp.\u00a0287\u2013292.\n\nIt is quite to be expected that the earliest experimenters in any\nbranch of science will overlook errors which afterwards become most\napparent. The Arabian astronomers determined the meridian by taking the\nmiddle point between the places of the sun when at equal altitudes on\nthe same day. They overlooked the fact that the sun has its own motion\nin the time between the observations. Newton thought that the mutual\ndisturbances of the planets might be disregarded, excepting perhaps the\neffect of the mutual attraction of the greater planets, Jupiter and\nSaturn, near their conjunction.[382] The expansion of quicksilver was\nlong used as the measure of temperature, no clear idea being possessed\nof temperature apart from some of its more obvious effects. Rumford,\nin the first experiment leading to a determination of the mechanical\nequivalent of heat, disregarded the heat absorbed by the apparatus,\notherwise he would, in Dr. Joule\u2019s opinion, have come nearly to the\ncorrect result.\n\n  [382] *Principia*, bk. iii. Prop. 13.\n\nIt is surprising to learn the number of causes of error which enter\ninto the simplest experiment, when we strive to attain rigid accuracy.\nWe cannot accurately perform the simple experiment of compressing gas\nin a bent tube by a column of mercury, in order to test the truth\nof Boyle\u2019s Law, without paying regard to--(1) the variations of\natmospheric pressure, which are communicated to the gas through the\nmercury; (2) the compressibility of mercury, which causes the column\nof mercury to vary in density; (3) the temperature of the mercury\nthroughout the column; (4) the temperature of the gas, which is with\ndifficulty maintained invariable; (5) the expansion of the glass tube\ncontaining the gas. Although Regnault took all these circumstances\ninto account in his examination of the law,[383] there is no reason to\nsuppose that he exhausted the sources of inaccuracy.\n\n  [383] Jamin, *Cours de Physique*, vol.\u00a0i. pp.\u00a0282, 283.\n\nThe early investigations concerning the nature of waves in elastic\nmedia proceeded upon the assumption that waves of different lengths\nwould travel with equal speed. Newton\u2019s theory of sound led him to\nthis conclusion, and observation (p.\u00a0295) had verified the inference.\nWhen the undulatory theory came to be applied at the commencement of\nthis century to explain the phenomena of light, a great difficulty was\nencountered. The angle at which a ray of light is refracted in entering\na denser medium depends, according to that theory, on the velocity\nwith which the wave travels, so that if all waves of light were to\ntravel with equal velocity in the same medium, the dispersion of mixed\nlight by the prism and the production of the spectrum could not take\nplace. Some most striking phenomena were thus in direct conflict with\nthe theory. Cauchy first pointed out the explanation, namely, that all\nprevious investigators had made an arbitrary assumption for the sake\nof simplifying the calculations. They had assumed that the particles\nof the vibrating medium are so close together that the intervals are\ninconsiderable compared with the length of the wave. This hypothesis\nhappened to be approximately true in the case of air, so that no error\nwas discovered in experiments on sound. Had it not been so, the earlier\nanalysts would probably have failed to give any solution, and the\nprogress of the subject might have been retarded. Cauchy was able to\nmake a new approximation under the more difficult supposition, that\nthe particles of the vibrating medium are situated at considerable\ndistances, and act and react upon the neighbouring particles by\nattractive and repulsive forces. To calculate the rate of propagation\nof disturbance in such a medium is a work of excessive difficulty. The\ncomplete solution of the problem appears indeed to be beyond human\npower, so that we must be content, as in the case of the planetary\nmotions, to look forward to successive approximations. All that Cauchy\ncould do was to show that certain quantities, neglected in previous\ntheories, became of considerable amount under the new conditions of\nthe problem, so that there will exist a relation between the length\nof the wave, and the velocity at which it travels. To remove, then,\nthe difficulties in the way of the undulatory theory of light, a new\napproach to probable conditions was needed.[384]\n\n  [384] Lloyd\u2019s *Lectures on the Wave Theory*, pp.\u00a022, 23.\n\nIn a similar manner Fourier\u2019s theory of the conduction and radiation\nof heat was based upon the hypothesis that the quantity of heat\npassing along any line is simply proportional to the rate of change\nof temperature. But it has since been shown by Forbes that the\nconductivity of a body diminishes as its temperature increases. All\nthe details of Fourier\u2019s solution therefore require modification, and\nthe results are in the meantime to be regarded as only approximately\ntrue.[385]\n\n  [385] Tait\u2019s *Thermodynamics*, p.\u00a010.\n\nWe ought to distinguish between those problems which are physically and\nthose which are merely mathematically incomplete. In the latter case\nthe physical law is correctly seized, but the mathematician neglects,\nor is more often unable to follow out the law in all its results.\nThe law of gravitation and the principles of harmonic or undulatory\nmovement, even supposing the data to be correct, can never be followed\ninto all their ultimate results. Young explained the production of\nNewton\u2019s rings by supposing that the rays reflected from the upper and\nlower surfaces of a thin film of a certain thickness were in opposite\nphases, and thus neutralised each other. It was pointed out, however,\nthat as the light reflected from the nearer surface must be undoubtedly\na little brighter than that from the further surface, the two rays\nought not to neutralise each other so completely as they are observed\nto do. It was finally shown by Poisson that the discrepancy arose only\nfrom incomplete solution of the problem; for the light which has once\ngot into the film must be to a certain extent reflected backwards and\nforwards *ad infinitum*; and if we follow out this course of the light\nby perfect mathematical analysis, absolute darkness may be shown to\nresult from the interference of the rays.[386] In this case the natural\nlaws concerned, those of reflection and refraction, are accurately\nknown, and the only difficulty consists in developing their full\nconsequences.\n\n  [386] Lloyd\u2019s *Lectures on the Wave Theory*, pp.\u00a082, 83.\n\n\n*Discovery of Hypothetically Simple Laws.*\n\nIn some branches of science we meet with natural laws of a simple\ncharacter which are in a certain point of view exactly true and yet can\nnever be manifested as exactly true in natural phenomena. Such, for\ninstance, are the laws concerning what is called a *perfect gas*. The\ngaseous state of matter is that in which the properties of matter are\nexhibited in the simplest manner. There is much advantage accordingly\nin approaching the question of molecular mechanics from this side.\nBut when we ask the question--What is a gas? the answer must be a\nhypothetical one. Finding that gases *nearly* obey the law of Boyle\nand Mariotte; that they *nearly* expand by heat at the uniform rate of\none part in 272\u00b79 of their volume at 0\u00b0 for each degree centigrade;\nand that they *more nearly* fulfil these conditions the more distant\nthe point of temperature at which we examine them from the liquefying\npoint, we pass by the principle of continuity to the conception of a\nperfect gas. Such a gas would probably consist of atoms of matter at\nso great a distance from each other as to exert no attractive forces\nupon each other; but for this condition to be fulfilled the distances\nmust be infinite, so that an absolutely perfect gas cannot exist. But\nthe perfect gas is not merely a limit to which we may approach, it is a\nlimit passed by at least one real gas. It has been shown by Despretz,\nPouillet, Dulong, Arago, and finally Regnault, that all gases diverge\nfrom the Boylean law, and in nearly all cases the density of the gas\nincreases in a somewhat greater ratio than the pressure, indicating\na tendency on the part of the molecules to approximate of their own\naccord. In the more condensable gases such as sulphurous acid, ammonia,\nand cyanogen, this tendency is strongly apparent near the liquefying\npoint. Hydrogen, on the contrary, diverges from the law of a perfect\ngas in the opposite direction, that is, the density increases less than\nin the ratio of the pressure.[387] This is a singular exception, the\nbearing of which I am unable to comprehend.\n\n  [387] Jamin, *Cours de Physique*, vol.\u00a0i. pp.\u00a0283\u2013288.\n\nAll gases diverge again from the law of uniform expansion by heat, but\nthe divergence is less as the gas in question is less condensable, or\nexamined at a temperature more removed from its liquefying point. Thus\nthe perfect gas must have an infinitely high temperature. According\nto Dalton\u2019s law each gas in a mixture retains its own properties\nunaffected by the presence of any other gas.[388] This law is probably\ntrue only by approximation, but it is obvious that it would be true of\nthe perfect gas with infinitely distant particles.[389]\n\n  [388] Joule and Thomson, *Philosophical Transactions*, 1854,\n  vol.\u00a0cxliv. p. 337.\n\n  [389] The properties of a perfect gas have been described by Rankine,\n  *Transactions of the Royal Society of Edinburgh*, vol.\u00a0xxv. p.\u00a0561.\n\n\n*Mathematical Principles of Approximation.*\n\nThe approximate character of physical science will be rendered more\nplain if we consider it from a mathematical point of view. Throughout\nquantitative investigations we deal with the relation of one quantity\nto other quantities, of which it is a function; but the subject is\nsufficiently complicated if we view one quantity as a function of\none other. Now, as a general rule, a function can be developed or\nexpressed as the sum of quantities, the values of which depend upon the\nsuccessive powers of the variable quantity. If *y* be a function of *x*\nthen we may say that\n\n  *y*\u00a0=\u00a0A\u00a0+\u00a0B*x*\u00a0+\u00a0C*x*^{2}\u00a0+ D*x*^{3}\u00a0+ E*x*^{4}\u00a0....\n\nIn this equation, A, B, C, D, &c., are fixed quantities, of different\nvalues in different cases. The terms may be infinite in number or\nafter a time may cease to have any value. Any of the coefficients A,\nB, C, &c., may be zero or negative; but whatever they be they are\nfixed. The quantity *x* on the other hand may be made what we like,\nbeing variable. Suppose, in the first place, that *x* and *y* are both\nlengths. Let us assume that 1/10,000 part of an inch is the least that\nwe can take note of. Then when *x* is one hundredth of an inch, we have\n*x*^{2}\u00a0= 1/10,000, and if C be less than unity, the term C*x*^{2} will\nbe inappreciable, being less than we can measure. Unless any of the\nquantities D, E, &c., should happen to be very great, it is evident\nthat all the succeeding terms will also be inappreciable, because the\npowers of *x* become rapidly smaller in geometrical ratio. Thus when\n*x* is made small enough the quantity *y* seems to obey the equation\n\n  *y* = A + B*x*.\n\nIf *x* should be still less, if it should become as small, for\ninstance, as 1/1,000,000 of an inch, and B should not be very great,\nthen *y* would appear to be the fixed quantity A, and would not seem to\nvary with *x* at all. On the other hand, were x to grow greater, say\nequal to 1/10 inch, and C not be very small, the term C*x*^{2} would\nbecome appreciable, and the law would now be more complicated.\n\nWe can invert the mode of viewing this question, and suppose that\nwhile the quantity *y* undergoes variations depending on many powers\nof *x*, our power of detecting the changes of value is more or less\nacute. While our powers of observation remain very rude we may be\nunable to detect any change in the quantity at all, that is to say,\nB*x* may always be too small to come within our notice, just as in\nformer days the fixed stars were so called because they remained at\napparently fixed distances from each other. With the use of telescopes\nand micrometers we become able to detect the existence of some motion,\nso that the distance of one star from another may be expressed by\nA\u00a0+\u00a0B*x*, the term including *x*^{2} being still inappreciable. Under\nthese circumstances the star will seem to move uniformly, or in simple\nproportion to the time *x*. With much improved means of measurement it\nwill probably be found that this uniformity of motion is only apparent,\nand that there exists some acceleration or retardation. More careful\ninvestigation will show the law to be more and more complicated than\nwas previously supposed.\n\nThere is yet another way of explaining the apparent results of a\ncomplicated law. If we take any curve and regard a portion of it free\nfrom any kind of discontinuity, we may represent the character of such\nportion by an equation of the form\n\n  *y*\u00a0=\u00a0A\u00a0+\u00a0B*x*\u00a0+\u00a0C*x*^{2}\u00a0+ D*x*^{3}\u00a0+\u00a0....\n\nRestrict the attention to a very small portion of the curve, and the\neye will be unable to distinguish its difference from a straight\nline, which amounts to saying that in the portion examined the term\nC*x*^{2} has no value appreciable by the eye. Take a larger portion of\nthe curve and it will be apparent that it possesses curvature, but it\nwill be possible to draw a parabola or ellipse so that the curve shall\napparently coincide with a portion of that parabola or ellipse. In the\nsame way if we take larger and larger arcs of the curve it will assume\nthe character successively of a curve of the third, fourth, and perhaps\nhigher degrees; that is to say, it corresponds to equations involving\nthe third, fourth, and higher powers of the variable quantity.\n\nWe have arrived then at the conclusion that every phenomenon, when its\namount can only be rudely measured, will either be of fixed amount,\nor will seem to vary uniformly like the distance between two inclined\nstraight lines. More exact measurement may show the error of this first\nassumption, and the variation will then appear to be like that of the\ndistance between a straight line and a parabola or ellipse. We may\nafterwards find that a curve of the third or higher degrees is really\nrequired to represent the variation. I propose to call the variation\nof a quantity *linear*, *elliptic*, *cubic*, *quartic*, *quintic*, &c.,\naccording as it is discovered to involve the first, second, third,\nfourth, fifth, or higher powers of the variable. It is a general\nrule in quantitative investigation that we commence by discovering\nlinear, and afterwards proceed to elliptic or more complicated laws of\nvariation. The approximate curves which we employ are all, according\nto De Morgan\u2019s use of the name, parabolas of some order or other; and\nsince the common parabola of the second order is approximately the same\nas a very elongated ellipse, and is in fact an infinitely elongated\nellipse, it is convenient and proper to call variation of the second\norder *elliptic*. It might also be called *quadric* variation.\n\nAs regards many important phenomena we are yet only in the first stage\nof approximation. We know that the sun and many so-called fixed stars,\nespecially 61 Cygni, have a proper motion through space, and the\ndirection of this motion at the present time is known with some degree\nof accuracy. But it is hardly consistent with the theory of gravity\nthat the path of any body should really be a straight line. Hence, we\nmust regard a rectilinear path as only a provisional description of\nthe motion, and look forward to the time when its curvature will be\ndetected, though centuries perhaps must first elapse.\n\nWe are accustomed to assume that on the surface of the earth the force\nof gravity is uniform, because the variation is of so slight an amount\nthat we are scarcely able to detect it. But supposing we could measure\nthe variation, we should find it simply proportional to the height.\nTaking the earth\u2019s radius to be unity, let *h* be the height at which\nwe measure the force of gravity. Then by the well-known law of the\ninverse square, that force will be proportional to\n\n  *g*/(1\u00a0+\u00a0*h*)^{2}, or to *g*(1\u00a0-\u00a02*h*\u00a0+ 3*h*^{2}\u00a0- 4*h*^{3}\u00a0+\u00a0...).\n\nBut at all heights to which we can attain *h* will be so small a\nfraction of the earth\u2019s radius that 3*h*^{2} will be inappreciable, and\nthe force of gravity will seem to follow the law of linear variation,\nbeing proportional to 1\u00a0-\u00a02*h*.\n\nWhen the circumstances of an experiment are much altered, different\npowers of the variable may become prominent. The resistance of a liquid\nto a body moving through it may be approximately expressed as the sum\nof two terms respectively involving the first and second powers of the\nvelocity. At very low velocities the first power is of most importance,\nand the resistance, as Professor Stokes has shown, is nearly in simple\nproportion to the velocity. When the motion is rapid the resistance\nincreases in a still greater degree, and is more nearly proportional to\nthe square of the velocity.\n\n\n*Approximate Independence of Small Effects.*\n\nOne result of the theory of approximation possesses such importance\nin physical science, and is so often applied, that we may consider\nit separately. The investigation of causes and effects is immensely\nsimplified when we may consider each cause as producing its own effect\ninvariably, whether other causes are acting or not. Thus, if the\nbody P produces *x*, and Q produces *y*, the question is whether P\nand Q acting together will produce the sum of the separate effects,\n*x*\u00a0+\u00a0*y*. It is under this supposition that we treated the methods of\neliminating error (Chap. XV.), and errors of a less amount would still\nremain if the supposition was a forced one. There are probably some\nparts of science in which the supposition of independence of effects\nholds rigidly true. The mutual gravity of two bodies is entirely\nunaffected by the presence of other gravitating bodies. People do not\nusually consider that this important principle is involved in such a\nsimple thing as putting two pound weights in the scale of a balance.\nHow do we know that two pounds together will weigh twice as much as\none? Do we know it to be exactly so? Like other results founded on\ninduction we cannot prove it absolutely, but all the calculations of\nphysical astronomy proceed upon the assumption, so that we may consider\nit proved to a very high degree of approximation. Had not this been\ntrue, the calculations of physical astronomy would have been infinitely\nmore complex than they actually are, and the progress of knowledge\nwould have been much slower.\n\nIt is a general principle of scientific method that if effects be of\nsmall amount, comparatively to our means of observation, all joint\neffects will be of a higher order of smallness, and may therefore\nbe rejected in a first approximation. This principle was employed\nby Daniel Bernoulli in the theory of sound, under the title of *The\nPrinciple of the Coexistence of Small Vibrations*. He showed that if\na string is affected by two kinds of vibrations, we may consider each\nto be going on as if the other did not exist. We cannot perceive that\nthe sounding of one musical instrument prevents or even modifies the\nsound of another, so that all sounds would seem to travel through the\nair, and act upon the ear in independence of each other. A similar\nassumption is made in the theory of tides, which are great waves. One\nwave is produced by the attraction of the moon, and another by the\nattraction of the sun, and the question arises, whether when these\nwaves coincide, as at the time of spring tides, the joint wave will be\nsimply the sum of the separate waves. On the principle of Bernoulli\nthis will be so, because the tides on the ocean are very small compared\nwith the depth of the ocean.\n\nThe principle of Bernoulli, however, is only approximately true. A wave\nnever is exactly the same when another wave is interfering with it,\nbut the less the displacement of particles due to each wave, the less\nin a still higher degree is the effect of one wave upon the other. In\nrecent years Helmholtz was led to suspect that some of the phenomena\nof sound might after all be due to resultant effects overlooked by the\nassumption of previous physicists. He investigated the secondary waves\nwhich would arise from the interference of considerable disturbances,\nand was able to show that certain summation of resultant tones ought to\nbe heard, and experiments subsequently devised for the purpose showed\nthat they might be heard.\n\n[Illustration]\n\nThroughout the mechanical sciences the *Principle of the Superposition\nof Small Motions* is of fundamental importance,[390] and it may be\nthus explained. Suppose that two forces, acting from the points B and\nC, are simultaneously moving a body A. Let the force acting from B be\nsuch that in one second it would move A to *p*, and similarly let the\nsecond force, acting alone, move A to *r*. The question arises, then,\nwhether their joint action will urge A to *q* along the diagonal of the\nparallelogram. May we say that A will move the distance A*p* in the\ndirection AB, and A*r* in the direction AC, or, what is the same thing,\nalong the parallel line *pq*? In strictness we cannot say so; for when\nA has moved towards *p*, the force from C will no longer act along the\nline AC, and similarly the motion of A towards *r* will modify the\naction of the force from B. This interference of one force with the\nline of action of the other will evidently be greater the larger is\nthe extent of motion considered; on the other hand, as we reduce the\nparallelogram A*pqr*, compared with the distances AB and AC, the less\nwill be the interference of the forces. Accordingly mathematicians\navoid all error by considering the motions as infinitely small, so that\nthe interference becomes of a still higher order of infinite smallness,\nand may be entirely neglected. By the resources of the differential\ncalculus it is possible to calculate the motion of the particle A, as\nif it went through an infinite number of infinitely small diagonals\nof parallelograms. The great discoveries of Newton really arose from\napplying this method of calculation to the movements of the moon round\nthe earth, which, while constantly tending to move onward in a straight\nline, is also deflected towards the earth by gravity, and moves\nthrough an elliptic curve, composed as it were of the infinitely small\ndiagonals of infinitely numerous parallelograms. The mathematician,\nin his investigation of a curve, always treats it as made up of a\ngreat number of straight lines, and it may be doubted whether he could\ntreat it in any other manner. There is no error in the final results,\nbecause having obtained the formul\u00e6 flowing from this supposition, each\nstraight line is then regarded as becoming infinitely small, and the\npolygonal line becomes undistinguishable from a perfect curve.[391]\n\n  [390] Thomson and Tait\u2019s *Natural Philosophy*, vol.\u00a0i. p.\u00a060.\n\n  [391] Challis, *Notes on the Principles of Pure and Applied\n  Calculation*, 1869, p.\u00a083.\n\nIn abstract mathematical theorems the approximation to absolute truth\nis perfect, because we can treat of infinitesimals. In physical\nscience, on the contrary, we treat of the least quantities which are\nperceptible. Nevertheless, while carefully distinguishing between these\ntwo different cases, we may fearlessly apply to both the principle\nof the superposition of small effects. In physical science we have\nonly to take care that the effects really are so small that any joint\neffect will be unquestionably imperceptible. Suppose, for instance,\nthat there is some cause which alters the dimensions of a body in the\nratio of 1 to 1\u00a0+\u00a0\u03b1, and another cause which produces an alteration in\nthe ratio of 1 to 1\u00a0+\u00a0\u03b2. If they both act at once the change will be\nin the ratio of 1 to (1\u00a0+\u00a0\u03b1)(1\u00a0+\u00a0\u03b2), or as 1 to 1\u00a0+\u00a0\u03b1\u00a0+\u00a0\u03b2\u00a0+\u00a0\u03b1\u03b2. But if\n\u03b1 and \u03b2 be both very small fractions of the total dimensions, \u03b1\u03b2 will\nbe yet far smaller and may be disregarded; the ratio of change is then\napproximately that of 1 to 1\u00a0+\u00a0\u03b1\u00a0+\u00a0\u03b2, or the joint effect is the sum of\nthe separate effects. Thus if a body were subjected to three strains,\nat right angles to each other, the total change in the volume of the\nbody would be approximately equal to the sum of the changes produced\nby the separate strains, provided that these are very small. In like\nmanner not only is the expansion of every solid and liquid substance\nby heat approximately proportional to the change of temperature, when\nthis change is very small in amount, but the cubic expansion may also\nbe considered as being three times as great as the linear expansion.\nFor if the increase of temperature expands a bar of metal in the ratio\nof 1 to 1\u00a0+\u00a0\u03b1, and the expansion be equal in all directions, then a\ncube of the same metal would expand as 1 to (1\u00a0+\u00a0\u03b1)^{3}, or as 1 to\n1\u00a0+\u00a03\u03b1\u00a0+\u00a03\u03b1^{2}\u00a0+\u00a0\u03b1^{3}. When \u03b1 is a very small quantity the third term\n3\u03b1^{2} will be imperceptible, and still more so the fourth term \u03b1^{3}.\nThe coefficients of expansion of solids are in fact so small, and so\nimperfectly determined, that physicists seldom take into account their\nsecond and higher powers.\n\nIt is a result of these principles that all small errors may be assumed\nto vary in simple proportion to their causes--a new reason why, in\neliminating errors, we should first of all make them as small as\npossible. Let us suppose that there is a right-angled triangle of which\nthe two sides containing the right angle are really of the lengths 3\nand 4, so that the hypothenuse is \u221a(3^{2}\u00a0+\u00a04^{2}) or 5. Now,\nif in two measurements of the first side we commit slight errors,\nmaking it successively 4\u00b7001 and 4\u00b7002, then calculation will give the\nlengths of the hypothenuse as almost exactly 5\u00b70008 and 5\u00b70016, so that\nthe error in the hypothenuse will seem to vary in simple proportion\nto that of the side, although it does not really do so with perfect\nexactness. The logarithm of a number does not vary in proportion to\nthat number--nevertheless we find the difference between the logarithms\nof the numbers 100000 and 100001 to be almost exactly equal to that\nbetween the numbers 100001 and 100002. It is thus a general rule that\nvery small differences between successive values of a function are\napproximately proportional to the small differences of the variable\nquantity.\n\nOn these principles it is easy to draw up a series of rules such as\nthose given by Kohlrausch[392] for performing calculations in an\nabbreviated form when the variable quantity is very small compared with\nunity. Thus for 1\u00a0\u00f7\u00a0(1\u00a0+\u00a0\u03b1) we may substitute 1\u00a0-\u00a0\u03b1; for 1\u00a0\u00f7\u00a0(1\u00a0-\u00a0\u03b1) we\nmay put 1\u00a0+\u00a0\u03b1; 1\u00a0\u00f7\u00a0\u221a(1\u00a0+\u00a0\u03b1) becomes 1\u00a0-\u00a0(1/2)\u03b1, and so forth.\n\n  [392] *An Introduction to Physical Measurements*, translated by\n  Waller and Procter, 1873, p.\u00a010.\n\n\n*Four Meanings of Equality.*\n\nAlthough it might seem that there are few terms more free from\nambiguity than the term *equal*, yet scientific men do employ it with\nat least four meanings, which it is desirable to distinguish. These\nmeanings I may describe as\n\n  (1) Absolute Equality.\n  (2) Sub-equality.\n  (3) Apparent Equality.\n  (4) Probable Equality.\n\nBy *absolute equality* we signify that which is complete and perfect\nto the last degree; but it is obvious that we can only know such\nequality in a theoretical or hypothetical manner. The areas of two\ntriangles standing upon the same base and between the same parallels\nare absolutely equal. Hippocrates beautifully proved that the area\nof a lunula or figure contained between two segments of circles was\nabsolutely equal to that of a certain right-angled triangle. As a\ngeneral rule all geometrical and other elementary mathematical theorems\ninvolve absolute equality.\n\nDe Morgan proposed to describe as *sub-equal* those quantities which\nare equal within an infinitely small quantity, so that *x* is sub-equal\nto *x*\u00a0+\u00a0*dx*. The differential calculus may be said to arise out\nof the neglect of infinitely small quantities, and in mathematical\nscience other subtle distinctions may have to be drawn between kinds of\nequality, as De Morgan has shown in a remarkable memoir \u201cOn Infinity;\nand on the sign of Equality.\u201d[393]\n\n  [393] *Cambridge Philosophical Transactions* (1865), vol.\u00a0xi. Part I.\n\n*Apparent equality* is that with which physical science deals. Those\nmagnitudes are apparently equal which differ only by an imperceptible\nquantity. To the carpenter anything less than the hundredth part of\nan inch is non-existent; there are few arts or artists to which the\nhundred-thousandth of an inch is of any account. Since all coincidence\nbetween physical magnitudes is judged by one or other sense, we must be\nrestricted to a knowledge of apparent equality.\n\nIn reality even apparent equality is rarely to be expected. More\ncommonly experiments will give only *probable equality*, that is\nresults will come so near to each other that the difference may be\nascribed to unimportant disturbing causes. Physicists often assume\nquantities to be equal provided that they fall within the limits of\nprobable error of the processes employed. We cannot expect observations\nto agree with theory more closely than they agree with each other, as\nNewton remarked of his investigations concerning Halley\u2019s Comet.\n\n\n*Arithmetic of Approximate Quantities.*\n\nConsidering that almost all the quantities which we treat in physical\nand social science are approximate only, it seems desirable that\nattention should be paid in the teaching of arithmetic to the correct\ninterpretation and treatment of approximate numerical statements. We\nseem to need notation for expressing the approximateness or exactness\nof decimal numbers. The fraction \u00b7025 may mean either precisely one\n40th part, or it may mean anything between \u00b70245 and \u00b70255. I propose\nthat when a decimal fraction is completely and exactly given, a *small\ncipher* or circle should be added to indicate that there is nothing\nmore to come, as in \u00b7025\u25e6. When the first figure of the decimals\nrejected is 5 or more, the first figure retained should be raised by\na unit, according to a rule approved by De Morgan, and now generally\nrecognised. To indicate that the fraction thus retained is more than\nthe truth, a point has been placed over the last figure in some tables\nof logarithms; but a similar point is used to denote the period of a\nrepeating decimal, and I should therefore propose to employ a colon\n*after* the figure; thus \u00b7025: would mean that the true quantity lies\nbetween \u00b70245\u00b0 and \u00b7025\u00b0 inclusive of the lower but not the higher\nlimit. When the fraction is less than the truth, two dots might be\nplaced horizontally as in 025.. which would mean anything between \u00b7025\u00b0\nand \u00b70255\u00b0 not inclusive.\n\nWhen approximate numbers are added, subtracted, multiplied, or divided,\nit becomes a matter of some complexity to determine the degree of\naccuracy of the result. There are few persons who could assert off-hand\nthat the sum of the approximate numbers 34\u00b770, 52\u00b7693, 80\u00b71, is 167\u00b75\n*within less than* \u00b707. Mr. Sandeman has traced out the rules of\napproximate arithmetic in a very thorough manner, and his directions\nare worthy of careful attention.[394] The third part of Sonnenschein\nand Nesbitt\u2019s excellent book on arithmetic[395] describes fully all\nkinds of approximate calculations, and shows both how to avoid needless\nlabour and how to take proper account of inaccuracy in operating with\napproximate decimal fractions. A simple investigation of the subject is\nto be found in Sonnet\u2019s *Alg\u00e8bre El\u00e9mentaire* (Paris, 1848) chap. xiv.,\n\u201cDes Approximations Absolues et Relatives.\u201d There is also an American\nwork on the subject.[396]\n\n  [394] Sandeman, *Pelicotetics*, p.\u00a0214.\n\n  [395] *The Science and Art of Arithmetic for the Use of Schools.*\n  (Whitaker and Co.)\n\n  [396] *Principles of Approximate Calculations*, by J.\u00a0J. Skinner,\n  C.E. (New York, Henry Holt), 1876.\n\nAlthough the accuracy of measurement has so much advanced since the\ntime of Leslie, it is not superfluous to repeat his protest against\nthe unfairness of affecting by a display of decimal fractions a\ngreater degree of accuracy than the nature of the case requires and\nadmits.[397] I have known a scientific man to register the barometer\nto a second of time when the nearest quarter of an hour would have\nbeen amply sufficient. Chemists often publish results of analysis to\nthe ten-thousandth or even the millionth part of the whole, when in\nall probability the processes employed cannot be depended on beyond\nthe hundredth part. It is seldom desirable to give more than one\nplace of figures of uncertain amount; but it must be allowed that a\nnice perception of the degree of accuracy possible and desirable is\nrequisite to save misapprehension and needless computation on the one\nhand, and to secure all attainable exactness on the other hand.\n\n  [397] Leslie, *Inquiry into the Nature of Heat*, p.\u00a0505.\n\n\n\n\nCHAPTER XXII.\n\nQUANTITATIVE INDUCTION.\n\n\nWe have not yet formally considered any processes of reasoning\nwhich have for their object to disclose laws of nature expressed in\nquantitative equations. We have been inquiring into the modes by which\na phenomenon may be measured, and, if it be a composite phenomenon, may\nbe resolved, by the aid of several measurements, into its component\nparts. We have also considered the precautions to be taken in the\nperformance of observations and experiments in order that we may know\nwhat phenomena we really do measure, but we must remember that, no\nnumber of facts and observations can by themselves constitute science.\nNumerical facts, like other facts, are but the raw materials of\nknowledge, upon which our reasoning faculties must be exerted in order\nto draw forth the principles of nature. It is by an inverse process of\nreasoning that we can alone discover the mathematical laws to which\nvarying quantities conform. By well-conducted experiments we gain a\nseries of values of a variable, and a corresponding series of values\nof a variant, and we now want to know what mathematical function the\nvariant is as regards the variable. In the usual progress of a science\nthree questions will have to be answered as regards every important\nquantitative phenomenon:--\n\n(1) Is there any constant relation between a variable and a variant?\n\n(2) What is the empirical formula expressing this relation?\n\n(3) What is the rational formula expressing the law of nature involved?\n\n\n*Probable Connection of Varying Quantities.*\n\nWe find it stated by Mill,[398] that \u201cWhatever phenomenon varies in any\nmanner whenever another phenomenon varies in some particular manner, is\neither a cause or an effect of that phenomenon, or is connected with\nit through some fact of causation.\u201d This assertion may be considered\ntrue when it is interpreted with sufficient caution; but it might\notherwise lead us into error. There is nothing whatever in the nature\nof things to prevent the existence of two variations which should\napparently follow the same law, and yet have no connection with each\nother. One binary star might be going through a revolution which, so\nfar as we could tell, was of equal period with that of another binary\nstar, and according to the above rule the motion of one would be the\ncause of the motion of the other, which would not be really the case.\nTwo astronomical clocks might conceivably be made so nearly perfect\nthat, for several years, no difference could be detected, and we might\nthen infer that the motion of one clock was the cause or effect of the\nmotion of the other. This matter requires careful discrimination. We\nmust bear in mind that the continuous quantities of space, time, force,\n&c., which we measure, are made up of an infinite number of infinitely\nsmall units. We may then meet with two variable phenomena which follow\nlaws so nearly the same, that in no part of the variations open to\nour observation can any discrepancy be discovered. I grant that if\ntwo clocks could be shown to have kept *exactly* the same time during\nany finite interval, the probability would become infinitely high\nthat there was a connection between their motions. But we can never\nabsolutely prove such coincidences to exist. Allow that we may observe\na difference of one-tenth of a second in their time, yet it is possible\nthat they were independently regulated so as to go together within\nless than that quantity of time. In short, it would require either an\ninfinitely long time of observation, or infinitely acute powers of\nmeasuring discrepancy, to decide positively whether two clocks were or\nwere not in relation with each other.\n\n  [398] *System of Logic*, bk. iii. chap. viii \u00a7 6.\n\nA similar question actually occurs in the case of the moon\u2019s motion. We\nhave no record that any other portion of the moon was ever visible to\nmen than such as we now see. This fact sufficiently proves that within\nthe historical period the rotation of the moon on its own axis has\ncoincided with its revolutions round the earth. Does this coincidence\nprove a relation of cause and effect to exist? The answer must be in\nthe negative, because there might have been so slight a discrepancy\nbetween the motions that there has not yet been time to produce any\nappreciable effect. There may nevertheless be a high probability of\nconnection.\n\nThe whole question of the relation of quantities thus resolves itself\ninto one of probability. When we can only rudely measure a quantitative\nresult, we can assign but slight importance to any correspondence.\nBecause the brightness of two stars seems to vary in the same manner,\nthere is no considerable probability that they have any relation with\neach other. Could it be shown that their periods of variation were\nthe same to infinitely small quantities it would be certain, that\nis infinitely probable, that they were connected, however unlikely\nthis might be on other grounds. The general mode of estimating such\nprobabilities is identical with that applied to other inductive\nproblems. That any two periods of variation should by chance become\n*absolutely equal* is infinitely improbable; hence if, in the case of\nthe moon or other moving bodies, we could prove absolute coincidence\nwe should have certainty of connection.[399] With approximate\nmeasurements, which alone are within our power, we must hope for\napproximate certainty at the most.\n\n  [399] Laplace, *System of the World*, translated by Harte, vol.\u00a0ii.\n  p.\u00a0366.\n\nThe principles of inference and probability, according to which we\ntreat causes and effects varying in amount, are exactly the same as\nthose by which we treated simple experiments. Continuous quantity,\nhowever, affords us an infinitely more extensive sphere of observation,\nbecause every different amount of cause, however little different,\nought to be followed by a different amount of effect. If we can measure\ntemperature to the one-hundredth part of a degree centigrade, then\nbetween 0\u00b0 and 100\u00b0 we have 10,000 possible trials. If the precision\nof our measurements is increased, so that the one-thousandth part of\na degree can be appreciated, our trials may be increased tenfold. The\nprobability of connection will be proportional to the accuracy of our\nmeasurements.\n\nWhen we can vary the quantity of a cause at will it is easy to discover\nwhether a certain effect is due to that cause or not. We can then\nmake as many irregular changes as we like, and it is quite incredible\nthat the supposed effect should by chance go through exactly the\ncorresponding series of changes except by dependence. If we have a\nbell ringing *in vacuo*, the sound increases as we let in the air, and\nit decreases again as we exhaust the air. Tyndall\u2019s singing flames\nevidently obeyed the directions of his own voice; and Faraday when he\ndiscovered the relation of magnetism and light found that, by making\nor breaking or reversing the current of the electro-magnet, he had\ncomplete command over a ray of light, proving beyond all reasonable\ndoubt the dependence of cause and effect. In such cases it is the\nperfect coincidence in time between the change in the effect and that\nin the cause which raises a high improbability of casual coincidence.\n\nIt is by a simple case of variation that we infer the existence of\na material connection between two bodies moving with exactly equal\nvelocity, such as the locomotive engine and the train which follows\nit. Elaborate observations were requisite before astronomers could all\nbe convinced that the red hydrogen flames seen during solar eclipses\nbelonged to the sun, and not to the moon\u2019s atmosphere as Flamsteed\nassumed. As early as 1706, Stannyan noticed a blood-red streak in\nan eclipse which he witnessed at Berne, and he asserted that it\nbelonged to the sun; but his opinion was not finally established until\nphotographs of the eclipse in 1860, taken by Mr. De la Rue, showed that\nthe moon\u2019s dark body gradually covered the red prominences on one side,\nand uncovered those on the other; in short, that these prominences\nmoved precisely as the sun moved, and not as the moon moved.\n\nEven when we have no means of accurately measuring the variable\nquantities we may yet be convinced of their connection, if one always\nvaries perceptibly at the same time as the other. Fatigue increases\nwith exertion; hunger with abstinence from food; desire and degree of\nutility decrease with the quantity of commodity consumed. We know that\nthe sun\u2019s heating power depends upon his height of the sky; that the\ntemperature of the air falls in ascending a mountain; that the earth\u2019s\ncrust is found to be perceptibly warmer as we sink mines into it; we\ninfer the direction in which a sound comes from the change of loudness\nas we approach or recede. The facility with which we can time after\ntime observe the increase or decrease of one quantity with another\nsufficiently shows the connection, although we may be unable to assign\nany precise law of relation. The probability in such cases depends upon\nfrequent coincidence in time.\n\n\n*Empirical Mathematical Laws.*\n\nIt is important to acquire a clear comprehension of the part which\nis played in scientific investigation by empirical formul\u00e6 and laws.\nIf we have a table containing certain values of a variable and the\ncorresponding values of the variant, there are mathematical processes\nby which we can infallibly discover a mathematical formula yielding\nnumbers in more or less exact agreement with the table. We may\ngenerally assume that the quantities will approximately conform to a\nlaw of the form\n\n  *y* = A + B*x* + C*x*^{2},\n\nin which *x* is the variable and *y* the variant. We can then select\nfrom the table three values of *y*, and the corresponding values of\n*x*; inserting them in the equation, we obtain three equations by the\nsolution of which we gain the values of A, B, and C. It will be found\nas a general rule that the formula thus obtained yields the other\nnumbers of the table to a considerable degree of approximation.\n\nIn many cases even the second power of the variable will be\nunnecessary; Regnault found that the results of his elaborate inquiry\ninto the latent heat of steam at different pressures were represented\nwith sufficient accuracy by the empirical formula\n\n  \u03bb = 606\u00b75 + 0\u00b7305 *t*,\n\nin which \u03bb is the total heat of the steam, and *t* the\ntemperature.[400] In other cases it may be requisite to include the\nthird power of the variable. Thus physicists assume the law of the\ndilatation of liquids to be of the form\n\n  \u03b4_{t} = *at* + *bt*^{2} + *ct*^{3},\n\n  [400] *Chemical Reports and Memoirs*, Cavendish Society, p.\u00a0294.\n\nand they calculate from results of observation the values of the\nthree constants *a*, *b*, *c*, which are usually small quantities\nnot exceeding one-hundredth part of a unit, but requiring to be\ndetermined with great accuracy.[401] Theoretically speaking, this\nprocess of empirical representation might be applied with any degree of\naccuracy; we might include still higher powers in the formula, and with\nsufficient labour obtain the values of the constants, by using an equal\nnumber of experimental results. The method of least squares may also be\nemployed to obtain the most probable values of the constants.\n\n  [401] Jamin, *Cours de Physique*, vol.\u00a0ii. p.\u00a038.\n\nIn a similar manner all periodic variations may be represented with any\nrequired degree of accuracy by formul\u00e6 involving the sines and cosines\nof angles and their multiples. The form of any tidal or other wave may\nthus be expressed, as Sir G.\u00a0B. Airy has explained.[402] Almost all\nthe phenomena registered by meteorologists are periodic in character,\nand when freed from disturbing causes may be embodied in empirical\nformul\u00e6. Bessel has given a rule by which from any regular series of\nobservations we may, on the principle of the method of least squares,\ncalculate out with a moderate amount of labour a formula expressing the\nvariation of the quantity observed, in the most probable manner. In\nmeteorology three or four terms are usually sufficient for representing\nany periodic phenomenon, but the calculation might be carried to any\nhigher degree of accuracy. As the details of the process have been\ndescribed by Herschel in his treatise on Meteorology,[403] I need not\nfurther enter into them.\n\n  [402] *On Tides and Waves*, Encyclop\u00e6dia Metropolitana, p.\u00a0366*.\n\n  [403] *Encyclop\u00e6dia Britannica*, art. *Meteorology*. Reprint, \u00a7\u00a7\n  152\u2013156.\n\nThe reader might be tempted to think that in these processes of\ncalculation we have an infallible method of discovering inductive\nlaws, and that my previous statements (Chap. VII.) as to the purely\ntentative and inverse character of the inductive process are negatived.\nWere there indeed any general method of inferring laws from facts it\nwould overturn my statement, but it must be carefully observed that\nthese empirical formul\u00e6 do not coincide with natural laws. They are\nonly approximations to the results of natural laws founded upon the\ngeneral principles of approximation. It has already been pointed out\nthat however complicated be the nature of a curve, we may examine so\nsmall a portion of it, or we may examine it with such rude means of\nmeasurement, that its divergence from an elliptic curve will not be\napparent. As a still ruder approximation a portion of a straight line\nwill always serve our purpose; but if we need higher precision a curve\nof the third or fourth degree will almost certainly be sufficient. Now\nempirical formul\u00e6 really represent these approximate curves, but they\ngive us no information as to the precise nature of the curve itself to\nwhich we are approximating. We do not learn what function the variant\nis of the variable, but we obtain another function which, within the\nbounds of observation, gives nearly the same values.\n\n\n*Discovery of Rational Formul\u00e6.*\n\nLet us now proceed to consider the modes in which from numerical\nresults we can establish the actual relation between the quantity of\nthe cause and that of the effect. What we want is a *rational* formula\nor function, which will exhibit the *reason* or exact nature and\norigin of the law in question. There is no word more frequently used\nby mathematicians than the word *function*, and yet it is difficult\nto define its meaning with perfect accuracy. Originally it meant\nperformance or execution, being equivalent to the Greek \u03bb\u03b5\u03b9\u03c4\u03bf\u03c5\u03c1\u03b3\u03af\u03b1\nor \u03c4\u03ad\u03bb\u03b5\u03c3\u03bc\u03b1. Mathematicians at first used it to mean *any power of a\nquantity*, but afterwards generalised it so as to include \u201cany quantity\nformed in any manner whatsoever from another quantity.\u201d[404] Any\nquantity, then, which depends upon and varies with another quantity may\nbe called a function of it, and either may be considered a function of\nthe other.\n\n  [404] Lagrange, *Le\u00e7ons sur le Calcul des Fonctions*, 1806, p.\u00a04.\n\nGiven the quantities, we want the function of which they are the\nvalues. Simple inspection of the numbers cannot as a general rule\ndisclose the function. In an earlier chapter (p.\u00a0124) I put before the\nreader certain numbers, and requested him to point out the law which\nthey obey, and the same question will have to be asked in every case of\nquantitative induction. There are perhaps three methods, more or less\ndistinct, by which we may hope to obtain an answer:\n\n(1) By purely haphazard trial.\n\n(2) By noting the general character of the variation of the quantities,\nand trying by preference functions which give a similar form of\nvariation.\n\n(3) By deducing from previous knowledge the form of the function which\nis most likely to suit.\n\nHaving numerical results we are always at liberty to invent any kind\nof mathematical formula we like, and then try whether, by the suitable\nselection of values for the unknown constant quantities, we can make it\ngive the required results. If ever we fall upon a formula which does\nso, to a fair degree of approximation, there is a presumption in favour\nof its being the true function, although there is no certainty whatever\nin the matter. In this way I discovered a simple mathematical law which\nclosely agreed with the results of my experiments on muscular exertion.\nThis law was afterwards shown by Professor Haughton to be the true\nrational law according to his theory of muscular action.[405]\n\n  [405] Haughton, *Principles of Animal Mechanics*, 1873, pp.\u00a0444\u2013450.\n  Jevons, *Nature*, 30th of June, 1870, vol.\u00a0ii. p.\u00a0158. See also the\n  experiments of Professor Nipher, of Washington University, St. Louis,\n  in *American Journal of Science*, vol.\u00a0ix. p.\u00a0130, vol.\u00a0x. p.\u00a01;\n  *Nature*, vol.\u00a0xi. pp.\u00a0256, 276.\n\nBut the chance of succeeding in this manner is small. The number of\npossible functions is infinite, and even the number of comparatively\nsimple functions is so large that the probability of falling upon the\ncorrect one by mere chance is very slight. Even when we obtain the law\nit is by a deductive process, not by showing that the numbers give the\nlaw, but that the law gives the numbers.\n\nIn the second way, we may, by a survey of the numbers, gain a general\nnotion of the kind of law they are likely to obey, and we may be much\nassisted in this process by drawing them out in the form of a curve.\nWe can in this way ascertain with some probability whether the curve\nis likely to return into itself, or whether it has infinite branches;\nwhether such branches are asymptotic, that is, approach infinitely\ntowards straight lines; whether it is logarithmic in character, or\ntrigonometric. This indeed we can only do if we remember the results of\nprevious investigations. The process is still inversely deductive, and\nconsists in noting what laws give particular curves, and then inferring\ninversely that such curves belong to such laws. If we can in this way\ndiscover the class of functions to which the required law belongs,\nour chances of success are much increased, because our haphazard\ntrials are now reduced within a narrower sphere. But, unless we have\nalmost the whole curve before us, the identification of its character\nmust be a matter of great uncertainty; and if, as in most physical\ninvestigations, we have a mere fragment of the curve, the assistance\ngiven would be quite illusory. Curves of almost any character can be\nmade to approximate to each other for a limited extent, so that it is\nonly by a kind of *divination* that we fall upon the actual function,\nunless we have theoretical knowledge of the kind of function applicable\nto the case.\n\nWhen we have once obtained what we believe to be the correct form of\nfunction, the remainder of the work is mere mathematical computation to\nbe performed infallibly according to fixed rules,[406] which include\nthose employed in the determination of empirical formul\u00e6 (p.\u00a0487).\nThe function will involve two or three or more unknown constants, the\nvalues of which we need to determine by our experimental results.\nSelecting some of our results widely apart and nearly equidistant,\nwe form by means of them as many equations as there are constant\nquantities to be determined. The solution of these equations will then\ngive us the constants required, and having now the actual function we\ncan try whether it gives with sufficient accuracy the remainder of\nour experimental results. If not, we must either make a new selection\nof results to give a new set of equations, and thus obtain a new set\nof values for the constants, or we must acknowledge that our form\nof function has been wrongly chosen. If it appears that the form of\nfunction has been correctly ascertained, we may regard the constants\nas only approximately accurate and may proceed by the Method of Least\nSquares (p.\u00a0393) to determine the most probable values as given by the\nwhole of the experimental results.\n\n  [406] Jamin, *Cours de Physique*, vol.\u00a0ii. p.\u00a050.\n\nIn most cases we shall find ourselves obliged to fall back upon the\nthird mode, that is, anticipation of the form of the law to be expected\non the ground of previous knowledge. Theory and analogical reasoning\nmust be our guides. The general nature of the phenomenon will often\nindicate the kind of law to be looked for. If one form of energy or one\nkind of substance is being converted into another, we may expect the\nlaw of direct simple proportion. In one distinct class of cases the\neffect already produced influences the amount of the ensuing effect,\nas for instance in the cooling of a heated body, when the law will be\nof an exponential form. When the direction of a force influences its\naction, trigonometrical functions enter. Any influence which spreads\nfreely through tridimensional space will be subject to the law of\nthe inverse square of the distance. From such considerations we may\nsometimes arrive deductively and analogically at the general nature of\nthe mathematical law required.\n\n\n*The Graphical Method.*\n\nIn endeavouring to discover the mathematical law obeyed by\nexperimental results it is often desirable to call in the aid of\nspace-representations. Every equation involving two variable quantities\ncorresponds to some kind of plane curve, and every plane curve may\nbe represented symbolically in an equation containing two unknown\nquantities. Now in an experimental research we obtain a number of\nvalues of the variant corresponding to an equal number of values of the\nvariable; but all the numbers are affected by more or less error, and\nthe values of the variable will often be irregularly disposed. Even if\nthe numbers were absolutely correct and disposed at regular intervals,\nthere is, as we have seen, no direct mode of discovering the law, but\nthe difficulty of discovery is much increased by the uncertainty and\nirregularity of the results.\n\nUnder such circumstances, the best mode of proceeding is to prepare a\npaper divided into equal rectangular spaces, a convenient size for the\nspaces being one-tenth of an inch square. The values of the variable\nbeing marked off on the lowest horizontal line, a point is marked for\neach corresponding value of the variant perpendicularly above that of\nthe variable, and at such a height as corresponds to the value of the\nvariant.\n\nThe exact scale of the drawing is not of much importance, but it may\nrequire to be adjusted according to circumstances, and different values\nmust often be attributed to the upright and horizontal divisions, so\nas to make the variations conspicuous but not excessive. If a curved\nline be drawn through all the points or ends of the ordinates, it will\nprobably exhibit irregular inflections, owing to the errors which\naffect the numbers. But, when the results are numerous, it becomes\napparent which results are more divergent than others, and guided by a\nso-called *sense of continuity*, it is possible to trace a line among\nthe points which will approximate to the true law more nearly than the\npoints themselves. The accompanying figure sufficiently explains itself.\n\n[Illustration]\n\nPerkins employed this graphical method with much care in exhibiting\nthe results of his experiments on the compression of water.[407] The\nnumerical results were marked upon a sheet of paper very exactly ruled\nat intervals of one-tenth of an inch, and the original marks were left\nin order that the reader might judge of the correctness of the curve\ndrawn, or choose another for himself. Regnault carried the method to\nperfection by laying off the points with a screw dividing engine;[408]\nand he then formed a table of results by drawing a continuous curve,\nand measuring its height for equidistant values of the variable. Not\nonly does a curve drawn in this manner enable us to infer numerical\nresults more free from accidental errors than any of the numbers\nobtained directly from experiment, but the form of the curve sometimes\nindicates the class of functions to which our results belong.\n\n  [407] *Philosophical Transactions*, 1826, p.\u00a0544.\n\n  [408] Jamin, *Cours de Physique*, vol.\u00a0ii. p.\u00a024, &c.\n\nEngraved sheets of paper prepared for the drawing of curves may be\nobtained from Mr. Stanford at Charing Cross, Messrs. W. and A.\u00a0K.\nJohnston, of London and Edinburgh, Waterlow and Sons, Letts and Co.,\nand probably other publishers. When we do not require great accuracy,\npaper ruled by the common machine-ruler into equal squares of about\none-fifth or one-sixth of an inch square will serve well enough. I\nhave met with engineers\u2019 and surveyors\u2019 memorandum books ruled with\none-twelfth inch squares. When a number of curves have to be drawn, I\nhave found it best to rule a good sheet of drawing paper with lines\ncarefully adjusted at the most convenient distances, and then to\nprick the points of the curve through it upon another sheet fixed\nunderneath. In this way we obtain an accurate curve upon a blank sheet,\nand need only introduce such division lines as are requisite to the\nunderstanding of the curve.\n\nIn some cases our numerical results will correspond, not to the\nheight of single ordinates, but to the area of the curve between two\nordinates, or the average height of ordinates between certain limits.\nIf we measure, for instance, the quantities of heat absorbed by water\nwhen raised in temperature from 0\u00b0 to 5\u00b0, from 5\u00b0 to 10\u00b0, and so on,\nthese quantities will really be represented by *areas* of the curve\ndenoting the specific heat of water; and since the specific heat varies\ncontinuously between every two points of temperature, we shall not get\nthe correct curve by simply laying off the quantities of heat at the\nmean temperatures, namely 2-1/2\u00b0, and 7-1/2\u00b0, and so on. Lord Rayleigh\nhas shown that if we have drawn such an incorrect curve, we can with\nlittle trouble correct it by a simple geometrical process, and obtain\nto a close approximation the true ordinates instead of those denoting\nareas.[409]\n\n  [409] J.\u00a0W. Strutt, *On a correction sometimes required in curves\n  professing to represent the connexion between two physical\n  magnitudes*. Philosophical Magazine, 4th Series, vol.\u00a0xlii. p.\u00a0441.\n\n\n*Interpolation and Extrapolation.*\n\nWhen we have by experiment obtained two or more numerical results,\nand endeavour, without further experiment, to calculate intermediate\nresults, we are said to *interpolate*. If we wish to assign by\nreasoning results lying beyond the limits of experiment, we may be\nsaid, using an expression of Sir George Airy, to *extrapolate*. These\ntwo operations are the same in principle, but differ in practicability.\nIt is a matter of great scientific importance to apprehend precisely\nhow far we can practise interpolation or extrapolation, and on what\ngrounds we proceed.\n\nIn the first place, if the interpolation is to be more than empirical,\nwe must have not only the experimental results, but the laws which they\nobey--we must in fact go through the complete process of scientific\ninvestigation. Having discovered the laws of nature applying to the\ncase, and verified them by showing that they agree with the experiments\nin question, we are then in a position to anticipate the results of\nsimilar experiments. Our knowledge even now is not certain, because we\ncannot completely prove the truth of any assumed law, and we cannot\npossibly exhaust all the circumstances which may affect the result. At\nthe best then our interpolations will partake of the want of certainty\nand precision attaching to all our knowledge of nature. Yet, having the\nsupposed laws, our results will be as sure and accurate as any we can\nattain to. But such a complete procedure is more than we commonly mean\nby interpolation, which usually denotes some method of estimating in a\nmerely approximate manner the results which might have been expected\nindependently of a theoretical investigation.\n\nRegarded in this light, interpolation is in reality an indeterminate\nproblem. From given values of a function it is impossible to determine\nthat function; for we can invent an infinite number of functions which\nwill give those values if we are not restricted by any conditions, just\nas through a given series of points we can draw an infinite number of\ncurves, if we may diverge between or beyond the points into bends and\ncusps as we think fit.[410] In interpolation we must in fact be guided\nmore or less by *\u00e0 priori* considerations; we must know, for instance,\nwhether or not periodical fluctuations are to be expected. Supposing\nthat the phenomenon is non-periodic, we proceed to assume that the\nfunction can be expressed in a limited series of the powers of the\nvariable. The number of powers which can be included depends upon the\nnumber of experimental results available, and must be at least one less\nthan this number. By processes of calculation, which have been already\nalluded to in the section on empirical formul\u00e6, we then calculate the\ncoefficients of the powers, and obtain an empirical formula which will\ngive the required intermediate results. In reality, then, we return\nto the methods treated under the head of approximation and empirical\nformul\u00e6; and interpolation, as commonly understood, consists in\nassuming that a curve of simple character is to pass through certain\ndetermined points. If we have, for instance, two experimental results,\nand only two, we assume that the curve is a straight line; for the\nparabolas which can be passed through two points are infinitely various\nin magnitude, and quite indeterminate. One straight line alone can\npass through two points, and it will have an equation of the form,\n*y*\u00a0=\u00a0*mx*\u00a0+\u00a0*n*, the constant quantities of which can be determined\nfrom two results. Thus, if the two values for *x*, 7 and 11, give\nthe values for *y*, 35 and 53, the solution of two equations gives\n*y*\u00a0=\u00a04\u00b75\u00a0\u00d7\u00a0*x*\u00a0+\u00a03\u00b75 as the equation, and for any other value of *x*,\nfor instance 10, we get a value of *y*, that is 48\u00b75. When we take a\nmean value of *x*, namely 9, this process yields a simple mean result,\nnamely 44. Three experimental results being given, we assume that they\nfall upon a portion of a parabola and algebraic calculation gives the\nposition of any intermediate point upon the parabola. Concerning the\nprocess of interpolation as practised in the science of meteorology\nthe reader will find some directions in the French edition of Ka\u00ebmtz\u2019s\nMeteorology.[411]\n\n  [410] Herschel: Lacroix\u2019 *Differential Calculus*, p.\u00a0551.\n\n  [411] *Cours complet de M\u00e9t\u00e9orologie*, Note A, p.\u00a0449.\n\nWhen we have, either by direct experiment or by the use of a curve,\na series of values of the variant for equidistant values of the\nvariable, it is instructive to take the differences between each\nvalue of the variant and the next, and then the differences between\nthose differences, and so on. If any series of differences approaches\nclosely to zero it is an indication that the numbers may be correctly\nrepresented by a finite empirical formula; if the *n*th differences\nare zero, then the formula will contain only the first *n*\u00a0-\u00a01 powers\nof the variable. Indeed we may sometimes obtain by the calculus of\ndifferences a correct empirical formula; for if *p* be the first term\nof the series of values, and \u0394*p*, \u0394^{2}*p*, \u0394^{3}*p*, be the first\nnumber in each column of differences, then the *m*th term of the series\nof values will be\n\n  *p*\u00a0+\u00a0*m*\u0394*p*\u00a0+\u00a0*m*[(*m*\u00a0-\u00a01)/2]\u0394^{2}*p*\u00a0+\n  *m*[(*m*\u00a0-\u00a01)/2][(*m*\u00a0-\u00a02)/3]\u0394^{3}*p*\u00a0+\u00a0&c.\n\nA closely equivalent but more practicable formula for interpolation\nby differences, as devised by Lagrange, will be found in Thomson and\nTait\u2019s *Elements of Natural Philosophy*, p.\u00a0115.\n\nIf no column of differences shows any tendency to become zero\nthroughout, it is an indication that the law is of a more complicated,\nfor instance of an exponential character, so that it requires different\ntreatment. Dr. J. Hopkinson has suggested a method of arithmetical\ninterpolation,[412] which is intended to avoid much that is arbitrary\nin the graphical method. His process will yield the same results in all\nhands.\n\n  [412] *On the Calculation of Empirical Formul\u00e6. The Messenger of\n  Mathematics*, New Series, No. 17, 1872.\n\nSo far as we can infer the results likely to be obtained by variations\nbeyond the limits of experiment, we must proceed upon the same\nprinciples. If possible we must detect the exact laws in action,\nand then trust to them as a guide when we have no experience. If\nnot, an empirical formula of the same character as those employed in\ninterpolation is our only resource. But to extend our inference far\nbeyond the limits of experience is exceedingly unsafe. Our knowledge is\nat the best only approximate, and takes no account of small tendencies.\nNow it usually happens that tendencies small within our limits of\nobservation become perceptible or great under extreme circumstances.\nWhen the variable in our empirical formula is small, we are justified\nin overlooking the higher powers, and taking only two or three lower\npowers. But as the variable increases, the higher powers gain in\nimportance, and in time yield the principal part of the value of the\nfunction.\n\nThis is no mere theoretical inference. Excepting the few primary\nlaws of nature, such as the law of gravity, of the conservation of\nenergy, &c., there is hardly any natural law which we can trust in\ncircumstances widely different from those with which we are practically\nacquainted. From the expansion or contraction, fusion or vaporisation\nof substances by heat at the surface of the earth, we can form a most\nimperfect notion of what would happen near the centre of the earth,\nwhere the pressure almost infinitely exceeds anything possible in our\nexperiments. The physics of the earth give us a feeble, and probably a\nmisleading, notion of a body like the sun, in which an inconceivably\nhigh temperature is united with an inconceivably high pressure. If\nthere are in the realms of space nebul\u00e6 consisting of incandescent\nand unoxidised vapours of metals and other elements, so highly heated\nperhaps that chemical composition is out of the question, we are\nhardly able to treat them as subjects of scientific inference. Hence\narises the great importance of experiments in which we investigate the\nproperties of substances under extreme circumstances of cold or heat,\ndensity or rarity, intense electric excitation, &c. This insecurity\nin extending our inferences arises from the approximate character\nof our measurements. Had we the power of appreciating infinitely\nsmall quantities, we should by the principle of continuity discover\nsome trace of every change which a substance could undergo under\nunattainable circumstances. By observing, for instance, the tension of\naqueous vapour between 0\u00b0 and 100\u00b0\u00a0C., we ought theoretically to be\nable to infer its tension at every other temperature; but this is out\nof the question practically because we cannot really ascertain the law\nprecisely between those temperatures.\n\nMany instances might be given to show that laws which appear to\nrepresent correctly the results of experiments within certain limits\naltogether fail beyond those limits. The experiments of Roscoe and\nDittmar, on the absorption of gases in water[413] afford interesting\nillustrations, especially in the case of hydrochloric acid, the\nquantity of which dissolved in water under different pressures\nfollows very closely a linear law of variation, from which however it\ndiverges widely at low pressures.[414] Herschel, having deduced from\nobservations of the double star \u03b3 Virginis an elliptic orbit for the\nmotion of one component round the centre of gravity of both, found that\nfor a time the motion of the star agreed very well with this orbit.\nNevertheless divergence began to appear and after a time became so\ngreat that an entirely new orbit, of more than double the dimensions of\nthe old one, had ultimately to be adopted.[415]\n\n  [413] Watts\u2019 *Dictionary of Chemistry*, vol.\u00a0ii. p.\u00a0790.\n\n  [414] *Quarterly Journal of the Chemical Society*, vol.\u00a0viii. p.\u00a015.\n\n  [415] *Results of Observations at the Cape of Good Hope*, p.\u00a0293.\n\n\n*Illustrations of Empirical Quantitative Laws.*\n\nAlthough our object in quantitative inquiry is to discover the exact or\nrational formul\u00e6, expressing the laws which apply to the subject, it\nis instructive to observe in how many important branches of science,\nno precise laws have yet been detected. The tension of aqueous vapour\nat different temperatures has been determined by a succession of\neminent experimentalists--Dalton, Ka\u00ebmtz, Dulong, Arago, Magnus, and\nRegnault--and by the last mentioned the measurements were conducted\nwith extraordinary care. Yet no incontestable general law has been\nestablished. Several functions have been proposed to express the\nelastic force of the vapour as depending on the temperature. The first\nform is that of Young, namely F\u00a0=\u00a0(*a*\u00a0+\u00a0*b t*)^{m}, in which *a*,\n*b*, and *m* are unknown quantities to be determined by observation.\nRoche proposed, on theoretical grounds, a complicated formula of an\nexponential form, and a third form of function is that of Biot,[416] as\nfollows--log\u00a0F\u00a0=\u00a0*a*\u00a0+\u00a0*b*\u03b1^{t}\u00a0+\u00a0*c*\u03b2^{t}. I mention these formul\u00e6,\nbecause they well illustrate the feeble powers of empirical inquiry.\nNone of the formul\u00e6 can be made to correspond closely with experimental\nresults, and the two last forms correspond almost equally well. There\nis very little probability that the real law has been reached, and\nit is unlikely that it will be discovered except by deduction from\nmechanical theory.\n\n  [416] Jamin, *Cours de Physique*, vol.\u00a0ii. p.\u00a0138.\n\nMuch ingenious labour has been spent upon the discovery of some general\nlaw of atmospheric refraction. Tycho Brahe and Kepler commenced the\ninquiry: Cassini first formed a table of refractions, calculated on\ntheoretical grounds: Newton entered into some profound investigations\nupon the subject: Brooke Taylor, Bouguer, Simpson, Bradley, Mayer,\nand Kramp successively attacked the question, which is of the highest\npractical importance as regards the correction of astronomical\nobservations. Laplace next laboured on the subject without exhausting\nit, and Brinkley and Ivory have also treated it. The true law is\nyet undiscovered. A closely connected problem, that regarding the\nrelation between the pressure and elevation in different strata of\nthe atmosphere, has received the attention of a long succession of\nphysicists and was most carefully investigated by Laplace. Yet no\ninvariable and general law has been detected. The same may be said\nconcerning the law of human mortality; abundant statistics on this\nsubject are available, and many hypotheses more or less satisfactory\nhave been put forward as to the form of the curve of mortality, but it\nseems to be impossible to discover more than an approximate law.\n\nIt may perhaps be urged that in such subjects no single invariable\nlaw can be expected. The atmosphere may be divided into several\nvariable strata which by their unconnected changes frustrate the exact\ncalculations of astronomers. Human life may be subject at different\nages to a succession of different influences incapable of reduction\nunder any one law. The results observed may in fact be aggregates of an\nimmense number of separate results each governed by its own separate\nlaws, so that the subjects may be complicated beyond the possibility of\ncomplete resolution by empirical methods. This is certainly true of the\nmathematical functions which must some time or other be introduced into\nthe science of political economy.\n\n\n*Simple Proportional Variation.*\n\nWhen we first treat numerical results in any novel kind of\ninvestigation, our impression will probably be that one quantity\nvaries in *simple proportion* to another, so as to obey the law\n*y*\u00a0=\u00a0*mx*\u00a0+\u00a0*n*. We must learn to distinguish carefully between the\ncases where this proportionality is really, and where it is only\napparently true. In considering the principles of approximation we\nfound that a small portion of any curve will appear to be a straight\nline. When our modes of measurement are comparatively rude, we must\nexpect to be unable to detect the curvature. Kepler made meritorious\nattempts to discover the law of refraction, and he approximated to it\nwhen he observed that the angles of incidence and refraction *if small*\nbear a constant ratio to each other. Angles when small are nearly as\ntheir sines, so that he reached an approximate result of the true law.\nCardan assumed, probably as a mere guess, that the force required to\nsustain a body on an inclined plane was simply proportional to the\nangle of elevation of the plane. This is approximately the case when\nthe angle is small, but in reality the law is much more complicated,\nthe power required being proportional to the sine of the angle. The\nearly thermometer-makers were unaware whether the expansion of mercury\nwas proportional or not to the heat communicated to it, and it is only\nin the present century that we have learnt it to be not so. We now\nknow that even gases obey the law of uniform expansion by heat only\nin an approximate manner. Until reason to the contrary is shown, we\nshould do well to look upon every law of simple proportion as only\nprovisionally true.\n\nNevertheless many important laws of nature are in the form of simple\nproportions. Wherever a cause acts in independence of its previous\neffects, we may expect this relation. An accelerating force acts\nequally upon a moving and a motionless body. Hence the velocity\nproduced is in simple proportion to the force, and to the duration of\nits uniform action. As gravitating bodies never interfere with each\nother\u2019s gravity, this force is in direct simple proportion to the\nmass of each of the attracting bodies, the mass being measured by, or\nproportional to inertia. Similarly, in all cases of \u201cdirect unimpeded\naction,\u201d as Herschel has remarked,[417] we may expect simple proportion\nto manifest itself. In such cases the equation expressing the relation\nmay have the simple form *y*\u00a0=\u00a0*mx*.\n\n  [417] *Preliminary Discourse*, &c., p.\u00a0152.\n\nA similar relation holds true when there is conversion of one substance\nor form of energy into another. The quantity of a compound is equal\nto the quantity of the elements which combine. The heat produced in\nfriction is exactly proportional to the mechanical energy absorbed.\nIt was experimentally proved by Faraday that \u201cthe chemical power of\nthe current of electricity is in direct proportion to the quantity of\nelectricity which passes.\u201d When an electric current is produced, the\nquantity of electric energy is simply proportional to the weight of\nmetal dissolved. If electricity is turned into heat, there is again\nsimple proportion. Wherever, in fact, one thing is but another thing\nwith a new aspect, we may expect to find the law of simple proportion.\nBut it is only in the most elementary cases that this simple relation\nwill hold true. Simple conditions do not, generally speaking, produce\nsimple results. The planets move in approximate circles round the sun,\nbut the apparent motions, as seen from the earth, are very various. All\nthose motions, again, are summed up in the law of gravity, of no great\ncomplexity; yet men never have been, and never will be, able to exhaust\nthe complications of action and reaction arising from that law, even\namong a small number of planets. We should be on our guard against a\ntendency to assume that the connection of cause and effect is one of\ndirect proportion. Bacon reminds us of the woman in \u00c6sop\u2019s fable, who\nexpected that her hen, with a double measure of barley, would lay two\neggs a day instead of one, whereas it grew fat, and ceased to lay any\neggs at all. It is a wise maxim that the half is often better than the\nwhole.\n\n\n\n\nCHAPTER XXIII.\n\nTHE USE OF HYPOTHESIS.\n\n\nIf the views upheld in this work be correct, all inductive\ninvestigation consists in the marriage of hypothesis and experiment.\nWhen facts are in our possession, we frame an hypothesis to explain\ntheir relations, and by the success of this explanation is the value\nof the hypothesis to be judged. In the invention and treatment of\nsuch hypotheses, we must avail ourselves of the whole body of science\nalready accumulated, and when once we have obtained a probable\nhypothesis, we must not rest until we have verified it by comparison\nwith new facts. We must endeavour by deductive reasoning to anticipate\nsuch phenomena, especially those of a singular and exceptional nature,\nas would happen if the hypothesis be true. Out of the infinite number\nof experiments which are possible, theory must lead us to select those\ncritical ones which are suitable for confirming or negativing our\nanticipations.\n\nThis work of inductive investigation cannot be guided by any system of\nprecise and infallible rules, like those of deductive reasoning. There\nis, in fact, nothing to which we can apply rules of method, because\nthe laws of nature must be in our possession before we can treat them.\nIf there were any rule of inductive method, it would direct us to make\nan exhaustive arrangement of facts in all possible orders. Given the\nspecimens in a museum, we might arrive at the best classification by\ngoing systematically through all possible classifications, and, were\nwe endowed with infinite time and patience, this would be an effective\nmethod. It is the method by which the first simple steps are taken in\nan incipient branch of science. Before the dignified name of science is\napplicable, some coincidences will force themselves upon the attention.\nBefore there was a science of meteorology observant persons learned\nto associate clearness of the atmosphere with coming rain, and a\ncolourless sunset with fine weather. Knowledge of this kind is called\n*empirical*, as seeming to come directly from experience; and there is\na considerable portion of knowledge which bears this character.\n\nWe may be obliged to trust to the casual detection of coincidences\nin those branches of knowledge where we are deprived of the aid of\nany guiding notions; but a little reflection will show the utter\ninsufficiency of haphazard experiment, when applied to investigations\nof a complicated nature. At the best, it will be the simple identity,\nor partial identity, of classes, as illustrated in pages 127 or 134,\nwhich can be thus detected. It was pointed out that, even when a law\nof nature involves only two circumstances, and there are one hundred\ndistinct circumstances which may possibly be connected, there will be\nno less than 4,950 pairs of circumstances between which coincidence may\nexist. When a law involves three or more circumstances, the possible\nnumber of relations becomes vastly greater. When considering the\nsubject of combinations and permutations, it became apparent that we\ncould never cope with the possible variety of nature. An exhaustive\nexamination of the possible metallic alloys, or chemical compounds, was\nfound to be out of the question (p.\u00a0191).\n\nIt is on such considerations that we can explain the very small\nadditions made to our knowledge by the alchemists. Many of them were\nmen of the greatest acuteness, and their indefatigable labours were\npursued through many centuries. A few things were discovered by them,\nbut a true insight into nature, now enables chemists to discover more\nuseful facts in a year than were yielded by the alchemists during\nmany centuries. There can be no doubt that Newton was an alchemist,\nand that he often laboured night and day at alchemical experiments.\nBut in trying to discover the secret by which gross metals might be\nrendered noble, his lofty powers of deductive investigation were wholly\nuseless. Deprived of all guiding clues, his experiments were like\nthose of all the alchemists, purely tentative and haphazard. While his\nhypothetical and deductive investigations have given us the true system\nof the Universe, and opened the way in almost all the great branches of\nnatural philosophy, the whole results of his tentative experiments are\ncomprehended in a few happy guesses, given in his celebrated \u201cQueries.\u201d\n\nEven when we are engaged in apparently passive observation of a\nphenomenon, which we cannot modify experimentally, it is advantageous\nthat our attention should be guided by theoretical anticipations. A\nphenomenon which seems simple is, in all probability, really complex,\nand unless the mind is actively engaged in looking for particular\ndetails, it is likely that the critical circumstances will be passed\nover. Bessel regretted that no distinct theory of the constitution\nof comets had guided his observations of Halley\u2019s comet;[418] in\nattempting to verify or refute a hypothesis, not only would there be a\nchance of establishing a true theory, but if confuted, the confutation\nwould involve a store of useful observations.\n\n  [418] Tyndall, *On Cometary Theory*, Philosophical Magazine, April\n  1869. 4th Series, vol.\u00a0xxxvii. p.\u00a0243.\n\nIt would be an interesting work, but one which I cannot undertake, to\ntrace out the gradual reaction which has taken place in recent times\nagainst the purely empirical or Baconian theory of induction. Francis\nBacon, seeing the futility of the scholastic logic, which had long\nbeen predominant, asserted that the accumulation of facts and the\norderly abstraction of axioms, or general laws from them, constituted\nthe true method of induction. Even Bacon was not wholly unaware of\nthe value of hypothetical anticipation. In one or two places he\nincidentally acknowledges it, as when he remarks that the subtlety of\nnature surpasses that of reason, adding that \u201caxioms abstracted from\nparticular facts in a careful and orderly manner, readily suggest and\nmark out new particulars.\u201d\n\nNevertheless Bacon\u2019s method, as far as we can gather the meaning of\nthe main portions of his writings, would correspond to the process of\nempirically collecting facts and exhaustively classifying them, to\nwhich I alluded. The value of this method may be estimated historically\nby the fact that it has not been followed by any of the great masters\nof science. Whether we look to Galileo, who preceded Bacon, to Gilbert,\nhis contemporary, or to Newton and Descartes, Leibnitz and Huyghens,\nhis successors, we find that discovery was achieved by the opposite\nmethod to that advocated by Bacon. Throughout Newton\u2019s works, as\nI shall show, we find deductive reasoning wholly predominant, and\nexperiments are employed, as they should be, to confirm or refute\nhypothetical anticipations of nature. In my \u201cElementary Lessons in\nLogic\u201d (p.\u00a0258), I stated my belief that there was no kind of reference\nto Bacon in Newton\u2019s works. I have since found that Newton does once or\ntwice employ the expression *experimentum crucis* in his \u201cOpticks,\u201d but\nthis is the only expression, so far as I am aware, which could indicate\non the part of Newton direct or indirect acquaintance with Bacon\u2019s\nwritings.[419]\n\n  [419] See *Philosophical Transactions*, abridged by Lowthorp. 4th\n  edit. vol.\u00a0i. p.\u00a0130. I find that opinions similar to those in the\n  text have been briefly expressed by De Morgan in his remarkable\n  preface to *From Matter to Spirit*, by C.D., pp. xxi. xxii.\n\nOther great physicists of the same age were equally prone to the use\nof hypotheses rather than the blind accumulation of facts in the\nBaconian manner. Hooke emphatically asserts in his posthumous work\non Philosophical Method, that the first requisite of the Natural\nPhilosopher is readiness at guessing the solution of phenomena and\nmaking queries. \u201cHe ought to be very well skilled in those several\nkinds of philosophy already known, to understand their several\nhypotheses, suppositions, collections, observations, &c., their various\nways of ratiocinations and proceedings, the several failings and\ndefects, both in their way of raising and in their way of managing\ntheir several theories: for by this means the mind will be somewhat\nmore ready at guessing at the solution of many phenomena almost at\nfirst sight, and thereby be much more prompt at making queries, and at\ntracing the subtlety of Nature, and in discovering and searching into\nthe true reason of things.\u201d\n\nWe find Horrocks, again, than whom no one was more filled with the\nscientific spirit, telling us how he tried theory after theory in order\nto discover one which was in accordance with the motions of Mars.[420]\nHuyghens, who possessed one of the most perfect philosophical\nintellects, followed the deductive process combined with continual\nappeal to experiment, with a skill closely analogous to that of Newton.\nAs to Descartes and Leibnitz, they fell into excess in the use of\nhypothesis, since they sometimes adopted hypothetical reasoning to\nthe exclusion of experimental verification. Throughout the eighteenth\ncentury science was supposed to be advancing by the pursuance of the\nBaconian method, but in reality hypothetical investigation was the\nmain instrument of progress. It is only in the present century that\nphysicists began to recognise this truth. So much opprobrium had been\nattached by Bacon to the use of hypotheses, that we find Young speaking\nof them in an apologetic tone. \u201cThe practice of advancing general\nprinciples and applying them to particular instances is so far from\nbeing fatal to truth in all sciences, that when those principles are\nadvanced on sufficient grounds, it constitutes the essence of true\nphilosophy;\u201d[421] and he quotes cases in which Davy trusted to his\ntheories rather than his experiments.\n\n  [420] Horrocks, *Opera Posthuma* (1673), p.\u00a0276.\n\n  [421] Young\u2019s *Works*, vol.\u00a0i. p.\u00a0593.\n\nHerschel, who was both a practical physicist and an abstract logician,\nentertained the deepest respect for Bacon, and made the \u201cNovum\nOrganum\u201d as far as possible the basis of his own admirable *Discourse\non the Study of Natural Philosophy*. Yet we find him in Chapter VII.\nrecognising the part which the formation and verification of theories\ntakes in the higher and more general investigations of physical\nscience. J.\u00a0S. Mill carried on the reaction by describing the Deductive\nMethod in which ratiocination, that is deductive reasoning, is employed\nfor the discovery of new opportunities of testing and verifying an\nhypothesis. Nevertheless throughout the other parts of his system\nhe inveighed against the value of the deductive process, and even\nasserted that empirical inference from particulars to particulars is\nthe true type of reasoning. The irony of fate will probably decide\nthat the most original and valuable part of Mill\u2019s System of Logic is\nirreconcilable with those views of the syllogism and of the nature of\ninference which occupy the main part of the treatise, and are said to\nhave effected a revolution in logical science. Mill would have been\nsaved from much confusion of thought had he not failed to observe that\nthe inverse use of deduction constitutes induction. In later years\nProfessor Huxley has strongly insisted upon the value of hypothesis.\nWhen he advocates the use of \u201cworking hypotheses\u201d he means no doubt\nthat any hypothesis is better that none, and that we cannot avoid being\nguided in our observations by some hypothesis or other. Professor\nTyndall\u2019s views as to the use of the Imagination in the pursuit of\nScience put the same truth in another light.\n\nIt ought to be pointed out that Neil in his *Art of Reasoning*, a\npopular but able exposition of the principles of Logic, published\nin 1853, fully recognises in Chapter XI. the value and position of\nhypothesis in the discovery of truth. He endeavours to show, too\n(p.\u00a0109), that Francis Bacon did not object to the use of hypothesis.\n\nThe true course of inductive procedure is that which has yielded\nall the more lofty results of science. It consists in *Anticipating\nNature*, in the sense of forming hypotheses as to the laws which are\nprobably in operation; and then observing whether the combinations\nof phenomena are such as would follow from the laws supposed. The\ninvestigator begins with facts and ends with them. He uses facts to\nsuggest probable hypotheses; deducing other facts which would happen if\na particular hypothesis is true, he proceeds to test the truth of his\nnotion by fresh observations. If any result prove different from what\nhe expects, it leads him to modify or to abandon his hypothesis; but\nevery new fact may give some new suggestion as to the laws in action.\nEven if the result in any case agrees with his anticipations, he does\nnot regard it as finally confirmatory of his theory, but proceeds to\ntest the truth of the theory by new deductions and new trials.\n\nIn such a process the investigator is assisted by the whole body of\nscience previously accumulated. He may employ analogy, as I shall\npoint out, to guide him in the choice of hypotheses. The manifold\nconnections between one science and another give him clues to the kind\nof laws to be expected, and out of the infinite number of possible\nhypotheses he selects those which are, as far as can be foreseen at the\nmoment, most probable. Each experiment, therefore, which he performs\nis that most likely to throw light upon his subject, and even if it\nfrustrate his first views, it tends to put him in possession of the\ncorrect clue.\n\n\n*Requisites of a good Hypothesis.*\n\nThere is little difficulty in pointing out to what condition an\nhypothesis must conform in order to be accepted as probable and\nvalid. That condition, as I conceive, is the single one of enabling\nus to infer the existence of phenomena which occur in our experience.\n*Agreement with fact is the sole and sufficient test of a true\nhypothesis.*\n\nHobbes has named two conditions which he considers requisite in an\nhypothesis, namely (1) That it should be conceivable and not absurd;\n(2) That it should allow of phenomena being necessarily inferred.\nBoyle, in noticing Hobbes\u2019 views, proposed to add a third condition,\nto the effect that the hypothesis should not be inconsistent with any\nother truth on phenomenon of nature.[422] I think that of these three\nconditions, the first cannot be accepted, unless by *inconceivable* and\n*absurd* we mean self-contradictory or inconsistent with the laws of\nthought and nature. I shall have to point out that some satisfactory\ntheories involve suppositions which are wholly *inconceivable* in\na certain sense of the word, because the mind cannot sufficiently\nextend its ideas to frame a notion of the actions supposed to take\nplace. That the force of gravity should act instantaneously between\nthe most distant parts of the planetary system, or that a ray of\nviolet light should consist of about 700 billions of vibrations in a\nsecond, are statements of an inconceivable and absurd character in\none sense; but they are so far from being opposed to fact that we\ncannot on any other suppositions account for phenomena observed. But\nif an hypothesis involve self-contradiction, or is inconsistent with\nknown laws of nature, it is self-condemned. We cannot even apply\ndeductive reasoning to a self-contradictory notion; and being opposed\nto the most general and certain laws known to us, the primary laws of\nthought, it thereby conspicuously fails to agree with facts. Since\nnature, again, is never self-contradictory, we cannot at the same time\naccept two theories which lead to contradictory results. If the one\nagrees with nature, the other cannot. Hence if there be a law which we\nbelieve with high probability to be verified by observation, we must\nnot frame an hypothesis in conflict with it, otherwise the hypothesis\nwill necessarily be in disagreement with observation. Since no law or\nhypothesis is proved, indeed, with absolute certainty, there is always\na chance, however slight, that the new hypothesis may displace the\nold one; but the greater the probability which we assign to that old\nhypothesis, the greater must be the evidence required in favour of the\nnew and conflicting one.\n\n  [422] Boyle\u2019s *Physical Examen*, p.\u00a084.\n\nI assert, then, that there is but one test of a good hypothesis,\nnamely, *its conformity with observed facts*; but this condition may\nbe said to involve three constituent conditions, nearly equivalent to\nthose suggested by Hobbes and Boyle, namely:--\n\n(1) That it allow of the application of deductive reasoning and the\ninference of consequences capable of comparison with the results of\nobservation.\n\n(2) That it do not conflict with any laws of nature, or of mind, which\nwe hold to be true.\n\n(3) That the consequences inferred do agree with facts of observation.\n\n\n*Possibility of Deductive Reasoning.*\n\nAs the truth of an hypothesis is to be proved by its conformity with\nfact, the first condition is that we be able to apply methods of\ndeductive reasoning, and learn what should happen according to such\nan hypothesis. Even if we could imagine an object acting according to\nlaws hitherto wholly unknown it would be useless to do so, because we\ncould never decide whether it existed or not. We can only infer what\nwould happen under supposed conditions by applying the knowledge of\nnature we possess to those conditions. Hence, as Boscovich truly said,\nwe are to understand by hypotheses \u201cnot fictions altogether arbitrary,\nbut suppositions conformable to experience or analogy.\u201d It follows that\nevery hypothesis worthy of consideration must suggest some likeness,\nanalogy, or common law, acting in two or more things. If, in order to\nexplain certain facts, *a*, *a\u2032*, *a\u2033*, &c., we invent a cause A, then\nwe must in some degree appeal to experience as to the mode in which A\nwill act. As the laws of nature are not known to the mind intuitively,\nwe must point out some other cause, B, which supplies the requisite\nnotions, and all we do is to invent a fourth term to an analogy. As B\nis to its effects *b*, *b\u2032*, *b\u2033*, &c., so is A to its effects *a*,\n*a\u2032*, *a\u2033*, &c. When we attempt to explain the passage of light and\nheat radiations through space unoccupied by matter, we imagine the\nexistence of the so-called *ether*. But if this ether were wholly\ndifferent from anything else known to us, we should in vain try to\nreason about it. We must apply to it at least the laws of motion, that\nis we must so far liken it to matter. And as, when applying those laws\nto the elastic medium air, we are able to infer the phenomena of sound,\nso by arguing in a similar manner concerning ether we are able to infer\nthe existence of light phenomena corresponding to what do occur. All\nthat we do is to take an elastic substance, increase its elasticity\nimmensely, and denude it of gravity and some other properties of\nmatter, but we must retain sufficient likeness to matter to allow of\ndeductive calculations.\n\nThe force of gravity is in some respects an incomprehensible existence,\nbut in other respects entirely conformable to experience. We observe\nthat the force is proportional to mass, and that it acts in entire\nindependence of other matter which may be present or intervening.\nThe law of the decrease of intensity, as the square of the distance\nincreases, is observed to hold true of light, sound, and other\ninfluences emanating from a point, and spreading uniformly through\nspace. The law is doubtless connected with the properties of space, and\nis so far in agreement with our necessary ideas.\n\nIt may be said, however, that no hypothesis can be so much as framed\nin the mind unless it be more or less conformable to experience. As\nthe material of our ideas is derived from sensation we cannot figure\nto ourselves any agent, but as endowed with some of the properties of\nmatter. All that the mind can do in the creation of new existences is\nto alter combinations, or the intensity of sensuous properties. The\nphenomenon of motion is familiar to sight and touch, and different\ndegrees of rapidity are also familiar; we can pass beyond the limits of\nsense, and imagine the existence of rapid motion, such as our senses\ncould not observe. We know what is elasticity, and we can therefore\nin a way figure to ourselves elasticity a thousand or a million times\ngreater than any which is sensuously known to us. The waves of the\nocean are many times higher than our own bodies; other waves, are\nmany times less; continue the proportion, and we ultimately arrive\nat waves as small as those of light. Thus it is that the powers of\nmind enable us from a sensuous basis to reason concerning agents and\nphenomena different in an unlimited degree. If no hypothesis then can\nbe absolutely opposed to sense, accordance with experience must always\nbe a question of degree.\n\nIn order that an hypothesis may allow of satisfactory comparison with\nexperience, it must possess definiteness and in many cases mathematical\nexactness allowing of the precise calculation of results. We must\nbe able to ascertain whether it does or does not agree with facts.\nThe theory of vortices is an instance to the contrary, for it did\nnot present any mode of calculating the exact relations between the\ndistances and periods of the planets and satellites; it could not,\ntherefore, undergo that rigorous testing to which Newton scrupulously\nsubmitted his theory of gravity before its promulgation. Vagueness\nand incapability of precise proof or disproof often enable a false\ntheory to live; but with those who love truth, vagueness should\nexcite suspicion. The upholders of the ancient doctrine of Nature\u2019s\nabhorrence of a vacuum, had been unable to anticipate the important\nfact that water would not rise more than 33 feet in a common suction\npump. Nor when the fact was pointed out could they explain it, except\nby introducing a special alteration of the theory to the effect that\nNature\u2019s abhorrence of a vacuum was limited to 33 feet.\n\n\n*Consistency with the Laws of Nature.*\n\nIn the second place an hypothesis must not be contradictory to\nwhat we believe to be true concerning Nature. It must not involve\nself-inconsistency which is opposed to the highest and simplest laws,\nnamely, those of Logic. Neither ought it to be irreconcilable with\nthe simple laws of motion, of gravity, of the conservation of energy,\nnor any parts of physical science which we consider to be established\nbeyond reasonable doubt. Not that we are absolutely forbidden to\nentertain such an hypothesis, but if we do so we must be prepared to\ndisprove some of the best demonstrated truths in the possession of\nmankind. The fact that conflict exists means that the consequences of\nthe theory are not verified if previous discoveries are correct, and we\nmust therefore show that previous discoveries are incorrect before we\ncan verify our theory.\n\nAn hypothesis will be exceedingly improbable, not to say absurd, if\nit supposes a substance to act in a manner unknown in other cases;\nfor it then fails to be verified in our knowledge of that substance.\nSeveral physicists, especially Euler and Grove, have supposed that\nwe might dispense with an ethereal basis of light, and infer from\nthe interstellar passage of rays that there was a kind of rare gas\noccupying space. But if so, that gas must be excessively rare, as we\nmay infer from the apparent absence of an atmosphere around the moon,\nand from other facts known to us concerning gases and the atmosphere;\nyet it must possess an elastic force at least a billion times as great\nas atmospheric air at the earth\u2019s surface, in order to account for the\nextreme rapidity of light rays. Such an hypothesis then is inconsistent\nwith our knowledge concerning gases.\n\nProvided that there be no clear and absolute conflict with known\nlaws of nature, there is no hypothesis so improbable or apparently\ninconceivable that it may not be rendered probable, or even\napproximately certain, by a sufficient number of concordances. In\nfact the two best founded and most successful theories in physical\nscience involve the most absurd suppositions. Gravity is a force which\nappears to act between bodies through vacuous space; it is in positive\ncontradiction to the old dictum that nothing can act but through\nsome medium. It is even more puzzling that the force acts in perfect\nindifference to intervening obstacles. Light in spite of its extreme\nvelocity shows much respect to matter, for it is almost instantaneously\nstopped by opaque substances, and to a considerable extent absorbed\nand deflected by transparent ones. But to gravity all media are, as\nit were, absolutely transparent, nay non-existent; and two particles\nat opposite points of the earth affect each other exactly as if the\nglobe were not between. The action is, so far as we can observe,\ninstantaneous, so that every particle of the universe is at every\nmoment in separate cognisance, as it were, of the relative position of\nevery other particle throughout the universe at that same moment of\ntime. Compared with such incomprehensible conditions, the theory of\nvortices deals with commonplace realities. Newton\u2019s celebrated saying\n*hypotheses non fingo*, bears the appearance of irony; and it was not\nwithout apparent grounds that Leibnitz and the continental philosophers\ncharged Newton with re-introducing occult powers and qualities.\n\nThe undulatory theory of light presents almost equal difficulties\nof conception. We are asked by physical philosophers to give up our\nprepossessions, and to believe that interstellar space which seems\nempty is not empty at all, but filled with *something* immensely\nmore solid and elastic than steel. As Young himself remarked,[423]\n\u201cthe luminiferous ether, pervading all space, and penetrating almost\nall substances, is not only highly elastic, but absolutely solid!!!\u201d\nHerschel calculated the force which may be supposed, according to the\nundulatory theory of light, to be constantly exerted at each point in\nspace, and finds it to be 1,148,000,000,000 times the elastic force\nof ordinary air at the earth\u2019s surface, so that the pressure of ether\nper square inch must be about seventeen billions of pounds.[424] Yet\nwe live and move without appreciable resistance through this medium,\nimmensely harder and more elastic than adamant. All our ordinary\nnotions must be laid aside in contemplating such an hypothesis; yet it\nis no more than the observed phenomena of light and heat force us to\naccept. We cannot deny even the strange suggestion of Young, that there\nmay be independent worlds, some possibly existing in different parts\nof space, but others perhaps pervading each other unseen and unknown\nin the same space.[425] For if we are bound to admit the conception\nof this adamantine firmament, it is equally easy to admit a plurality\nof such. We see, then, that mere difficulties of conception must not\ndiscredit a theory which otherwise agrees with facts, and we must only\nreject hypotheses which are inconceivable in the sense of breaking\ndistinctly the primary laws of thought and nature.\n\n  [423] Young\u2019s *Works*, vol.\u00a0i. p.\u00a0415.\n\n  [424] *Familiar Lectures on Scientific Subjects*, p.\u00a0282.\n\n  [425] Young\u2019s *Works*, vol.\u00a0i. p.\u00a0417.\n\n\n*Conformity with Facts.*\n\nBefore we accept a new hypothesis it must be shown to agree not only\nwith the previously known laws of nature, but also with the particular\nfacts which it is framed to explain. Assuming that these facts are\nproperly established, it must agree with all of them. A single absolute\nconflict between fact and hypothesis, is fatal to the hypothesis;\n*falsa in uno, falsa in omnibus*.\n\nSeldom, indeed, shall we have a theory free from difficulties and\napparent inconsistency with facts. Though one real inconsistency\nwould overturn the most plausible theory, yet there is usually some\nprobability that the fact may be misinterpreted, or that some supposed\nlaw of nature, on which we are relying, may not be true. It may be\nexpected, moreover, that a good hypothesis, besides agreeing with\nfacts already noticed, will furnish us with distinct credentials by\nenabling us to anticipate deductively series of facts which are not\nalready connected and accounted for by any equally probable hypothesis.\nWe cannot lay down any precise rule as to the number of accordances\nwhich can establish the truth of an hypothesis, because the accordances\nwill vary much in value. While, on the one hand, no finite number\nof accordances will give entire certainty, the probability of the\nhypothesis will increase very rapidly with the number of accordances.\nAlmost every problem in science thus takes the form of a balance of\nprobabilities. It is only when difficulty after difficulty has been\nsuccessfully explained away, and decisive *experimenta crucis* have,\ntime after time, resulted in favour of our theory, that we can venture\nto assert the falsity of all objections.\n\nThe sole real test of an hypothesis is its accordance with fact.\nDescartes\u2019 celebrated system of vortices is exploded, not because it\nwas intrinsically absurd and inconceivable, but because it could not\ngive results in accordance with the actual motions of the heavenly\nbodies. The difficulties of conception involved in the apparatus of\nvortices, are child\u2019s play compared with those of gravitation and\nthe undulatory theory already described. Vortices are on the whole\nplausible suppositions; for planets and satellites bear at first sight\nmuch resemblance to objects carried round in whirlpools, an analogy\nwhich doubtless suggested the theory. The failure was in the first and\nthird requisites; for, as already remarked, the theory did not allow\nof precise calculation of planetary motions, and was thus incapable of\nrigorous verification. But so far as we can institute a comparison,\nfacts are entirely against the vortices. Newton did not ridicule the\ntheory as absurd, but showed[426] that it was \u201cpressed with many\ndifficulties.\u201d He carefully pointed out that the Cartesian theory was\ninconsistent with the laws of Kepler, and would represent the planets\nas moving more rapidly at their aphelia than at their perihelia.[427]\nThe rotatory motion of the sun and planets on their own axes is in\nstriking conflict with the revolutions of the satellites carried round\nthem; and comets, the most flimsy of bodies, calmly pursue their\ncourses in elliptic paths, irrespective of the vortices which they pass\nthrough. We may now also point to the interlacing orbits of the minor\nplanets as a new and insuperable difficulty in the way of the Cartesian\nideas.\n\n  [426] *Principia*, bk. iii. Prop. 43. General Scholium.\n\n  [427] Ibid. bk. ii. Sect. ix. Prop. 53.\n\nNewton, though he established the best of theories, was also capable\nof proposing one of the worst; and if we want an instance of a theory\ndecisively contradicted by facts, we have only to turn to his views\nconcerning the origin of natural colours. Having analysed, with\nincomparable skill, the origin of the colours of thin plates, he\nsuggests that the colours of all bodies are determined in like manner\nby the size of their ultimate particles. A thin plate of a definite\nthickness will reflect a definite colour; hence, if broken up into\nfragments it will form a powder of the same colour. But, if this be\na sufficient explanation of coloured substances, then every coloured\nfluid ought to reflect the complementary colour of that which it\ntransmits. Colourless transparency arises, according to Newton, from\nparticles being too minute to reflect light; but if so, every black\nsubstance should be transparent. Newton himself so acutely felt this\nlast difficulty as to suggest that true blackness is due to some\ninternal refraction of the rays to and fro, and an ultimate stifling\nof them, which he did not attempt to explain further. Unless some\nother process comes into operation, neither refraction nor reflection,\nhowever often repeated, will destroy the energy of light. The theory\ntherefore gives no account, as Brewster shows, of 24 parts out of 25 of\nthe light which falls upon a black coal, and the remaining part which\nis reflected from the lustrous surface is equally inconsistent with the\ntheory, because fine coal-dust is almost entirely devoid of reflective\npower.[428] It is now generally believed that the colours of natural\nbodies are due to the unequal absorption of rays of light of different\nrefrangibility.\n\n  [428] Brewster\u2019s *Life of Newton*, 1st edit. chap. vii.\n\n\n*Experimentum Crucis.*\n\nAs we deduce more and more conclusions from a theory, and find them\nverified by trial, the probability of the theory increases in a rapid\nmanner; but we never escape the risk of error altogether. Absolute\ncertainty is beyond the powers of inductive investigation, and the\nmost plausible supposition may ultimately be proved false. Such is the\ngroundwork of similarity in nature, that two very different conditions\nmay often give closely similar results. We sometimes find ourselves\ntherefore in possession of two or more hypotheses which both agree\nwith so many experimental facts as to have great appearance of truth.\nUnder such circumstances we have need of some new experiment, which\nshall give results agreeing with one hypothesis but not with the other.\n\nAny such experiment which decides between two rival theories may be\ncalled an *Experimentum Crucis*, an Experiment of the Finger Post.\nWhenever the mind stands, as it were, at cross-roads and knows not\nwhich way to select, it needs some decisive guide, and Bacon therefore\nassigned great importance and authority to instances which serve in\nthis capacity. The name given by Bacon has become familiar; it is\nalmost the only one of Bacon\u2019s figurative expressions which has passed\ninto common use. Even Newton, as I have mentioned (p.\u00a0507), used the\nname.\n\nI do not think, indeed, that the common use of the word at all agrees\nwith that intended by Bacon. Herschel says that \u201cwe make an experiment\nof the crucial kind when we form combinations, and put in action\ncauses from which some particular one shall be deliberately excluded,\nand some other purposely admitted.\u201d[429] This, however, seems to be\nthe description of any special experiment not made at haphazard.\nPascal\u2019s experiment of causing a barometer to be carried to the top of\nthe Puy-de-D\u00f4me has often been considered as a perfect *experimentum\ncrucis*, if not the first distinct one on record;[430] but if so, we\nmust dignify the doctrine of Nature\u2019s abhorrence of a vacuum with\nthe position of a rival theory. A crucial experiment must not simply\nconfirm one theory, but must negative another; it must decide a mind\nwhich is in equilibrium, as Bacon says,[431] between two equally\nplausible views. \u201cWhen in search of any nature, the understanding comes\nto an equilibrium, as it were, or stands suspended as to which of two\nor more natures the cause of nature inquired after should be attributed\nor assigned, by reason of the frequent and common occurrence of several\nnatures, then these Crucial Instances show the true and inviolable\nassociation of one of these natures to the nature sought, and the\nuncertain and separable alliance of the other, whereby the question\nis decided, the former nature admitted for the cause, and the other\nrejected. These instances, therefore, afford great light, and have a\nkind of overruling authority, so that the course of interpretation will\nsometimes terminate in them, or be finished by them.\u201d\n\n  [429] *Discourse on the Study of Natural Philosophy*, p.\u00a0151.\n\n  [430] Ibid. p.\u00a0229.\n\n  [431] *Novum Organum*, bk. ii. Aphorism 36.\n\nThe long-continued strife between the Corpuscular and Undulatory\ntheories of light forms the best possible illustration of an\nExperimentum Crucis. It is remarkable in how plausible a manner both\nthese theories agreed with the ordinary laws of geometrical optics,\nrelating to reflection and refraction. According to the first law of\nmotion a moving particle proceeds in a perfectly straight line, when\nundisturbed by extraneous forces. If the particle being perfectly\nelastic, strike a perfectly elastic plane, it will bound off in such\na path that the angles of incidence and reflection will be equal.\nNow a ray of light proceeds in a straight line, or appears to do\nso, until it meets a reflecting body, when its path is altered in a\nmanner exactly similar to that of the elastic particle. Here is a\nremarkable correspondence which probably suggested to Newton\u2019s mind\nthe hypothesis that light consists of minute elastic particles moving\nwith excessive rapidity in straight lines. The correspondence was found\nto extend also to the law of simple refraction; for if particles of\nlight be supposed capable of attracting matter, and being attracted\nby it at insensibly small distances, then a ray of light, falling on\nthe surface of a transparent medium, will suffer an increase in its\nvelocity perpendicular to the surface, and the law of sines is the\nconsequence. This remarkable explanation of the law of refraction had\ndoubtless a very strong effect in leading Newton to entertain the\ncorpuscular theory, and he appears to have thought that the analogy\nbetween the propagation of rays of light and the motion of bodies was\nperfectly exact, whatever might be the actual nature of light.[432]\nIt is highly remarkable, again, that Newton was able to give by his\ncorpuscular theory, a plausible explanation of the inflection of light\nas discovered by Grimaldi. The theory would indeed have been a very\nprobable one could Newton\u2019s own law of gravity have applied; but this\nwas out of the question, because the particles of light, in order that\nthey may move in straight lines, must be devoid of any influence upon\neach other.\n\n  [432] *Principia*, bk. i. Sect. xiv. Prop. 96. Scholium. *Opticks*,\n  Prop. vi. 3rd edit. p.\u00a070.\n\nThe Huyghenian or Undulatory theory of light was also able to explain\nthe same phenomena, but with one remarkable difference. If the\nundulatory theory be true, light must move more slowly in a dense\nrefracting medium than in a rarer one; but the Newtonian theory assumed\nthat the attraction of the dense medium caused the particles of light\nto move more rapidly than in the rare medium. On this point, then,\nthere was complete discrepancy between the theories, and observation\nwas required to show which theory was to be preferred. Now by simply\ncutting a uniform plate of glass into two pieces, and slightly\ninclining one piece so as to increase the length of the path of a ray\npassing through it, experimenters were able to show that light does\nmove more slowly in glass than in air.[433] More recently Fizeau and\nFoucault independently measured the velocity of light in air and in\nwater, and found that the velocity is greater in air.[434]\n\n  [433] Airy\u2019s *Mathematical Tracts*, 3rd edit. pp.\u00a0286\u2013288.\n\n  [434] Jamin, *Cours de Physique*, vol.\u00a0iii. p.\u00a0372.\n\nThere are a number of other points at which experience decides against\nNewton, and in favour of Huyghens and Young. Laplace pointed out that\nthe attraction supposed to exist between matter and the corpuscular\nparticles of light would cause the velocity of light to vary with the\nsize of the emitting body, so that if a star were 250 times as great\nin diameter as our sun, its attraction would prevent the emanation of\nlight altogether.[435] But experience shows that the velocity of light\nis uniform, and independent of the magnitude of the emitting body,\nas it should be according to the undulatory theory. Lastly, Newton\u2019s\nexplanation of diffraction or inflection fringes of colours was only\n*plausible*, and not true; for Fresnel ascertained that the dimensions\nof the fringes are not what they would be according to Newton\u2019s theory.\n\n  [435] Young\u2019s *Lectures on Natural Philosophy* (1845), vol.\u00a0i. p.\u00a0361.\n\nAlthough the Science of Light presents us with the most beautiful\nexamples of crucial experiments and observations, instances are\nnot wanting in other branches of science. Copernicus asserted, in\nopposition to the ancient Ptolemaic theory, that the earth moved round\nthe sun, and he predicted that if ever the sense of sight could be\nrendered sufficiently acute and powerful, we should see phases in\nMercury and Venus. Galileo with his telescope was able, in 1610 to\nverify the prediction as regards Venus, and subsequent observations\nof Mercury led to a like conclusion. The discovery of the aberration\nof light added a new proof, still further strengthened by the more\nrecent determination of the parallax of fixed stars. Hooke proposed\nto prove the existence of the earth\u2019s diurnal motion by observing the\ndeviation of a falling body, an experiment successfully accomplished by\nBenzenberg; and Foucault\u2019s pendulum has since furnished an additional\nindication of the same motion, which is indeed also apparent in the\ntrade winds. All these are crucial facts in favour of the Copernican\ntheory.\n\n\n*Descriptive Hypotheses.*\n\nThere are hypotheses which we may call *descriptive hypotheses*,\nand which serve for little else than to furnish convenient names.\nWhen a phenomenon is of an unusual kind, we cannot even speak of\nit without using some analogy. Every word implies some resemblance\nbetween the thing to which it is applied, and some other thing, which\nfixes the meaning of the word. If we are to speak of what constitutes\nelectricity, we must search for the nearest analogy, and as electricity\nis characterised by the rapidity and facility of its movements, the\nnotion of a fluid of a very subtle character presents itself as\nappropriate. There is the single-fluid and the double-fluid theory of\nelectricity, and a great deal of discussion has been uselessly spent\nupon them. The fact is, that if these theories be understood as more\nthan convenient modes of describing the phenomena, they are altogether\ninvalid. The analogy extends only to the rapidity of motion, or rather\nthe fact that a phenomenon occurs successively at different points of\nthe body. The so-called electric fluid adds nothing to the weight of\nthe conductor, and to suppose that it really consists of particles of\nmatter is even more absurd than to reinstate the corpuscular theory\nof light. A far closer analogy exists between electricity and light\nundulations, which are about equally rapid in propagation. We shall\nprobably continue for a long time to talk of the *electric fluid*, but\nthere can be no doubt that this expression represents merely a phase of\nmolecular motion, a wave of disturbance. The invalidity of these fluid\ntheories is shown moreover in the fact that they have not led to the\ninvention of a single new experiment.\n\nAmong these merely descriptive hypotheses I should place Newton\u2019s\ntheory of Fits of Easy Reflection and Refraction. That theory did not\ndo more than describe what took place. It involved no analogy to other\nphenomena of nature, for Newton could not point to any other substance\nwhich went through these extraordinary fits. We now know that the true\nanalogy would have been waves of sound, of which Newton had acquired in\nother respects so complete a comprehension. But though the notion of\ninterference of waves had distinctly occurred to Hooke, Newton failed\nto see how the periodic phenomena of light could be connected with the\nperiodic character of waves. His hypothesis fell because it was out\nof analogy with everything else in nature, and it therefore did not\nallow him, as in other cases, to descend by mathematical deduction to\nconsequences which could be verified or refuted.\n\nWe are at freedom to imagine the existence of a new agent, and to\ngive it an appropriate name, provided there are phenomena incapable\nof explanation from known causes. We may speak of *vital force* as\noccasioning life, provided that we do not take it to be more than a\nname for an undefined something giving rise to inexplicable facts, just\nas the French chemists called Iodine the Substance X, so long as they\nwere unaware of its real character and place in chemistry.[436] Encke\nwas quite justified in speaking of the *resisting medium* in space so\nlong as the retardation of his comet could not be otherwise accounted\nfor. But such hypotheses will do much harm whenever they divert us from\nattempts to reconcile the facts with known laws, or when they lead us\nto mix up discrete things. Because we speak of vital force we must not\nassume that it is a really existing physical force like electricity; we\ndo not know what it is. We have no right to confuse Encke\u2019s supposed\nresisting medium with the basis of light without distinct evidence of\nidentity. The name protoplasm, now so familiarly used by physiologists,\nis doubtless legitimate so long as we do not mix up different\nsubstances under it, or imagine that the name gives us any knowledge\nof the obscure origin of life. To name a substance protoplasm no more\nexplains the infinite variety of forms of life which spring out of the\nsubstance, than does the *vital force* which may be supposed to reside\nin the protoplasm. Both expressions are mere names for an inexplicable\nseries of causes which out of apparently similar conditions produce the\nmost diverse results.\n\n  [436] Paris, *Life of Davy*, p.\u00a0274.\n\nHardly to be distinguished from descriptive hypotheses are certain\nimaginary objects which we frame for the ready comprehension of\na subject. The mathematician, in treating abstract questions of\nprobability, finds it convenient to represent the conditions by a\nconcrete hypothesis in the shape of a ballot-box. Poisson proved the\nprinciple of the inverse method of probabilities by imagining a number\nof ballot-boxes to have their contents mixed in one great ballot-box\n(p.\u00a0244). Many such devices are used by mathematicians. The Ptolemaic\ntheory of *cycles* and *epi-cycles* was no grotesque and useless work\nof the imagination, but a perfectly valid mode of analysing the motions\nof the heavenly bodies; in reality it is used by mathematicians at the\npresent day. Newton employed the pendulum as a means of representing\nthe nature of an undulation. Centres of gravity, oscillation, &c.,\npoles of the magnet, lines of force, are other imaginary existences\nemployed to assist our thoughts (p.\u00a0364). Such devices may be called\n*Representative Hypotheses*, and they are only permissible so far as\nthey embody analogies. Their further consideration belongs either to\nthe subject of Analogy, or to that of language and representation,\nfounded upon analogy.\n\n\n\n\nCHAPTER XXIV.\n\nEMPIRICAL KNOWLEDGE, EXPLANATION, AND PREDICTION.\n\n\nInductive investigation, as we have seen, consists in the union of\nhypothesis and experiment, deductive reasoning being the link by which\nexperimental results are made to confirm or confute the hypothesis. Now\nwhen we consider this relation between hypothesis and experiment it is\nobvious that we may classify our knowledge under four heads.\n\n(1) We may be acquainted with facts which have not yet been brought\ninto accordance with any hypothesis. Such facts constitute what is\ncalled *Empirical Knowledge*.\n\n(2) Another extensive portion of our knowledge consists of facts which\nhaving been first observed empirically, have afterwards been brought\ninto accordance with other facts by an hypothesis concerning the\ngeneral laws applying to them. This portion of our knowledge may be\nsaid to be *explained*, *reasoned*, or *generalised*.\n\n(3) In the third place comes the collection of facts, minor in number,\nbut most important as regards their scientific interest, which have\nbeen anticipated by theory and afterwards verified by experiment.\n\n(4) Lastly, there exists knowledge which is accepted solely on the\nground of theory, and is incapable of experimental confirmation, at\nleast with the instrumental means in our possession.\n\nIt is a work of much interest to compare and illustrate the relative\nextent and value of these four groups of knowledge. We shall observe\nthat as a general rule a great branch of science originates in facts\nobserved accidentally, or without distinct consciousness of what is to\nbe expected. As a science progresses, its power of foresight rapidly\nincreases, until the mathematician in his library acquires the power of\nanticipating nature, and predicting what will happen in circumstances\nwhich the eye of man has never examined.\n\n\n*Empirical Knowledge.*\n\nBy empirical knowledge we mean such as is derived directly from the\nexamination of detached facts, and rests entirely on those facts,\nwithout corroboration from other branches of knowledge. It is\ncontrasted with generalised and theoretical knowledge, which embraces\nmany series of facts under a few comprehensive principles, so that each\nseries serves to throw light upon each other series of facts. Just\nas, in the map of a half-explored country, we see detached bits of\nrivers, isolated mountains, and undefined plains, not connected into\nany complete plan, so a new branch of knowledge consists of groups of\nfacts, each group standing apart, so as not to allow us to reason from\none to another.\n\nBefore the time of Descartes, and Newton, and Huyghens, there was\nmuch empirical knowledge of the phenomena of light. The rainbow had\nalways struck the attention of the most careless observers, and there\nwas no difficulty in perceiving that its conditions of occurrence\nconsisted in rays of the sun shining upon falling drops of rain. It\nwas impossible to overlook the resemblance of the ordinary rainbow\nto the comparatively rare lunar rainbow, to the bow which appears\nupon the spray of a waterfall, or even upon beads of dew suspended on\ngrass and spiders\u2019 webs. In all these cases the uniform conditions\nare rays of light and round drops of water. Roger Bacon had noticed\nthese conditions, as well as the analogy of the rainbow colours to\nthose produced by crystals.[437] But the knowledge was empirical until\nDescartes and Newton showed how the phenomena were connected with facts\nconcerning the refraction of light.\n\n  [437] *Opus Majus.* Edit. 1733. Cap. x. p.\u00a0460.\n\nThere can be no better instance of an empirical truth than that\ndetected by Newton concerning the high refractive powers of combustible\nsubstances. Newton\u2019s chemical notions were almost as vague as those\nprevalent in his day, but he observed that certain \u201cfat, sulphureous,\nunctuous bodies,\u201d as he calls them, such as camphor, oils spirit of\nturpentine, amber, &c., have refractive powers two or three times\ngreater than might be anticipated from their densities.[438] The\nenormous refractive index of diamond, led him with great sagacity to\nregard this substance as of the same unctuous or inflammable nature,\nso that he may be regarded as predicting the combustibility of the\ndiamond, afterwards demonstrated by the Florentine Academicians\nin 1694. Brewster having entered into a long investigation of the\nrefractive powers of different substances, confirmed Newton\u2019s\nassertions, and found that the three elementary combustible substances,\ndiamond, phosphorus, and sulphur, have, in comparison with their\ndensities, by far the highest known refractive indices,[439] and\nthere are only a few substances, such as chromate of lead or glass of\nantimony, which exceed them in absolute power of refraction. The oils\nand hydrocarbons generally possess excessive indices. But all this\nknowledge remains to the present day purely empirical, no connection\nhaving been pointed out between this coincidence of inflammability\nand high refractive power, with other laws of chemistry or optics.\nIt is worth notice, as pointed out by Brewster, that if Newton had\nargued concerning two minerals, Greenockite and Octahedrite, as he did\nconcerning diamond, his predictions would have proved false, showing\nsufficiently that he did not make any sure induction on the subject. In\nthe present day, the relation of the refractive index to the density\nand atomic weight of a substance is becoming a matter of theory; yet\nthere remain specific differences of refracting power known only on\nempirical grounds, and it is curious that in hydrogen an abnormally\nhigh refractive power has been found to be joined to inflammability.\n\n  [438] Newton\u2019s *Opticks*. Third edit. p.\u00a0249.\n\n  [439] Brewster. *Treatise on New Philosophical Instruments*, p.\u00a0266,\n  &c.\n\nThe science of chemistry, however much its theory may have progressed,\nstill presents us with a vast body of empirical knowledge. Not only\nis it as yet hopeless to attempt to account for the particular group\nof qualities belonging to each element, but there are multitudes of\nparticular facts of which no further account can be given. Why should\nthe sulphides of many metals be intensely black? Why should a slight\namount of phosphoric acid have so great a power of interference with\nthe crystallisation of vanadic acid?[440] Why should the compound\nsilicates of alkalies and alkaline metals be transparent? Why should\ngold be so highly ductile, and gold and silver the only two sensibly\ntranslucent metals? Why should sulphur be capable of so many peculiar\nchanges into allotropic modifications?\n\n  [440] Roscoe, Bakerian Lecture, *Philosophical Transactions* (1868),\n  vol.\u00a0clviii. p.\u00a06.\n\nThere are whole branches of chemical knowledge which are mere\ncollections of disconnected facts. The properties of alloys are\noften remarkable; but no laws have yet been detected, and the laws\nof combining proportions seem to have no clear application.[441] Not\nthe slightest explanation can be given of the wonderful variations of\nthe qualities of iron, according as it contains more or less carbon\nand silicon, nay, even the facts of the case are often involved in\nuncertainty. Why, again, should the properties of steel be remarkably\naffected by the presence of a little tungsten or manganese? All\nthat was determined by Matthiessen concerning the conducting powers\nof copper, was of a purely empirical character.[442] Many animal\nsubstances cannot be shown to obey the laws of combining proportions.\nThus for the most part chemistry is yet an empirical science occupied\nwith the registration of immense numbers of disconnected facts, which\nmay at some future time become the basis of a greatly extended theory.\n\n  [441] *Life of Faraday*, vol.\u00a0ii. p.\u00a0104.\n\n  [442] Watts, *Dictionary of Chemistry*, vol.\u00a0ii, p.\u00a039, &c.\n\nWe must not indeed suppose that any science will ever entirely cease\nto be empirical. Multitudes of phenomena have been explained by the\nundulatory theory of light; but there yet remain many facts to be\ntreated. The natural colours of bodies and the rays given off by them\nwhen heated, are unexplained, and yield few empirical coincidences.\nThe theory of electricity is partially understood, but the conditions\nof the production of frictional electricity defy explanation, although\nthey have been studied for two centuries. I shall subsequently point\nout that even the establishment of a wide and true law of nature is\nbut the starting-point for the discovery of exceptions and divergences\ngiving a new scope to empirical discovery.\n\nThere is probably no science, I have said, which is entirely free\nfrom empirical and unexplained facts. Logic approaches most nearly to\nthis position, as it is merely a deductive development of the laws\nof thought and the principle of substitution. Yet some of the facts\nestablished in the investigation of the inverse logical problem may\nbe considered empirical. That a proposition of the form A\u00a0=\u00a0BC\u00a0\ua58c\u00a0*b\nc* possesses the least number of distinct logical variations, and\nthe greatest number of logical equivalents of the same form among\npropositions involving three classes (p.\u00a0141), is a case in point. So\nalso is the fact discovered by Professor Clifford that in regard to\nstatements involving four classes, there is only one example of two\ndissimilar statements having the same distances (p.\u00a0144). Mathematical\nscience often yields empirical truths. Why, for instance, should the\nvalue of \u03c0, when expressed to a great number of figures, contain the\ndigit 7 much less frequently than any other digit?[443] Even geometry\nmay allow of empirical truths, when the matter does not involve\nquantities of space, but numerical results and the positive or negative\ncharacter of quantities, as in De Morgan\u2019s theorem concerning negative\nareas.\n\n  [443] De Morgan\u2019s *Budget of Paradoxes*, p.\u00a0291.\n\n\n*Accidental Discovery.*\n\nThere are not a few cases where almost pure accident has determined\nthe moment when a new branch of knowledge was to be created. The laws\nof the structure of crystals were not discovered until Ha\u00fcy happened\nto drop a beautiful crystal of calc-spar upon a stone pavement. His\nmomentary regret at destroying a choice specimen was quickly removed\nwhen, in attempting to join the fragments together, he observed\nregular geometrical faces, which did not correspond with the external\nfacets of the crystals. A great many more crystals were soon broken\nintentionally, to observe the planes of cleavage, and the discovery of\nthe internal structure of crystalline substances was the result. Here\nwe see how much more was due to the reasoning power of the philosopher,\nthan to an accident which must often have happened to other persons.\n\nIn a similar manner, a fortuitous occurrence led Malus to discover\nthe polarisation of light by reflection. The phenomena of double\nrefraction had been long known, and when engaged in Paris in 1808, in\ninvestigating the character of light thus polarised, Malus chanced to\nlook through a double refracting prism at the light of the setting\nsun, reflected from the windows of the Luxembourg Palace. In turning\nthe prism round, he was surprised to find that the ordinary image\ndisappeared at two opposite positions of the prism. He remarked that\nthe reflected light behaved like light which had been polarised by\npassing through another prism. He was induced to test the character of\nlight reflected under other circumstances, and it was eventually proved\nthat polarisation is invariably connected with reflection. Some of the\ngeneral laws of optics, previously unsuspected, were thus discovered by\npure accident. In the history of electricity, accident has had a large\npart. For centuries some of the more common effects of magnetism and\nof frictional electricity had presented themselves as unaccountable\ndeviations from the ordinary course of Nature. Accident must have\nfirst directed attention to such phenomena, but how few of those who\nwitnessed them had any conception of the all-pervading character of the\npower manifested. The very existence of galvanism, or electricity of\nlow tension, was unsuspected until Galvani accidentally touched the leg\nof a frog with pieces of metal. The decomposition of water by voltaic\nelectricity also was accidentally discovered by Nicholson in 1801, and\nDavy speaks of this discovery as the foundation of all that had since\nbeen done in electro-chemical science.\n\nIt is otherwise with the discovery of electro-magnetism. Oersted, in\ncommon with many others, had suspected the existence of some relation\nbetween the magnet and electricity, and he appears to have tried to\ndetect its exact nature. Once, as we are told by Hansteen, he had\nemployed a strong galvanic battery during a lecture, and at the close\nit occurred to him to try the effect of placing the conducting wire\nparallel to a magnetic needle, instead of at right angles, as he had\npreviously done. The needle immediately moved and took up a position\nnearly at right angles to the wire; he inverted the direction of the\ncurrent, and the needle deviated in a contrary direction. The great\ndiscovery was made, and if by accident, it was such an accident as\nhappens, as Lagrange remarked of Newton, only to those who deserve\nit.[444] There was, in fact, nothing accidental, except that, as in\nall totally new discoveries, Oersted did not know what to look for. He\ncould not infer from previous knowledge the nature of the relation,\nand it was only repeated trial in different modes which could lead him\nto the right combination. High and happy powers of inference, and not\naccident, subsequently led Faraday to reverse the process, and to show\nthat the motion of the magnet would occasion an electric current in the\nwire.\n\n  [444] *Life of Faraday*, vol.\u00a0ii p.\u00a0396.\n\nSufficient investigation would probably show that almost every branch\nof art and science had an accidental beginning. In historical times\nalmost every important new instrument as the telescope, the microscope,\nor the compass, was probably suggested by some accidental occurrence.\nIn pre-historic times the germs of the arts must have arisen still more\nexclusively in the same way. Cultivation of plants probably arose, in\nMr. Darwin\u2019s opinion, from some such accident as the seeds of a fruit\nfalling upon a heap of refuse, and producing an unusually fine variety.\nEven the use of fire must, some time or other, have been discovered in\nan accidental manner.\n\nWith the progress of a branch of science, the element of chance becomes\nmuch reduced. Not only are laws discovered which enable results to be\npredicted, as we shall see, but the systematic examination of phenomena\nand substances often leads to discoveries which can in no sense be said\nto be accidental. It has been asserted that the an\u00e6sthetic properties\nof chloroform were disclosed by a little dog smelling at a saucerful\nof the liquid in a chemist\u2019s shop in Linlithgow, the singular effects\nupon the dog being reported to Simpson, who turned the incident to\ngood account. This story, however, has been shown to be a fabrication,\nthe fact being that Simpson had for many years been endeavouring to\ndiscover a better an\u00e6sthetic than those previously employed, and that\nhe tested the properties of chloroform, among other substances, at\nthe suggestion of Waldie, a Liverpool chemist. The valuable powers\nof chloral hydrate have since been discovered in a like manner, and\nsystematic inquiries are continually being made into the therapeutic or\neconomic values of new chemical compounds.\n\nIf we must attempt to draw a conclusion concerning the part which\nchance plays in scientific discovery, it must be allowed that it more\nor less affects the success of all inductive investigation, but becomes\nless important with the progress of science. Accident may bring a new\nand valuable combination to the notice of some person who had never\nexpressly searched for a discovery of the kind, and the probabilities\nare certainly in favour of a discovery being occasionally made in this\nmanner. But the greater the tact and industry with which a physicist\napplies himself to the study of nature, the greater is the probability\nthat he will meet with fortunate accidents, and will turn them to good\naccount. Thus it comes to pass that, in the refined investigations\nof the present day, genius united to extensive knowledge, cultivated\npowers, and indomitable industry, constitute the characteristics of the\nsuccessful discoverer.\n\n\n*Empirical Observations subsequently Explained.*\n\nThe second great portion of scientific knowledge consists of facts\nwhich have been first learnt in a purely empirical manner, but have\nafterwards been shown to follow from some law of nature, that is,\nfrom some highly probable hypothesis. Facts are said to be explained\nwhen they are thus brought into harmony with other facts, or bodies\nof general knowledge. There are few words more familiarly used in\nscientific phraseology than this word *explanation*, and it is\nnecessary to decide exactly what we mean by it, since the question\ntouches the deepest points concerning the nature of science. Like most\nterms referring to mental actions, the verbs *to explain*, or *to\nexplicate*, involve material similes. The action is *ex plicis plana\nreddere*, to take out the folds, and render a thing plain or even.\nExplanation thus renders a thing clearly comprehensible in all its\npoints, so that there is nothing left outstanding or obscure.\n\nEvery act of explanation consists in pointing out a resemblance\nbetween facts, or in showing that similarity exists between apparently\ndiverse phenomena. This similarity may be of any extent and depth; it\nmay be a general law of nature, which harmonises the motions of all\nthe heavenly bodies by showing that there is a similar force which\ngoverns all those motions, or the explanation may involve nothing more\nthan a single identity, as when we explain the appearance of shooting\nstars by showing that they are identical with portions of a comet.\nWherever we detect resemblance, there is a more or less explanation.\nThe mind is disquieted when it meets a novel phenomenon, one which is\n*sui generis*; it seeks at once for parallels which may be found in\nthe memory of past sensations. The so-called sulphurous smell which\nattends a stroke of lightning often excited attention, and it was not\nexplained until the exact similarity of the smell to that of ozone was\npointed out. The marks upon a flagstone are explained when they are\nshown to correspond with the feet of an extinct animal, whose bones\nare elsewhere found. Explanation, in fact, generally commences by the\ndiscovery of some simple resemblance; the theory of the rainbow began\nas soon as Antonio de Dominis pointed out the resemblance between its\ncolours and those presented by a ray of sunlight passing through a\nglass globe full of water.\n\nThe nature and limits of explanation can only be fully considered,\nafter we have entered upon the subjects of generalisation and analogy.\nIt must suffice to remark, in this place, that the most important\nprocess of explanation consists in showing that an observed fact is one\ncase of a general law or tendency. Iron is always found combined with\nsulphur, when it is in contact with coal, whereas in other parts of\nthe carboniferous strata it always occurs as a carbonate. We explain\nthis empirical fact as being due to the reducing power of carbon and\nhydrogen, which prevents the iron from combining with oxygen, and\nleaves it open to the affinity of sulphur. The uniform strength and\ndirection of the trade-winds were long familiar to mariners, before\nthey were explained by Halley on hydrostatical principles. The winds\nwere found to arise from the action of gravity, which causes a heavier\nbody to displace a lighter one, while the direction from east to west\nwas explained as a result of the earth\u2019s rotation. Whatever body in\nthe northern hemisphere changes its latitude, whether it be a bird, or\na railway train, or a body of air, must tend towards the right hand.\nDove\u2019s law of the winds is that the winds tend to veer in the northern\nhemisphere in the direction N.E.S.W., and in the southern hemisphere\nin the direction N.W.S.E. This tendency was shown by him to be the\nnecessary effect of the same conditions which apply to the trade winds.\nWhenever, then, any fact is connected by resemblance, law, theory, or\nhypothesis, with other facts, it is explained.\n\nAlthough the great mass of recorded facts must be empirical, and\nawaiting explanation, such knowledge is of minor value, because it does\nnot admit of safe and extensive inference. Each recorded result informs\nus exactly what will be experienced again in the same circumstances,\nbut has no bearing upon what will happen in other circumstances.\n\n\n*Overlooked Results of Theory.*\n\nWe must by no means suppose that, when a scientific truth is in our\npossession, all its consequences will be foreseen. Deduction is certain\nand infallible, in the sense that each step in deductive reasoning will\nlead us to some result, as certain as the law itself. But it does not\nfollow that deduction will lead the reasoner to every result of a law\nor combination of laws. Whatever road a traveller takes, he is sure to\narrive somewhere, but unless he proceeds in a systematic manner, it is\nunlikely that he will reach every place to which a network of roads\nwill conduct him.\n\nIn like manner there are many phenomena which were virtually within\nthe reach of philosophers by inference from their previous knowledge,\nbut were never discovered until accident or systematic empirical\nobservation disclosed their existence.\n\nThat light travels with a uniform high velocity was proved by Roemer\nfrom observations of the eclipses of Jupiter\u2019s satellites. Corrections\nwere thenceforward made in all astronomical observations requiring it,\nfor the difference of absolute time at which an event happened, and\nthat at which it would be seen on the earth. But no person happened\nto remark that the motion of light compounded with that of the earth\nin its orbit would occasion a small apparent displacement of the\ngreater part of the heavenly bodies. Fifty years elapsed before Bradley\nempirically discovered this effect, called by him aberration, when\nreducing his observations of the fixed stars.\n\nWhen once the relation between an electric current and a magnet had\nbeen detected by Oersted and Faraday, it ought to have been possible\nfor them to foresee the diverse results which must ensue in different\ncircumstances. If, for instance, a plate of copper were placed beneath\nan oscillating magnetic needle, it should have been seen that the\nneedle would induce currents in the copper, but as this could not take\nplace without a certain reaction against the needle, it ought to have\nbeen seen that the needle would come to rest more rapidly than in the\nabsence of the copper. This peculiar effect was accidentally discovered\nby Gambey in 1824. Arago acutely inferred from Gambey\u2019s experiment that\nif the copper were set in rotation while the needle was stationary the\nmotion would gradually be communicated to the needle. The phenomenon\nnevertheless puzzled the whole scientific world, and it required\nthe deductive genius of Faraday to show that it was a result of the\nprinciples of electro-magnetism.[445]\n\n  [445] *Experimental Researches in Electricity*, 1st Series, pp.\u00a024\u201344.\n\nMany other curious facts might be mentioned which when once noticed\nwere explained as the effects of well-known laws. It was accidentally\ndiscovered that the navigation of canals of small depth could be\nfacilitated by increasing the speed of the boats, the resistance being\nactually reduced by this increase of speed, which enables the boat\nto ride as it were upon its own forced wave. Now mathematical theory\nmight have predicted this result had the right application of the\nformul\u00e6 occurred to any one.[446] Giffard\u2019s injector for supplying\nsteam boilers with water by the force of their own steam, was, I\nbelieve, accidentally discovered, but no new principles of mechanics\nare involved in it, so that it might have been theoretically invented.\nThe same may be said of the curious experiment in which a stream of\nair or steam issuing from a pipe is made to hold a free disc upon the\nend of the pipe and thus obstruct its own outlet. The possession then\nof a true theory does not by any means imply the foreseeing of all the\nresults. The effects of even a few simple laws may be manifold, and\nsome of the most curious and useful effects may remain undetected until\naccidental observation brings them to our notice.\n\n  [446] Airy, *On Tides and Waves*, Encyclop\u00e6dia Metropolitana, p.\u00a0348*.\n\n\n*Predicted Discoveries.*\n\nThe most interesting of the four classes of facts specified in p.\u00a0525,\nis probably the third, containing those the occurrence of which has\nbeen first predicted by theory and then verified by observation. There\nis no more convincing proof of the soundness of knowledge than that it\nconfers the gift of foresight. Auguste Comte said that \u201cPrevision is\nthe test of true theory;\u201d I should say that it is *one test* of true\ntheory, and that which is most likely to strike the public attention.\nCoincidence with fact is the test of true theory, but when the result\nof theory is announced before-hand, there can be no doubt as to the\nunprejudiced spirit in which the theorist interprets the results of his\nown theory.\n\nThe earliest instance of scientific prophecy is naturally furnished\nby the science of Astronomy, which was the earliest in development.\nHerodotus[447] narrates that, in the midst of a battle between the\nMedes and Lydians, the day was suddenly turned into night, and the\nevent had been foretold by Thales, the Father of Philosophy. A\ncessation of the combat and peace confirmed by marriages were the\nconsequences of this happy scientific effort. Much controversy has\ntaken place concerning the date of this occurrence, Baily assigning\nthe year 610 B.C., but Airy has calculated that the exact day was\nthe 28th of May, 584 B.C. There can be no doubt that this and other\npredictions of eclipses attributed to ancient philosophers were due to\na knowledge of the Metonic Cycle, a period of 6,585 days, or 223 lunar\nmonths, or about 19 years, after which a nearly perfect recurrence of\nthe phases and eclipses of the moon takes place; but if so, Thales must\nhave had access to long series of astronomical records of the Egyptians\nor the Chaldeans. There is a well-known story as to the happy use which\nColumbus made of the power of predicting eclipses in overawing the\nislanders of Jamaica who refused him necessary supplies of food for his\nfleet. He threatened to deprive them of the moon\u2019s light. \u201cHis threat\nwas treated at first with indifference, but when the eclipse actually\ncommenced, the barbarians vied with each other in the production of the\nnecessary supplies for the Spanish fleet.\u201d\n\n  [447] Lib. i. cap. 74.\n\nExactly the same kind of awe which the ancients experienced at the\nprediction of eclipses, has been felt in modern times concerning the\nreturn of comets. Seneca asserted in distinct terms that comets would\nbe found to revolve in periodic orbits and return to sight. The ancient\nChaldeans and the Pythagoreans are also said to have entertained a\nlike opinion. But it was not until the age of Newton and Halley that\nit became possible to calculate the path of a comet in future years. A\ngreat comet appeared in 1682, a few years before the first publication\nof the *Principia*, and Halley showed that its orbit corresponded\nwith that of remarkable comets recorded to have appeared in the years\n1531 and 1607. The intervals of time were not quite equal, but Halley\nconceived the bold idea that this difference might be due to the\ndisturbing power of Jupiter, near which the comet had passed in the\ninterval 1607\u20131682. He predicted that the comet would return about\nthe end of 1758 or the beginning of 1759, and though Halley did not\nlive to enjoy the sight, it was actually detected on the night of\nChristmas-day, 1758. A second return of the comet was witnessed in 1835\nnearly at the anticipated time.\n\nIn recent times the discovery of Neptune has been the most remarkable\ninstance of prevision in astronomical science. A full account of this\ndiscovery may be found in several works, as for instance Herschel\u2019s\n*Outlines of Astronomy*, and *Grant\u2019s History of Physical Astronomy*,\nChapters XII and XIII.\n\n\n*Predictions in the Science of Light.*\n\nNext after astronomy the science of physical optics has furnished the\nmost beautiful instances of the prophetic power of correct theory.\nThese cases are the more striking because they proceed from the\nprofound application of mathematical analysis and show an insight\ninto the mysterious workings of matter which is surprising to all,\nbut especially to those who are unable to comprehend the methods\nof research employed. By its power of prevision the truth of the\nundulatory theory of light has been conspicuously proved, and the\ncontrast in this respect between the undulatory and Corpuscular\ntheories is remarkable. Even Newton could get no aid from his\ncorpuscular theory in the invention of new experiments, and to his\nfollowers who embraced that theory we owe little or nothing in the\nscience of light. Laplace did not derive from the theory a single\ndiscovery. As Fresnel remarks:[448]\n\n  [448] Taylor\u2019s *Scientific Memoirs*, vol.\u00a0v. p.\u00a0241.\n\n\u201cThe assistance to be derived from a good theory is not to be confined\nto the calculation of the forces when the laws of the phenomena\nare known. There are certain laws so complicated and so singular,\nthat observation alone, aided by analogy, could never lead to their\ndiscovery. To divine these enigmas we must be guided by theoretical\nideas founded on a *true* hypothesis. The theory of luminous vibrations\npresents this character, and these precious advantages; for to it\nwe owe the discovery of optical laws the most complicated and most\ndifficult to divine.\u201d\n\nPhysicists who embraced the corpuscular theory had nothing but their\nown quickness of observation to rely upon. Fresnel having once seized\nthe conditions of the true undulatory theory, as previously stated\nby Young, was enabled by the mere manipulation of his mathematical\nsymbols to foresee many of the complicated phenomena of light. Who\ncould possibly suppose, that by stopping a portion of the rays\npassing through a circular aperture, the illumination of a point upon\na screen behind the aperture might be many times multiplied. Yet this\nparadoxical effect was predicted by Fresnel, and verified both by\nhimself, and in a careful repetition of the experiment, by Billet.\nFew persons are aware that in the middle of the shadow of an opaque\ncircular disc is a point of light sensibly as bright as if no disc\nhad been interposed. This startling fact was deduced from Fresnel\u2019s\ntheory by Poisson, and was then verified experimentally by Arago.\nAiry, again, was led by pure theory to predict that Newton\u2019s rings\nwould present a modified appearance if produced between a lens of\nglass and a plate of metal. This effect happened to have been observed\nfifteen years before by Arago, unknown to Airy. Another prediction of\nAiry, that there would be a further modification of the rings when\nmade between two substances of very different refractive indices, was\nverified by subsequent trial with a diamond. A reversal of the rings\ntakes place when the space intervening between the plates is filled\nwith a substance of intermediate refractive power, another phenomenon\npredicted by theory and verified by experiment. There is hardly a limit\nto the number of other complicated effects of the interference of rays\nof light under different circumstances which might be deduced from\nthe mathematical expressions, if it were worth while, or which, being\npreviously observed, can be explained. An interesting case was observed\nby Herschel and explained by Airy.[449]\n\n  [449] Airy\u2019s *Mathematical Tracts*, 3rd edit. p.\u00a0312.\n\nBy a somewhat different effort of scientific foresight, Fresnel\ndiscovered that any solid transparent medium might be endowed with the\npower of double refraction by mere compression. As he attributed the\ndouble refracting power of crystals to unequal elasticity in different\ndirections, he inferred that unequal elasticity, if artificially\nproduced, would give similar phenomena. With a powerful screw and a\npiece of glass, he then produced not only the colours due to double\nrefraction, but the actual duplication of images. Thus, by a great\nscientific generalisation, are the remarkable properties of Iceland\nspar shown to belong to all transparent substances under certain\nconditions.[450]\n\n  [450] Young\u2019s *Works*, vol.\u00a0i. p.\u00a0412.\n\nAll other predictions in optical science are, however, thrown into the\nshade by the theoretical discovery of conical refraction by the late\nSir W.\u00a0R. Hamilton, of Dublin. In investigating the passage of light\nthrough certain crystals, Hamilton found that Fresnel had slightly\nmisinterpreted his own formul\u00e6, and that, when rightly understood,\nthey indicated a phenomenon of a kind never witnessed. A small ray of\nlight sent into a crystal of arragonite in a particular direction,\nbecomes spread out into an infinite number of rays, which form a hollow\ncone within the crystal, and a hollow cylinder when emerging from the\nopposite side. In another case, a different, but equally strange,\neffect is produced, a ray of light being spread out into a hollow cone\nat the point where it quits the crystal. These phenomena are peculiarly\ninteresting, because cones and cylinders of light are not produced\nin any other cases. They are opposed to all analogy, and constitute\nsingular exceptions, of a kind which we shall afterwards consider more\nfully. Their strangeness rendered them peculiarly fitted to test the\ntruth of the theory by which they were discovered; and when Professor\nLloyd, at Hamilton\u2019s request, succeeded, after considerable difficulty,\nin witnessing the new appearances, no further doubt could remain of\nthe validity of the wave theory which we owe to Huyghens, Young, and\nFresnel.[451]\n\n  [451] Lloyd\u2019s *Wave Theory*, Part ii. pp.\u00a052\u201358. Babbage, *Ninth\n  Bridgewater Treatise*, p.\u00a0104, quoting Lloyd, *Transactions of the\n  Royal Irish Academy*, vol.\u00a0xvii. Clifton, *Quarterly Journal of Pure\n  and Applied Mathematics*, January 1860.\n\n\n*Predictions from the Theory of Undulations.*\n\nIt is curious that the undulations of light, although inconceivably\nrapid and small, admit of more accurate measurement than waves of any\nother kind. But so far as we can carry out exact experiments on other\nkinds of waves, we find the phenomena of interference repeated, and\nanalogy gives considerable power of prediction. Herschel was perhaps\nthe first to suggest that two sounds might be made to destroy each\nother by interference.[452] For if one-half of a wave travelling\nthrough a tube could be separated, and conducted by a longer passage,\nso as, on rejoining the other half, to be one-quarter of a vibration\nbehind-hand, the two portions would exactly neutralise each other.\nThis experiment has been performed with success. The interference\narising between the waves from the two prongs of a tuning-fork was\nalso predicted by theory, and proved to exist by Weber; indeed it may\nbe observed by merely holding a vibrating fork close to the ear and\nturning it round.[453]\n\n  [452] *Encyclop\u00e6dia Metropolitana*, art. *Sound*, p.\u00a0753.\n\n  [453] Tyndall\u2019s *Sound*, pp.\u00a0261, 273.\n\nIt is a result of the theory of sound that, if we move rapidly towards\na sounding body, or if it move rapidly towards us, the pitch of the\nsound will be a little more acute; and, *vice vers\u00e2*, when the relative\nmotion is in the opposite direction, the pitch will be more grave. This\narises from the less or greater intervals of time elapsing between the\nsuccessive strokes of waves upon the auditory nerve, according as the\near moves towards or from the source of sound relatively speaking.\nThis effect was predicted by theory, and afterwards verified by the\nexperiments of Buys Ballot, on Dutch railways, and of Scott Russell, in\nEngland. Whenever one railway train passes another, on the locomotive\nof which the whistle is being sounded, the drop in the acuteness of\nthe sound may be noticed at the moment of passing. This change gives\nthe sound a peculiar howling character, which many persons must have\nnoticed. I have calculated that with two trains travelling thirty miles\nan hour, the effect would amount to rather more than half a tone, and\nwith some express trains it would amount to a tone. A corresponding\neffect is produced in the case of light undulations, when the eye and\nthe luminous body approach or recede from each other. It is shown by\na slight change in the refrangibility of the rays of light, and a\nconsequent change in the place of the lines of the spectrum, which has\nbeen made to give important and unexpected information concerning the\nrelative approach or recession of stars.\n\nTides are vast waves, and were the earth\u2019s surface entirely covered\nby an ocean of uniform depth, they would admit of exact theoretical\ninvestigation. The irregular form of the seas introduces unknown\nquantities and complexities with which theory cannot cope.\nNevertheless, Whewell, observing that the tides of the German Ocean\nconsist of interfering waves, which arrive partly round the North of\nScotland and partly through the British Channel, was enabled to predict\nthat at a point about midway between Brill on the coast of Holland,\nand Lowestoft no tides would be found to exist. At that point the two\nwaves would be of the same amount, but in opposite phases, so as to\nneutralise each other. This prediction was verified by a surveying\nvessel of the British navy.[454]\n\n  [454] Whewell\u2019s *History of the Inductive Sciences*, vol.\u00a0ii. p.\u00a0471.\n  Herschel\u2019s *Physical Geography*, \u00a7 77.\n\n\n*Prediction in other Sciences.*\n\nGenerations, or even centuries, may elapse before mankind are\nin possession of a mathematical theory of the constitution of\nmatter as complete as the theory of gravitation. Nevertheless,\nmathematical physicists have in recent years acquired a hold of some\nof the relations of the physical forces, and the proof is found in\nanticipations of curious phenomena which had never been observed.\nProfessor James Thomson deduced from Carnot\u2019s theory of heat that the\napplication of pressure would lower the melting-point of ice. He even\nventured to assign the amount of this effect, and his statement was\nafterwards verified by Sir W. Thomson.[455] \u201cIn this very remarkable\nspeculation, an entirely novel physical phenomenon was *predicted*, in\nanticipation of any direct experiments on the subject; and the actual\nobservation of the phenomenon was pointed out as a highly interesting\nobject for experimental research.\u201d Just as liquids which expand in\nsolidifying will have the temperature of solidification lowered by\npressure, so liquids which contract in solidifying will exhibit the\nreverse effect. They will be assisted in solidifying, as it were,\nby pressure, so as to become solid at a higher temperature, as the\npressure is greater. This latter result was verified by Bunsen and\nHopkins, in the case of paraffin, spermaceti, wax, and stearin. The\neffect upon water has more recently been carried to such an extent by\nMousson, that under the vast pressure of 1300 atmospheres, water did\nnot freeze until cooled down to -18\u00b0C. Another remarkable prediction\nof Professor Thomson was to the effect that, if a metallic spring\nbe weakened by a rise of temperature, work done against the spring\nin bending it will cause a cooling effect. Although the effect to\nbe expected in a certain apparatus was only about four-thousandths\nof a degree Centigrade, Dr. Joule[456] succeeded in measuring it to\nthe extent of three-thousandths of a degree, such is the delicacy of\nmodern heat measurements. I cannot refrain from quoting Dr. Joule\u2019s\nreflections upon this fact. \u201cThus even in the above delicate case,\u201d he\nsays, \u201cis the formula of Professor Thomson completely verified. The\nmathematical investigation of the thermo-elastic qualities of metals\nhas enabled my illustrious friend to predict with certainty a whole\nclass of highly interesting phenomena. To him especially do we owe\nthe important advance which has been recently made to a new era in\nthe history of science, when the famous philosophical system of Bacon\nwill be to a great extent superseded, and when, instead of arriving at\ndiscovery by induction from experiment, we shall obtain our largest\naccessions of new facts by reasoning deductively from fundamental\nprinciples.\u201d\n\n  [455] Maxwell\u2019s *Theory of Heat*, p.\u00a0174. *Philosophical Magazine*,\n  August 1850. Third Series, vol.\u00a0xxxvii. p.\u00a0123.\n\n  [456] *Philosophical Transactions*, 1858, vol.\u00a0cxlviii. p.\u00a0127.\n\nThe theory of electricity is a necessary part of the general theory of\nmatter, and is rapidly acquiring the power of prevision. As soon as\nWheatstone had proved experimentally that the conduction of electricity\noccupies time, Faraday remarked in 1838, with wonderful sagacity,\nthat if the conducting wires were connected with the coatings of a\nlarge Leyden jar, the rapidity of conduction would be lessened. This\nprediction remained unverified for sixteen years, until the submarine\ncable was laid beneath the Channel. A considerable retardation of the\nelectric spark was then detected, and Faraday at once pointed out that\nthe wire surrounded by water resembles a Leyden jar on a large scale,\nso that each message sent through the cable verified his remark of\n1838.[457]\n\n  [457] Tyndall\u2019s *Faraday*, pp.\u00a073, 74; *Life of Faraday*, vol.\u00a0ii.\n  pp.\u00a082, 83.\n\nThe joint relations of heat and electricity to the metals constitute a\nnew science of thermo-electricity by which Sir W. Thomson was enabled\nto anticipate the following curious effect, namely, that an electric\ncurrent passing in an iron bar from a hot to a cold part produces a\ncooling effect, but in a copper bar the effect is exactly opposite in\ncharacter, that is, the bar becomes heated.[458] The action of crystals\nwith regard to heat and electricity was partly foreseen on the grounds\nof theory by Poisson.\n\n  [458] Tait\u2019s *Thermodynamics*, p.\u00a077.\n\nChemistry, although to a great extent an empirical science, has not\nbeen without prophetic triumphs. The existence of the metals potassium\nand sodium was foreseen by Lavoisier, and their elimination by Davy was\none of the chief *experimenta crucis* which established Lavoisier\u2019s\nsystem. The existence of many other metals which eye had never seen\nwas a natural inference, and theory has not been at fault. In the\nabove cases the compounds of the metal were well known, and it was the\nresult of decomposition that was foretold. The discovery in 1876 of the\nmetal gallium is peculiarly interesting because the existence of this\nmetal, previously wholly unknown, had been inferred from theoretical\nconsiderations by M. Mendelief, and some of its properties had been\ncorrectly predicted. No sooner, too, had a theory of organic compounds\nbeen conceived by Professor A.\u00a0W. Williamson than he foretold the\nformation of a complex substance consisting of water in which both\natoms of hydrogen are replaced by atoms of acetyle. This substance,\nknown as the acetic anhydride, was afterwards produced by Gerhardt. In\nthe subsequent progress of organic chemistry occurrences of this kind\nhave become common. The theoretical chemist by the classification of\nhis specimens and the manipulation of his formul\u00e6 can plan out whole\nseries of unknown oils, acids, and alcohols, just as a designer might\ndraw out a multitude of patterns. Professor Cayley has even calculated\nfor certain cases the possible numbers of chemical compounds.[459] The\nformation of many such substances is a matter of course; but there is\nan interesting prediction given by Hofmann, concerning the possible\nexistence of new compounds of sulphur and selenium, and even oxides of\nammonium, which it remains for chemists to verify.[460]\n\n  [459] *On the Analytical Forms called Trees, with Application to the\n  Theory of Chemical Combinations.* Report of the British Association,\n  1875, p.\u00a0257.\n\n  [460] Hofmann\u2019s *Introduction to Chemistry*, pp.\u00a0224, 225.\n\n\n*Prediction by Inversion of Cause and Effect.*\n\nThere is one process of experiment which has so often led to important\ndiscoveries as to deserve separate illustration--I mean the inversion\nof Cause and Effect. Thus if A and B in one experiment produce C as a\nconsequent, then antecedents of the nature of B and C may usually be\nmade to produce a consequent of the nature of A inverted in direction.\nWhen we apply heat to a gas it tends to expand; hence if we allow the\ngas to expand by its own elastic force, cold is the result; that is,\nB (air) and C (expansion) produce the negative of A (heat). Again, B\n(air) and compression, the negative of C, produce A (heat). Similar\nresults may be expected in a multitude of cases. It is a familiar\nlaw that heat expands iron. What may be expected, then, if instead\nof increasing the length of an iron bar by heat we use mechanical\nforce and stretch the bar? Having the bar and the former consequent,\nexpansion, we should expect the negative of the former antecedent,\nnamely cold. The truth of this inference was proved by Dr. Joule, who\ninvestigated the amount of the effect with his usual skill.[461]\n\n  [461] *Philosophical Transactions* (1855), vol.\u00a0cxlv. pp.\u00a0100, &c.\n\nThis inversion of cause and effect in the case of heat may be itself\ninverted in a highly curious manner. It happens that there are a\nfew substances which are unexplained exceptions to the general law\nof expansion by heat. India-rubber especially is remarkable for\n*contracting* when heated. Since, then, iron and india-rubber are\noppositely related to heat, we may expect that as distension of the\niron produced cold, distension of the india-rubber will produce heat.\nThis is actually found to be the case, and anyone may detect the effect\nby suddenly stretching an india-rubber band while the middle part is\nin the mouth. When being stretched it grows slightly warm, and when\nrelaxed cold.\n\nThe reader will see that some of the scientific predictions mentioned\nin preceding sections were due to the principle of inversion; for\ninstance, Thomson\u2019s speculations on the relation between pressure and\nthe melting-point. But many other illustrations could be adduced.\nThe usual agent by which we melt a substance is heat; but if we can\nmelt a substance without heat, then we may expect the negative of\nheat as an effect. This is the foundation of all freezing mixtures.\nThe affinity of salt for water causes it to melt ice, and we may thus\nreduce the temperature to Fahrenheit\u2019s zero. Calcium chloride has so\nmuch higher an attraction for water that a temperature of -45\u00b0\u00a0C. may\nbe attained by its use. Even the solution of a certain alloy of lead,\ntin, and bismuth in mercury, may be made to reduce the temperature\nthrough 27\u00b0\u00a0C. All the other modes of producing cold are inversions of\nmore familiar uses of heat. Carr\u00e9\u2019s freezing machine is an inverted\ndistilling apparatus, the distillation being occasioned by chemical\naffinity instead of heat. Another kind of freezing machine is the exact\ninverse of the steam-engine.\n\nA very paradoxical effect is due to another inversion. It is hard to\nbelieve that a current of steam at 100\u00b0\u00a0C. can raise a body of liquid\nto a higher temperature than the steam itself possesses. But Mr. Spence\nhas pointed out that if the boiling-point of a saline solution be\nabove 100\u00b0, it will continue, on account of its affinity for water, to\ncondense steam when above 100\u00b0 in temperature. It will condense the\nsteam until heated to the point at which the tension of its vapour is\nequal to that of the atmosphere, that is, its own boiling-point.[462]\nAgain, since heat melts ice, we might expect to produce heat by the\ninverse change from water into ice. This is accomplished in the\nphenomenon of suspended freezing. Water may be cooled in a clean glass\nvessel many degrees below the freezing-point, and yet retained in the\nliquid condition. But if disturbed, and especially if brought into\ncontact with a small particle of ice, it instantly solidifies and rises\nin temperature to 0\u00b0\u00a0C. The effect is still better displayed in the\nlecture-room experiment of the suspended crystallisation of a solution\nof sodium sulphate, in which a sudden rise of temperature of 15\u00b0 or\n20\u00b0\u00a0C. is often manifested.\n\n  [462] *Proceedings of the Manchester Philosophical Society*, Feb.\n  1870.\n\nThe science of electricity is full of most interesting cases of\ninversion. As Professor Tyndall has remarked, Faraday had a profound\nbelief in the reciprocal relations of the physical forces. The great\nstarting-point of his researches, the discovery of electro-magnetism,\nwas clearly an inversion. Oersted and Amp\u00e8re had proved that with\nan electric current and a magnet in a particular position as\nantecedents, motion is the consequent. If then a magnet, a wire and\nmotion be the antecedents, an *opposite* electric current will be the\nconsequent. It would be an endless task to trace out the results of\nthis fertile relationship. Another part of Faraday\u2019s researches was\noccupied in ascertaining the direct and inverse relations of magnetic\nand diamagnetic, amorphous and crystalline substances in various\ncircumstances. In all other relations of electricity the principle of\ninversion holds. The voltameter or the electro-plating cell is the\ninverse of the galvanic battery. As heat applied to a junction of\nantimony and bismuth bars produces electricity, it follows that an\nelectric current passed through such a junction will produce cold. But\nit is now sufficiently apparent that inversion of cause and effect is a\nmost fertile means of discovery and prediction.\n\n\n*Facts known only by Theory.*\n\nOf the four classes of facts enumerated in p.\u00a0525 the last remains\nunconsidered. It includes the unverified predictions of science.\nScientific prophecy arrests the attention of the world when it refers\nto such striking events as an eclipse, the appearance of a great comet,\nor any phenomenon which people can verify with their own eyes. But\nit is surely a matter for greater wonder that a physicist describes\nand measures phenomena which eye cannot see, nor sense of any kind\ndetect. In most cases this arises from the effect being too small in\namount to affect our organs of sense, or come within the powers of our\ninstruments as at present constructed. But there is a class of yet more\nremarkable cases, in which a phenomenon cannot possibly be observed,\nand yet we can say what it would be if it were observed.\n\nIn astronomy, systematic aberration is an effect of the sun\u2019s proper\nmotion almost certainly known to exist, but which we have no hope of\ndetecting by observation in the present age of the world. As the\nearth\u2019s motion round the sun combined with the motion of light causes\nthe stars to deviate apparently from their true positions to the extent\nof about 18\u2033 at the most, so the motion of the whole planetary system\nthrough space must occasion a similar displacement of at most 5\u2033. The\nordinary aberration can be readily detected with modern astronomical\ninstruments, because it goes through a yearly change in direction or\namount; but systematic aberration is constant so long as the planetary\nsystem moves uniformly in a sensibly straight line. Only then in the\ncourse of ages, when the curvature of the sun\u2019s path becomes apparent,\ncan we hope to verify the existence of this kind of aberration. A\ncurious effect must also be produced by the sun\u2019s proper motion upon\nthe apparent periods of revolution of the binary stars.\n\nTo my mind, some of the most interesting truths in the whole range of\nscience are those which have not been, and in many cases probably never\ncan be, verified by trial. Thus the chemist assigns, with a very high\ndegree of probability, the vapour densities of such elements as carbon\nand silicon, which have never been observed separately in a state of\nvapour. The chemist is also familiar with the vapour densities of\nelements at temperatures at which the elements in question never have\nbeen, and probably never can be, submitted to experiment in the form of\nvapour.\n\nJoule and others have calculated the actual velocity of the molecules\nof a gas, and even the number of collisions which must take place per\nsecond during their constant circulation. Physicists have not yet given\nus the exact magnitudes of the particles of matter, but they have\nascertained by several methods the limits within which their magnitudes\nmust lie. Such scientific results must be for ever beyond the power of\nverification by the senses. I have elsewhere had occasion to remark\nthat waves of light, the intimate processes of electrical changes,\nthe properties of the ether which is the base of all phenomena, are\nnecessarily determined in a hypothetical, but not therefore a less\ncertain manner.\n\nThough only two of the metals, gold and silver, have ever been\nobserved to be transparent, we know on the grounds of theory that\nthey are all more or less so; we can even estimate by theory their\nrefractive indices, and prove that they are exceedingly high. The\nphenomena of elliptic polarisation, and perhaps also those of internal\nradiation,[463] depend upon the refractive index, and thus, even when\nwe cannot observe any refracted rays, we can indirectly learn how they\nwould be refracted.\n\n  [463] Balfour Stewart, *Elementary Treatise on Heat*, 1st edit.\n  p.\u00a0198.\n\nIn many cases large quantities of electricity must be produced, which\nwe cannot observe because it is instantly discharged. In the common\nelectric machine the cylinder and rubber are made of non-conductors,\nso that we can separate and accumulate the electricity. But a little\ndamp, by serving as a conductor, prevents this separation from enduring\nany sensible time. Hence there is no doubt that when we rub two good\nconductors against each other, for instance two pieces of metals,\nmuch electricity is produced, but instantaneously converted into some\nother form of energy. Joule believes that all the heat of friction is\ntransmuted electricity.\n\nAs regards phenomena of insensible amount, nature is absolutely\nfull of them. We must regard those changes which we can observe as\nthe comparatively rare aggregates of minuter changes. On a little\nreflection we must allow that no object known to us remains for two\ninstants of exactly the same temperature. If so, the dimensions of\nobjects must be in a perpetual state of variation. The minor planetary\nand lunar perturbations are infinitely numerous, but usually too\nsmall to be detected by observation, although their amounts may be\nassigned by theory. There is every reason to believe that chemical\nand electric actions of small amount are constantly in progress. The\nhardest substances, if reduced to extremely small particles, and\ndiffused in pure water, manifest oscillatory movements which must be\ndue to chemical and electric changes, so slight that they go on for\nyears without affecting appreciably the weight of the particles.[464]\nThe earth\u2019s magnetism must more or less affect every object which we\nhandle. As Tyndall remarks, \u201cAn upright iron stone influenced by the\nearth\u2019s magnetism becomes a magnet, with its bottom a north and its\ntop a south pole. Doubtless, though in an immensely feebler degree,\nevery erect marble statue is a true diamagnet, with its head a north\npole and its feet a south pole. The same is certainly true of man as\nhe stands upon the earth\u2019s surface, for all the tissues of the human\nbody are diamagnetic.\u201d[465] The sun\u2019s light produces a very quick and\nperceptible effect upon the photographic plate; in all probability it\nhas a less effect upon a great variety of substances. We may regard\nevery phenomenon as an exaggerated and conspicuous case of a process\nwhich is, in infinitely numerous cases, beyond the means of observation.\n\n  [464] Jevons, *Proceedings of the Manchester Literary and\n  Philosophical Society*, 25th January, 1870, vol.\u00a0ix. p.\u00a078.\n\n  [465] *Philosophical Transactions*, vol.\u00a0cxlvi. p.\u00a0249.\n\n\n\n\nCHAPTER XXV.\n\nACCORDANCE OF QUANTITATIVE THEORIES.\n\n\nIn the preceding chapter we found that facts may be classed under\nfour heads as regards their connection with theory, and our powers\nof explanation or prediction. The facts hitherto considered were\ngenerally of a qualitative rather than a quantitative nature; but when\nwe look exclusively to the quantity of a phenomenon, and the various\nmodes in which we may determine its amount, nearly the same system of\nclassification will hold good. There will, however, be five possible\ncases:--\n\n(1) We may directly and empirically measure a phenomenon, without being\nable to explain why it should have any particular quantity, or to\nconnect it by theory with other quantities.\n\n(2) In a considerable number of cases we can theoretically predict the\nexistence of a phenomenon, but are unable to assign its amount, except\nby direct measurement, or to explain the amount theoretically when thus\nascertained.\n\n(3) We may measure a quantity, and afterwards explain it as related to\nother quantities, or as governed by known quantitative laws.\n\n(4) We may predict the quantity of an effect on theoretical grounds,\nand afterwards confirm the prediction by direct measurement.\n\n(5) We may indirectly determine the quantity of an effect without being\nable to verity it by experiment.\n\nThese classes of quantitative facts might be illustrated by an immense\nnumber of interesting points in the history of physical science. Only\na few instances of each class can be given here.\n\n\n*Empirical Measurements.*\n\nUnder the first head of purely empirical measurements, which have\nnot been brought under any theoretical system, may be placed the\ngreat bulk of quantitative facts recorded by scientific observers.\nThe tables of numerical results which abound in books on chemistry\nand physics, the huge quartos containing the observations of public\nobservatories, the multitudinous tables of meteorological observations,\nwhich are continually being published, the more abstruse results\nconcerning terrestrial magnetism--such results of measurement, for\nthe most part, remain empirical, either because theory is defective,\nor the labour of calculation and comparison is too formidable. In\nthe Greenwich Observatory, indeed, the salutary practice has been\nmaintained by the present Astronomer Royal, of always reducing the\nobservations, and comparing them with the theories of the several\nbodies. The divergences from theory thus afford material for the\ndiscovery of errors or of new phenomena; in short, the observations\nhave been turned to the use for which they were intended. But it is to\nbe feared that other establishments are too often engaged in merely\nrecording numbers of which no real use is made, because the labour\nof reduction and comparison with theory is too great for private\ninquirers to undertake. In meteorology, especially, great waste of\nlabour and money is taking place, only a small fraction of the results\nrecorded being ever used for the advancement of the science. For one\nmeteorologist like Quetelet, Dove, or Baxendell, who devotes himself to\nthe truly useful labour of reducing other people\u2019s observations, there\nare hundreds who labour under the delusion that they are advancing\nscience by loading our book-shelves with numerical tables. It is to\nbe feared, in like manner, that almost the whole bulk of statistical\nnumbers, whether commercial, vital, or moral, is of little scientific\nvalue. Purely empirical measurements may have a direct practical value,\nas when tables of the specific gravity, or strength of materials,\nassist the engineer; the specific gravities of mixtures of water with\nacids, alcohols, salts, &c., are useful in chemical manufactories,\ncustom-house gauging, &c.; observations of rainfall are requisite for\nquestions of water supply; the refractive index of various kinds of\nglass must be known in making achromatic lenses; but in all such cases\nthe use made of the measurements is not scientific but practical. It\nmay be asserted, that no number which remains isolated, and uncompared\nby theory with other numbers, is of scientific value. Having tried\nthe tensile strength of a piece of iron in a particular condition, we\nknow what will be the strength of the same kind of iron in a similar\ncondition, provided we can ever meet with that exact kind of iron\nagain; but we cannot argue from piece to piece, nor lay down any laws\nexactly connecting the strength of iron with the quantity of its\nimpurities.\n\n\n*Quantities indicated by Theory, but Empirically Measured.*\n\nIn many cases we are able to foresee the existence of a quantitative\neffect, on the ground of general principles, but are unable, either\nfrom the want of numerical data, or from the entire absence of any\nmathematical theory, to assign the amount of such effect. We then\nhave recourse to direct experiment to determine its amount. Whether\nwe argued from the oceanic tides by analogy, or deductively from the\ntheory of gravitation, there could be no doubt that atmospheric tides\nof some amount must occur in the atmosphere. Theory, however, even\nin the hands of Laplace, was not able to overcome the complicated\nmechanical conditions of the atmosphere, and predict the amounts of\nsuch tides; and, on the other hand, these amounts were so small, and\nwere so masked by far larger undulations arising from the heating power\nof the sun, and from other meteorological disturbances, that they would\nprobably have never been discovered by purely empirical observations.\nTheory having, however, indicated their existence and their periods, it\nwas easy to make series of barometrical observations in places selected\nso as to be as free as possible from casual fluctuations, and then, by\nthe suitable application of the method of means, to detect the small\neffects in question. The principal lunar atmospheric tide was thus\nproved to amount to between \u00b7003 and \u00b7004 inch.[466]\n\n  [466] Grant\u2019s *History of Physical Astronomy*, p.\u00a0162.\n\nTheory yields the greatest possible assistance in applying the method\nof means. For if we have a great number of empirical measurements, each\nrepresenting the joint effect of a number of causes, our object will\nbe to take the mean of all those in which the effect to be measured is\npresent, and compare it with the mean of the remainder in which the\neffect is absent, or acts in the opposite direction. The difference\nwill then represent the amount of the effect, or double the amount\nrespectively. Thus, in the case of the atmospheric tides, we take\nthe mean of all the observations when the moon was on the meridian,\nand compare it with the mean of all observations when she was on the\nhorizon. In this case we trust to chance that all other effects will\nlie about as often in one direction as the other, and will neutralise\nthemselves in the drawing of each mean. It is a great advantage,\nhowever, to be able to decide by theory when each principal disturbing\neffect is present or absent; for the means may then be drawn so as to\nseparate each such effect, leaving only minor and casual divergences\nto the law of error. Thus, if there be three principal effects, and\nwe draw means giving respectively the sum of all three, the sum of\nthe first two, and the sum of the last two, then we gain three simple\nequations, by the solution of which each quantity is determined.\n\n\n*Explained Results of Measurement.*\n\nThe second class of measured phenomena contains those which, after\nbeing determined in a direct and purely empirical application of\nmeasuring instruments, are afterwards shown to agree with some\nhypothetical explanation. Such results are turned to their proper\nuse, and several advantages may arise from the comparison. The\ncorrespondence with theory will seldom or never be precise; and, even\nif it be so, the coincidence must be regarded as accidental.\n\nIf the divergences between theory and experiment be comparatively\nsmall, and variable in amount and direction, they may often be safely\nattributed to inconsiderable sources of error in the experimental\nprocesses. The strict method of procedure is to calculate the probable\nerror of the mean of the observed results (p.\u00a0387), and then observe\nwhether the theoretical result falls within the limits of probable\nerror. If it does, and if the experimental results agree as well\nwith theory as they agree with each other, then the probability of\nthe theory is much increased, and we may employ the theory with more\nconfidence in the anticipation of further results. The probable error,\nit should be remembered, gives a measure only of the effects of\nincidental and variable sources of error, but in no degree indicates\nthe amount of fixed causes of error. Thus, if the mean results of\ntwo modes of determining a quantity are so far apart that the limits\nof probable error do not overlap, we may infer the existence of some\noverlooked source of fixed error in one or both modes. We will further\nconsider in a subsequent section the discordance of measurements.\n\n\n*Quantities determined by Theory and verified by Measurement.*\n\nOne of the most satisfactory tests of a theory consists in its\napplication not only to predict the nature of a phenomenon, and\nthe circumstances in which it may be observed, but also to assign\nthe precise quantity of the phenomenon. If we can subsequently\napply accurate instruments and measure the amount of the phenomenon\nwitnessed, we have an excellent opportunity of verifying or negativing\nthe theory. It was in this manner that Newton first attempted to verify\nhis theory of gravitation. He knew approximately the velocity produced\nin falling bodies at the earth\u2019s surface, and if the law of the inverse\nsquare of the distance held true, and the reputed distance of the\nmoon was correct, he could infer that the moon ought to fall towards\nthe earth at the rate of fifteen feet in one minute. Now, the actual\ndivergence of the moon from the tangent of its orbit appeared to amount\nonly to thirteen feet in one minute, and there was a discrepancy of\ntwo feet in fifteen, which caused Newton to lay \u201caside at that time\nany further thoughts of this matter.\u201d Many years afterwards, probably\nfifteen or sixteen years, Newton obtained more precise data from which\nhe could calculate the size of the moon\u2019s orbit, and he then found the\ndiscrepancy to be inconsiderable.\n\nHis theory of gravitation was thus verified as far as the moon was\nconcerned; but this was to him only the beginning of a long course of\ndeductive calculations, each ending in a verification. If the earth\nand moon attract each other, and also the sun and the earth, there\nis reason to expect that the sun and moon should attract each other.\nNewton followed out the consequences of this inference, and showed\nthat the moon would not move as if attracted by the earth only, but\nsometimes faster and sometimes slower. Comparison with Flamsteed\u2019s\nobservations of the moon showed that such was the case. Newton argued\nagain, that as the waters of the ocean are not rigidly attached to\nthe earth, they might attract the moon, and be attracted in return,\nindependently of the rest of the earth. Certain daily motions\nresembling the tides would then be caused, and there were the tides to\nverify the reasoning. It was the extraordinary power with which Newton\ntraced out geometrically the consequences of his theory, and submitted\nthem to repeated comparison with experience, which constitutes his\npre-eminence over all physicists.\n\n\n*Quantities determined by Theory and not verified.*\n\nIt will continually happen that we are able, from certain measured\nphenomena and a correct theory, to determine the amount of some other\nphenomenon which we may either be unable to measure at all, or to\nmeasure with an accuracy corresponding to that required to verify the\nprediction. Thus Laplace having worked out a theory of the motions of\nJupiter\u2019s satellites on the hypothesis of gravitation, found that these\nmotions were greatly affected by the spheroidal form of Jupiter. The\nmotions of the satellites can be observed with great accuracy owing to\ntheir frequent eclipses and transits, and from these motions he was\nable to argue inversely, and assign the ellipticity of the planet. The\nratio of the polar and equatorial axes thus determined was very nearly\nthat of 13 to 14; and it agrees well with such direct micrometrical\nmeasurements of the planet as have been made; but Laplace believed\nthat the theory gave a more accurate result than direct observation\ncould yield, so that the theory could hardly be said to admit of direct\nverification.\n\nThe specific heat of air was believed on the grounds of direct\nexperiment to amount to 0\u00b72669, the specific heat of water being taken\nas unity; but the methods of experiment were open to considerable\ncauses of error. Rankine showed in 1850 that it was possible\nto calculate from the mechanical equivalent of heat and other\nthermodynamic data, what this number should be, and he found it to\nbe 0\u00b72378. This determination was at the time accepted as the most\nsatisfactory result, although not verified; subsequently in 1853\nRegnault obtained by direct experiment the number 0\u00b72377, proving that\nthe prediction had been well grounded.\n\nIt is readily seen that in quantitative questions verification is a\nmatter of degree and probability. A less accurate method of measurement\ncannot verify the results of a more accurate method, so that if we\narrive at a determination of the same physical quantity in several\ndistinct modes it is often a delicate matter to decide which result\nis most reliable, and should be used for the indirect determination\nof other quantities. For instance, Joule\u2019s and Thomson\u2019s ingenious\nexperiments upon the thermal phenomena of fluids in motion[467]\ninvolved, as one physical constant, the mechanical equivalent of\nheat; if requisite, then, they might have been used to determine that\nimportant constant. But if more direct methods of experiment give\nthe mechanical equivalent of heat with superior accuracy, then the\nexperiments on fluids will be turned to a better use in determining\nvarious quantities relating to the theory of fluids. We will further\nconsider questions of this kind in succeeding sections.\n\n  [467] *Philosophical Transactions* (1854), vol.\u00a0cxliv. p.\u00a0364.\n\nThere are of course many quantities assigned on theoretical grounds\nwhich we are quite unable to verify with corresponding accuracy. The\nthickness of a film of gold leaf, the average depths of the oceans,\nthe velocity of a star\u2019s approach to or regression from the earth as\ninferred from spectroscopic data (pp.\u00a0296\u201399), are cases in point; but\nmany others might be quoted where direct verification seems impossible.\nNewton and subsequent physicists have measured light undulations, and\nby several methods we learn the velocity with which light travels.\nSince an undulation of the middle green is about five ten-millionths\nof a metre in length, and travels at the rate of nearly 300,000,000\nof metres per second, it follows that about 600,000,000,000,000\nundulations must strike in one second the retina of an eye which\nperceives such light. But how are we to verify such an astounding\ncalculation by directly counting pulses which recur six hundred\nbillions of times in a second?\n\n\n*Discordance of Theory and Experiment.*\n\nWhen a distinct want of accordance is found to exist between the\nresults of theory and direct measurement, interesting questions arise\nas to the mode in which we can account for this discordance. The\nultimate explanation of the discrepancy may be accomplished in at least\nfour ways as follows:--\n\n(1) The direct measurement may be erroneous owing to various sources of\ncasual error.\n\n(2) The theory may be correct as far as regards the general form of the\nsupposed laws, but some of the constant numbers or other quantitative\ndata employed in the theoretical calculations may be inaccurate.\n\n(3) The theory may be false, in the sense that the forms of the\nmathematical equations assumed to express the laws of nature are\nincorrect.\n\n(4) The theory and the involved quantities may be approximately\naccurate, but some regular unknown cause may have interfered, so that\nthe divergence may be regarded as a *residual effect* representing\npossibly a new and interesting phenomenon.\n\nNo precise rules can be laid down as to the best mode of proceeding to\nexplain the divergence, and the experimentalist will have to depend\nupon his own insight and knowledge; but the following recommendations\nmay be made.\n\nIf the experimental measurements are not numerous, repeat them and\ntake a more extensive mean result, the probable accuracy of which,\nas regards casual errors, will increase as the square root of the\nnumber of experiments. Supposing that no considerable modification\nof the result is thus effected, we may suspect the existence of more\ndeep-seated sources of error in our method of measurement. The next\nresource will be to change the size and form of the apparatus employed,\nand to introduce various modifications in the materials employed or the\ncourse of procedure, in the hope (p.\u00a0396) that some cause of constant\nerror may thus be removed. If the inconsistency with theory still\nremains unreduced we may attempt to invent some widely different mode\nof arriving at the same physical quantity, so that we may be almost\nsure that the same cause of error will not affect both the new and old\nresults. In some cases it is possible to find five or six essentially\ndifferent modes of arriving at the same determination.\n\nSupposing that the discrepancy still exists we may begin to suspect\nthat our direct measurements are correct, and that the data employed\nin the theoretical calculations are inaccurate. We must now review the\ngrounds on which these data depend, consisting as they must ultimately\ndo of direct measurements. A comparison of the recorded data will\nshow the degree of probability attaching to the mean result employed;\nand if there is any ground for imagining the existence of error, we\nshould repeat the observations, and vary the forms of experiment just\nas in the case of the previous direct measurements. The continued\nexistence of the discrepancy must show that we have not attained to\na complete acquaintance with the theory of the causes in action, but\ntwo different cases still remain. We may have misunderstood the action\nof those causes which we know to exist, or we may have overlooked the\nexistence of one or more other causes. In the first case our hypothesis\nappears to be wrongly chosen and inapplicable; but whether we are to\nreject it will depend upon whether we can form another hypothesis which\nyields a more accurate accordance. The probability of an hypothesis,\nit will be remembered (p.\u00a0243), is to be judged, in the absence of *\u00e0\npriori* grounds of judgment, by the probability that if the supposed\ncauses exist the observed result follows; but as there is now little\nprobability of reconciling the original hypothesis with our direct\nmeasurements the field is open for new hypotheses, and any one which\ngives a closer accordance with measurement will so far have better\nclaims to attention. Of course we must never estimate the probability\nof an hypothesis merely by its accordance with a few results only. Its\ngeneral analogy and accordance with other known laws of nature, and\nthe fact that it does not conflict with other probable theories, must\nbe taken into account, as we shall see in the next book. The requisite\ncondition of a good hypothesis, that it must admit of the deduction\nof facts verified in observation, must be interpreted in the widest\nmanner, as including all ways in which there may be accordance or\ndiscordance. All our attempts at reconciliation having failed, the only\nconclusion we can come to is that some unknown cause of a new character\nexists. If the measurements be accurate and the theory probable,\nthen there remains a *residual phenomenon*, which, being devoid of\ntheoretical explanation, must be set down as a new empirical fact\nworthy of further investigation. Outstanding residual discrepancies\nhave often been found to involve new discoveries of the greatest\nimportance.\n\n\n*Accordance of Measurements of Astronomical Distances.*\n\nOne of the most instructive instances which we can meet, of the\nmanner in which different measurements confirm or check each other,\nis furnished by the determination of the velocity of light, and the\ndimensions of the planetary system. Roemer first discovered that light\nrequires time to travel, by observing that the eclipses of Jupiter\u2019s\nsatellites, although they occur at fixed moments of absolute time, are\nvisible at different moments in different parts of the earth\u2019s orbit,\naccording to the distance between the earth and Jupiter. The time\noccupied by light in traversing the mean semi-diameter of the earth\u2019s\norbit is found to be about eight minutes. The mean distance of the sun\nand earth was long assumed by astronomers as being about 95,274,000\nmiles, this result being deduced by Bessel from the observations of the\ntransit of Venus, which occurred in 1769, and which were found to give\nthe solar parallax, or which is the same thing, the apparent angular\nmagnitude of the earth seen from the sun, as equal to 8\u2033\u00b7578. Dividing\nthe mean distance of the sun and earth by the number of seconds in\n8^{m}. 13^{s}.3 we find the velocity of light to be about 192,000 miles\nper second.\n\nNearly the same result was obtained in what seems a different manner.\nThe aberration of light is the apparent change in the direction of a\nray of light owing to the composition of its motion with that of the\nearth\u2019s motion round the sun. If we know the amount of aberration and\nthe mean velocity of the earth, we can estimate that of light, which\nis thus found to be 191,100 miles per second. Now this determination\ndepends upon a new physical quantity, that of aberration, which is\nascertained by direct observation of the stars, so that the close\naccordance of the estimates of the velocity of light as thus arrived\nat by different methods might seem to leave little room for doubt, the\ndifference being less than one per cent.\n\nNevertheless, experimentalists were not satisfied until they had\nsucceeded in measuring the velocity of light by direct experiments\nperformed upon the earth\u2019s surface. Fizeau, by a rapidly revolving\ntoothed wheel, estimated the velocity at 195,920 miles per second.\nAs this result differed by about one part in sixty from estimates\npreviously accepted, there was thought to be room for further\ninvestigation. The revolving mirror, used by Wheatstone in measuring\nthe velocity of electricity, was now applied in a more refined manner\nby Fizeau and by Foucault to determine the velocity of light. The\nlatter physicist came to the startling conclusion that the velocity\nwas not really more than 185,172 miles per second. No repetition of\nthe experiment would shake this result, and there was accordingly a\ndiscrepancy between the astronomical and the experimental results\nof about 7,000 miles per second. The latest experiments, those of\nM. Cornu, only slightly raise the estimate, giving 186,660 miles\nper second. A little consideration shows that both the astronomical\ndeterminations involve the magnitude of the earth\u2019s orbit as one datum,\nbecause our estimate of the earth\u2019s velocity in its orbit depends upon\nour estimate of the sun\u2019s mean distance. Accordingly as regards this\nquantity the two astronomical results count only for one. Though the\ntransit of Venus had been considered to give the best data for the\ncalculation of the sun\u2019s parallax, yet astronomers had not neglected\nless favourable opportunities. Hansen, calculating from certain\ninequalities in the moon\u2019s motion, had estimated it at 8\u2033\u00b7916; Winneke,\nfrom observations of Mars, at 8\u2033\u00b7964; Leverrier, from the motions\nof Mars, Venus, and the moon, at 8\u2033\u00b7950. These independent results\nagree much better with each other than with that of Bessel (8\u2033\u00b7578)\npreviously received, or that of Encke (8\u2033\u00b758) deduced from the transits\nof Venus in 1761 and 1769, and though each separately might be worthy\nof less credit, yet their close accordance renders their mean result\n(8\u2033\u00b7943) comparable in probability with that of Bessel. It was further\nfound that if Foucault\u2019s value for the velocity of light were assumed\nto be correct, and the sun\u2019s distance were inversely calculated from\nthat, the sun\u2019s parallax would be 8\u2033\u00b7960, which closely agreed with\nthe above mean result. This further correspondence of independent\nresults threw the balance of probability strongly against the results\nof the transit of Venus, and rendered it desirable to reconsider\nthe observations made on that occasion. Mr. E.\u00a0J. Stone, having\nre-discussed those observations,[468] found that grave oversights had\nbeen made in the calculations, which being corrected would alter the\nestimate of parallax to 8\u2033\u00b791, a quantity in such comparatively close\naccordance with the other results that astronomers did not hesitate\nat once to reduce their estimate of the sun\u2019s mean distance from\n95,274,000 to 91,771,000, miles, although this alteration involved a\ncorresponding correction in the assumed magnitudes and distances of\nmost of the heavenly bodies. The solar parallax is now (1875) believed\nto be about 8\u2033\u00b7878, the number deduced from Cornu\u2019s experiments on the\nvelocity of light. This result agrees very closely with 8\u2033\u00b7879, the\nestimate obtained from new observations on the transit of Venus, by the\nFrench observers, and with 8\u2033\u00b7873, the result of Galle\u2019s observations\nof the planet Flora. When all the observations of the late transit of\nVenus are fully discussed the sun\u2019s distance will probably be known to\nless than one part in a thousand, if not one part in ten thousand.[469]\n\n  [468] *Monthly Notices of the Royal Astronomical Society*,\n  vol.\u00a0xxviii. p. 264.\n\n  [469] It would seem to be absurd to repeat the profuse expenditure of\n  1874 at the approaching transit in 1882. The aggregate sum spent in\n  1874 by various governments and individuals can hardly be less than\n  \u00a3200,000, a sum which, wisely expended on scientific investigations,\n  would give a hundred important results.\n\nIn this question the theoretical relations between the velocity of\nlight, the constant of aberration, the sun\u2019s parallax, and the sun\u2019s\nmean distance, are of the simplest character, and can hardly be\nopen to any doubt, so that the only doubt was as to which result of\nobservation was the most reliable. Eventually the chief discrepancy was\nfound to arise from misapprehension in the reduction of observations,\nbut we have a satisfactory example of the value of different methods\nof estimation in leading to the detection of a serious error. Is it\nnot surprising that Foucault by measuring the velocity of light when\npassing through the space of a few yards, should lead the way to a\nchange in our estimates of the magnitudes of the whole universe?\n\n\n*Selection of the best Mode of Measurement.*\n\nWhen we once obtain command over a question of physical science\nby comprehending the theory of the subject, we often have a wide\nchoice opened to us as regards the methods of measurement, which may\nthenceforth be made to give the most accurate results. If we can\nmeasure one fundamental quantity very precisely we may be able by\ntheory to determine accurately many other quantitative results. Thus,\nif we determine satisfactorily the atomic weights of certain elements,\nwe do not need to determine with equal accuracy the composition and\natomic weights of their several compounds. Having learnt the relative\natomic weights of oxygen and sulphur, we can calculate the composition\nby weight of the several oxides of sulphur. Chemists accordingly select\nwith the greatest care that compound of two elements which seems to\nallow of the most accurate analysis, so as to give the ratio of their\natomic weights. It is obvious that we only need the ratio of the atomic\nweight of each element to that of some common element, in order to\ncalculate, that of each to each. Moreover the atomic weight stands\nin simple relation to other quantitative facts. The weights of equal\nvolumes of elementary gases at equal temperature and pressure have\nthe same ratios as the atomic weights; now, as nitrogen under such\ncircumstances weighs 14\u00b706 times as much as hydrogen, we may infer that\nthe atomic weight of nitrogen is about 14\u00b706, or more probably 14\u00b700,\nthat of hydrogen being unity. There is much evidence, again, that the\nspecific heats of elements are inversely as their atomic weights, so\nthat these two classes of quantitative data throw light mutually upon\neach other. In fact the atomic weight, the atomic volume, and the\natomic heat of an element, are quantities so closely connected that\nthe determination of one will lead to that of the others. The chemist\nhas to solve a complicated problem in deciding in the case of each\nof 60 or 70 elements which mode of determination is most accurate.\nModern chemistry presents us with an almost infinitely extensive web of\nnumerical ratios developed out of a few fundamental ratios.\n\nIn hygrometry we have a choice among at least four modes of measuring\nthe quantity of aqueous vapour contained in a given bulk of air. We\ncan extract the vapour by absorption in sulphuric acid, and directly\nweigh its amount; we can place the air in a barometer tube and observe\nhow much the absorption of the vapour alters the elastic force of the\nair; we can observe the dew-point of the air, that is the temperature\nat which the vapour becomes saturated; or, lastly, we can insert a dry\nand wet bulb thermometer and observe the temperature of an evaporating\nsurface. The results of each mode can be connected by theory with\nthose of the other modes, and we can select for each experiment that\nmode which is most accurate or most convenient. The chemical method\nof direct measurement is capable of the greatest accuracy, but is\ntroublesome; the dry and wet bulb thermometer is sufficiently exact for\nmeteorological purposes and is most easy to use.\n\n\n*Agreement of Distinct Modes of Measurement.*\n\nMany illustrations might be given of the accordance which has been\nfound to exist in some cases between the results of entirely different\nmethods of arriving at the measurement of a physical quantity. While\nsuch accordance must, in the absence of information to the contrary,\nbe regarded as the best possible proof of the approximate correctness\nof the mean result, yet instances have occurred to show that we can\nnever take too much trouble in confirming results of great importance.\nWhen three or even more distinct methods have given nearly coincident\nnumbers, a new method has sometimes disclosed a discrepancy which it is\nyet impossible to explain.\n\nThe ellipticity of the earth is known with considerable approach to\ncertainty and accuracy, for it has been estimated in three independent\nways. The most direct mode is to measure long arcs extending north and\nsouth upon the earth\u2019s surface, by means of trigonometrical surveys,\nand then to compare the lengths of these arcs with their curvature as\ndetermined by observations of the altitude of certain stars at the\nterminal points. The most probable ellipticity of the earth deduced\nfrom all measurements of this kind was estimated by Bessel at 1/300,\nthough subsequent measurements might lead to a slightly different\nestimate. The divergence from a globular form causes a small variation\nin the force of gravity at different parts of the earth\u2019s surface,\nso that exact pendulum observations give the data for an independent\nestimate of the ellipticity, which is thus found to be 1/320. In the\nthird place the spheroidal protuberance about the earth\u2019s equator leads\nto a certain inequality in the moon\u2019s motion, as shown by Laplace;\nand from the amount of that inequality, as given by observations,\nLaplace was enabled to calculate back to the amount of its cause.\nHe thus inferred that the ellipticity is 1/305, which lies between\nthe two numbers previously given, and was considered by him the most\nsatisfactory determination. In this case the accordance is undisturbed\nby subsequent results, so that we are obliged to accept Laplace\u2019s\nresult as a highly probable one.\n\nThe mean density of the earth is a constant of high importance, because\nit is necessary for the determination of the masses of all the other\nheavenly bodies. Astronomers and physicists accordingly have bestowed\na great deal of labour upon the exact estimation of this constant.\nThe method of procedure consists in comparing the gravitation of the\nglobe with that of some body of matter of which the mass is known in\nterms of the assumed unit of mass. This body of matter, serving as\nan intermediate term of comparison, may be variously chosen; it may\nconsist of a mountain, or a portion of the earth\u2019s crust, or a heavy\nball of metal. The method of experiment varies so much according as\nwe select one body or the other, that we may be said to have three\nindependent modes of arriving at the desired result.\n\nThe mutual gravitation of two balls is so exceedingly small compared\nwith their gravitation towards the immense mass of the earth, that it\nis usually quite imperceptible, and although asserted by Newton to\nexist, on the ground of theory, was never observed until the end of the\n18th century. Michell attached two small balls to the extremities of\na delicately suspended torsion balance, and then bringing heavy balls\nof lead alternately to either side of these small balls was able to\ndetect a slight deflection of the torsion balance. He thus furnished a\nnew verification of the theory of gravitation. Cavendish carried out\nthe experiment with more care, and estimated the gravitation of the\nballs by treating the torsion balance as a pendulum; then taking into\naccount the respective distances of the balls from each other and from\nthe centre of the earth, he was able to assign 5\u00b748 (or as re-computed\nby Baily, 5\u00b7448) as the probable mean density of the earth. Newton\u2019s\nsagacious guess to the effect that the density of the earth was between\nfive and six times that of water, was thus remarkably confirmed. The\nsame kind of experiment repeated by Reich gave 5\u00b7438. Baily having\nagain performed the experiment with every possible refinement obtained\na slightly higher number, 5\u00b7660.\n\nA different method of procedure consisted in ascertaining the effect\nof a mountain mass in deflecting the plumb-line; for, assuming that\nwe can determine the dimensions and mean density of the mountain,\nthe plumb-line enables us to compare its mass with that of the whole\nearth. The mountain Schehallien was selected for the experiment, and\nobservations and calculations performed by Maskelyne, Hutton, and\nPlayfair, gave as the most probable result 4\u00b7713. The difference from\nthe experimental results already mentioned is considerable and is\nimportant, because the instrumental operations are of an entirely\ndifferent character from those of Cavendish and Baily\u2019s experiments.\nSir Henry James\u2019 similar determination from the attraction of Arthur\u2019s\nSeat gave 5\u00b714.\n\nA third distinct method consists in determining the force of gravity\nat points elevated above the surface of the earth on mountain ranges,\nor sunk below it in mines. Carlini experimented with a pendulum at the\nhospice of Mont Cenis, 6,375 feet above the sea, and by comparing the\nattractive forces of the earth and the Alps, found the density to be\nstill smaller, namely, 4\u00b739, or as corrected by Giulio, 4\u00b7950. Lastly,\nthe Astronomer Royal has on two occasions adopted the opposite method\nof observing a pendulum at the bottom of a deep mine, so as to compare\nthe density of the strata penetrated with the density of the whole\nearth. On the second occasion he carried his method into effect at the\nHarton Colliery, 1,260 feet deep; all that could be done by skill in\nmeasurement and careful consideration of all the causes of error, was\naccomplished in this elaborate series of observations[470] (p.\u00a0291).\nNo doubt Sir George Airy was much perplexed when he found that his\nnew result considerably exceeded that obtained by any other method,\nbeing no less than 6\u00b7566, or 6\u00b7623 as finally corrected. In this\ncase we learn an impressive lesson concerning the value of repeated\ndeterminations by distinct methods in disabusing our minds of the\nreliance which we are only too apt to place in results which show a\ncertain degree of coincidence.\n\n  [470] *Philosophical Transactions* (1856), vol.\u00a0cxlvi. p.\u00a0342.\n\nIn 1844 Herschel remarked in his memoir of Francis Baily,[471]\n\u201cthat the mean specific gravity of this our planet is, in all\nhuman probability, quite as well determined as that of an ordinary\nhand-specimen in a mineralogical cabinet,--a marvellous result, which\nshould teach us to despair of nothing which lies within the compass of\nnumber, weight and measure.\u201d But at the same time he pointed out that\nBaily\u2019s final result, of which the probable error was only 0\u00b70032, was\nthe highest of all determinations then known, and Airy\u2019s investigation\nhas since given a much higher result, quite beyond the limits of\nprobable error of any of the previous experiments. If we treat all\ndeterminations yet made as of equal weight, the simple mean is about\n5\u00b745, the mean error nearly 0\u00b75, and the probable error almost 0\u00b72, so\nthat it is as likely as not that the truth lies between 5\u00b765 and 5\u00b725\non this view of the matter. But it is remarkable that the two most\nrecent and careful series of observations by Baily and Airy,[472] lie\nbeyond these limits, and as with the increase of care the estimate\nrises, it seems requisite to reject the earlier results, and look upon\nthe question as still requiring further investigation. Physicists\noften take 5-2/3 or 5\u00b767 as the best guess at the truth, but it is\nevident that new experiments are much required. I cannot help thinking\nthat a portion of the great sums of money which many governments and\nprivate individuals spent upon the transit of Venus expeditions in\n1874, and which they will probably spend again in 1882 (p.\u00a0562), would\nbe better appropriated to new determinations of the earth\u2019s density.\nIt seems desirable to repeat Baily\u2019s experiment in a vacuous case,\nand with the greater mechanical refinements which the progress of the\nlast forty years places at the disposal of the experimentalist. It\nwould be desirable, also, to renew the pendulum experiments of Airy\nin some other deep mine. It might even be well to repeat upon some\nsuitable mountain the observations performed at Schehallien. All these\noperations might be carried out for the cost of one of the superfluous\ntransit expeditions.\n\n  [471] *Monthly Notices of the Royal Astronomical Society*, for 8th\n  Nov. 1844, No. X. vol.\u00a0vi. p.\u00a089.\n\n  [472] *Philosophical Magazine*, 2nd Series, vol.\u00a0xxvi. p.\u00a061.\n\nSince the establishment of the dynamical theory of heat it has become\na matter of the greatest importance to determine with accuracy the\nmechanical equivalent of heat, or the quantity of energy which must\nbe given, or received, in a definite change of temperature effected\nin a definite quantity of a standard substance, such as water. No\nless than seven almost entirely distinct modes of determining this\nconstant have been tried. Dr. Joule first ascertained by the friction\nof water that to raise the temperature of one kilogram of water through\none degree centigrade, we must employ energy sufficient to raise\n424 kilograms through the height of one metre against the force of\ngravity at the earth\u2019s surface. Joule, Mayer, Clausius,[473] Favre\nand other experimentalists have made determinations by less direct\nmethods. Experiments on the mechanical properties of gases give 426\nkilogrammetres as the constant; the work done by a steam-engine\ngives 413; from the heat evolved in electrical experiments several\ndeterminations have been obtained; thus from induced electric currents\nwe get 452; from the electro-magnetic engine 443; from the circuit of a\nbattery 420; and, from an electric current, the lowest result of all,\nnamely, 400.[474]\n\n  [473] Clausius in *Philosophical Magazine*, 4th Series, vol.\u00a0ii.\n  p.\u00a0119.\n\n  [474] Watts\u2019 *Dictionary of Chemistry*, vol.\u00a0iii. p.\u00a0129.\n\nConsidering the diverse and in many cases difficult methods of\nobservation, these results exhibit satisfactory accordance, and their\nmean (423\u00b79) comes very close to the number derived by Dr. Joule from\nthe apparently most accurate method. The constant generally assumed as\nthe most probable result is 423\u00b755 kilogrammetres.\n\n\n*Residual Phenomena.*\n\nEven when the experimental data employed in the verification of a\ntheory are sufficiently accurate, and the theory itself is sound, there\nmay exist discrepancies demanding further investigation. Herschel\npointed out the importance of such outstanding quantities, and called\nthem *residual phenomena*.[475] Now if the observations and the theory\nbe really correct, such discrepancies must be due to the incompleteness\nof our knowledge of the causes in action, and the ultimate explanation\nmust consist in showing that there is in action, either\n\n  [475] *Preliminary Discourse*, \u00a7\u00a7 158, 174. *Outlines of Astronomy*,\n  4th edit. \u00a7 856.\n\n(1) Some agent of known nature whose presence was not suspected;\n\nOr (2) Some new agent of unknown nature.\n\nIn the first case we can hardly be said to make a new discovery, for\nour ultimate success consists merely in reconciling the theory with\nknown facts when our investigation is more comprehensive. But in\nthe second case we meet with a totally new fact, which may lead us\nto realms of new discovery. Take the instance adduced by Herschel.\nThe theory of Newton and Halley concerning comets was that they\nwere gravitating bodies revolving round the sun in elliptic orbits,\nand the return of Halley\u2019s Comet, in 1758, verified this theory.\nBut, when accurate observations of Encke\u2019s Comet came to be made,\nthe verification was not found to be exact. Encke\u2019s Comet returned\neach time a little sooner than it ought to do, the period regularly\ndecreasing from 1212\u00b779 days, between 1786 and 1789, to 1210\u00b744 between\n1855 and 1858; and the hypothesis has been started that there is a\nresisting medium filling the space through which the comet passes.\nThis hypothesis is a *deus ex machin\u00e2* for explaining this solitary\nphenomenon, and cannot possess much probability unless it can be\nshown that other phenomena are deducible from it. Many persons have\nidentified this medium with that through which light undulations pass,\nbut I am not aware that there is anything in the undulatory theory of\nlight to show that the medium would offer resistance to a moving body.\nIf Professor Balfour Stewart can prove that a rotating disc would\nexperience resistance in a vacuous receiver, here is an experimental\nfact which distinctly supports the hypothesis. But in the mean time\nit is open to question whether other known agents, for instance\nelectricity, may not be brought in, and I have tried to show that if,\nas is believed, the tail of a comet is an electrical phenomenon, it\nis a necessary result of the conservation of energy that the comet\nshall exhibit a loss of energy manifested in a diminution of its mean\ndistance from the sun and its period of revolution.[476] It should\nbe added that if Professor Tait\u2019s theory be correct, as seems very\nprobable, and comets consist of swarms of small meteors, there is no\ndifficulty in accounting for the retardation. It has long been known\nthat a collection of small bodies travelling together in an orbit round\na central body will tend to fall towards it. In either case, then, this\nresidual phenomenon seems likely to be reconciled with known laws of\nnature.\n\n  [476] *Proceedings of the Manchester Literary and Philosophical\n  Society*, 28th November, 1871, vol.\u00a0xi. p.\u00a033. Since the above\n  remarks were written, Professor Balfour Stewart has pointed out to\n  me his paper in the *Proceedings of the Manchester Literary and\n  Philosophical Society* for 15th November, 1870 (vol.\u00a0x. p.\u00a032),\n  in which he shows that a body moving in an enclosure of uniform\n  temperature would probably experience resistance independently\n  of the presence of a ponderable medium, such as gas, between the\n  moving body and the enclosure. The proof is founded on the theory\n  of the dissipation of energy, and this view is said to be accepted\n  by Professors Thomson and Tait. The enclosure is used in this case\n  by Professor Stewart simply as a means of obtaining a proof, just\n  as it was used by him on a previous occasion to obtain a proof of\n  certain consequences of the Theory of Exchanges. He is of opinion\n  that in both of these cases when once the proof has been obtained,\n  the enclosure may be dispensed with. We know, for instance, that\n  the relation between the inductive and absorptive powers of\n  bodies--although this relation may have been proved by means of an\n  enclosure, does not depend upon its presence, and Professor Stewart\n  thinks that in like manner two bodies, or at least two bodies\n  possessing heat such as the sun and the earth in motion relative to\n  each other, will have the differential motion retarded until perhaps\n  it is ultimately destroyed.\n\nIn other cases residual phenomena have involved important inferences\nnot recognised at the time. Newton showed how the velocity of sound in\nthe atmosphere could be calculated by a theory of pulses or undulations\nfrom the observed tension and density of the air. He inferred that\nthe velocity in the ordinary state of the atmosphere at the earth\u2019s\nsurface would be 968 feet per second, and rude experiments made by\nhim in the cloisters of Trinity College seemed to show that this was\nnot far from the truth. Subsequently it was ascertained by other\nexperimentalists that the velocity of sound was more nearly 1,142 feet,\nand the discrepancy being one-sixth part of the whole was far too much\nto attribute to casual errors in the numerical data. Newton attempted\nto explain away this discrepancy by hypotheses as to the reactions of\nthe molecules of air, but without success.\n\nNew investigations having been made from time to time concerning the\nvelocity of sound, both as observed experimentally and as calculated\nfrom theory, it was found that each of Newton\u2019s results was inaccurate,\nthe theoretical velocity being 916 feet per second, and the real\nvelocity about 1,090 feet. The discrepancy, nevertheless, remained\nas serious as ever, and it was not until the year 1816 that Laplace\nshowed it to be due to the heat developed by the sudden compression\nof the air in the passage of the wave, this heat having the effect of\nincreasing the elasticity of the air and accelerating the impulse. It\nis now perceived that this discrepancy really involves the doctrine\nof the equivalence of heat and energy, and it was applied by Mayer,\nat least by implication, to give an estimate of the mechanical\nequivalent of heat. The estimate thus derived agrees satisfactorily\nwith direct determinations by Dr. Joule and other physicists, so that\nthe explanation of the residual phenomenon which exercised Newton\u2019s\ningenuity is now complete, and forms an important part of the new\nscience of thermodynamics.\n\nAs Herschel observed, almost all great astronomical discoveries\nhave been disclosed in the form of residual differences. It is the\npractice at well-conducted observatories to compare the positions of\nthe heavenly bodies as actually observed with what might have been\nexpected theoretically. This practice was introduced by Halley when\nAstronomer Royal, and his reduction of the lunar observations gave a\nseries of residual errors from 1722 to 1739, by the examination of\nwhich the lunar theory was improved. Most of the greater astronomical\nvariations arising from nutation, aberration, planetary perturbation\nwere discovered in the same manner. The precession of the equinox was\nperhaps the earliest residual difference observed; the systematic\ndivergence of Uranus from its calculated places was one of the latest,\nand was the clue to the remarkable discovery of Neptune. We may also\nclass under residual phenomena all the so-called *proper motions* of\nthe stars. A complete star catalogue, such as that of the British\nAssociation, gives a greater or less amount of proper motion for almost\nevery star, consisting in the apparent difference of position of the\nstar as derived from the earliest and latest good observations. But\nthese apparent motions are often due, as explained by Baily,[477] the\nauthor of the catalogue, to errors of observation and reduction. In\nmany cases the best astronomical authorities have differed as to the\nvery direction of the supposed proper motion of stars, and as regards\nthe amount of the motion, for instance of \u03b1 Polaris, the most different\nestimates have been formed. Residual quantities will often be so small\nthat their very existence is doubtful. Only the gradual progress of\ntheory and of measurement will show clearly whether a discrepancy\nis to be referred to casual errors of observation or to some new\nphenomenon. But nothing is more requisite for the progress of science\nthan the careful recording and investigation of such discrepancies.\nIn no part of physical science can we be free from exceptions and\noutstanding facts, of which our present knowledge can give no account.\nIt is among such anomalies that we must look for the clues to new\nrealms of facts worthy of discovery. They are like the floating waifs\nwhich led Columbus to suspect the existence of the new world.\n\n  [477] *British Association Catalogue of Stars*, p.\u00a049.\n\n\n\n\nCHAPTER XXVI.\n\nCHARACTER OF THE EXPERIMENTALIST.\n\n\nIn the present age there seems to be a tendency to believe that the\nimportance of individual genius is less than it was--\n\n  \u201cThe individual withers, and the world is more and more.\u201d\n\nSociety, it is supposed, has now assumed so highly developed a form,\nthat what was accomplished in past times by the solitary exertions of a\ngreat intellect, may now be worked out by the united labours of an army\nof investigators. Just as the well-organised power of a modern army\nsupersedes the single-handed bravery of the medi\u00e6val knights, so we are\nto believe that the combination of intellectual labour has superseded\nthe genius of an Archimedes, a Newton, or a Laplace. So-called original\nresearch is now regarded as a profession, adopted by hundreds of\nmen, and communicated by a system of training. All that we need to\nsecure additions to our knowledge of nature is the erection of great\nlaboratories, museums, and observatories, and the offering of pecuniary\nrewards to those who can invent new chemical compounds, detect new\nspecies, or discover new comets. Doubtless this is not the real meaning\nof the eminent men who are now urging upon Government the endowment of\nphysical research. They can only mean that the greater the pecuniary\nand material assistance given to men of science, the greater the result\nwhich the available genius of the country may be expected to produce.\nMoney and opportunities of study can no more produce genius than\nsunshine and moisture can generate living beings; the inexplicable\ngerm is wanting in both cases. But as, when the germ is present, the\nplant will grow more or less vigorously according to the circumstances\nin which it is placed, so it may be allowed that pecuniary assistance\nmay favour development of intellect. Public opinion however is not\ndiscriminating, and is likely to interpret the agitation for the\nendowment of science as meaning that science can be had for money.\n\nAll such notions are erroneous. In no branch of human affairs,\nneither in politics, war, literature, industry, nor science, is the\ninfluence of genius less considerable than it was. It is possible\nthat the extension and organisation of scientific study, assisted by\nthe printing-press and the accelerated means of communication, has\nincreased the rapidity with which new discoveries are made known, and\ntheir details worked out by many heads and hands. A Darwin now no\nsooner propounds original ideas concerning the evolution of living\ncreatures, than those ideas are discussed and illustrated, and\napplied by naturalists in every part of the world. In former days his\ndiscoveries would have been hidden for decades of years in scarce\nmanuscripts, and generations would have passed away before his theory\nhad enjoyed the same amount of criticism and corroboration as it has\nalready received. The result is that the genius of Darwin is more\nvaluable, not less valuable, than it would formerly have been. The\nadvance of military science and the organisation of enormous armies\nhas not decreased the value of a skilful general; on the contrary,\nthe rank and file are still more in need than they used to be of the\nguiding power of a far-seeing intellect. The swift destruction of the\nFrench military power was not due alone to the perfection of the German\narmy, nor to the genius of Moltke; it was due to the combination of a\nwell-disciplined multitude with a leader of the highest powers. So in\nevery branch of human affairs the influence of the individual is not\nwithering, but is growing with the extent of the material resources\nwhich are at his command.\n\nTurning to our own subject, it is a work of undiminished interest to\nreflect upon those qualities of mind which lead to great advances in\nnatural knowledge. Nothing, indeed, is less amenable than genius to\nscientific analysis and explanation. Even definition is out of the\nquestion. Buffon said that \u201cgenius is patience,\u201d and certainly patience\nis one of its most requisite components. But no one can suppose that\npatient labour alone will invariably lead to those conspicuous results\nwhich we attribute to genius. In every branch of science, literature,\nart, or industry, there are thousands of men and women who work with\nunceasing patience, and thereby ensure moderate success; but it would\nbe absurd to suppose that equal amounts of intellectual labour yield\nequal results. A Newton may modestly attribute his discoveries to\nindustry and patient thought, and there is reason to believe that\ngenius is unconscious and unable to account for its own peculiar\npowers. As genius is essentially creative, and consists in divergence\nfrom the ordinary grooves of thought and action, it must necessarily be\na phenomenon beyond the domain of the laws of nature. Nevertheless, it\nis always an interesting and instructive work to trace out, as far as\npossible, the characteristics of mind by which great discoveries have\nbeen achieved, and we shall find in the analysis much to illustrate the\nprinciples of scientific method.\n\n\n*Error of the Baconian Method.*\n\nHundreds of investigators may be constantly engaged in experimental\ninquiry; they may compile numberless note-books full of scientific\nfacts, and endless tables of numerical results; but, if the views of\ninduction here maintained be true, they can never by such work alone\nrise to new and great discoveries. By a system of research they may\nwork out deductively the details of a previous discovery, but to arrive\nat a new principle of nature is another matter. Francis Bacon spread\nabroad the notion that to advance science we must begin by accumulating\nfacts, and then draw from them, by a process of digestion, successive\nlaws of higher and higher generality. In protesting against the false\nmethod of the scholastic logicians, he exaggerated a partially true\nphilosophy, until it became as false as that which preceded it. His\nnotion of scientific method was a kind of scientific bookkeeping.\nFacts were to be indiscriminately gathered from every source, and\nposted in a ledger, from which would emerge in time a balance of\ntruth. It is difficult to imagine a less likely way of arriving at\ngreat discoveries. The greater the array of facts, the less is the\nprobability that they will by any routine system of classification\ndisclose the laws of nature they embody. Exhaustive classification in\nall possible orders is out of the question, because the possible orders\nare practically infinite in number.\n\nIt is before the glance of the philosophic mind that facts must display\ntheir meaning, and fall into logical order. The natural philosopher\nmust therefore have, in the first place, a mind of impressionable\ncharacter, which is affected by the slightest exceptional phenomenon.\nHis associating and identifying powers must be great, that is, a\nstrange fact must suggest to his mind whatever of like nature has\npreviously come within his experience. His imagination must be active,\nand bring before his mind multitudes of relations in which the\nunexplained facts may possibly stand with regard to each other, or to\nmore common facts. Sure and vigorous powers of deductive reasoning\nmust then come into play, and enable him to infer what will happen\nunder each supposed condition. Lastly, and above all, there must be the\nlove of certainty leading him diligently and with perfect candour, to\ncompare his speculations with the test of fact and experiment.\n\n\n*Freedom of Theorising.*\n\nIt would be an error to suppose that the great discoverer seizes at\nonce upon the truth, or has any unerring method of divining it. In all\nprobability the errors of the great mind exceed in number those of the\nless vigorous one. Fertility of imagination and abundance of guesses at\ntruth are among the first requisites of discovery; but the erroneous\nguesses must be many times as numerous as those which prove well\nfounded. The weakest analogies, the most whimsical notions, the most\napparently absurd theories, may pass through the teeming brain, and no\nrecord remain of more than the hundredth part. There is nothing really\nabsurd except that which proves contrary to logic and experience. The\ntruest theories involve suppositions which are inconceivable, and no\nlimit can really be placed to the freedom of hypothesis.\n\nKepler is an extraordinary instance to this effect. No minor laws\nof nature are more firmly established than those which he detected\nconcerning the orbits and motions of planetary masses, and on these\nempirical laws the theory of gravitation was founded. Did we not learn\nfrom his own writings the multitude of errors into which he fell, we\nmight have imagined that he had some special faculty of seizing on the\ntruth. But, as is well known, he was full of chimerical notions; his\nfavourite and long-studied theory was founded on a fanciful analogy\nbetween the planetary orbits and the regular solids. His celebrated\nlaws were the outcome of a lifetime of speculation, for the most part\nvain and groundless. We know this because he had a curious pleasure\nin dwelling upon erroneous and futile trains of reasoning, which\nmost persons consign to oblivion. But Kepler\u2019s name was destined to\nbe immortal, on account of the patience with which he submitted his\nhypotheses to comparison with observation, the candour with which he\nacknowledged failure after failure, and the perseverance and ingenuity\nwith which he renewed his attack upon the riddles of nature.\n\nNext after Kepler perhaps Faraday is the physical philosopher who has\ngiven us the best insight into the progress of discovery, by recording\nerroneous as well as successful speculations. The recorded notions,\nindeed, are probably but a tithe of the fancies which arose in his\nactive brain. As Faraday himself said--\u201cThe world little knows how\nmany of the thoughts and theories which have passed through the mind\nof a scientific investigator, have been crushed in silence and secrecy\nby his own severe criticism and adverse examination; that in the most\nsuccessful instances not a tenth of the suggestions, the hopes, the\nwishes, the preliminary conclusions have been realised.\u201d\n\nNevertheless, in Faraday\u2019s researches, published in the *Philosophical\nTransactions*, in minor papers, in manuscript note-books, or in other\nmaterials, made known in his interesting life by Dr. Bence Jones, we\nfind invaluable lessons for the experimentalist. These writings are\nfull of speculations which we must not judge by the light of subsequent\ndiscovery. It may perhaps be said that Faraday committed to the\nprinting press crude ideas which a friend would have counselled him to\nkeep back. There was occasionally even a wildness and vagueness in his\nnotions, which in a less careful experimentalist would have been fatal\nto the attainment of truth. This is especially apparent in a curious\npaper concerning Ray-vibrations; but fortunately Faraday was aware of\nthe shadowy character of his speculations, and expressed the feeling in\nwords which must be quoted. \u201cI think it likely,\u201d he says,[478] \u201cthat\nI have made many mistakes in the preceding pages, for even to myself\nmy ideas on this point appear only as the shadow of a speculation, or\nas one of those impressions upon the mind, which are allowable for a\ntime as guides to thought and research. He who labours in experimental\ninquiries knows how numerous these are, and how often their apparent\nfitness and beauty vanish before the progress and development of real\nnatural truth.\u201d If, then, the experimentalist has no royal road to the\ndiscovery of the truth, it is an interesting matter to consider by what\nlogical procedure he attains the truth.\n\n  [478] *Experimental Researches in Chemistry and Physics*, p.\u00a0372.\n  *Philosophical Magazine*, 3rd Series, May 1846, vol.\u00a0xxviii. p.\u00a0350.\n\nIf I have taken a correct view of logical method, there is really no\nsuch thing as a distinct process of induction. The probability is\ninfinitely small that a collection of complicated facts will fall\ninto an arrangement capable of exhibiting directly the laws obeyed\nby them. The mathematician might as well expect to integrate his\nfunctions by a ballot-box, as the experimentalist to draw deep truths\nfrom haphazard trials. All induction is but the inverse application\nof deduction, and it is by the inexplicable action of a gifted mind\nthat a multitude of heterogeneous facts are ranged in luminous order\nas the results of some uniformly acting law. So different, indeed,\nare the qualities of mind required in different branches of science,\nthat it would be absurd to attempt to give an exhaustive description\nof the character of mind which leads to discovery. The labours of\nNewton could not have been accomplished except by a mind of the utmost\nmathematical genius; Faraday, on the other hand, has made the most\nextensive additions to human knowledge without passing beyond common\narithmetic. I do not remember meeting in Faraday\u2019s writings with a\nsingle algebraic formula or mathematical problem of any complexity.\nProfessor Clerk Maxwell, indeed, in the preface to his new *Treatise\non Electricity*, has strongly recommended the reading of Faraday\u2019s\nresearches by all students of science, and has given his opinion\nthat though Faraday seldom or never employed mathematical formul\u00e6,\nhis methods and conceptions were not the less mathematical in their\nnature.[479] I have myself protested against the prevailing confusion\nbetween a mathematical and an exact science,[480] yet I certainly think\nthat Faraday\u2019s experiments were for the most part qualitative, and that\nhis mathematical ideas were of a rudimentary character. It is true that\nhe could not possibly investigate such a subject as magne-crystallic\naction without involving himself in geometrical relations of some\ncomplexity. Nevertheless I think that he was deficient in mathematical\ndeductive power, that power which is so highly developed by the modern\nsystem of mathematical training at Cambridge.\n\n  [479] See also *Nature*, September 18, 1873; vol.\u00a0viii. p.\u00a0398.\n\n  [480] *Theory of Political Economy*, pp.\u00a03\u201314.\n\nFaraday was acquainted with the forms of his celebrated lines of force,\nbut I am not aware that he ever entered into the algebraic nature\nof those curves, and I feel sure that he could not have explained\ntheir forms as depending on the resultant attractions of all the\nmagnetic particles. There are even occasional indications that he did\nnot understand some of the simpler mathematical doctrines of modern\nphysical science. Although he so clearly foresaw the correlation of the\nphysical forces, and laboured so hard with his own hands to connect\ngravity with other forces, it is doubtful whether he understood the\ndoctrine of the conservation of energy as applied to gravitation.\nFaraday was probably equal to Newton in experimental skill, and in that\npeculiar kind of deductive power which leads to the invention of simple\nqualitative experiments; but it must be allowed that he exhibited\nlittle of that mathematical power which enabled Newton to follow out\nintuitively the quantitative results of a complicated problem with such\nwonderful facility. Two instances, Newton and Faraday, are sufficient\nto show that minds of widely different conformation will meet with\nsuitable regions of research. Nevertheless, there are certain traits\nwhich we may discover in all the highest scientific minds.\n\n\n*The Newtonian Method, the True Organum.*\n\nLaplace was of opinion that the *Principia* and the *Opticks* of\nNewton furnished the best models then available of the delicate art\nof experimental and theoretical investigation. In these, as he says,\nwe meet with the most happy illustrations of the way in which, from\na series of inductions, we may rise to the causes of phenomena, and\nthence descend again to all the resulting details.\n\nThe popular notion concerning Newton\u2019s discoveries is that in early\nlife, when driven into the country by the Great Plague, a falling apple\naccidentally suggested to him the existence of gravitation, and that,\navailing himself of this hint, he was led to the discovery of the law\nof gravitation, the explanation of which constitutes the *Principia*.\nIt is difficult to imagine a more ludicrous and inadequate picture of\nNewton\u2019s labours. No originality, or at least priority, was claimed\nby Newton as regards the discovery of the law of the inverse square,\nso closely associated with his name. In a well-known Scholium[481]\nhe acknowledges that Sir Christopher Wren, Hooke, and Halley, had\nseverally observed the accordance of Kepler\u2019s third law of motion with\nthe principle of the inverse square.\n\n  [481] *Principia*, bk. i. Prop. iv.\n\nNewton\u2019s work was really that of developing the methods of deductive\nreasoning and experimental verification, by which alone great\nhypotheses can be brought to the touchstone of fact. Archimedes was the\ngreatest of ancient philosophers, for he showed how mathematical theory\ncould be wedded to physical experiments; and his works are the first\ntrue Organum. Newton is the modern Archimedes, and the *Principia*\nforms the true Novum Organum of scientific method. The laws which he\nestablished are great, but his example of the manner of establishing\nthem is greater still. Excepting perhaps chemistry and electricity,\nthere is hardly a progressive branch of physical and mathematical\nscience, which has not been developed from the germs of true scientific\nprocedure which he disclosed in the *Principia* or the *Opticks*.\nOvercome by the success of his theory of universal gravitation, we\nare apt to forget that in his theory of sound he originated the\nmathematical investigation of waves and the mutual action of particles;\nthat in his corpuscular theory of light, however mistaken, he first\nventured to apply mathematical calculation to molecular attractions\nand repulsions; that in his prismatic experiments he showed how far\nexperimental verification could be pushed; that in his examination\nof the coloured rings named after him, he accomplished the most\nremarkable instance of minute measurement yet known, a mere practical\napplication of which by Fizeau was recently deemed worthy of a medal\nby the Royal Society. We only learn by degrees how complete was his\nscientific insight; a few words in his third law of motion display his\nacquaintance with the fundamental principles of modern thermodynamics\nand the conservation of energy, while manuscripts long overlooked prove\nthat in his inquiries concerning atmospheric refraction he had overcome\nthe main difficulties of applying theory to one of the most complex of\nphysical problems.\n\nAfter all, it is only by examining the way in which he effected\ndiscoveries, that we can rightly appreciate his greatness. The\n*Principia* treats not of gravity so much as of forces in general, and\nthe methods of reasoning about them. He investigates not one hypothesis\nonly, but mechanical hypotheses in general. Nothing so much strikes\nthe reader of the work as the exhaustiveness of his treatment, and the\nunbounded power of his insight. If he treats of central forces, it\nis not one law of force which he discusses, but many, or almost all\nimaginable laws, the results of each of which he sketches out in a few\npregnant words. If his subject is a resisting medium, it is not air or\nwater alone, but resisting media in general. We have a good example\nof his method in the scholium to the twenty-second proposition of the\nsecond book, in which he runs rapidly over many suppositions as to\nthe laws of the compressing forces which might conceivably act in an\natmosphere of gas, a consequence being drawn from each case, and that\none hypothesis ultimately selected which yields results agreeing with\nexperiments upon the pressure and density of the terrestrial atmosphere.\n\nNewton said that he did not frame hypotheses, but, in reality, the\ngreater part of the *Principia* is purely hypothetical, endless\nvarieties of causes and laws being imagined which have no counterpart\nin nature. The most grotesque hypotheses of Kepler or Descartes were\nnot more imaginary. But Newton\u2019s comprehension of logical method was\nperfect; no hypothesis was entertained unless it was definite in\nconditions, and admitted of unquestionable deductive reasoning; and the\nvalue of each hypothesis was entirely decided by the comparison of its\nconsequences with facts. I do not entertain a doubt that the general\ncourse of his procedure is identical with that view of the nature of\ninduction, as the inverse application of deduction, which I advocate\nthroughout this book. Francis Bacon held that science should be founded\non experience, but he mistook the true mode of using experience,\nand, in attempting to apply his method, ludicrously failed. Newton\ndid not less found his method on experience, but he seized the true\nmethod of treating it, and applied it with a power and success never\nsince equalled. It is a great mistake to say that modern science is\nthe result of the Baconian philosophy; it is the Newtonian philosophy\nand the Newtonian method which have led to all the great triumphs of\nphysical science, and I repeat that the *Principia* forms the true\n\u201cNovum Organum.\u201d\n\nIn bringing his theories to a decisive experimental verification,\nNewton showed, as a general rule, exquisite skill and ingenuity.\nIn his hands a few simple pieces of apparatus were made to give\nresults involving an unsuspected depth of meaning. His most beautiful\nexperimental inquiry was that by which he proved the differing\nrefrangibility of rays of light. To suppose that he originally\ndiscovered the power of a prism to break up a beam of white light\nwould be a mistake, for he speaks of procuring a glass prism to\ntry the \u201ccelebrated phenomena of colours.\u201d But we certainly owe to\nhim the theory that white light is a mixture of rays differing in\nrefrangibility, and that lights which differ in colour, differ also in\nrefrangibility. Other persons might have conceived this theory; in\nfact, any person regarding refraction as a quantitative effect must see\nthat different parts of the spectrum have suffered different amounts\nof refraction. But the power of Newton is shown in the tenacity with\nwhich he followed his theory into every consequence, and tested each\nresult by a simple but conclusive experiment. He first shows that\ndifferent coloured spots are displaced by different amounts when viewed\nthrough a prism, and that their images come to a focus at different\ndistances from the lense, as they should do, if the refrangibility\ndiffered. After excluding by many experiments a variety of indifferent\ncircumstances, he fixes his attention upon the question whether the\nrays are merely shattered, disturbed, and spread out in a chance\nmanner, as Grimaldi supposed, or whether there is a constant relation\nbetween the colour and the refrangibility.\n\nIf Grimaldi was right, it might be expected that a part of the spectrum\ntaken separately, and subjected to a second refraction, would suffer a\nnew breaking up, and produce some new spectrum. Newton inferred from\nhis own theory that a particular ray of the spectrum would have a\nconstant refrangibility, so that a second prism would merely bend it\nmore or less, but not further disperse it in any considerable degree.\nBy simply cutting off most of the rays of the spectrum by a screen,\nand allowing the remaining narrow ray to fall on a second prism, he\nproved the truth of this conclusion; and then slowly turning the first\nprism, so as to vary the colour of the ray falling on the second\none, he found that the spot of light formed by the twice-refracted\nray travelled up and down, a palpable proof that the amount of\nrefrangibility varies with the colour. For his further satisfaction,\nhe sometimes refracted the light a third or fourth time, and he found\nthat it might be refracted upwards or downwards or sideways, and yet\nfor each colour there was a definite amount of refraction through each\nprism. He completed the proof by showing that the separated rays may\nagain be gathered together into white light by an inverted prism, so\nthat no number of refractions alters the character of the light. The\nconclusion thus obtained serves to explain the confusion arising in the\nuse of a common lense; he shows that with homogeneous light there is\none distinct focus, with mixed light an infinite number of foci, which\nprevent a clear view from being obtained at any point.\n\nWhat astonishes the reader of the *Opticks* is the persistence with\nwhich Newton follows out the consequences of a preconceived theory, and\ntests the one notion by a wonderful variety of simple comparisons with\nfact. The ease with which he invents new combinations, and foresees\nthe results, subsequently verified, produces an insuperable conviction\nin the reader that he has possession of the truth. And it is certainly\nthe theory which leads him to the experiments, most of which could\nhardly be devised by accident. Newton actually remarks that it was by\nmathematically determining all kinds of phenomena of colours which\ncould be produced by refraction that he had \u201cinvented\u201d almost all the\nexperiments in the book, and he promises that others who shall \u201cargue\ntruly,\u201d and try the experiments with care, will not be disappointed in\nthe results.[482]\n\n  [482] *Opticks*, bk. i. part ii. Prop. 3. 3rd ed. p.\u00a0115.\n\nThe philosophic method of Huyghens was the same as that of Newton,\nand Huyghens\u2019 investigation of double refraction furnishes almost\nequally beautiful instances of theory guiding experiment. So far as\nwe know double refraction was first discovered by accident, and was\ndescribed by Erasmus Bartholinus in 1669. The phenomenon then appeared\nto be entirely exceptional, and the laws governing the two paths of\nthe refracted rays were so unapparent and complicated, that Newton\naltogether misunderstood the phenomenon, and it was only at the latter\nend of the last century that scientific men began to comprehend its\nlaws.\n\nNevertheless, Huyghens had, with rare genius, arrived at the true\ntheory as early as 1678. He regarded light as an undulatory motion of\nsome medium, and in his *Trait\u00e9 de la Lumi\u00e8re* he pointed out that,\nin ordinary refraction, the velocity of propagation of the wave is\nequal in all directions, so that the front of an advancing wave is\nspherical, and reaches equal distances in equal times. But in crystals,\nas he supposed, the medium would be of unequal elasticity in different\ndirections, so that a disturbance would reach unequal distances in\nequal times, and the wave produced would have a spheroidal form.\nHuyghens was not satisfied with an unverified theory. He calculated\nwhat might be expected to happen when a crystal of calc-spar was cut\nin various directions, and he says: \u201cI have examined in detail the\nproperties of the extraordinary refraction of this crystal, to see if\neach phenomenon which is deduced from theory would agree with what is\nreally observed. And this being so, it is no slight proof of the truth\nof our suppositions and principles; but what I am going to add here\nconfirms them still more wonderfully; that is, the different modes\nof cutting this crystal, in which the surfaces produced give rise to\nrefraction exactly such as they ought to be, and as I had foreseen\nthem, according to the preceding theory.\u201d\n\nNewton\u2019s mistaken corpuscular theory of light caused the theories and\nexperiments of Huyghens to be disregarded for more than a century; but\nit is not easy to imagine a more beautiful or successful application of\nthe true method of inductive investigation, theory guiding experiment,\nand yet wholly relying on experiment for confirmation.\n\n\n*Candour and Courage of the Philosophic Mind.*\n\nPerfect readiness to reject a theory inconsistent with fact is a\nprimary requisite of the philosophic mind. But it would be a mistake\nto suppose that this candour has anything akin to fickleness; on the\ncontrary, readiness to reject a false theory may be combined with a\npeculiar pertinacity and courage in maintaining an hypothesis as long\nas its falsity is not actually apparent. There must, indeed, be no\nprejudice or bias distorting the mind, and causing it to pass over the\nunwelcome results of experiment. There must be that scrupulous honesty\nand flexibility of mind, which assigns adequate value to all evidence;\nindeed, the more a man loves his theory, the more scrupulous should be\nhis attention to its faults. It is common in life to meet with some\ntheorist, who, by long cogitation over a single theory, has allowed it\nto mould his mind, and render him incapable of receiving anything but\nas a contribution to the truth of his one theory. A narrow and intense\ncourse of thought may sometimes lead to great results, but the adoption\nof a wrong theory at the outset is in such a mind irretrievable.\nThe man of one idea has but a single chance of truth. The fertile\ndiscoverer, on the contrary, chooses between many theories, and is\nnever wedded to any one, unless impartial and repeated comparison has\nconvinced him of its validity. He does not choose and then compare; but\nhe compares time after time, and then chooses.\n\nHaving once deliberately chosen, the philosopher may rightly entertain\nhis theory with the strongest fidelity. He will neglect no objection;\nfor he may chance at any time to meet a fatal one; but he will bear\nin mind the inconsiderable powers of the human mind compared with the\ntasks it has to undertake. He will see that no theory can at first be\nreconciled with all objections, because there may be many interfering\ncauses, and the very consequences of the theory may have a complexity\nwhich prolonged investigation by successive generations of men may not\nexhaust. If, then, a theory exhibit a number of striking coincidences\nwith fact, it must not be thrown aside until at least one *conclusive\ndiscordance* is proved, regard being had to possible error in\nestablishing that discordance. In science and philosophy something must\nbe risked. He who quails at the least difficulty will never establish a\nnew truth, and it was not unphilosophic in Leslie to remark concerning\nhis own inquiries into the nature of heat--\n\n\u201cIn the course of investigation, I have found myself compelled to\nrelinquish some preconceived notions; but I have not abandoned them\nhastily, nor, till after a warm and obstinate defence, I was driven\nfrom every post.\u201d[483]\n\n  [483] *Experimental Inquiry into the Nature of Heat.* Preface, p. xv.\n\nFaraday\u2019s life, again, furnishes most interesting illustrations of this\ntenacity of the philosophic mind. Though so candid in rejecting some\ntheories, there were others to which he clung through everything. One\nof his favourite notions resulted in a brilliant discovery; another\nremains in doubt to the present day.\n\n\n*The Philosophic Character of Faraday.*\n\nIn Faraday\u2019s researches concerning the connection of magnetism and\nlight, we find an excellent instance of the pertinacity with which a\nfavourite theory may be pursued, so long as the results of experiment\ndo not clearly negative the notions entertained. In purely quantitative\nquestions, as we have seen, the absence of apparent effect can seldom\nbe regarded as proving the absence of all effect. Now Faraday was\nconvinced that some mutual relation must exist between magnetism and\nlight. As early as 1822, he attempted to produce an effect upon a ray\nof polarised light, by passing it through water placed between the\npoles of a voltaic battery; but he was obliged to record that not the\nslightest effect was observable. During many years the subject, we are\ntold,[484] rose again and again to his mind, and no failure could make\nhim relinquish his search after this unknown relation. It was in the\nyear 1845 that he gained the first success; on August 30th he began\nto work with common electricity, vainly trying glass, quartz, Iceland\nspar, &c. Several days of labour gave no result; yet he did not desist.\nHeavy glass, a transparent medium of great refractive powers, composed\nof borate of lead, was now tried, being placed between the poles of a\npowerful electro-magnet while a ray of polarised light was transmitted\nthrough it. When the poles of the electro-magnet were arranged in\ncertain positions with regard to the substance under trial, no effects\nwere apparent; but at last Faraday happened fortunately to place a\npiece of heavy glass so that contrary magnetic poles were on the same\nside, and now an effect was witnessed. The glass was found to have the\npower of twisting the plane of polarisation of the ray of light.\n\n  [484] Bence Jones, *Life of Faraday*, vol.\u00a0i. p.\u00a0362.\n\nAll Faraday\u2019s recorded thoughts upon this great experiment are replete\nwith curious interest. He attributes his success to the opinion,\nalmost amounting to a conviction, that the various forms, under which\nthe forces of matter are made manifest, have one common origin,\nand are so directly related and mutually dependent that they are\nconvertible. \u201cThis strong persuasion,\u201d he says,[485] \u201cextended to the\npowers of light, and led to many exertions having for their object\nthe discovery of the direct relation of light and electricity. These\nineffectual exertions could not remove my strong persuasion, and I\nhave at last succeeded.\u201d He describes the phenomenon in somewhat\nfigurative language as *the magnetisation of a ray of light*, and\nalso as *the illumination of a magnetic curve or line of force*. He\nhas no sooner got the effect in one case, than he proceeds, with his\ncharacteristic comprehensiveness of research, to test the existence\nof a like phenomenon in all the substances available. He finds that\nnot only heavy glass, but solids and liquids, acids and alkalis, oils,\nwater, alcohol, ether, all possess this power; but he was not able to\ndetect its existence in any gaseous substance. His thoughts cannot be\nrestrained from running into curious speculations as to the possible\nresults of the power in certain cases. \u201cWhat effect,\u201d he says, \u201cdoes\nthis force have in the earth where the magnetic curves of the earth\ntraverse its substance? Also what effect in a magnet?\u201d And then he\nfalls upon the strange notion that perhaps this force tends to make\niron and oxide of iron transparent, a phenomenon never observed. We can\nmeet with nothing more instructive as to the course of mind by which\ngreat discoveries are made, than these records of Faraday\u2019s patient\nlabours, and his varied success and failure. Nor are his unsuccessful\nexperiments upon the relation of gravity and electricity less\ninteresting, or less worthy of study.\n\n  [485] Ibid. vol.\u00a0ii. p.\u00a0199.\n\nThroughout a large part of his life, Faraday was possessed by the idea\nthat gravity cannot be unconnected with the other forces of nature. On\nMarch 19th, 1849, he wrote in his laboratory book,--\u201cGravity. Surely\nthis force must be capable of an experimental relation to electricity,\nmagnetism, and the other forces, so as to bind it up with them in\nreciprocal action and equivalent effect?\u201d[486] He filled twenty\nparagraphs or more with reflections and suggestions, as to the mode\nof treating the subject by experiment. He anticipated that the mutual\napproach of two bodies would develop electricity in them, or that a\nbody falling through a conducting helix would excite a current changing\nin direction as the motion was reversed. \u201c*All this is a dream*,\u201d\nhe remarks; \u201cstill examine it by a few experiments. Nothing is too\nwonderful to be true, if it be consistent with the laws of nature;\nand in such things as these, experiment is the best test of such\nconsistency.\u201d\n\n  [486] See also his more formal statement in the *Experimental\n  Researches in Electricity*, 24th Series, \u00a7 2702, vol.\u00a0iii. p.\u00a0161.\n\nHe executed many difficult and tedious experiments, which are described\nin the 24th Series of Experimental Researches. The result was *nil*,\nand yet he concludes: \u201cHere end my trials for the present. The results\nare negative; they do not shake my strong feeling of the existence of\na relation between gravity and electricity, though they give no proof\nthat such a relation exists.\u201d\n\nHe returned to the work when he was ten years older, and in 1858\u20139\nrecorded many remarkable reflections and experiments. He was much\nstruck by the fact that electricity is essentially a *dual force*,\nand it had always been a conviction of Faraday that no body could be\nelectrified positively without some other body becoming electrified\nnegatively; some of his researches had been simple developments of\nthis relation. But observing that between two mutually gravitating\nbodies there was no apparent circumstance to determine which should be\npositive and which negative, he does not hesitate to call in question\nan old opinion. \u201cThe evolution of *one* electricity would be a new\nand very remarkable thing. The idea throws a doubt on the whole; but\nstill try, for who knows what is possible in dealing with gravity?\u201d\nWe cannot but notice the candour with which he thus acknowledges in\nhis laboratory book the doubtfulness of the whole thing, and is yet\nprepared as a forlorn hope to frame experiments in opposition to\nall his previous experience of the course of nature. For a time his\nthoughts flow on as if the strange detection were already made, and he\nhad only to trace out its consequences throughout the universe. \u201cLet us\nencourage ourselves by a little more imagination prior to experiment,\u201d\nhe says; and then he reflects upon the infinity of actions in nature,\nin which the mutual relations of electricity and gravity would come\ninto play; he pictures to himself the planets and the comets charging\nthemselves as they approach the sun; cascades, rain, rising vapour,\ncirculating currents of the atmosphere, the fumes of a volcano, the\nsmoke in a chimney become so many electrical machines. A multitude of\nevents and changes in the atmosphere seem to be at once elucidated by\nsuch actions; for a moment his reveries have the vividness of fact.\n\u201cI think we have been dull and blind not to have suspected some such\nresults,\u201d and he sums up rapidly the consequences of his great but\nimaginary theory; an entirely new mode of exciting heat or electricity,\nan entirely new relation of the natural forces, an analysis of\ngravitation, and a justification of the conservation of force.\n\nSuch were Faraday\u2019s fondest dreams of what might be, and to many a\nphilosopher they would have been sufficient basis for the writing of\na great book. But Faraday\u2019s imagination was within his full control;\nas he himself says, \u201cLet the imagination go, guarding it by judgment\nand principle, and holding it in and directing it by experiment.\u201d His\ndreams soon took a very practical form, and for many days he laboured\nwith ceaseless energy, on the staircase of the Royal Institution, in\nthe clock tower of the Houses of Parliament, or at the top of the Shot\nTower in Southwark, raising and lowering heavy weights, and combining\nelectrical helices and wires in every conceivable way. His skill and\nlong experience in experiment were severely taxed to eliminate the\neffects of the earth\u2019s magnetism, and time after time he saved himself\nfrom accepting mistaken indications, which to another man might have\nseemed conclusive verifications of his theory. When all was done there\nremained absolutely no results. \u201cThe experiments,\u201d he says, \u201cwere well\nmade, but the results are negative;\u201d and yet, he adds, \u201cI cannot accept\nthem as conclusive.\u201d In this position the question remains to the\npresent day; it may be that the effect was too slight to be detected,\nor it may be that the arrangements adopted were not suited to develop\nthe particular relation which exists, just as Oersted could not detect\nelectro-magnetism, so long as his wire was perpendicular to the plane\nof motion of his needle. But these are not matters which concern us\nfurther here. We have only to notice the profound conviction in the\nunity of natural laws, the active powers of inference and imagination,\nthe unbounded licence of theorising, combined above all with the utmost\ndiligence in experimental verification which this remarkable research\nexhibits.\n\n\n*Reservation of Judgment.*\n\nThere is yet another characteristic needed in the philosophic mind; it\nis that of suspending judgment when the data are insufficient. Many\npeople will express a confident opinion on almost any question which is\nput before them, but they thereby manifest not strength, but narrowness\nof mind. To see all sides of a complicated subject, and to weigh all\nthe different facts and probabilities correctly, require no ordinary\npowers of comprehension. Hence it is most frequently the philosophic\nmind which is in doubt, and the ignorant mind which is ready with a\npositive decision. Faraday has himself said, in a very interesting\nlecture:[487] \u201cOccasionally and frequently the exercise of the judgment\nought to end in *absolute reservation*. It may be very distasteful, and\ngreat fatigue, to suspend a conclusion; but as we are not infallible,\nso we ought to be cautious; we shall eventually find our advantage, for\nthe man who rests in his position is not so far from right as he who,\nproceeding in a wrong direction, is ever increasing his distance.\u201d\n\n  [487] Printed in *Modern Culture*, edited by Youmans, p.\u00a0219.\n\nArago presented a conspicuous example of this high quality of mind, as\nFaraday remarks; for when he made known his curious discovery of the\nrelation of a magnetic needle to a revolving copper plate, a number\nof supposed men of science in different countries gave immediate and\nconfident explanations of it, which were all wrong. But Arago, who\nhad both discovered the phenomenon and personally investigated its\nconditions, declined to put forward publicly any theory at all.\n\nAt the same time we must not suppose that the truly philosophic mind\ncan tolerate a state of doubt, while a chance of decision remains open.\nIn science nothing like compromise is possible, and truth must be one.\nHence, doubt is the confession of ignorance, and involves a painful\nfeeling of incapacity. But doubt lies between error and truth, so that\nif we choose wrongly we are further away than ever from our goal.\n\nSumming up, then, it would seem as if the mind of the great discoverer\nmust combine contradictory attributes. He must be fertile in\ntheories and hypotheses, and yet full of facts and precise results of\nexperience. He must entertain the feeblest analogies, and the merest\nguesses at truth, and yet he must hold them as worthless till they are\nverified in experiment. When there are any grounds of probability he\nmust hold tenaciously to an old opinion, and yet he must be prepared\nat any moment to relinquish it when a clearly contradictory fact is\nencountered. \u201cThe philosopher,\u201d says Faraday,[488] \u201cshould be a man\nwilling to listen to every suggestion, but determined to judge for\nhimself. He should not be biased by appearances; have no favourite\nhypothesis; be of no school; and in doctrine have no master. He should\nnot be a respecter of persons, but of things. Truth should be his\nprimary object. If to these qualities be added industry, he may indeed\nhope to walk within the veil of the temple of nature.\u201d\n\n  [488] *Life of Faraday*, vol.\u00a0i. p.\u00a0225.\n\n\n\n\nBOOK V.\n\nGENERALISATION, ANALOGY, AND CLASSIFICATION.\n\n\n\n\nCHAPTER XXVII.\n\nGENERALISATION.\n\n\nI have endeavoured to show in preceding chapters that all inductive\nreasoning is an inverse application of deductive reasoning, and\nconsists in demonstrating that the consequences of certain assumed laws\nagree with facts of nature gathered by active or passive observation.\nThe fundamental process of reasoning, as stated in the outset, consists\nin inferring of a thing what we know of similar objects, and it is on\nthis principle that the whole of deductive reasoning, whether simply\nlogical or mathematico-logical, is founded. All inductive reasoning\nmust be founded on the same principle. It might seem that by a plain\nuse of this principle we could avoid the complicated processes of\ninduction and deduction, and argue directly from one particular case\nto another, as Mill proposed. If the Earth, Venus, Mars, Jupiter,\nand other planets move in elliptic orbits, cannot we dispense with\nelaborate precautions, and assert that Neptune, Ceres, and the last\ndiscovered planet must do so likewise? Do we not know that Mr.\nGladstone must die, because he is like other men? May we not argue\nthat because some men die therefore he must? Is it requisite to ascend\nby induction to the general proposition \u201call men must die,\u201d and then\ndescend by deduction from that general proposition to the case of Mr.\nGladstone? My answer undoubtedly is that we must ascend to general\npropositions. The fundamental principle of the substitution of similars\ngives us no warrant in affirming of Mr. Gladstone what we know of\nother men, because we cannot be sure that Mr. Gladstone is exactly\nsimilar to other men. Until his death we cannot be perfectly sure that\nhe possesses all the attributes of other men; it is a question of\nprobability, and I have endeavoured to explain the mode in which the\ntheory of probability is applied to calculate the probability that from\na series of similar events we may infer the recurrence of like events\nunder identical circumstances. There is then no such process as that\nof inferring from particulars to particulars. A careful analysis of\nthe conditions under which such an inference appears to be made, shows\nthat the process is really a general one, and that what is inferred of\na particular case might be inferred of all similar cases. All reasoning\nis essentially general, and all science implies generalisation. In\nthe very birth-time of philosophy this was held to be so: \u201cNulla\nscientia est de individuis, sed de solis universalibus,\u201d was the\ndoctrine of Plato, delivered by Porphyry. And Aristotle[489] held a\nlike opinion--\u039f\u1f50\u03b4\u03b5\u03bc\u03af\u03b1 \u03b4\u1f72 \u03c4\u03ad\u03c7\u03bd\u03b7 \u03c3\u03ba\u03bf\u03c0\u03b5\u020b \u03c4\u1f78 \u03ba\u03b1\u03b8\u2019 \u1f15\u03ba\u03b1\u03c3\u03c4\u03bf\u03bd ... \u03c4\u1f78 \u03b4\u1f72 \u03ba\u03b1\u03b8\u2019\n\u1f15\u03ba\u03b1\u03c3\u03c4\u03bf\u03bd \u1f04\u03c0\u03b5\u03b9\u03c1\u03bf\u03bd \u03ba\u03b1\u1f76 \u03bf\u1f50\u03ba \u1f10\u03c0\u03b9\u03c3\u03c4\u03b7\u03c4\u03cc\u03bd. \u201cNo art treats of particular cases;\nfor particulars are infinite and cannot be known.\u201d No one who holds the\ndoctrine that reasoning may be from particulars to particulars, can\nbe supposed to have the most rudimentary notion of what constitutes\nreasoning and sc\u00edence.\n\n  [489] Aristotle\u2019s *Rhetoric*, Liber I. 2. 11.\n\nAt the same time there can be no doubt that practically what we find\nto be true of many similar objects will probably be true of the next\nsimilar object. This is the result to which an analysis of the Inverse\nMethod of Probabilities leads us, and, in the absence of precise data\nfrom which we may calculate probabilities, we are usually obliged to\nmake a rough assumption that similars in some respects are similars\nin other respects. Thus it comes to pass that a large part of the\nreasoning processes in which scientific men are engaged, consists in\ndetecting similarities between objects, and then rudely assuming that\nthe like similarities will be detected in other cases.\n\n\n*Distinction of Generalisation and Analogy.*\n\nThere is no distinction but that of degree between what is known as\nreasoning by *generalisation* and reasoning by *analogy*. In both\ncases from certain observed resemblances we infer, with more or less\nprobability, the existence of other resemblances. In generalisation\nthe resemblances have great extension and usually little intension,\nwhereas in analogy we rely upon the great intension, the extension\nbeing of small amount (p.\u00a026). If we find that the qualities A and B\nare associated together in a great many instances, and have never been\nfound separate, it is highly probable that on the next occasion when we\nmeet with A, B will also be present, and *vice vers\u00e2*. Thus wherever we\nmeet with an object possessing gravity, it is found to possess inertia\nalso, nor have we met with any material objects possessing inertia\nwithout discovering that they also possess gravity. The probability\nhas therefore become very great, as indicated by the rules founded on\nthe Inverse Method of Probabilities (p.\u00a0257), that whenever in the\nfuture we meet an object possessing either of the properties of gravity\nand inertia, it will be found on examination to possess the other\nof these properties. This is a clear instance of the employment of\ngeneralisation.\n\nIn analogy, on the other hand, we reason from likeness in many points\nto likeness in other points. The qualities or points of resemblance are\nnow numerous, not the objects. At the poles of Mars are two white spots\nwhich resemble in many respects the white regions of ice and snow at\nthe poles of the earth. There probably exist no other similar objects\nwith which to compare these, yet the exactness of the resemblance\nenables us to infer, with high probability, that the spots on Mars\nconsist of ice and snow. In short, many points of resemblance imply\nmany more. From the appearance and behaviour of those white spots we\ninfer that they have all the chemical and physical properties of\nfrozen water. The inference is of course only probable, and based upon\nthe improbability that aggregates of many qualities should be formed in\na like manner in two or more cases, without being due to some uniform\ncondition or cause.\n\nIn reasoning by analogy, then, we observe that two objects ABCDE....\nand A\u2032B\u2032C\u2032D\u2032E\u2032.... have many like qualities, as indicated by the\nidentity of the letters, and we infer that, since the first has another\nquality, X, we shall discover this quality in the second case by\nsufficiently close examination. As Laplace says,--\u201cAnalogy is founded\non the probability that similar things have causes of the same kind,\nand produce the same effects. The more perfect this similarity, the\ngreater is this probability.\u201d[490] The nature of analogical inference\nis aptly described in the work on Logic attributed to Kant, where the\nrule of ordinary induction is stated in the words, \u201c*Eines in vielen,\nalso in allen*,\u201d one quality in many things, therefore in all; and\nthe rule of analogy is \u201c*Vieles in einem, also auch das \u00fcbrige in\ndemselben*,\u201d[491] many (qualities) in one, therefore also the remainder\nin the same. It is evident that there may be intermediate cases in\nwhich, from the identity of a moderate number of objects in several\nproperties, we may infer to other objects. Probability must rest either\nupon the number of instances or the depth of resemblance, or upon the\noccurrence of both in sufficient degrees. What there is wanting in\nextension must be made up by intension, and *vice vers\u00e2*.\n\n  [490] *Essai Philosophique sur les Probabilit\u00e9s*, p.\u00a086.\n\n  [491] Kant\u2019s *Logik*, \u00a7 84, K\u00f6nigsberg, 1800, p.\u00a0207.\n\n\n*Two Meanings of Generalisation.*\n\nThe term generalisation, as commonly used, includes two processes which\nare of different character, but are often closely associated together.\nIn the first place, we generalise when we recognise even in two objects\na common nature. We cannot detect the slightest similarity without\nopening the way to inference from one case to the other. If we compare\na cubical crystal with a regular octahedron, there is little apparent\nsimilarity; but, as soon as we perceive that either can be produced\nby the symmetrical modification of the other, we discover a groundwork\nof similarity in the crystals, which enables us to infer many things\nof one, because they are true of the other. Our knowledge of ozone\ntook its rise from the time when the similarity of smell, attending\nelectric sparks, strokes of lightning, and the slow combustion of\nphosphorus, was noticed by Sch\u00f6nbein. There was a time when the rainbow\nwas an inexplicable phenomenon--a portent, like a comet, and a cause of\nsuperstitious hopes and fears. But we find the true spirit of science\nin Roger Bacon, who desires us to consider the objects which present\nthe same colours as the rainbow; he mentions hexagonal crystals from\nIreland and India, but he bids us not suppose that the hexagonal form\nis essential, for similar colours may be detected in many transparent\nstones. Drops of water scattered by the oar in the sun, the spray from\na water-wheel, the dewdrops lying on the grass in the summer morning,\nall display a similar phenomenon. No sooner have we grouped together\nthese apparently diverse instances, than we have begun to generalise,\nand have acquired a power of applying to one instance what we can\ndetect of others. Even when we do not apply the knowledge gained to\nnew objects, our comprehension of those already observed is greatly\nstrengthened and deepened by learning to view them as particular cases\nof a more general property.\n\nA second process, to which the name of generalisation is often\ngiven, consists in passing from a fact or partial law to a multitude\nof unexamined cases, which we believe to be subject to the same\nconditions. Instead of merely recognising similarity as it is brought\nbefore us, we predict its existence before our senses can detect it, so\nthat generalisation of this kind endows us with a prophetic power of\nmore or less probability. Having observed that many substances assume,\nlike water and mercury, the three states of solid, liquid, and gas, and\nhaving assured ourselves by frequent trial that the greater the means\nwe possess of heating and cooling, the more substances we can vaporise\nand freeze, we pass confidently in advance of fact, and assume that\nall substances are capable of these three forms. Such a generalisation\nwas accepted by Lavoisier and Laplace before many of the corroborative\nfacts now in our possession were known. The reduction of a single\ncomet beneath the sway of gravity was considered sufficient indication\nthat all comets obey the same power. Few persons doubted that the law\nof gravity extended over the whole heavens; certainly the fact that\na few stars out of many millions manifest the action of gravity, is\nnow held to be sufficient evidence of its general extension over the\nvisible universe.\n\n\n*Value of Generalisation.*\n\nIt might seem that if we know particular facts, there can be little\nuse in connecting them together by a general law. The particulars\nmust be more full of useful information than an abstract general\nstatement. If we know, for instance, the properties of an ellipse, a\ncircle, a parabola, and hyperbola, what is the use of learning all\nthese properties over again in the general theory of curves of the\nsecond degree? If we understand the phenomena of sound and light and\nwater-waves separately, what is the need of erecting a general theory\nof waves, which, after all, is inapplicable to practice until resolved\nagain into particular cases? But, in reality, we never do obtain an\nadequate knowledge of particulars until we regard them as cases of\nthe general. Not only is there a singular delight in discovering the\nmany in the one, and the one in the many, but there is a constant\ninterchange of light and knowledge. Properties which are unapparent\nin the hyperbola may be readily observed in the ellipse. Most of the\ncomplex relations which old geometers discovered in the circle will\nbe reproduced *mutatis mutandis* in the other conic sections. The\nundulatory theory of light might have been unknown at the present day,\nhad not the theory of sound supplied hints by analogy. The study of\nlight has made known many phenomena of interference and polarisation,\nthe existence of which had hardly been suspected in the case of\nsound, but which may now be sought out, and perhaps found to possess\nunexpected interest. The careful study of water-waves shows how waves\nalter in form and velocity with varying depth of water. Analogous\nchanges may some time be detected in sound waves. Thus there is mutual\ninterchange of aid.\n\n\u201cEvery study of a generalisation or extension,\u201d De Morgan has well\nsaid,[492] \u201cgives additional power over the particular form by which\nthe generalisation is suggested. Nobody who has ever returned to\nquadratic equations after the study of equations of all degrees,\nor who has done the like, will deny my assertion that \u03bf\u1f50 \u03b2\u03bb\u03ad\u03c0\u03b5\u03b9\n\u03b2\u03bb\u03ad\u03c0\u03c9\u03bd may be predicated of any one who studies a branch or a case,\nwithout afterwards making it part of a larger whole. Accordingly\nit is always worth while to generalise, were it only to give power\nover the *particular*. This principle, of daily familiarity to the\nmathematician, is almost unknown to the logician.\u201d\n\n  [492] *Syllabus of a Proposed System of Logic*, p.\u00a034.\n\n\n*Comparative Generality of Properties.*\n\nMuch of the value of science depends upon the knowledge which we\ngradually acquire of the different degrees of generality of properties\nand phenomena of various kinds. The use of science consists in enabling\nus to act with confidence, because we can foresee the result. Now this\nforesight must rest upon the knowledge of the powers which will come\ninto play. That knowledge, indeed, can never be certain, because it\nrests upon imperfect induction, and the most confident beliefs and\npredictions of the physicist may be falsified. Nevertheless, if we\nalways estimate the probability of each belief according to the due\nteaching of the data, and bear in mind that probability when forming\nour anticipations, we shall ensure the minimum of disappointment. Even\nwhen he cannot exactly apply the theory of probabilities, the physicist\nmay acquire the habit of making judgments in general agreement with its\nprinciples and results.\n\nSuch is the constitution of nature, that the physicist learns to\ndistinguish those properties which have wide and uniform extension,\nfrom those which vary between case and case. Not only are certain laws\ndistinctly laid down, with their extension carefully defined, but a\nscientific training gives a kind of tact in judging how far other laws\nare likely to apply under any particular circumstances. We learn by\ndegrees that crystals exhibit phenomena depending upon the directions\nof the axes of elasticity, which we must not expect in uniform solids.\nLiquids, compared even with non-crystalline solids, exhibit laws of\nfar less complexity and variety; and gases assume, in many respects,\nan aspect of nearly complete uniformity. To trace out the branches of\nscience in which varying degrees of generality prevail, would be an\ninquiry of great interest and importance; but want of space, if there\nwere no other reason, would forbid me to attempt it, except in a very\nslight manner.\n\nGases, so far as they are really gaseous, not only have exactly the\nsame properties in all directions of space, but one gas exactly\nresembles other gases in many qualities. All gases expand by heat,\naccording to the same law, and by nearly the same amount; the specific\nheats of equivalent weights are equal, and the densities are exactly\nproportional to the atomic weights. All such gases obey the general\nlaw, that the volume multiplied by the pressure, and divided by the\nabsolute temperature, is constant or nearly so. The laws of diffusion\nand transpiration are the same in all cases, and, generally speaking,\nall physical laws, as distinguished from chemical laws, apply\nequally to all gases. Even when gases differ in chemical or physical\nproperties, the differences are minor in degree. Thus the differences\nof viscosity are far less marked than in the liquid and solid states.\nNearly all gases, again, are colourless, the exceptions being chlorine,\nthe vapours of iodine, bromine, and a few other substances.\n\nOnly in one single point, so far as I am aware, do gases present\ndistinguishing marks unknown or nearly so, in the solid and liquid\nstates. I mean as regards the light given off when incandescent.\nEach gas when sufficiently heated, yields its own peculiar series\nof rays, arising from the free vibrations of the constituent parts\nof the molecules. Hence the possibility of distinguishing gases by\nthe spectroscope. But the molecules of solids and liquids appear to\nbe continually in conflict with each other, so that only a confused\n*noise* of atoms is produced, instead of a definite series of luminous\nchords. At the same temperature, accordingly, all solids and liquids\ngive off nearly the same rays when strongly heated, and we have in this\ncase an exception to the greater generality of properties in gases.\n\nLiquids are in many ways intermediate in character between gases\nand solids. While incapable of possessing different elasticity in\ndifferent directions, and thus denuded of the rich geometrical\ncomplexity of solids, they retain the variety of density, colour\ndegrees of transparency, great diversity in surface tension, viscosity,\ncoefficients of expansion, compressibility, and many other properties\nwhich we observe in solids, but not for the most part in gases. Though\nour knowledge of the physical properties of liquids is much wanting in\ngenerality at present, there is ground to hope that by degrees laws\nconnecting and explaining the variations may be traced out.\n\nSolids are in every way contrasted to gases. Each solid substance\nhas its own peculiar degree of density, hardness, compressibility,\ntransparency, tenacity, elasticity, power of conducting heat and\nelectricity, magnetic properties, capability of producing frictional\nelectricity, and so forth. Even different specimens of the same kind\nof substance will differ widely, according to the accidental treatment\nreceived. And not only has each substance its own specific properties,\nbut, when crystallised, its properties vary in each direction with\nregard to the axes of crystallisation. The velocity of radiation, the\nrate of conduction of heat, the coefficients of expansibility and\ncompressibility, the thermo-electric properties, all vary in different\ncrystallographic directions.\n\nIt is probable that many apparent differences between liquids, and\neven between solids, will be explained when we learn to regard them\nunder exactly corresponding circumstances. The extreme generality of\nthe properties of gases is in reality only true at an infinitely high\ntemperature, when they are all equally remote from their condensing\npoints. Now, it is found that if we compare liquids--for instance,\ndifferent kinds of alcohols--not at equal temperatures, but at points\nequally distant from their respective boiling points, the laws and\ncoefficients of expansion are nearly equal. The vapour-tensions of\nliquids also are more nearly equal, when compared at corresponding\npoints, and the boiling-points appear in many cases to be simply\nrelated to the chemical composition. No doubt the progress of\ninvestigation will enable us to discover generality, where at present\nwe only see variety and puzzling complexity.\n\nIn some cases substances exhibit the same physical properties in\nthe liquid as in the solid state. Lead has a high refractive power,\nwhether in solution, or in solid salts, crystallised or vitreous.\nThe magnetic power of iron is conspicuous, whatever be its chemical\ncondition; indeed, the magnetic properties of substances, though\nvarying with temperature, seem not to be greatly affected by other\nphysical changes. Colour, absorptive power for heat or light rays,\nand a few other properties are also often the same in liquids and\ngases. Iodine and bromine possess a deep colour whenever they are\nchemically uncombined. Nevertheless, we can seldom argue safely from\nthe properties of a substance in one condition to those in another\ncondition. Ice is an insulator, water a conductor of electricity, and\nthe same contrast exists in most other substances. The conducting power\nof a liquid for electricity increases with the temperature, while that\nof a solid decreases. By degrees we may learn to distinguish between\nthose properties of matter which depend upon the intimate construction\nof the chemical molecule, and those which depend upon the contact,\nconflict, mutual attraction, or other relations of distinct molecules.\nThe properties of a substance with respect to light seem generally to\ndepend upon the molecule; thus, the power of certain substances to\ncause the plane of polarisation of a ray of light to rotate, is exactly\nthe same whatever be its degree of density, or the diluteness of the\nsolution in which it is contained. Taken as a whole, the physical\nproperties of substances and their quantitative laws, present a problem\nof infinite complexity, and centuries must elapse before any moderately\ncomplete generalisations on the subject become possible.\n\n\n*Uniform Properties of all Matter.*\n\nSome laws are held to be true of all matter in the universe absolutely,\nwithout exception, no instance to the contrary having ever been\nnoticed. This is the case with the laws of motion, as laid down\nby Galileo and Newton. It is also conspicuously true of the law\nof universal gravitation. The rise of modern physical science may\nperhaps be considered as beginning at the time when Galileo showed,\nin opposition to the Aristotelians, that matter is equally affected\nby gravity, irrespective of its form, magnitude, or texture. All\nobjects fall with equal rapidity, when disturbing causes, such as the\nresistance of the air, are removed or allowed for. That which was\nrudely demonstrated by Galileo from the leaning tower of Pisa, was\nproved by Newton to a high degree of approximation, in an experiment\nwhich has been mentioned (p.\u00a0443).\n\nNewton formed two pendulums, as nearly as possible the same in outward\nshape and size by taking two equal round wooden boxes, and suspending\nthem by equal threads, eleven feet long. The pendulums were therefore\nequally subject to the resistance of the air. He filled one box with\nwood, and in the centre of oscillation of the other he placed an equal\nweight of gold. The pendulums were then equal in weight as well as in\nsize; and, on setting them simultaneously in motion, Newton found that\nthey vibrated for a length of time with equal vibrations. He tried the\nsame experiment with silver, lead, glass, sand, common salt, water,\nand wheat, in place of the gold, and ascertained that the motion of\nhis pendulum was exactly the same whatever was the kind of matter\ninside.[493] He considered that a difference of a thousandth part\nwould have been apparent. The reader must observe that the pendulums\nwere made of equal weight only in order that they might suffer equal\nretardation from the air. The meaning of the experiment is that all\nsubstances manifest exactly equal acceleration from the force of\ngravity, and that therefore the inertia or resistance of matter to\nforce, which is the only independent measure of mass known to us, is\nalways proportional to gravity.\n\n  [493] *Principia*, bk. iii. Prop. VI. Motte\u2019s translation, vol.\u00a0ii.\n  p.\u00a0220.\n\nThese experiments of Newton were considered conclusive up to very\nrecent times, when certain discordances between the theory and\nobservations of the movements of planets led Nicolai, in 1826, to\nsuggest that the equal gravitation of different kinds of matter might\nnot be absolutely exact. It is perfectly philosophical thus to call\nin question, from time to time, some of the best accepted laws. On\nthis occasion Bessel carefully repeated the experiments of Newton with\npendulums composed of ivory, glass, marble, quartz, meteoric stones,\n&c., but was unable to detect the least difference. This conclusion\nis also confirmed by the ultimate agreement of all the calculations\nof physical astronomy based upon it. Whether the mass of Jupiter be\ncalculated from the motion of its own satellites, from the effect\nupon the small planets, Vesta, Juno, &c., or from the perturbation\nof Encke\u2019s Comet, the results are closely accordant, showing that\nprecisely the same law of gravity applies to the most different bodies\nwhich we can observe. The gravity of a body, again, appears to be\nentirely independent of its other physical conditions, being totally\nunaffected by any alteration in the temperature, density, electric or\nmagnetic condition, or other physical properties of the substance.\n\nOne paradoxical result of the law of equal gravitation is the theorem\nof Torricelli, to the effect that all liquids of whatever density\nfall or flow with equal rapidity. If there be two equal cisterns\nrespectively filled with mercury and water, the mercury, though\nthirteen times as heavy, would flow from an aperture neither more\nrapidly nor more slowly than the water, and the same would be true of\nether, alcohol, and other liquids, allowance being made, however, for\nthe resistance of the air, and the differing viscosities of the liquids.\n\nIn its exact equality and its perfect independence of all\ncircumstances, except mass and distance, the force of gravity stands\napart from all the other forces and phenomena of nature, and has\nnot yet been brought into any relation with them except through the\ngeneral principle of the conservation of energy. Magnetic attraction,\nas remarked by Newton, follows very different laws, depending upon the\nchemical quality and molecular structure of each particular substance.\n\nWe must remember that in saying \u201call matter gravitates,\u201d we exclude\nfrom the term matter the basis of light-undulations, which is\nimmensely more extensive in amount, and obeys in many respects the\nlaws of mechanics. This adamantine substance appears, so far as can be\nascertained, to be perfectly uniform in its properties when existing\nin space unoccupied by matter. Light and heat are conveyed by it with\nequal velocity in all directions, and in all parts of space so far as\nobservation informs us. But the presence of gravitating matter modifies\nthe density and mechanical properties of the so-called ether in a way\nwhich is yet quite unexplained.[494]\n\nLeaving gravity, it is somewhat difficult to discover other laws\nwhich are equally true of all matter. Boerhaave was considered to\nhave established that all bodies expand by heat; but not only is the\nexpansion very different in different substances, but we now know\npositive exceptions. Many liquids and a few solids contract by heat\nat certain temperatures. There are indeed other relations of heat to\nmatter which seem to be universal and uniform; all substances begin\nto give off rays of light at the same temperature, according to the\nlaw of Draper; and gases will not be an exception if sufficiently\ncondensed, as in the experiments of Frankland. Grove considers it to\nbe universally true that all bodies in combining produce heat; with\nthe doubtful exception of sulphur and selenium, all solids in becoming\nliquids, and all liquids in becoming gases, absorb heat; but the\nquantities of heat absorbed vary with the chemical qualities of the\nmatter. Carnot\u2019s Thermodynamic Law is held to be exactly true of all\nmatter without distinction; it expresses the fact that the amount of\nmechanical energy which might be theoretically obtained from a certain\namount of heat energy depends only upon the change of the temperatures,\nso that whether an engine be worked by water, air, alcohol, ammonia, or\nany other substance, the result would theoretically be the same, if the\nboiler and condenser were maintained at similar temperatures.\n\n  [494] Professor Lovering has pointed out how obscure and uncertain\n  the ideas of scientific men about this ether are, in his interesting\n  Presidential Address before the American Association at Hartford,\n  1874. *Silliman\u2019s Journal*, October 1874, p.\u00a0297. *Philosophical\n  Magazine*, vol.\u00a0xlviii. p.\u00a0493.\n\n\n*Variable Properties of Matter.*\n\nI have enumerated some of the few properties of matter, which are\nmanifested in exactly the same manner by all substances, whatever be\ntheir differences of chemical or physical constitution. But by far\nthe greater number of qualities vary in degree; substances are more\nor less dense, more or less transparent, more or less compressible,\nmore or less magnetic, and so on. One common result of the progress of\nscience is to show that qualities supposed to be entirely absent from\nmany substances are present only in so low a degree of intensity that\nthe means of detection were insufficient. Newton believed that most\nbodies were quite unaffected by the magnet; Faraday and Tyndall have\nrendered it very doubtful whether any substance whatever is wholly\ndevoid of magnetism, including under that term diamagnetism. We are\nrapidly learning to believe that there are no substances absolutely\nopaque, or non-conducting, non-electric, non-elastic, non-viscous,\nnon-compressible, insoluble, infusible, or non-volatile. All tends\nto become a matter of degree, or sometimes of direction. There may\nbe some substances oppositely affected to others, as ferro-magnetic\nsubstances are oppositely affected to diamagnetics, or as substances\nwhich contract by heat are opposed to those which expand; but the\ntendency is certainly for every affection of one kind of matter to be\nrepresented by something similar in other kinds. On this account one of\nNewton\u2019s rules of philosophising seems to lose all validity; he said,\n\u201cThose qualities of bodies which are not capable of being heightened,\nand remitted, and which are found in all bodies on which experiment can\nbe made, must be considered as universal qualities of all bodies.\u201d As\nfar as I can see, the contrary is more probable, namely, that qualities\nvariable in degree will be found in every substance in a greater or\nless degree.\n\nIt is remarkable that Newton whose method of investigation was\nlogically perfect, seemed incapable of generalising and describing\nhis own procedure. His celebrated \u201cRules of Reasoning in Philosophy,\u201d\ndescribed at the commencement of the third book of the *Principia*, are\nof questionable truth, and still more questionable value.\n\n\n*Extreme Instances of Properties.*\n\nAlthough substances usually differ only in degree, great interest\nmay attach to particular substances which manifest a property in a\nconspicuous and intense manner. Every branch of physical science\nhas usually been developed from the attention forcibly drawn to some\nsingular substance. Just as the loadstone disclosed magnetism and\namber frictional electricity, so did Iceland spar show the existence\nof double refraction, and sulphate of quinine the phenomenon of\nfluorescence. When one such startling instance has drawn the attention\nof the scientific world, numerous less remarkable cases of the\nphenomenon will be detected, and it will probably prove that the\nproperty in question is actually universal to all matter. Nevertheless,\nthe extreme instances retain their interest, partly in a historical\npoint of view, partly because they furnish the most convenient\nsubstances for experiment.\n\nFrancis Bacon was fully aware of the value of such examples, which he\ncalled *Ostensive Instances* or Light-giving, Free and Predominant\nInstances. \u201cThey are those,\u201d he says,[495] \u201cwhich show the nature\nunder investigation naked, in an exalted condition, or in the highest\ndegree of power; freed from impediments, or at least by its strength\npredominating over and suppressing them.\u201d He mentions quicksilver as\nan ostensive instance of weight or density, thinking it not much less\ndense than gold, and more remarkable than gold as joining density\nto liquidity. The magnet is mentioned as an ostensive instance of\nattraction. It would not be easy to distinguish clearly between these\nostensive instances and those which he calls *Instantiae Monodicae*,\nor *Irregulares*, or *Heteroclitae*, under which he places whatever\nis extravagant in its properties or magnitude, or exhibits least\nsimilarity to other things, such as the sun and moon among the heavenly\nbodies, the elephant among animals, the letter *s* among letters, or\nthe magnet among stones.[496]\n\n  [495] *Novum Organum*, bk. ii. Aphorisms, 24, 25.\n\n  [496] Ibid. Aph. 28.\n\nIn optical science great use has been made of the high dispersive\npower of the transparent compounds of lead, that is, the power of\ngiving a long spectrum (p.\u00a0432). Dollond, having noticed this peculiar\ndispersive power in lenses made of flint glass, employed them to\nproduce an achromatic arrangement. The element strontium presents a\ncontrast to lead in this respect, being characterised by a remarkably\nlow dispersive power; but I am not aware that this property has yet\nbeen turned to account.\n\nCompounds of lead have both a high dispersive and a high refractive\nindex, and in the latter respect they proved very useful to Faraday.\nHaving spent much labour in preparing various kinds of optical glass,\nFaraday happened to form a compound of lead, silica, and boracic\nacid, now known as *heavy glass*, which possessed an intensely high\nrefracting power. Many years afterwards in attempting to discover the\naction of magnetism upon light he failed to detect any effect, as has\nbeen already mentioned, (p.\u00a0588), until he happened to test a piece of\nthe heavy glass. The peculiar refractive power of this medium caused\nthe magnetic strain to be apparent, and the rotation of the plane of\npolarisation was discovered.\n\nIn almost every part of physical science there is some substance\nof powers pre-eminent for the special purpose to which it is put.\nRock-salt is invaluable for its extreme diathermancy or transparency to\nthe least refrangible rays of the spectrum. Quartz is equally valuable\nfor its transparency, as regards the ultra-violet or most refrangible\nrays. Diamond is the most highly refracting substance which is at the\nsame time transparent; were it more abundant and easily worked it would\nbe of great optical importance. Cinnabar is distinguished by possessing\na power of rotating the plane of polarisation of light, from 15 to 17\ntimes as much as quartz. In electric experiments copper is employed for\nits high conducting powers and exceedingly low magnetic properties;\niron is of course indispensable for its enormous magnetic powers; while\nbismuth holds a like place as regards its diamagnetic powers, and was\nof much importance in Tyndall\u2019s decisive researches upon the polar\ncharacter of the diamagnetic force.[497] In regard to magne-crystallic\naction the mineral cyanite is highly remarkable, being so powerfully\naffected by the earth\u2019s magnetism, that, when delicately suspended, it\nassumes a constant position with regard to the magnetic meridian, and\nmay almost be used like the compass needle. Sodium is distinguished\nby its unique light-giving powers, which are so extraordinary that\nprobably one half of the whole number of stars in the heavens have a\nyellow tinge in consequence.\n\n  [497] *Philosophical Transactions* (1856) vol.\u00a0cxlvi. p.\u00a0246.\n\nIt is remarkable that water, though the most common of all fluids,\nis distinguished in almost every respect by extreme qualities. Of\nall known substances water has the highest specific heat, being thus\npeculiarly fitted for the purpose of warming and cooling, to which\nit is often put. It rises by capillary attraction to a height more\nthan twice that of any other liquid. In the state of ice it is nearly\ntwice as dilatable by heat as any other known solid substance.[498] In\nproportion to its density it has a far higher surface tension than any\nother substance, being surpassed in absolute tension only by mercury;\nand it would not be difficult to extend considerably the list of its\nremarkable and useful properties.\n\n  [498] *Philosophical Magazine*, 4th Series, January 1870, vol.\u00a0xxxix.\n  p.\u00a02.\n\nUnder extreme instances we may include cases of remarkably low powers\nor qualities. Such cases seem to correspond to what Bacon calls\n*Clandestine Instances*, which exhibit a given nature in the least\nintensity, and as it were in a rudimentary state.[499] They may often\nbe important, he thinks, as allowing the detection of the cause of\nthe property by difference. I may add that in some cases they may be\nof use in experiments. Thus hydrogen is the least dense of all known\nsubstances, and has the least atomic weight. Liquefied nitrous oxide\nhas the lowest refractive index of all known fluids.[500] The compounds\nof strontium have the lowest dispersive power. It is obvious that\na property of very low degree may prove as curious and valuable a\nphenomenon as a property of very high degree.\n\n  [499] *Novum Organum*, bk. ii. Aphorism 25.\n\n  [500] Faraday\u2019s *Experimental Researches in Chemistry and Physics*,\n  p. 93.\n\n\n*The Detection of Continuity.*\n\nWe should bear in mind that phenomena which are in reality of a\nclosely similar or even identical nature, may present to the senses\nvery different appearances. Without a careful analysis of the changes\nwhich take place, we may often be in danger of widely separating facts\nand processes, which are actually instances of the same law. Extreme\ndifference of degree or magnitude is a frequent cause of error. It is\ntruly difficult at the first moment to recognise any similarity between\nthe gradual rusting of a piece of iron, and the rapid combustion of a\nheap of straw. Yet Lavoisier\u2019s chemical theory was founded upon the\nsimilarity of the oxydising process in one case and the other. We have\nonly to divide the iron into excessively small particles to discover\nthat it is really the more combustible of the two, and that it actually\ntakes fire spontaneously and burns like tinder. It is the excessive\nslowness of the process in the case of a massive piece of iron which\ndisguises its real character.\n\nIf Xenophon reports truly, Socrates was misled by not making sufficient\nallowance for extreme differences of degree and quantity. Anaxagoras\nheld that the sun is a fire, but Socrates rejected this opinion, on the\nground that we can look at a fire, but not at the sun, and that plants\ngrow by sunshine while they are killed by fire. He also pointed out\nthat a stone heated in a fire is not luminous, and soon cools, whereas\nthe sun ever remains equally luminous and hot.[501] All such mistakes\nevidently arise from not perceiving that difference of quantity may be\nso extreme as to assume the appearance of difference of quality. It is\nthe least creditable thing we know of Socrates, that after pointing\nout these supposed mistakes of earlier philosophers, he advised his\nfollowers not to study astronomy.\n\n  [501] *Memorabilia*, iv. 7.\n\nMasses of matter of very different size may be expected to exhibit\napparent differences of conduct, arising from the various intensity of\nthe forces brought into play. Many persons have thought it requisite\nto imagine occult forces producing the suspension of the clouds, and\nthere have even been absurd theories representing cloud particles as\nminute water-balloons buoyed up by the warm air within them. But we\nhave only to take proper account of the enormous comparative resistance\nwhich the air opposes to the fall of minute particles, to see that\nall cloud particles are probably constantly falling through the air,\nbut so slowly that there is no apparent effect. Mineral matter again\nis always regarded as inert and incapable of spontaneous movement. We\nare struck by astonishment on observing in a powerful microscope, that\nevery kind of solid matter suspended in extremely minute particles\nin pure water, acquires an oscillatory movement, often so marked as\nto resemble dancing or skipping. I conceive that this movement is due\nto the comparatively vast intensity of chemical action when exerted\nupon minute particles, the effect being 5,000 or 10,000 greater in\nproportion to the mass than in fragments of an inch diameter (p.\u00a0406).\n\nMuch that was formerly obscure in the science of electricity arose from\nthe extreme differences of intensity and quantity in which this form of\nenergy manifests itself. Between the brilliant explosive discharge of a\nthunder-cloud and the gentle continuous current produced by two pieces\nof metal and some dilute acid, there is no apparent analogy whatever.\nIt was therefore a work of great importance when Faraday demonstrated\nthe identity of the forces in action, showing that common frictional\nelectricity would decompose water like that from the voltaic battery.\nThe relation of the phenomena became plain when he succeeded in showing\nthat it would require 800,000 discharges of his large Leyden battery\nto decompose one single grain of water. Lightning was now seen to be\nelectricity of excessively high tension, but extremely small quantity,\nthe difference being somewhat analogous to that between the force of\none million gallons of water falling through one foot, and one gallon\nof water falling through one million feet. Faraday estimated that one\ngrain of water acting on four grains of zinc, would yield electricity\nenough for a great thunderstorm.\n\nIt was long believed that electrical conductors and insulators belonged\nto two opposed classes of substances. Between the inconceivable\nrapidity with which the current passes through pure copper wire,\nand the apparently complete manner in which it is stopped by a\nthin partition of gutta-percha or gum-lac, there seemed to be no\nresemblance. Faraday again laboured successfully to show that these\nwere but the extreme cases of a chain of substances varying in all\ndegrees in their powers of conduction. Even the best conductors, such\nas pure copper or silver, offer resistance to the electric current. The\nother metals have considerably higher powers of resistance, and we pass\ngradually down through oxides and sulphides. The best insulators, on\nthe other hand, allow of an atomic induction which is the necessary\nantecedent of conduction. Hence Faraday inferred that whether we can\nmeasure the effect or not, all substances discharge electricity more\nor less.[502] One consequence of this doctrine must be, that every\ndischarge of electricity produces an induced current. In the case of\nthe common galvanic current we can readily detect the induced current\nin any parallel wire or other neighbouring conductor, and can separate\nthe opposite currents which arise at the moments when the original\ncurrent begins and ends. But a discharge of high tension electricity\nlike lightning, though it certainly occupies time and has a beginning\nand an end, yet lasts so minute a fraction of a second, that it would\nbe hopeless to attempt to detect and separate the two opposite induced\ncurrents, which are nearly simultaneous and exactly neutralise each\nother. Thus an apparent failure of analogy is explained away, and\nwe are furnished with another instance of a phenomenon incapable of\nobservation and yet theoretically known to exist.[503]\n\n  [502] *Experimental Researches in Electricity*, Series xii. vol.\u00a0i.\n  p.\u00a0420.\n\n  [503] *Life of Faraday*, vol.\u00a0ii. p.\u00a07.\n\nPerhaps the most extraordinary case of the detection of unsuspected\ncontinuity is found in the discovery of Cagniard de la Tour and\nProfessor Andrews, that the liquid and gaseous conditions of matter\nare only remote points in a continuous course of change. Nothing is\nat first sight more apparently distinct than the physical condition\nof water and aqueous vapour. At the boiling-point there is an\nentire breach of continuity, and the gas produced is subject to\nlaws incomparably more simple than the liquid from which it arose.\nBut Cagniard de la Tour showed that if we maintain a liquid under\nsufficient pressure its boiling point may be indefinitely raised, and\nyet the liquid will ultimately assume the gaseous condition with but\na small increase of volume. Professor Andrews, recently following out\nthis course of inquiry, has shown that liquid carbonic acid may, at\na particular temperature (30\u00b0\u00b792\u00a0C.), and under the pressure of 74\natmospheres, be at the same time in a state indistinguishable from\nthat of liquid and gas. At higher pressures carbonic acid may be\nmade to pass from a palpably liquid state to a truly gaseous state\nwithout any abrupt change whatever. As the pressure is greater the\nabruptness of the change from liquid to gas gradually decreases, and\nfinally vanishes. Similar phenomena or an approximation to them have\nbeen observed in other liquids, and there is little doubt that we may\nmake a wide generalisation, and assert that, under adequate pressure,\nevery liquid might be made to pass into a gas without breach of\ncontinuity.[504] The liquid state, moreover, is considered by Professor\nAndrews to be but an intermediate step between the solid and gaseous\nconditions. There are various indications that the process of melting\nis not perfectly abrupt; and could experiments be made under adequate\npressures, it is believed that every solid could be made to pass by\ninsensible degrees into the state of liquid, and subsequently into that\nof gas.\n\n  [504] *Nature*, vol.\u00a0ii. p.\u00a0278.\n\nThese discoveries appear to open the way to most important and\nfundamental generalisations, but it is probable that in many other\ncases phenomena now regarded as discrete may be shown to be different\ndegrees of the same process. Graham was of opinion that chemical\naffinity differs but in degree from the ordinary attraction which\nholds different particles of a body together. He found that sulphuric\nacid continued to evolve heat when mixed even with the fiftieth\nequivalent of water, so that there seemed to be no distinct limit to\nchemical affinity. He concludes, \u201cThere is reason to believe that\nchemical affinity passes in its lowest degree into the attraction of\naggregation.\u201d[505]\n\n  [505] *Journal of the Chemical Society*, vol.\u00a0viii. p.\u00a051.\n\nThe atomic theory is well established, but its limits are not marked\nout. As Grove points out, we may by selecting sufficiently high\nmultipliers express any combination or mixture of elements in terms\nof their equivalent weights.[506] Sir W. Thomson has suggested that\nthe power which vegetable fibre, oatmeal, and other substances possess\nof attracting and condensing aqueous vapour is probably continuous,\nor, in fact, identical with capillary attraction, which is capable\nof interfering with the pressure of aqueous vapour and aiding its\ncondensation.[507] There are many cases of so-called catalytic or\nsurface action, such as the extraordinary power of animal charcoal\nfor attracting organic matter, or of spongy platinum for condensing\nhydrogen, which can only be considered as exalted cases of a more\ngeneral power of attraction. The number of substances which are\ndecomposed by light in a striking manner is very limited; but many\nother substances, such as vegetable colours, are affected by long\nexposure; on the principle of continuity we might expect to find that\nall kinds of matter are more or less susceptible of change by the\nincidence of light rays.[508] It is the opinion of Grove that wherever\nan electric current passes there is a tendency to decomposition, a\nstrain on the molecules, which when sufficiently intense leads to\ndisruption. Even a metallic conducting wire may be regarded as tending\nto decomposition. Davy was probably correct in describing electricity\nas chemical affinity acting on masses, or rather, as Grove suggests,\ncreating a disturbance through a chain of particles.[509] Laplace went\nso far as to suggest that all chemical phenomena may be results of\nthe Newtonian law of attraction, applied to atoms of various mass and\nposition; but the time is probably far distant when the progress of\nmolecular philosophy and of mathematical methods will enable such a\ngeneralisation to be verified or refuted.\n\n  [506] *Correlation of Physical Forces*, 3rd edit. p.\u00a0184.\n\n  [507] *Philosophical Magazine*, 4th Series, vol.\u00a0xlii. p.\u00a0451.\n\n  [508] Grove, *Correlation of Physical Forces*, 3rd edit. p.\u00a0118.\n\n  [509] Ibid. pp.\u00a0166, 199, &c.\n\n\n*The Law of Continuity.*\n\nUnder the title of the Law of Continuity we may place many applications\nof the general principle of reasoning, that what is true of one case\nwill be true of similar cases, and probably true of what are probably\nsimilar. Whenever we find that a law or similarity is rigorously\nfulfilled up to a certain point in time or space, we expect with a high\ndegree of probability that it will continue to be fulfilled at least a\nlittle further. If we see part only of a circle, we naturally expect\nthat the circular form will be continued in the part hidden from us. If\na body has moved uniformly over a certain space, we expect that it will\ncontinue to move uniformly. The ground of such inferences is doubtless\nidentical with that of other inductive inferences. In continuous\nmotion every infinitely small space passed over constitutes a separate\nconstituent fact, and had we perfect powers of observation the smallest\nfinite motion would include an infinity of information, which, by the\nprinciples of the inverse method of probabilities, would enable us to\ninfer with certainty to the next infinitely small portion of the path.\nBut when we attempt to infer from one finite portion of a path to\nanother finite portion, inference will be only more or less probable,\naccording to the comparative lengths of the portions and the accuracy\nof observation; the longer our experience is, the more probable our\ninference will be; the greater the length of time or space over which\nthe inference extends, the less probable.\n\nThis principle of continuity presents itself in nature in a great\nvariety of forms and cases. It is familiarly expressed in the dictum\n*Natura non agit per saltum*. As Graham expressed the maxim, there are\nin nature no abrupt transitions, and the distinctions of class are\nnever absolute.[510] There is always some notice--some forewarning\nof every phenomenon, and every change begins by insensible degrees,\ncould we observe it with perfect accuracy. The cannon ball, indeed, is\nforced from the cannon in an inappreciable portion of time; the trigger\nis pulled, the fuze fired, the powder inflamed, the ball expelled,\nall simultaneously to our senses. But there is no doubt that time is\noccupied by every part of the process, and that the ball begins to\nmove at first with infinite slowness. Captain Noble is able to measure\nby his chronoscope the progress of the shot in a 300-pounder gun, and\nfinds that the whole motion within the barrel takes place in something\nless than one 200th part of a second. It is certain that no finite\nforce can produce motion, except in a finite space of time. The amount\nof momentum communicated to a body is proportional to the accelerating\nforce multiplied by the time during which it acts uniformly. Thus a\nslight force produces a great velocity only by long-continued action.\nIn a powerful shock, like that of a railway collision, the stroke of a\nhammer on an anvil, or the discharge of a gun, the time is very short,\nand therefore the accelerating forces brought into play are exceedingly\ngreat, but never infinite. In the case of a large gun the powder in\nexploding is said to exert for a moment a force equivalent to at least\n2,800,000 horses.\n\n  [510] *Philosophical Transactions*, 1861. *Chemical and Physical\n  Researches*, p. 598.\n\nOur belief in some of the fundamental laws of nature rests upon the\nprinciple of continuity. Galileo is held to be the first philosopher\nwho consciously employed this principle in his arguments concerning\nthe nature of motion, and it is certain that we can never by mere\nexperience assure ourselves of the truth even of the first law of\nmotion. *A material particle*, we are told, *when not acted on by\nextraneous forces will continue in the same state of rest or motion.*\nThis may be true, but as we can find no body which is free from the\naction of extraneous causes, how are we to prove it? Only by observing\nthat the less the amount of those forces the more nearly is the law\nfound to be true. A ball rolled along rough ground is soon stopped;\nalong a smooth pavement it continues longer in movement. A delicately\nsuspended pendulum is almost free from friction against its supports,\nbut it is gradually stopped by the resistance of the air; place it\nin the vacuous receiver of an air-pump and we find the motion much\nprolonged. A large planet like Jupiter experiences almost infinitely\nless friction, in comparison to its vast momentum, than we can produce\nexperimentally, and we find in such a case that there is not the least\nevidence of the falsity of the law. Experience, then, informs us that\nwe may approximate indefinitely to a uniform motion by sufficiently\ndecreasing the disturbing forces. It is an act of inference which\nenables us to travel on beyond experience, and assert that, in the\ntotal absence of any extraneous force, motion would be absolutely\nuniform. The state of rest, again, is a limiting case in which motion\nis infinitely small or zero, to which we may attain, on the principle\nof continuity, by successively considering cases of slower and slower\nmotion. There are many classes of phenomena, in which, by gradually\npassing from the apparent to the obscure, we can assure ourselves of\nthe nature of phenomena which would otherwise be a matter of great\ndoubt. Thus we can sufficiently prove in the manner of Galileo, that\na musical sound consists of rapid uniform pulses, by causing strokes\nto be made at intervals which we gradually diminish until the separate\nstrokes coalesce into a uniform hum or note. With great advantage we\napproach, as Tyndall says, the sonorous through the grossly mechanical.\nIn listening to a great organ we cannot fail to perceive that the\nlongest pipes, or their partial tones, produce a tremor and fluttering\nof the building. At the other extremity of the scale, there is no fixed\nlimit to the acuteness of sounds which we can hear; some individuals\ncan hear sounds too shrill for other ears, and as there is nothing in\nthe nature of the atmosphere to prevent the existence of undulations\nfar more rapid than any of which we are conscious, we may infer, by the\nprinciple of continuity, that such undulations probably exist.\n\nThere are many habitual actions which we perform we know not how. So\nrapidly are acts of minds accomplished that analysis seems impossible.\nWe can only investigate them when in process of formation, observing\nthat the best formed habit is slowly and continuously acquired, and\nit is in the early stages that we can perceive the rationale of the\nprocess.\n\nLet it be observed that this principle of continuity must be held of\nmuch weight only in exact physical laws, those which doubtless repose\nultimately upon the simple laws of motion. If we fearlessly apply\nthe principle to all kinds of phenomena, we may often be right in\nour inferences, but also often wrong. Thus, before the development\nof spectrum analysis, astronomers had observed that the more they\nincreased the powers of their telescopes the more nebul\u00e6 they could\nresolve into distinct stars. This result had been so often found\ntrue that they almost irresistibly assumed that all nebul\u00e6 would be\nultimately resolved by telescopes of sufficient power; yet Huggins has\nin recent years proved by the spectroscope, that certain nebul\u00e6 are\nactually gaseous, and in a truly nebulous state.\n\nThe principle of continuity must have been continually employed in the\ninquiries of Galileo, Newton, and other experimental philosophers,\nbut it appears to have been distinctly formulated for the first time\nby Leibnitz. He at least claims to have first spoken of \u201cthe law of\ncontinuity\u201d in a letter to Bayle, printed in the *Nouvelles de la\nR\u00e9publique des Lettres*, an extract from which is given in Erdmann\u2019s\nedition of Leibnitz\u2019s works, p.\u00a0104, under the title \u201cSur un Principe\nG\u00e9n\u00e9ral utile \u00e0 l\u2019explication des Lois de la Nature.\u201d[511] It has\nindeed been asserted that the doctrine of the *latens processus* of\nFrancis Bacon involves the principle of continuity,[512] but I think\nthat this doctrine, like that of the *natures* of substances, is merely\na vague statement of the principle of causation.\n\n  [511] *Life of Sir W. Hamilton*, p.\u00a0439.\n\n  [512] Powell\u2019s *History of Natural Philosophy*, p.\u00a0201. *Novum\n  Organum*, bk. ii. Aphorisms 5\u20137.\n\n\n*Failure of the Law of Continuity.*\n\nThere are certain cautions which must be given as to the application of\nthe principle of continuity. In the first place, where this principle\nreally holds true, it may seem to fail owing to our imperfect means\nof observation. Though a physical law may not admit of perfectly\nabrupt change, there is no limit to the approach which it may make to\nabruptness. When we warm a piece of very cold ice, the absorption of\nheat, the temperature, and the dilatation of the ice vary according\nto apparently simple laws until we come to the zero of the Centigrade\nscale. Everything is then changed; an enormous absorption of heat\ntakes place without any rise of temperature, and the volume of the ice\ndecreases as it changes into water. Unless carefully investigated, this\nchange appears to be perfectly abrupt; but accurate observation seems\nto show that there is a certain forewarning; the ice does not turn into\nwater all at once, but through a small fraction of a degree the change\nis gradual. All the phenomena concerned, if measured very exactly,\nwould be represented not by angular lines, but continuous curves,\nundergoing rapid flexures; and we may probably assert with safety that\nbetween whatever points of temperature we examine ice, there would be\nfound some indication, though almost infinitesimally small, of the\napparently abrupt change which was to occur at a higher temperature.\nIt might also be pointed out that the important and apparently simple\nphysical laws, such as those of Boyle and Mariotte, Dalton and\nGay-Lussac, &c., are only approximately true, and the divergences\nfrom the simple laws are forewarnings of abrupt changes, which would\notherwise break the law of continuity.\n\nSecondly, it must be remembered that mathematical laws of some\ncomplexity will probably present singular cases or negative results,\nwhich may bear the appearance of discontinuity, as when the law of\nretraction suddenly yields us with perfect abruptness the phenomenon of\ntotal internal reflection. In the undulatory theory, however, there is\nno real change of law between refraction and reflection. Faraday in the\nearlier part of his career found so many substances possessing magnetic\npower, that he ventured on a great generalisation, and asserted that\nall bodies shared in the magnetic property of iron. His mistake, as he\nafterwards discovered, consisted in overlooking the fact that though\nmagnetic in a certain sense, some substances have negative magnetism,\nand are repelled instead of being attracted by the magnet.\n\nThirdly, where we might expect to find a uniform mathematical law\nprevailing, the law may undergo abrupt change at singular points,\nand actual discontinuity may arise. We may sometimes be in danger of\ntreating under one law phenomena which really belong to different laws.\nFor instance, a spherical shell of uniform matter attracts an external\nparticle of matter with a force varying inversely as the square of the\ndistance from the centre of the sphere. But this law only holds true\nso long as the particle is external to the shell. Within the shell\nthe law is wholly different, and the aggregate gravity of the sphere\nbecomes zero, the force in every direction being neutralised by an\nexactly equal opposite force. If an infinitely small particle be in the\nsuperficies of a sphere, the law is again different, and the attractive\npower of the shell is half what it would be with regard to particles\ninfinitely close to the surface of the shell. Thus in approaching the\ncentre of a shell from a distance, the force of gravity shows double\ndiscontinuity in passing through the shell.[513]\n\n  [513] Thomson and Tait, *Treatise on Natural Philosophy*, vol.\u00a0i. pp.\n  346\u2013351.\n\nIt may admit of question, too, whether discontinuity is really unknown\nin nature. We perpetually do meet with events which are real breaks\nupon the previous law, though the discontinuity may be a sign that\nsome independent cause has come into operation. If the ordinary\ncourse of the tides is interrupted by an enormous irregular wave, we\nattribute it to an earthquake, or some gigantic natural disturbance.\nIf a meteoric stone falls upon a person and kills him, it is clearly a\ndiscontinuity in his life, of which he could have had no anticipation.\nA sudden sound may pass through the air neither preceded nor followed\nby any continuous effect. Although, then, we may regard the Law of\nContinuity as a principle of nature holding rigorously true in many of\nthe relations of natural forces, it seems to be a matter of difficulty\nto assign the limits within which the law is verified. Much caution is\nrequired in its application.\n\n\n*Negative Arguments on the Principle of Continuity.*\n\nUpon the principle of continuity we may sometimes found arguments of\ngreat force which prove an hypothesis to be impossible, because it\nwould involve a continual repetition of a process *ad infinitum*, or\nelse a purely arbitrary breach at some point. Bonnet\u2019s famous theory\nof reproduction represented every living creature as containing\ngerms which were perfect representatives of the next generation, so\nthat on the same principle they necessarily included germs of the\nnext generation, and so on indefinitely. The theory was sufficiently\nrefuted when once clearly stated, as in the following poem called the\nUniverse,[514] by Henry Baker:--\n\n  \u201cEach seed includes a plant: that plant, again,\n   Has other seeds, which other plants contain:\n   Those other plants have all their seeds, and those\n   More plants again, successively inclose.\n\n  \u201cThus, ev\u2019ry single berry that we find,\n   Has, really, in itself whole forests of its kind,\n   Empire and wealth one acorn may dispense,\n   By fleets to sail a thousand ages hence.\u201d\n\n  [514] *Philosophical Transactions* (1740), vol.\u00a0xli. p.\u00a0454.\n\nThe general principle of inference, that what we know of one case must\nbe true of similar cases, so far as they are similar, prevents our\nasserting anything which we cannot apply time after time under the same\ncircumstances. On this principle Stevinus beautifully demonstrated\nthat weights resting on two inclined planes and balancing each other\nmust be proportional to the lengths of the planes between their apex\nand a horizontal plane. He imagined a uniform endless chain to be hung\nover the planes, and to hang below in a symmetrical festoon. If the\nchain were ever to move by gravity, there would be the same reason\nfor its moving on for ever, and thus producing a perpetual motion. As\nthis is absurd, the portions of the chain lying on the planes, and\nequal in length to the planes, must balance each other. On similar\ngrounds we may disprove the existence of any *self-moving machine*;\nfor if it could once alter its own state of motion or rest, in however\nsmall a degree, there is no reason why it should not do the like\ntime after time *ad infinitum*. Newton\u2019s proof of his third law of\nmotion, in the case of gravity, is of this character. For he remarks\nthat if two gravitating bodies do not exert exactly equal forces in\nopposite directions, the one exerting the strongest pull will carry\nboth away, and the two bodies will move off into space together with\nvelocity increasing *ad infinitum*. But though the argument might\nseem sufficiently convincing, Newton in his characteristic way made\nan experiment with a loadstone and iron floated upon the surface of\nwater.[515] In recent years the very foundation of the principle of\nconservation of energy has been placed on the assumption that it is\nimpossible by any combination of natural bodies to produce force\ncontinually from nothing.[516] The principle admits of application in\nvarious subtle forms.\n\n  [515] *Principia*, bk. i. Law iii. Corollary 6.\n\n  [516] Helmholtz, Taylor\u2019s *Scientific Memoirs* (1853), vol.\u00a0vi.\n  p.\u00a0118.\n\nLucretius attempted to prove, by a most ingenious argument of this\nkind, that matter must be indestructible. For if a finite quantity,\nhowever small, were to fall out of existence in any finite time, an\nequal quantity might be supposed to lapse in every equal interval of\ntime, so that in the infinity of past time the universe must have\nceased to exist.[517] But the argument, however ingenious, seems to\nfail at several points. If past time be infinite, why may not matter\nhave been created infinite also? It would be most reasonable, again,\nto suppose the matter destroyed in any time to be proportional to the\nmatter then remaining, and not to the original quantity; under this\nhypothesis even a finite quantity of original matter could never wholly\ndisappear from the universe. For like reasons we cannot hold that the\ndoctrine of the conservation of energy is really proved, or can ever be\nproved to be absolutely true, however probable it may be regarded.\n\n  [517] *Lucretius*, bk. i. lines 232\u2013264.\n\n\n*Tendency to Hasty Generalisation.*\n\nIn spite of all the powers and advantages of generalisation, men\nrequire no incitement to generalise; they are too apt to draw hasty and\nill-considered inferences. As Francis Bacon said, our intellects want\nnot wings, but rather weights of lead to moderate their course.[518]\nThe process is inevitable to the human mind; it begins with childhood\nand lasts through the second childhood. The child that has once been\nhurt fears the like result on all similar occasions, and can with\ndifficulty be made to distinguish between case and case. It is caution\nand discrimination in the adoption of conclusions that we have chiefly\nto learn, and the whole experience of life is one continued lesson\nto this effect. Baden Powell has excellently described this strong\nnatural propensity to hasty inference, and the fondness of the human\nmind for tracing resemblances real or fanciful. \u201cOur first inductions,\u201d\nhe says,[519] \u201care always imperfect and inconclusive; we advance\ntowards real evidence by successive approximations; and accordingly we\nfind false generalisation the besetting error of most first attempts\nat scientific research. The faculty to generalise accurately and\nphilosophically requires large caution and long training, and is not\nfully attained, especially in reference to more general views, even\nby some who may properly claim the title of very accurate scientific\nobservers in a more limited field. It is an intellectual habit which\nacquires immense and accumulating force from the contemplation of wider\nanalogies.\u201d\n\n  [518] *Novum Organum*, bk. 1 Aphorism 104.\n\n  [519] *The Unity of Worlds and of Nature*, 2nd edit. p.\u00a0116.\n\nHasty and superficial generalisations have always been the bane\nof science, and there would be no difficulty in finding endless\nillustrations. Between things which are the same in number there is a\ncertain resemblance, namely in number; but in the infancy of science\nmen could not be persuaded that there was not a deeper resemblance\nimplied in that of number. Pythagoras was not the inventor of a\nmystical science of number. In the ancient Oriental religions the seven\nmetals were connected with the seven planets, and in the seven days\nof the week we still have, and probably always shall have, a relic of\nthe septiform system ascribed by Dio Cassius to the ancient Egyptians.\nThe disciples of Pythagoras carried the doctrine of the number seven\ninto great detail. Seven days are mentioned in Genesis; infants acquire\ntheir teeth at the end of seven months; they change them at the end\nof seven years; seven feet was the limit of man\u2019s height; every\nseventh year was a climacteric or critical year, at which a change\nof disposition took place. Then again there were the seven sages of\nGreece, the seven wonders of the world, the seven rites of the Grecian\ngames, the seven gates of Thebes, and the seven generals destined to\nconquer that city.\n\nIn natural science there were not only the seven planets, and the seven\nmetals, but also the seven primitive colours, and the seven tones\nof music. So deep a hold did this doctrine take that we still have\nits results in many customs, not only in the seven days of the week,\nbut the seven years\u2019 apprenticeship, puberty at fourteen years, the\nsecond climacteric, and legal majority at twenty-one years, the third\nclimacteric. The idea was reproduced in the seven sacraments of the\nRoman Catholic Church, and the seven year periods of Comte\u2019s grotesque\nsystem of domestic worship. Even in scientific matters the loftiest\nintellects have occasionally yielded, as when Newton was misled by the\nanalogy between the seven tones of music and the seven colours of his\nspectrum. Other numerical analogies, though rejected by Galileo, held\nKepler in thraldom; no small part of Kepler\u2019s labours during seventeen\nyears was spent upon numerical and geometrical analogies of the most\nbaseless character; and he gravely held that there could not be more\nthan six planets, because there were not more than five regular solids.\nEven the genius of Huyghens did not prevent him from inferring that\nbut one satellite could belong to Saturn, because, with those of\nJupiter and the Earth, it completed the perfect number of six. A whole\nseries of other superstitions and fallacies attach to the numbers six\nand nine.\n\nIt is by false generalisation, again, that the laws of nature have\nbeen supposed to possess that perfection which we attribute to simple\nforms and relations. The heavenly bodies, it was held, must move in\ncircles, for the circle was the perfect figure. Newton seemed to adopt\nthe questionable axiom that nature always proceeds in the simplest way;\nin stating his first rule of philosophising, he adds:[520] \u201cTo this\npurpose the philosophers say, that nature does nothing in vain, when\nless will serve; for nature is pleased with simplicity, and affects\nnot the pomp of superfluous causes.\u201d Keill lays down[521] as an axiom\nthat \u201cThe causes of natural things are such, as are the most simple,\nand are sufficient to explain the phenomena: for nature always proceeds\nin the simplest and most expeditious method; because by this manner of\noperating the Divine Wisdom displays itself the more.\u201d If this axiom\nhad any clear grounds of truth, it would not apply to proximate laws;\nfor even when the ultimate law is simple the results may be infinitely\ndiverse, as in the various elliptic, hyperbolic, parabolic, or circular\norbits of the heavenly bodies. Simplicity is naturally agreeable to a\nmind of limited powers, but to an infinite mind all things are simple.\n\n  [520] *Principia*, bk. iii, *ad initium*.\n\n  [521] Keill, *Introduction to Natural Philosophy*, p.\u00a089.\n\nEvery great advance in science consists in a great generalisation,\npointing out deep and subtle resemblances. The Copernican system was\na generalisation, in that it classed the earth among the planets; it\nwas, as Bishop Wilkins expressed it, \u201cthe discovery of a new planet,\u201d\nbut it was opposed by a more shallow generalisation. Those who argued\nfrom the condition of things upon the earth\u2019s surface, thought that\nevery object must be attached to and rest upon something else. Shall\nthe earth, they said, alone be free? Accustomed to certain special\nresults of gravity they could not conceive its action under widely\ndifferent circumstances.[522] No hasty thinker could seize the deep\nanalogy pointed out by Horrocks between a pendulum and a planet,\ntrue in substance though mistaken in some details. All the advances\nof modern science rise from the conception of Galileo, that in the\nheavenly bodies, however apparently different their condition, we shall\nultimately recognise the same fundamental principles of mechanical\nscience which are true on earth.\n\n  [522] Jeremi\u00e6 Horroccii *Opera Posthuma* (1673), pp.\u00a026, 27.\n\nGeneralisation is the great prerogative of the intellect, but it\nis a power only to be exercised safely with much caution and after\nlong training. Every mind must generalise, but there are the widest\ndifferences in the depth of the resemblances discovered and the care\nwith which the discovery is verified. There seems to be an innate\npower of insight which a few men have possessed pre-eminently, and\nwhich enabled them, with no exemption indeed from labour or temporary\nerror, to discover the one in the many. Minds of excessive acuteness\nmay exist, which have yet only the powers of minute discrimination,\nand of storing up, in the treasure-house of memory, vast accumulations\nof words and incidents. But the power of discovery belongs to a\nmore restricted class of minds. Laplace said that, of all inventors\nwho had contributed the most to the advancement of human knowledge,\nNewton and Lagrange appeared to possess in the highest degree the\nhappy tact of distinguishing general principles among a multitude of\nobjects enveloping them, and this tact he conceived to be the true\ncharacteristic of scientific genius.[523]\n\n  [523] Young\u2019s *Works*, vol.\u00a0ii. p.\u00a0564.\n\n\n\n\nCHAPTER XXVIII.\n\nANALOGY.\n\n\nAs we have seen in the previous chapter, generalisation passes\ninsensibly into reasoning by analogy, and the difference is one of\ndegree. We are said to generalise when we view many objects as agreeing\nin a few properties, so that the resemblance is extensive rather\nthan deep. When we have only a few objects of thought, but are able\nto discover many points of resemblance, we argue by analogy that the\ncorrespondence will be even deeper than appears. It may not be true\nthat the words are always used in such distinct senses, and there is\ngreat vagueness in the employment of these and many logical terms; but\nif any clear discrimination can be drawn between generalisation and\nanalogy, it is as indicated above.\n\nIt has been said, indeed, that analogy denotes not a resemblance\nbetween things, but between the relations of things. A pilot is a very\ndifferent man from a prime minister, but he bears the same relation to\na ship that the minister does to the state, so that we may analogically\ndescribe the prime minister as the pilot of the state. A man differs\nstill more from a horse, nevertheless four men bear to three men the\nsame relation as four horses bear to three horses. There is a real\nanalogy between the tones of the Monochord, the Sages of Greece, and\nthe Gates of Thebes, but it does not extend beyond the fact that they\nwere all seven in number. Between the most discrete notions, as, for\ninstance, those of time and space, analogy may exist, arising from\nthe fact that the mathematical conditions of the lapse of time and\nof motion along a line are similar. There is no identity of nature\nbetween a word and the thing it signifies; the substance *iron* is\na heavy solid, the word *iron* is either a momentary disturbance of\nthe air, or a film of black pigment on white paper; but there is\nanalogy between words and their significates. The substance iron is\nto the substance iron-carbonate, as the name iron is to the name\niron-carbonate, when these names are used according to their scientific\ndefinitions. The whole structure of language and the whole utility\nof signs, marks, symbols, pictures, and representations of various\nkinds, rest upon analogy. I may hope perhaps to enter more fully upon\nthis important subject at some future time, and to attempt to show how\nthe invention of signs enables us to express, guide, and register our\nthoughts. It will be sufficient to observe here that the use of words\nconstantly involves analogies of a subtle kind; we should often be at\na loss how to describe a notion, were we not at liberty to employ in\na metaphorical sense the name of anything sufficiently resembling it.\nThere would be no expression for the sweetness of a melody, or the\nbrilliancy of an harangue, unless it were furnished by the taste of\nhoney and the brightness of a torch.\n\nA cursory examination of the way in which we popularly use the\nword analogy, shows that it includes all degrees of resemblance or\nsimilarity. The analogy may consist only in similarity of number or\nratio, or in like relations of time and space. It may also consist in\nsimple resemblance between physical properties. We should not be using\nthe word inconsistently with custom, if we said that there was an\nanalogy between iron, nickel, and cobalt, manifested in the strength of\ntheir magnetic powers. There is a still more perfect analogy between\niodine and chlorine; not that every property of iodine is identical\nwith the corresponding property of chlorine; for then they would\nbe one and the same kind of substance, and not two substances; but\nevery property of iodine resembles in all but degree some property of\nchlorine. For almost every substance in which iodine forms a component,\na corresponding substance may be discovered containing chlorine, so\nthat we may confidently infer from the compounds of the one to the\ncompounds of the other substance. Potassium iodide crystallises in\ncubes; therefore it is to be expected that potassium chloride will\nalso crystallise in cubes. The science of chemistry as now developed\nrests almost entirely upon a careful and extensive comparison of the\nproperties of substances, bringing deep-lying analogies to light.\nWhen any new substance is encountered, the chemist is guided in his\ntreatment of it by the analogies which it seems to present with\npreviously known substances.\n\nIn this chapter I cannot hope to illustrate the all-pervading influence\nof analogy in human thought and science. All science, it has been said,\nat the outset, arises from the discovery of identity, and analogy is\nbut one name by which we denote the deeper-lying cases of resemblance.\nI shall only try to point out at present how analogy between apparently\ndiverse classes of phenomena often serves as a guide in discovery. We\nthus commonly gain the first insight into the nature of an apparently\nunique object, and thus, in the progress of a science, we often\ndiscover that we are treating over again, in a new form, phenomena\nwhich were well known to us in another form.\n\n\n*Analogy as a Guide in Discovery.*\n\nThere can be no doubt that discovery is most frequently accomplished\nby following up hints received from analogy, as Jeremy Bentham\nremarked.[524] Whenever a phenomenon is perceived, the first impulse\nof the mind is to connect it with the most nearly similar phenomenon.\nIf we could ever meet a thing wholly *sui generis*, presenting no\nanalogy to anything else, we should be incapable of investigating its\nnature, except by purely haphazard trial. The probability of success\nby such a process is so slight, that it is preferable to follow up the\nfaintest clue. As I have pointed out already (p.\u00a0418), the possible\nexperiments are almost infinite in number, and very numerous also are\nthe hypotheses upon which we may proceed. Now it is self-evident that,\nhowever slightly superior the probability of success by one course of\nprocedure may be over another, the most probable one should always be\nadopted first.\n\n  [524] *Essay on Logic*, *Works*, vol.\u00a0viii. p.\u00a0276.\n\nThe chemist having discovered what he believes to be a new element,\nwill have before him an infinite variety of modes of treating and\ninvestigating it. If in any of its qualities the substance displays\na resemblance to an alkaline metal, for instance, he will naturally\nproceed to try whether it possesses other properties of the alkaline\nmetals. Even the simplest phenomenon presents so many points for notice\nthat we have a choice from among many hypotheses.\n\nIt would be difficult to find a more instructive instance of the way in\nwhich the mind is guided by analogy than in the description by Sir John\nHerschel of the course of thought by which he was led to anticipate in\ntheory one of Faraday\u2019s greatest discoveries. Herschel noticed that\na screw-like form, technically called helicoidal dissymmetry, was\nobserved in three cases, namely, in electrical helices, plagihedral\nquartz crystals, and the rotation of the plane of polarisation of\nlight. As he said,[525] \u201cI reasoned thus: Here are three phenomena\nagreeing in a *very strange peculiarity*. Probably, this peculiarity\nis a connecting link, physically speaking, among them. Now, in the\ncase of the crystals and the light, this probability has been turned\ninto certainty by my own experiments. Therefore, induction led me to\nconclude that a similar connection exists, and must turn up, somehow\nor other, between the electric current and polarised light, and that\nthe plane of polarisation would be deflected by magneto-electricity.\u201d\nBy this course of analogical thought Herschel had actually been led\nto anticipate Faraday\u2019s great discovery of the influence of magnetic\nstrain upon polarised light. He had tried in 1822\u201325 to discover the\ninfluence of electricity on light, by sending a ray of polarised light\nthrough a helix, or near a long wire conveying an electric current.\nSuch a course of inquiry, followed up with the persistency of Faraday,\nand with his experimental resources, would doubtless have effected the\ndiscovery. Herschel also suggests that the plagihedral form of quartz\ncrystals must be due to a screw-like strain during crystallisation; but\nthe notion remains unverified by experiment.\n\n  [525] *Life of Faraday*, by Bence Jones, vol.\u00a0ii. p.\u00a0206.\n\n\n*Analogy in the Mathematical Sciences.*\n\nWhoever wishes to acquire a deep acquaintance with Nature must observe\nthat there are analogies which connect whole branches of science in a\nparallel manner, and enable us to infer of one class of phenomena what\nwe know of another. It has thus happened on several occasions that the\ndiscovery of an unsuspected analogy between two branches of knowledge\nhas been the starting-point for a rapid course of discovery. The truths\nreadily observed in the one may be of a different character from those\nwhich present themselves in the other. The analogy, once pointed out,\nleads us to discover regions of one science yet undeveloped, to which\nthe key is furnished by the corresponding truths in the other science.\nAn interchange of aid most wonderful in its results may thus take\nplace, and at the same time the mind rises to a higher generalisation,\nand a more comprehensive view of nature.\n\nNo two sciences might seem at first sight more different in their\nsubject matter than geometry and algebra. The first deals with circles,\nsquares, parallelograms, and other forms in space; the latter with\nmere symbols of number. Prior to the time of Descartes, the sciences\nwere developed slowly and painfully in almost entire independence of\neach other. The Greek philosophers indeed could not avoid noticing\noccasional analogies, as when Plato in the Th\u00e6etetus describes a square\nnumber as *equally equal*, and a number produced by multiplying two\nunequal factors as *oblong*. Euclid, in the 7th and 8th books of his\nElements, continually uses expressions displaying a consciousness\nof the same analogies, as when he calls a number of two factors a\n*plane number*, \u1f10\u03c0\u03af\u03c0\u03b5\u03b4\u03bf\u03c2 \u1f00\u03c1\u03b9\u03b8\u03bc\u03cc\u03c2, and distinguishes a square number of\nwhich the two factors are equal as an equal-sided and plane number,\n\u1f30\u03c3\u03cc\u03c0\u03bb\u03b5\u03c5\u03c1\u03bf\u03c2 \u03ba\u03b1\u1f76 \u1f10\u03c0\u03af\u03c0\u03b5\u03b4\u03bf\u03c2 \u1f00\u03c1\u03b9\u03b8\u03bc\u03cc\u03c2. He also calls the root of a cubic\nnumber its side, \u03c0\u03bb\u03b5\u03c5\u03c1\u03ac. In the Diophantine algebra many problems of a\ngeometrical character were solved by algebraic or numerical processes;\nbut there was no general system, so that the solutions were of an\nisolated character. In general the ancients were far more advanced in\ngeometric than symbolic methods; thus Euclid in his 4th book gives the\nmeans of dividing a circle by purely geometric means into 2, 3, 4, 5,\n6, 8, 10, 12, 15, 16, 20, 24, 30 parts, but he was totally unacquainted\nwith the theory of the roots of unity exactly corresponding to this\ndivision of the circle.\n\nDuring the middle ages, on the contrary, algebra advanced beyond\ngeometry, and modes of solving equations were gradually discovered\nby those who had no notion that at every step they were implicitly\nsolving geometric problems. It is true that Regiomontanus, Tartaglia,\nBombelli, and possibly other early algebraists, solved isolated\ngeometrical problems by the aid of algebra, but particular numbers were\nalways used, and no consciousness of a general method was displayed.\nVieta in some degree anticipated the final discovery, and occasionally\nrepresented the roots of an equation geometrically, but it was\nreserved for Descartes to show, in the most general manner, that every\nequation may be represented by a curve or figure in space, and that\nevery bend, point, cusp, or other peculiarity in the curve indicates\nsome peculiarity in the equation. It is impossible to describe in any\nadequate manner the importance of this discovery. The advantage was\ntwo-fold: algebra aided geometry, and geometry gave reciprocal aid\nto algebra. Curves such as the well-known sections of the cone were\nfound to correspond to quadratic equations; and it was impossible\nto manipulate the equations without discovering properties of those\nall-important curves. The way was thus opened for the algebraic\ntreatment of motions and forces, without which Newton\u2019s *Principia*\ncould never have been worked out. Newton indeed was possessed by a\nstrong infatuation in favour of the ancient geometrical methods; but\nit is well known that he employed symbolic methods to discover his\ntheorems, and he now and then, by some accidental use of algebraic\nexpression, confessed its greater power and generality.\n\nGeometry, on the other hand, gave great assistance to algebra, by\naffording concrete representations of relations which would otherwise\nbe too abstract for easy comprehension. A curve of no great complexity\nmay give the whole history of the variations of value of a troublesome\nmathematical expression. As soon as we know, too, that every regular\ngeometrical curve represents some algebraic equation, we are presented\nby observation of mechanical movements with abundant suggestions\ntowards the discovery of mathematical problems. Every particle of a\ncarriage-wheel when moving on a level road is constantly describing\na cycloidal curve, the curious properties of which exercised the\ningenuity of all the most skilful mathematicians of the seventeenth\ncentury, and led to important advancements in algebraic power. It may\nbe held that the discovery of the Differential Calculus was mainly\ndue to geometrical analogy, because mathematicians, in attempting to\ntreat algebraically the tangent of a curve, were obliged to entertain\nthe notion of infinitely small quantities.[526] There can be no doubt\nthat Newton\u2019s fluxional, that is, geometrical mode of stating the\ndifferential calculus, however much it subsequently retarded its\nprogress in England, facilitated its apprehension at first, and I\nshould think it almost certain that Newton discovered the principles of\nthe calculus geometrically.\n\n  [526] Lacroix, *Trait\u00e9 \u00c9l\u00e9mentaire de Calcul Diff\u00e9rentiel et de\n  Calcul Int\u00e9gral*, 5^{me} \u00e9dit. p.\u00a0699.\n\nWe may accordingly look upon this discovery of analogy, this happy\nalliance, as Bossut calls it,[527] between geometry and algebra,\nas the chief source of discoveries which have been made for three\ncenturies past in mathematical methods. This is certainly the opinion\nof Lagrange, who says, \u201cSo long as algebra and geometry have been\nseparate, their progress was slow, and their employment limited; but\nsince these two sciences have been united, they have lent each other\nmutual strength, and have marched together with a rapid step towards\nperfection.\u201d\n\n  [527] *Histoire des Math\u00e9matiques*, vol.\u00a0i. p.\u00a0298.\n\nThe advancement of mechanical science has also been greatly aided by\nanalogy. An abstract and intangible existence like force demands much\npower of conception, but it has a perfect concrete representative in\na line, the end of which may denote the point of application, and\nthe direction the line of action of the force, while the length can\nbe made arbitrarily to denote the amount of the force. Nor does the\nanalogy end here; for the moment of the force about any point, or its\nproduct into the perpendicular distance of its line of action from the\npoint, is found to be represented by an area, namely twice the area\nof the triangle contained between the point and the ends of the line\nrepresenting the force. Of late years a great generalisation has been\neffected; the Double Algebra of De Morgan is true not only of space\nrelations, but of forces, so that the triangle of forces is reduced\nto a case of pure geometrical addition. Nay, the triangle of lines,\nthe triangle of velocities, the triangle of forces, the triangle of\ncouples, and perhaps other cognate theorems, are reduced by analogy to\none simple theorem, which amounts to this, that there are two ways of\ngetting from one angular point of a triangle to another, which ways,\nthough different in length, are identical in their final results.[528]\nIn the system of quaternions of the late Sir W.\u00a0R. Hamilton, these\nanalogies are embodied and carried out in the most general manner, so\nthat whatever problem involves the threefold dimensions of space, or\nrelations analogous to those of space, is treated by a symbolic method\nof the most comprehensive simplicity.\n\n  [528] See Goodwin, *Cambridge Philosophical Transactions* (1845),\n  vol. viii. p.\u00a0269. O\u2019Brien, \u201cOn Symbolical Statics,\u201d *Philosophical\n  Magazine*, 4th Series, vol.\u00a0i. pp.\u00a0491, &c. See also Professor\n  Clerk Maxwell\u2019s delightful *Manual of Elementary Science*, called\n  *Matter and Motion*, published by the Society for Promoting Christian\n  Knowledge. In this admirable little work some of the most advanced\n  results of mechanical and physical science are explained according\n  to the method of quaternions, but with hardly any use of algebraic\n  symbols.\n\nIt ought to be added that to the discovery of analogy between the forms\nof mathematical and logical expressions, we owe the greatest advance\nin logical science. Boole based his extension of logical processes\nupon the notion that logic is an algebra of two quantities 0 and 1.\nHis profound genius for symbolic investigation led him to perceive by\nanalogy that there must exist a general system of logical deduction, of\nwhich the old logicians had seized only a few fragments. Mistaken as\nhe was in placing algebra as a higher science than logic, no one can\ndeny that the development of the more complex and dependent science\nhad advanced far beyond that of the simpler science, and that Boole,\nin drawing attention to the connection, made one of the most important\ndiscoveries in the history of science. As Descartes had wedded algebra\nand geometry, so did Boole accomplish the marriage of logic and\nalgebra.\n\n\n*Analogy in the Theory of Undulations.*\n\nThere is no class of phenomena which more thoroughly illustrates alike\nthe power and weakness of analogy than the waves which agitate every\nkind of medium. All waves, whatsoever be the matter through which\nthey pass, obey the principles of rhythmical or harmonic motion,\nand the subject therefore presents a fine field for mathematical\ngeneralisation. Each kind of medium may allow of waves peculiar in\ntheir conditions, so that it is a beautiful exercise in analogical\nreasoning to decide how, in making inferences from one kind of medium\nto another, we must make allowance for difference of circumstances. The\nwaves of the ocean are large and visible, and there are the yet greater\ntidal waves which extend around the globe. From such palpable cases of\nrhythmical movement we pass to waves of sound, varying in length from\nabout 32 feet to a small fraction of an inch. We have but to imagine,\nif we can, the fortieth octave of the middle C of a piano, and we reach\nthe undulations of yellow light, the ultra-violet being about the\nforty-first octave. Thus we pass from the palpable and evident to that\nwhich is obscure, if not incomprehensible. Yet the same phenomena of\nreflection, interference, and refraction, which we find in some kinds\nof waves, may be expected to occur, *mutatis mutandis*, in other kinds.\n\nFrom the great to the small, from the evident to the obscure, is not\nonly the natural order of inference, but it is the historical order of\ndiscovery. The physical science of the Greek philosophers must have\nremained incomplete, and their theories groundless, because they did\nnot understand the nature of undulations. Their systems were based\nupon the notion of movement of translation from place to place. Modern\nscience tends to the opposite notion that all motion is alternating or\nrhythmical, energy flowing onwards but matter remaining comparatively\nfixed in position. Diogenes Laertius indeed correctly compared the\npropagation of sound with the spreading of waves on the surface of\nwater when disturbed by a stone, and Vitruvius displayed a more\ncomplete comprehension of the same analogy. It remained for Newton\nto create the theory of undulatory motion in showing by mathematical\ndeductive reasoning that the particles of an elastic fluid by vibrating\nbackwards and forwards, might carry a pulse or wave moving from\nthe source of disturbance, while the disturbed particles return to\ntheir place of rest. He was even able to make a first approximation\nby theoretical calculation to the velocity of sound-waves in the\natmosphere. His theory of sound formed a hardly less important epoch in\nscience than his far more celebrated theory of gravitation. It opened\nthe way to all the subsequent applications of mechanical principles\nto the insensible motion of molecules. He seems to have been, too,\nupon the brink of another application of the same principles which\nwould have advanced science by a century of progress, and made him the\nundisputed founder of all the theories of matter. He expressed opinions\nat various times that light might be due to undulatory movements of\na medium occupying space, and in one intensely interesting sentence\nremarks[529] that colours are probably vibrations of different lengths,\n\u201cmuch after the manner that, in the sense of hearing, nature makes\nuse of a\u00ebrial vibrations of several bignesses to generate sounds\nof divers tones, for the analogy of nature is to be observed.\u201d He\ncorrectly foresaw that red and yellow light would consist of the longer\nundulations, and blue and violet of the shorter, while white light\nwould be composed of an indiscriminate mixture of waves of various\nlengths. Newton almost overcame the strongest apparent difficulty of\nthe undulatory theory of light, namely, the propagation of light in\nstraight lines. For he observed that though waves of sound bend round\nan obstacle to some extent, they do not do so in the same degree as\nwater-waves.[530] He had but to extend the analogy proportionally to\nlight-waves, and not only would the difficulty have vanished, but the\ntrue theory of diffraction would have been open to him. Unfortunately\nhe had a preconceived theory that rays of light are bent from and\nnot towards the shadow of a body, a theory which for once he did not\nsufficiently compare with observation to detect its falsity. I am\nnot aware, too, that Newton has, in any of his works, displayed an\nunderstanding of the phenomena of interference without which his notion\nof waves must have been imperfect.\n\n  [529] Birch, *History of the Royal Society*, vol.\u00a0iii. p.\u00a0262, quoted\n  by Young, *Works*, vol.\u00a0i. p.\u00a0246.\n\n  [530] *Opticks*, Query 28, 3rd edit. p.\u00a0337.\n\nWhile the general principles of undulatory motion will be the same\nin whatever medium the motion takes place, the circumstances may be\nexcessively different. Between light travelling 186,000 miles per\nsecond and sound travelling in air only about 1,100 feet in the same\ntime, or almost 900,000 times as slowly, we cannot expect a close\noutward resemblance. There are great differences, too, in the character\nof the vibrations. Gases scarcely admit of transverse vibration, so\nthat sound travelling in air is a longitudinal wave, the particles of\nair moving backwards and forwards in the same line in which the wave\nmoves onwards. Light, on the other hand, appears to consist entirely\nin the movement of points of force transversely to the direction of\npropagation of the ray. The light-wave is partially analogous to the\nbending of a rod or of a stretched cord agitated at one end. Now this\nbending motion may take place in any one of an infinite number of\nplanes, and waves of which the planes are perpendicular to each other\ncannot interfere any more than two perpendicular forces can interfere.\nThe complicated phenomena of polarised light arise out of this\ntransverse character of the luminous wave, and we must not expect to\nmeet analogous phenomena in atmospheric sound-waves. It is conceivable\nthat in solids we might produce transverse sound undulations, in which\nphenomena of polarisation might be reproduced. But it would appear that\neven between transverse sound and light-waves the analogy holds true\nrather of the principles of harmonic motion than the circumstances of\nthe vibrating medium; from experiment and theory it is inferred that\nthe plane of polarisation in plane polarised light is perpendicular\nto instead of being coincident with the direction of vibration, as it\nwould be in the case of transverse sound undulations. If so the laws\nof elastic forces are essentially different in application to the\nluminiferous ether and to ordinary solid bodies.[531]\n\n  [531] Rankine, *Philosophical Transactions* (1856), vol.\u00a0cxlvi.\n  p.\u00a0282.\n\n\n*Analogy in Astronomy.*\n\nWe shall be much assisted in gaining a true appreciation of the value\nof analogy in its feebler degrees, by considering how much it has\ncontributed to the progress of astronomical science. Our point of\nobservation is so fixed with regard to the universe, and our means of\nexamining distant bodies are so restricted, that we are necessarily\nguided by limited and apparently feeble resemblances. In many cases the\nresult has been confirmed by subsequent direct evidence of the most\nforcible character.\n\nWhile the scientific world was divided in opinion between the\nCopernican and Ptolemaic systems, it was analogy which furnished the\nmost satisfactory argument. Galileo discovered, by the use of his new\ntelescope, the four small satellites which circulate round Jupiter,\nand make a miniature planetary world. These four Medicean Stars, as\nthey were called, were plainly seen to revolve round Jupiter in various\nperiods, but approximately in one plane, and astronomers irresistibly\ninferred that what might happen on the smaller scale might also be\nfound true of the greater planetary system. This discovery gave \u201cthe\nholding turn,\u201d as Herschel expressed it, to the opinions of mankind.\nEven Francis Bacon, who, little to the credit of his scientific\nsagacity, had previously opposed the Copernican views, now became\nconvinced, saying \u201cWe affirm the solisequium of Venus and Mercury;\nsince it has been found by Galileo that Jupiter also has attendants.\u201d\nNor did Huyghens think it superfluous to adopt the analogy as a valid\nargument.[532] Even in an advanced stage of physical astronomy, the\nJovian system has not lost its analogical interest; for the mutual\nperturbations of the four satellites pass through all their phases\nwithin a few centuries, and thus enable us to verify in a miniature\ncase the principles of stability, which Laplace established for the\ngreat planetary system. Oscillations or disturbances which in the\nmotions of the planets appear to be secular, because their periods\nextend over millions of years, can be watched, in the case of Jupiter\u2019s\nsatellites, through complete revolutions within the historical period\nof astronomy.[533]\n\n  [532] *Cosmotheoros* (1699), p.\u00a016.\n\n  [533] Laplace, *System of the World*, vol.\u00a0ii. p.\u00a0316.\n\nIn obtaining a knowledge of the stellar universe we must sometimes\ndepend upon precarious analogies. We still hold upon this ground the\nopinion, entertained by Bruno as long ago as 1591, that the stars may\nbe suns attended by planets like our earth. This is the most probable\nfirst assumption, and it is supported by spectrum observations, which\nshow the similarity of light derived from many stars with that of the\nsun. But at the same time we learn by the prism that there are nebul\u00e6\nand stars in conditions widely different from anything known in our\nsystem. In the course of time the analogy may perhaps be restored to\ncomparative completeness by the discovery of suns in various stages\nof nebulous condensation. The history of the evolution of our own\nworld may be traced back in bodies less developed, or traced forwards\nin systems more advanced towards the dissipation of energy, and the\nextinction of life. As in a great workshop, we may perhaps see the\nmaterial work of Creation as it has progressed through thousands of\nmillions of years.\n\nIn speculations concerning the physical condition of the planets and\ntheir satellites, we depend upon analogies of a weak character. We may\nbe said to know that the moon has mountains and valleys, plains and\nridges, volcanoes and streams of lava, and, in spite of the absence of\nair and water, the rocky surface of the moon presents so many familiar\nappearances that we do not hesitate to compare them with the features\nof our globe. We infer with high probability that Mars has polar snow\nand an atmosphere absorbing blue rays like our own; Jupiter undoubtedly\npossesses a cloudy atmosphere, possibly not unlike a magnified copy\nof that surrounding the earth, but our tendency to adopt analogies\nreceives a salutary correction in the recently discovered fact that the\natmosphere of Uranus contains hydrogen.\n\nPhilosophers have not stopped at these comparatively safe inferences,\nbut have speculated on the existence of living creatures in other\nplanets. Huyghens remarked that as we infer by analogy from the\ndissected body of a dog to that of a pig and ox or other animal of\nthe same general form, and as we expect to find the same viscera, the\nheart, stomach, lungs, intestines, &c., in corresponding positions,\nso when we notice the similarity of the planets in many respects,\nwe must expect to find them alike in other respects.[534] He even\nenters into an inquiry whether the inhabitants of other planets would\npossess reason and knowledge of the same sort as ours, concluding in\nthe affirmative. Although the power of intellect might be different,\nhe considers that they would have the same geometry if they had any\nat all, and that what is true with us would be true with them.[535]\nAs regards the sun, he wisely observes that every conjecture fails.\nLaplace entertained a strong belief in the existence of inhabitants on\nother planets. The benign influence of the sun gives birth to animals\nand plants upon the surface of the earth, and analogy induces us to\nbelieve that his rays would tend to have a similar effect elsewhere.\nIt is not probable that matter which is here so fruitful of life would\nbe sterile upon so great a globe as Jupiter, which, like the earth,\nhas its days and nights and years, and changes which indicate active\nforces. Man indeed is formed for the temperature and atmosphere in\nwhich he lives, and, so far as appears, could not live upon the other\nplanets. But there might be an infinity of organisations relative to\nthe diverse constitutions of the bodies of the universe. The most\nactive imagination cannot form any idea of such various creatures, but\ntheir existence is not unlikely.[536]\n\n  [534] *Cosmotheoros* (1699), p.\u00a017.\n\n  [535] Ibid. p.\u00a036.\n\n  [536] *System of the World*, vol.\u00a0ii. p.\u00a0326. *Essai Philosophique*,\n  p.\u00a087.\n\nWe now know that many metals and other elements never found in organic\nstructures are yet capable of forming compounds with substances of\nvegetable or animal origin. It is therefore just possible that at\ndifferent temperatures creatures formed of different yet analogous\ncompounds might exist, but it would seem indispensable that carbon\nshould form the basis of organic structures. We have no analogies to\nlead us to suppose that in the absence of that complex element life\ncan exist. Could we find globes surrounded by atmospheres resembling\nour own in temperature and composition, we should be almost forced to\nbelieve them inhabited, but the probability of any analogical argument\ndecreases rapidly as the condition of a globe diverges from that of our\nown. The Cardinal Nicholas de Cusa held long ago that the moon was\ninhabited, but the absence of any appreciable atmosphere renders the\nexistence of inhabitants highly improbable. Speculations resting upon\nweak analogies hardly belong to the scope of true science, and can only\nbe tolerated as an antidote to the far worse dogmas which assert that\nthe thousand million of persons on earth, or rather a small fraction\nof them, are the sole objects of care of the Power which designed this\nlimitless Universe.\n\n\n*Failures of Analogy.*\n\nSo constant is the aid which we derive from the use of analogy in all\nattempts at discovery or explanation, that it is most important to\nobserve in what cases it may lead us into difficulties. That which we\nexpect by analogy to exist\n\n(1) May be found to exist;\n\n(2) May seem not to exist, but nevertheless may really exist;\n\n(3) May actually be non-existent.\n\nIn the second case the failure is only apparent, and arises from\nour obtuseness of perception, the smallness of the phenomenon to be\nnoticed, or the disguised character in which it appears. I have already\npointed out that the analogy of sound and light seems to fail because\nlight does not apparently bend round a corner, the fact being that it\ndoes so bend in the phenomena of diffraction, which present the effect,\nhowever, in such an unexpected and minute form, that even Newton was\nmisled, and turned from the correct hypothesis of undulations which he\nhad partially entertained.\n\nIn the third class of cases analogy fails us altogether, and we\nexpect that to exist which really does not exist. Thus we fail to\ndiscover the phenomena of polarisation in sound travelling through the\natmosphere, since air is not capable of any appreciable transverse\nundulations. These failures of analogy are of peculiar interest,\nbecause they make the mind aware of its superior powers. There have\nbeen many philosophers who said that we can conceive nothing in the\nintellect which we have not previously received through the senses.\nThis is true in the sense that we cannot *image* them to the mind in\nthe concrete form of a shape or a colour; but we can speak of them\nand reason concerning them; in short, we often know them in everything\nbut a sensuous manner. Accurate investigation shows that all material\nsubstances retard the motion of bodies through them by subtracting\nenergy by impact. By the law of continuity we can frame the notion of\na vacuous space in which there is no resistance whatever, nor need we\nstop there; for we have only to proceed by analogy to the case where\na medium should accelerate the motion of bodies passing through it,\nsomewhat in the mode which Aristotelians attributed falsely to the air.\nThus we can frame the notion of *negative density*, and Newton could\nreason exactly concerning it, although no such thing exists.[537]\n\n  [537] *Principia*, bk. ii. Section ii. Prop. x.\n\nIn every direction of thought we may meet ultimately with similar\nfailures of analogy. A moving point generates a line, a moving line\ngenerates a surface, a moving surface generates a solid, but what does\na moving solid generate? When we compare a polyhedron, or many-sided\nsolid, with a polygon, or plane figure of many sides, the volume of the\nfirst is analogous to the area of the second; the face of the solid\nanswers to the side of the polygon; the edge of the solid to the point\nof the figure; but the corner, or junction of edges in the polyhedron,\nis left wholly unrepresented in the plane of the polygon. Even if\nwe attempted to draw the analogies in some other manner, we should\nstill find a geometrical notion embodied in the solid which has no\nrepresentative in the figure of two dimensions.[538]\n\n  [538] De Morgan, *Cambridge Philosophical Transactions*, vol.\u00a0xi.\n  Part ii. p.\u00a0246.\n\nFaraday was able to frame some notion of matter in a fourth condition,\nwhich should be to gas what gas is to liquid.[539] Such substance,\nhe thought, would not fall far short of *radiant matter*, by which\napparently he meant the supposed caloric or matter assumed to\nconstitute heat, according to the corpuscular theory. Even if we could\nframe the notion, matter in such a state cannot be known to exist, and\nrecent discoveries concerning the continuity of the solid, liquid, and\ngaseous states remove the basis of the speculation.\n\n  [539] *Life of Faraday*, vol.\u00a0i. p.\u00a0216.\n\nFrom these and many other instances which might be adduced, we learn\nthat analogical reasoning leads us to the conception of many things\nwhich, so far as we can ascertain, do not exist. In this way great\nperplexities have arisen in the use of language and mathematical\nsymbols. All language depends upon analogy; for we join and arrange\nwords so that they may represent the corresponding junctions or\narrangements of things and their equalities. But in the use of language\nwe are obviously capable of forming many combinations of words to which\nno corresponding meaning apparently exists. The same difficulty arises\nin the use of mathematical signs, and mathematicians have needlessly\npuzzled themselves about the square root of a negative quantity, which\nis, in many applications of algebraic calculation, simply a sign\nwithout any analogous meaning, there being a failure of analogy.\n\n\n\n\nCHAPTER XXIX.\n\nEXCEPTIONAL PHENOMENA.\n\n\nIf science consists in the detection of identity and the recognition\nof uniformity existing in many objects, it follows that the progress\nof science depends upon the study of exceptional phenomena. Such new\nphenomena are the raw material upon which we exert our faculties of\nobservation and reasoning, in order to reduce the new facts beneath\nthe sway of the laws of nature, either those laws already well known,\nor those to be discovered. Not only are strange and inexplicable\nfacts those which are on the whole most likely to lead us to some\nnovel and important discovery, but they are also best fitted to\narouse our attention. So long as events happen in accordance with\nour anticipations, and the routine of every-day observation is\nunvaried, there is nothing to impress upon the mind the smallness of\nits knowledge, and the depth of mystery, which may be hidden in the\ncommonest sights and objects. In early times the myriads of stars which\nremained in apparently fixed relative positions upon the heavenly\nsphere, received less notice from astronomers than those few planets\nwhose wandering and inexplicable motions formed a riddle. Hipparchus\nwas induced to prepare the first catalogue of stars, because a single\nnew star had been added to those nightly visible; and in the middle\nages two brilliant but temporary stars caused more popular interest\nin astronomy than any other events, and to one of them we owe all the\nobservations of Tycho Brahe, the medi\u00e6val Hipparchus.\n\nIn other sciences, as well as in that of the heavens, exceptional\nevents are commonly the points from which we start to explore new\nregions of knowledge. It has been beautifully said that Wonder is the\ndaughter of Ignorance, but the mother of Invention; and though the most\nfamiliar and slight events, if fully examined, will afford endless food\nfor wonder and for wisdom, yet it is the few peculiar and unlooked-for\nevents which most often lead to a course of discovery. It is true,\nindeed, that it requires much philosophy to observe things which are\ntoo near to us.\n\nThe high scientific importance attaching, then, to exceptions, renders\nit desirable that we should carefully consider the various modes in\nwhich an exception may be disposed of; while some new facts will be\nfound to confirm the very laws to which they seem at first sight\nclearly opposed, others will cause us to limit the generality of our\nprevious statements. In some cases the exception may be proved to be\nno exception; occasionally it will prove fatal to our previous most\nconfident speculations; and there are some new phenomena which, without\nreally destroying any of our former theories, open to us wholly new\nfields of scientific investigation. The study of this subject is\nespecially interesting and important, because, as I have before said\n(p.\u00a0587), no important theory can be built up complete and perfect all\nat once. When unexplained phenomena present themselves as objections\nto the theory, it will often demand the utmost judgment and sagacity\nto assign to them their proper place and force. The acceptance\nor rejection of a theory will depend upon discriminating the one\ninsuperable contradictory fact from many, which, however singular and\ninexplicable at first sight, may afterwards be shown to be results of\ndifferent causes, or possibly the most striking results of the very law\nwith which they stand in apparent conflict.\n\nI can enumerate at least eight classes or kinds of exceptional\nphenomena, to one or other of which any supposed exception to the known\nlaws of nature can usually be referred; they may be briefly described\nas below, and will be sufficiently illustrated in the succeeding\nsections.\n\n(1) Imaginary, or false exceptions, that is, facts, objects, or events\nwhich are not really what they are supposed to be.\n\n(2) Apparent, but congruent exceptions, which, though apparently in\nconflict with a law of nature, are really in agreement with it.\n\n(3) Singular exceptions, which really agree with a law of nature, but\nexhibit remarkable and unique results of it.\n\n(4) Divergent exceptions, which really proceed from the ordinary action\nof known processes of nature, but which are excessive in amount or\nmonstrous in character.\n\n(5) Accidental exceptions, arising from the interference of some\nentirely distinct but known law of nature.\n\n(6) Novel and unexplained exceptions, which lead to the discovery of a\nnew series of laws and phenomena, modifying or disguising the effects\nof previously known laws, without being inconsistent with them.\n\n(7) Limiting exceptions showing the falsity of a supposed law in some\ncases to which it had been extended, but not affecting its truth in\nother cases.\n\n(8) Contradictory or real exceptions which lead us to the conclusion\nthat a supposed hypothesis or theory is in opposition to the phenomena\nof nature, and must therefore be abandoned.\n\nIt ought to be clearly understood that in no case is a law of nature\nreally thwarted or prevented from being fulfilled. The effects of a\nlaw may be disguised and hidden from our view in some instances: in\nothers the law itself may be rendered inapplicable altogether; but if\na law is applicable it must be carried out. Every law of nature must\ntherefore be stated with the utmost generality of all the instances\nreally coming under it. Babbage proposed to distinguish between\n*universal principles*, which do not admit of a single exception, such\nas that every number ending in 5 is divisible by five, and *general\nprinciples* which are more frequently obeyed than violated, as that\n\u201cmen will be governed by what they believe to be their interest.\u201d[540]\nBut in a scientific point of view general principles must be universal\nas regards some distinct class of objects, or they are not principles\nat all. If a law to which exceptions exist is stated without allusion\nto those exceptions, the statement is erroneous. I have no right to\nsay that \u201cAll liquids expand by heat,\u201d if I know that water below\n4\u00b0\u00a0C. does not; I ought to say, \u201cAll liquids, except water below 4\u00b0\u00a0C.,\nexpand by heat;\u201d and every new exception discovered will falsify\nthe statement until inserted in it. To speak of some laws as being\n*generally* true, meaning not universally but in the majority of cases,\nis a hurtful abuse of the word, but is quite usual. *General* should\nmean that which is true of a whole *genus* or class, and every true\nstatement must be true of some assigned or assignable class.\n\n  [540] Babbage, *The Exposition of 1851*, p.\u00a01.\n\n\n*Imaginary or False Exceptions.*\n\nWhen a supposed exception to a law of nature is brought to our notice,\nthe first inquiry ought properly to be--Is there any breach of the law\nat all? It may be that the supposed exceptional fact is not a fact at\nall, but a mere figment of the imagination. When King Charles requested\nthe Royal Society to investigate the curious fact that a live fish put\ninto a bucket of water does not increase the weight of the bucket and\nits contents, the Royal Society wisely commenced their deliberations\nby inquiring whether the fact was so or not. Every statement, however\nfalse, must have some cause or prior condition, and the real question\nfor the Royal Society to investigate was, how the King came to think\nthat the fact was so. Mental conditions, as we have seen, enter into\nall acts of observation, and are often a worthy subject of inquiry. But\nthere are many instances in the history of science, in which trouble\nand error have been caused by false assertions carelessly made, and\ncarelessly accepted without verification.\n\nThe reception of the Copernican theory was much impeded by the\nobjection, that if the earth were moving, a stone dropped from the\ntop of a high tower should be left behind, and should appear to move\ntowards the west, just as a stone dropped from the mast-head of a\nmoving ship would fall behind, owing to the motion of the ship. The\nCopernicans attempted to meet this grave objection in every way but\nthe true one, namely, showing by trial that the asserted facts are not\ncorrect. In the first place, if a stone had been dropped with suitable\nprecautions from the mast-head of a moving ship, it would have fallen\nclose to the foot of the mast, because, by the first law of motion, it\nwould remain in the same state of horizontal motion communicated to it\nby the mast. As the anti-Copernicans had assumed the contrary result as\ncertain to ensue, their argument would of course have fallen through.\nHad the Copernicans next proceeded to test with great care the other\nassertion involved, they would have become still better convinced of\nthe truth of their own theory. A stone dropped from the top of a high\ntower, or into a deep well, would certainly not have been deflected\nfrom the vertical direction in the considerable degree required to\nagree with the supposed consequences of the Copernican views; but, with\nvery accurate observation, they might have discovered, as Benzenberg\nsubsequently did, a very small deflection towards the east, showing\nthat the eastward velocity is greater at the top than the bottom. Had\nthe Copernicans then been able to detect and interpret the meaning\nof the small divergence thus arising, they would have found in it\ncorroboration of their own views.\n\nMultitudes of cases might be cited in which laws of nature seem to\nbe evidently broken, but in which the apparent breach arises from a\nmisapprehension of the case. It is a general law, absolutely true\nof all crystals yet submitted to examination, that no crystal has\na re-entrant angle, that is an angle which towards the axis of the\ncrystal is greater than two right angles. Wherever the faces of a\ncrystal meet they produce a projecting edge, and wherever edges\nmeet they produce a corner. Many crystals, however, when carelessly\nexamined, present exceptions to this law, but closer observation\nalways shows that the apparently re-entrant angle really arises from\nthe oblique union of two distinct crystals. Other crystals seem to\npossess faces contradicting all the principles of crystallography;\nbut careful examination shows that the supposed faces are not true\nfaces, but surfaces produced by the orderly junction of an immense\nnumber of distinct thin crystalline plates, each plate being in fact\na separate crystal, in which the laws of crystallography are strictly\nobserved. The roughness of the supposed face, the stri\u00e6 detected by\nthe microscope, or inference by continuity from other specimens where\nthe true faces of the plates are clearly seen, prove the mistaken\ncharacter of the supposed exceptions. Again, four of the faces of a\nregular octahedron may become so enlarged in the crystallisation\nof iron pyrites and some other substances, that the other four\nfaces become imperceptible and a regular tetrahedron appears to be\nproduced, contrary to the laws of crystallographic symmetry. Many other\ncrystalline forms are similarly modified, so as to produce a series of\nwhat are called *hemihedral* forms.\n\nIn tracing out the isomorphic relations of the elements, great\nperplexity has often been caused by mistaking one substance for\nanother. It was pointed out that though arsenic was supposed to be\nisomorphous with phosphorus, the arseniate of soda crystallised in a\nform distinct from that of the corresponding phosphate. Some chemists\nheld this to be a fatal objection to the doctrine of isomorphism;\nbut it was afterwards pointed out by Clarke, that the arseniate and\nphosphate in question were not corresponding compounds, as they\ndiffered in regard to the water of crystallisation.[541] Vanadium again\nappeared to be an exception to the laws of isomorphism, until it was\nproved by Professor Roscoe, that what Berzelius supposed to be metallic\nvanadium was really an oxide of vanadium.[542]\n\n  [541] Daubeny\u2019s *Atomic Theory*, p.\u00a076.\n\n  [542] *Bakerian Lecture, Philosophical Transactions* (1868),\n  vol.\u00a0clviii. p. 2.\n\n\n*Apparent but Congruent Exceptions.*\n\nNot unfrequently a law of nature will present results in certain\ncircumstances which appear to be entirely in conflict with the law\nitself. Not only may the action of the law be much complicated and\ndisguised, but it may in various ways be reversed or inverted, so that\ncareless observers are misled. Ancient philosophers generally believed\nthat while some bodies were heavy by nature, others, such as flame,\nsmoke, bubbles, clouds, &c., were essentially light, or possessed a\ntendency to move upwards. So acute an inquirer as Aristotle failed to\nperceive the true nature of buoyancy, and the doctrine of intrinsic\nlightness, expounded in his works, became the accepted view for many\ncenturies. It is true that Lucretius was aware why flame tends to rise,\nholding that--\n\n  \u201cThe flame has weight, though highly rare,\n  Nor mounts but when compelled by heavier air.\u201d\n\nArchimedes also was so perfectly acquainted with the buoyancy of bodies\nimmersed in water, that he could not fail to perceive the existence\nof a parallel effect in air. Yet throughout the early middle ages\nthe light of true science could not contend with the glare of the\nPeripatetic doctrine. The genius of Galileo and Newton was required to\nconvince people of the simple truth that all matter is heavy, but that\nthe gravity of one substance may be overborne by that of another, as\none scale of a balance is carried up by the preponderating weight in\nthe opposite scale. It is curious to find Newton gravely explaining\nthe difference of absolute and relative gravity, as if it were a new\ndiscovery proceeding from his theory.[543] More than a century elapsed\nbefore other apparent exceptions to the Newtonian philosophy were\nexplained away.\n\n  [543] *Principia*, bk. ii. Prop. 20. Corollaries, 5 and 6.\n\nNewton himself allowed that the motion of the apsides of the moon\u2019s\norbit appeared to be irreconcilable with the law of gravity, and\nit remained for Clairaut to remove the difficulty by more complete\nmathematical analysis. There must always remain, in the motions of\nthe heavenly bodies, discrepancies of some amount between theory and\nobservation; but such discrepancies have so often yielded in past times\nto prolonged investigation that physicists now regard them as merely\napparent exceptions, which will afterwards be found to agree with the\nlaw of gravity.\n\nThe most beautiful instance of an apparent exception, is found in the\ntotal reflection of light, which occurs when a beam of light within\na medium falls very obliquely upon the boundary separating it from a\nrarer medium. The general law is that when a ray strikes the limit\nbetween two media of different refractive indices, part of the light\nis reflected and part is refracted; but when the obliquity of the ray\nwithin the denser medium passes beyond a certain point, there is a\nsudden apparent breach of continuity, and the whole of the light is\nreflected. A clear reason can be given for this exceptional conduct\nof the light. According to the law of refraction, the sine of the\nangle of incidence bears a fixed ratio to the sine of the angle of\nrefraction, so that the greater of the two angles, which is always that\nin the less dense medium, may increase up to a right angle; but when\nthe media differ in refractive power, the less angle cannot become a\nright angle, as this would require the sine of an angle to be greater\nthan the radius. It might seem that this is an exception of the kind\ndescribed below as a limiting exception, by which a law is shown to\nbe inapplicable beyond certain limits; but in the explanation of the\nexception according to the undulatory theory, we find that there is\nreally no breach of the general law. When an undulation strikes a point\nin a bounding surface, spherical waves are produced and spread from\nthe point. The refracted ray is the resultant of an infinite number of\nsuch spherical waves, and the bending of the ray at the common surface\nof two media depends upon the comparative velocities of propagation of\nthe undulations in those media. But if a ray falls very obliquely upon\nthe surface of a rarer medium, the waves proceeding from successive\npoints of the surface spread so rapidly as never to intersect, and\nno resultant wave will then be produced. We thus perceive that from\nsimilar mathematical conditions arise distinct apparent effects.\n\nThere occur from time to time failures in our best grounded\npredictions. A comet, of which the orbit has been well determined, may\nfail, like Lexell\u2019s Comet, to appear at the appointed time and place in\nthe heavens. In the present day we should not allow such an exception\nto our successful predictions to weigh against our belief in the theory\nof gravitation, but should assume that some unknown body had through\nthe action of gravitation deflected the comet. As Clairaut remarked,\nin publishing his calculations concerning the expected reappearance of\nHalley\u2019s Comet, a body which passes into regions so remote, and which\nis hidden from our view during such long periods, might be exposed to\nthe influence of forces totally unknown to us, such as the attraction\nof other comets, or of planets too far removed from the sun to be\never perceived by us. In the case of Lexell\u2019s Comet it was afterwards\nshown, curiously enough, that its appearance was not one of a regular\nseries of periodical returns within the sphere of our vision, but a\nsingle exceptional visit never to be repeated, and probably due to\nthe perturbing powers of Jupiter. This solitary visit became a strong\nconfirmation of the law of gravity with which it seemed to be in\nconflict.\n\n\n*Singular Exceptions.*\n\nAmong the most interesting of apparent exceptions are those which I\ncall *singular exceptions*, because they are more or less analogous to\nthe singular cases or solutions which occur in mathematical science.\nA general mathematical law embraces an infinite multitude of cases\nwhich perfectly agree with each other in a certain respect. It may\nnevertheless happen that a single case, while really obeying the\ngeneral law, stands out as apparently different from all the rest.\nThe rotation of the earth upon its axis gives to all the stars an\napparent motion of rotation from east to west; but while countless\nthousands obey the rule, the Pole Star alone seems to break it. Exact\nobservations indeed show that it also revolves in a small circle, but\na star might happen for a short time to exist so close to the pole\nthat no appreciable change of place would be caused by the earth\u2019s\nrotation. It would then constitute a perfect singular exception; while\nreally obeying the law, it would break the terms in which it is usually\nstated. In the same way the poles of every revolving body are singular\npoints.\n\nWhenever the laws of nature are reduced to a mathematical form we\nmay expect to meet with singular cases, and, as all the physical\nsciences will meet in the mathematical principles of mechanics, there\nis no part of nature where we may not encounter them. In mechanical\nscience the motion of rotation may be considered an exception to the\nmotion of translation. It is a general law that any number of parallel\nforces, whether acting in the same or opposite directions, will have\na resultant which may be substituted for them with like effect. This\nresultant will be equal to the algebraic sum of the forces, or the\ndifference of those acting in one direction and the other; it will pass\nthrough a point which is determined by a simple formula, and which may\nbe described as the mean point of all the points of application of the\nparallel forces (p.\u00a0364). Thus we readily determine the resultant of\nparallel forces except in one peculiar case, namely, when two forces\nare equal and opposite but not in the same straight line. Being equal\nand opposite the amount of the resultant is nothing, yet, as the forces\nare not in the same straight line, they do not balance each other.\nExamining the formula for the point of application of the resultant,\nwe find that it gives an infinitely great magnitude, so that the\nresultant is nothing at all, and acts at an infinite distance, which\nis practically the same as to say that there is no resultant. Two such\nforces constitute what is known in mechanical science as a *couple*,\nwhich occasions rotatory instead of rectilinear motion, and can only be\nneutralised by an equal and opposite couple of forces.\n\nThe best instances of singular exceptions are furnished by the science\nof optics. It is a general law that in passing through transparent\nmedia the plane of vibration of polarised light remains unchanged. But\nin certain liquids, some peculiar crystals of quartz, and transparent\nsolid media subjected to a magnetic strain, as in Faraday\u2019s experiment\n(pp.\u00a0588, 630), the plane of polarisation is rotated in a screw-like\nmanner. This effect is so entirely *sui generis*, so unlike any other\nphenomena in nature, as to appear truly exceptional; yet mathematical\nanalysis shows it to be only a single case of much more general laws.\nAs stated by Thomson and Tait,[544] it arises from the composition\nof two uniform circular motions. If while a point is moving round a\ncircle, the centre of that circle move upon another circle, a great\nvariety of curious curves will be produced according as we vary the\ndimensions of the circles, the rapidity or the direction of the\nmotions. When the two circles are exactly equal, the rapidities nearly\nso, and the directions opposite, the point will be found to move\ngradually round the centre of the stationary circle, and describe a\ncurious star-like figure connected with the molecular motions out of\nwhich the rotational power of the media rises. Among other singular\nexceptions in optics may be placed the conical refraction of light,\nalready noticed (p.\u00a0540), arising from the peculiar form assumed by a\nwave of light when passing through certain double-refracting crystals.\nThe laws obeyed by the wave are exactly the same as in other cases,\nyet the results are entirely *sui generis*. So far are such cases from\ncontradicting the law of ordinary cases, that they afford the best\nopportunities for verification.\n\n  [544] *Treatise on Natural Philosophy*, vol.\u00a0i. p.\u00a050.\n\nIn astronomy singular exceptions might occur, and in an approximate\nmanner they do occur. We may point to the rings of Saturn as objects\nwhich, though undoubtedly obeying the law of gravity, are yet unique,\nas far as our observation of the universe has gone. They agree, indeed,\nwith the other bodies of the planetary system in the stability of their\nmovements, which never diverge far from the mean position. There seems\nto be little doubt that these rings are composed of swarms of small\nmeteoric stones; formerly they were thought to be solid continuous\nrings, and mathematicians proved that if so constituted an entirely\nexceptional event might have happened under certain circumstances.\nHad the rings been exactly uniform all round, and with a centre of\ngravity coinciding for a moment with that of Saturn, a singular case of\nunstable equilibrium would have arisen, necessarily resulting in the\nsudden collapse of the rings, and the fall of their debris upon the\nsurface of the planet. Thus in one single case the theory of gravity\nwould give a result wholly unlike anything else known in the mechanism\nof the heavens.\n\nIt is possible that we might meet with singular exceptions in\ncrystallography. If a crystal of the second or dimetric system, in\nwhich the third axis is usually unequal to either of the other two,\nhappened to have the three axes equal, it might be mistaken for a\ncrystal of the cubic system, but would exhibit different faces and\ndissimilar properties. There is, again, a possible class of diclinic\ncrystals in which two axes are at right angles and the third axis\ninclined to the other two. This class is chiefly remarkable for its\nnon-existence, since no crystals have yet been proved to have such\naxes. It seems likely that the class would constitute only a singular\ncase of the more general triclinic system, in which all three axes are\ninclined to each other at various angles. Now if the diclinic form were\nmerely accidental, and not produced by any general law of molecular\nconstitution, its actual occurrence would be infinitely improbable,\njust as it is infinitely improbable that any star should indicate the\nNorth Pole with perfect exactness.\n\nIn the curves denoting the relation between the temperature and\npressure of water there is, as shown by Professor J. Thomson, one very\nremarkable point entirely unique, at which alone water can remain in\nthe three conditions of gas, liquid, and solid in the same vessel. It\nis the triple point at which three lines meet, namely (1) the steam\nline, which shows at what temperatures and pressures water is just\nupon the point of becoming gaseous; (2) the ice line, showing when ice\nis just about to melt; and (3) the hoar-frost line, which similarly\nindicates the pressures and temperatures at which ice is capable of\npassing directly into the state of gaseous vapour.[545]\n\n  [545] Maxwell\u2019s *Theory of Heat*, (1871), p.\u00a0175.\n\n\n*Divergent Exceptions.*\n\nClosely analogous to singular exceptions are those divergent\nexceptions, in which a phenomenon manifests itself in unusual magnitude\nor character, without becoming subject to peculiar laws. Thus in\nthrowing ten coins, it happened in four cases out of 2,048 throws,\nthat all the coins fell with heads uppermost (p.\u00a0208); these would\nusually be regarded as very singular events, and, according to the\ntheory of probabilities, they would be rare; yet they proceed only\nfrom an unusual conjunction of accidental events, and from no really\nexceptional causes. In all classes of natural phenomena we may expect\nto meet with similar divergencies from the average, sometimes due\nmerely to the principles of probability, sometimes to deeper reasons.\nAmong every large collection of persons, we shall probably find some\npersons who are remarkably large or remarkably small, giants or dwarfs,\nwhether in bodily or mental conformation. Such cases appear to be\nnot mere *lusus natur\u00e6*, since they occur with a frequency closely\naccordant with the law of error or divergence from an average, as shown\nby Quetelet and Mr. Galton.[546] The rise of genius, and the occurrence\nof extraordinary musical or mathematical faculties, are attributed by\nMr. Galton to the same principle of divergence.\n\n  [546] Galton, on the Height and Weight of Boys. *Journal of the\n  Anthropological Institute*, 1875, p.\u00a0174.\n\nWhen several distinct forces happen to concur together, we may have\nsurprising or alarming results. Great storms, floods, droughts, and\nother extreme deviations from the average condition of the atmosphere\nthus arise. They must be expected to happen from time to time, and will\nyet be very infrequent compared with minor disturbances. They are not\nanomalous but only extreme events, analogous to extreme runs of luck.\nThere seems, indeed, to be a fallacious impression in the minds of many\npersons, that the theory of probabilities necessitates uniformity in\nthe happening of events, so that in the same space of time there will\nalways be nearly the same number of railway accidents and murders.\nBuckle has superficially remarked upon the constancy of such events\nas ascertained by Quetelet, and some of his readers acquire the false\nnotion that there is a mysterious inexorable law producing uniformity\nin human affairs. But nothing can be more opposed to the teachings of\nthe theory of probability, which always contemplates the occurrence of\nunusual runs of luck. That theory shows the great improbability that\nthe number of railway accidents per month should be always equal, or\nnearly so. The public attention is strongly attracted to any unusual\nconjunction of events, and there is a fallacious tendency to suppose\nthat such conjunction must be due to a peculiar new cause coming\ninto operation. Unless it can be clearly shown that such unusual\nconjunctions occur more frequently than they should do according to\nthe theory of probabilities, we should regard them as merely divergent\nexceptions.\n\nEclipses and remarkable conjunctions of the heavenly bodies may also\nbe regarded as results of ordinary laws which nevertheless appear to\nbreak the regular course of nature, and never fail to excite surprise.\nSuch events vary greatly in frequency. One or other of the satellites\nof Jupiter is eclipsed almost every day, but the simultaneous eclipse\nof three satellites can only take place, according to the calculations\nof Wargentin, after the lapse of 1,317,900 years. The relations of the\nfour satellites are so remarkable, that it is actually impossible,\naccording to the theory of gravity, that they should all suffer eclipse\nsimultaneously. But it may happen that while some of the satellites are\nreally eclipsed by entering Jupiter\u2019s shadow, the others are either\nocculted or rendered invisible by passing over his disk. Thus on four\noccasions, in 1681, 1802, 1826, and 1843, Jupiter has been witnessed in\nthe singular condition of being apparently deprived of satellites. A\nclose conjunction of two planets always excites admiration, though such\nconjunctions must occur at intervals in the ordinary course of their\nmotions. We cannot wonder that when three or four planets approach\neach other closely, the event is long remembered. A most remarkable\nconjunction of Mars, Jupiter, Saturn, and Mercury, which took place in\nthe year 2446 B.C., was adopted by the Chinese Emperor, Chuen Hio, as a\nnew epoch for the chronology of his Empire, though there is some doubt\nwhether the conjunction was really observed, or was calculated from the\nsupposed laws of motion of the planets. It is certain that on the 11th\nNovember, 1524, the planets Venus, Jupiter, Mars, and Saturn were seen\nvery close together, while Mercury was only distant by about 16\u00b0 or\nthirty apparent diameters of the sun, this conjunction being probably\nthe most remarkable which has occurred in historical times.\n\nAmong the perturbations of the planets we find divergent exceptions\narising from the peculiar accumulation of effects, as in the case of\nthe long inequality of Jupiter and Saturn (p.\u00a0455). Leverrier has shown\nthat there is one place between the orbits of Mercury and Venus, and\nanother between those of Mars and Jupiter, in either of which, if a\nsmall planet happened to exist, it would suffer comparatively immense\ndisturbance in the elements of its orbit. Now between Mars and Jupiter\nthere do occur the minor planets, the orbits of which are in many cases\nexceptionally divergent.[547]\n\n  [547] Grant\u2019s *History of Physical Astronomy*, p.\u00a0116.\n\nUnder divergent exceptions we might place all or nearly all the\ninstances of substances possessing physical properties in a very high\nor low degree, which were described in the chapter on Generalisation\n(p.\u00a0607). Quicksilver is divergent among metals as regards its\nmelting point, and potassium and sodium as regards their specific\ngravities. Monstrous productions and variations, whether in the animal\nor vegetable kingdoms, should probably be assigned to this class of\nexceptions.\n\nIt is worthy of notice that even in such a subject as formal logic,\ndivergent exceptions seem to occur, not of course due to chance,\nbut exhibiting in an unusual degree a phenomenon which is more or\nless manifested in all other cases. I pointed out in p.\u00a0141 that\npropositions of the general type A\u00a0=\u00a0BC\u00a0\ua58c\u00a0*bc* are capable of\nexpression in six equivalent logical forms, so that they manifest in a\nhigher degree than any other proposition yet discovered the phenomenon\nof logical equivalence.\n\n\n*Accidental Exceptions.*\n\nThe third and largest class of exceptions contains those which arise\nfrom the casual interference of extraneous causes. A law may be in\noperation, and, if so, must be perfectly fulfilled; but, while we\nconceive that we are examining its results, we may have before us\nthe effects of a different cause, possessing no connexion with the\nsubject of our inquiry. The law is not really broken, but at the same\ntime the supposed exception is not illusory. It may be a phenomenon\nwhich cannot occur but under the condition of the law in question, yet\nthere has been such interference that there is an apparent failure of\nscience. There is, for instance, no subject in which more rigorous and\ninvariable laws have been established than in crystallography. As a\ngeneral rule, each chemical substance possesses its own definite form,\nby which it can be infallibly recognised; but the mineralogist has to\nbe on his guard against what are called *pseudomorphic* crystals. In\nsome circumstances a substance, having assumed its proper crystalline\nform, may afterwards undergo chemical change; a new ingredient may be\nadded, a former one removed, or one element may be substituted for\nanother. In calcium carbonate the carbonic acid is sometimes replaced\nby sulphuric acid, so that we find gypsum in the form of calcite;\nother cases are known where the change is inverted and calcite is\nfound in the form of gypsum. Mica, talc, steatite, hematite, are\nother minerals subject to these curious transmutations. Sometimes a\ncrystal embedded in a matrix is entirely dissolved away, and a new\nmineral is subsequently deposited in the cavity as in a mould. Quartz\nis thus found cast in many forms wholly unnatural to it. A still\nmore perplexing case sometimes occurs. Calcium carbonate is capable\nof assuming two distinct forms of crystallisation, in which it bears\nrespectively the names of calcite and arragonite. Now arragonite, while\nretaining its outward form unchanged, may undergo an internal molecular\nchange into calcite, as indicated by the altered cleavage. Thus we may\ncome across crystals apparently of arragonite, which seem to break all\nthe laws of crystallography, by possessing the cleavage of a different\nsystem of crystallisation.\n\nSome of the most invariable laws of nature are disguised by\ninterference of unlooked-for causes. While the barometer was yet a\nnew and curious subject of investigation, its theory, as stated by\nTorricelli and Pascal, seemed to be contradicted by the fact that\nin a well-constructed instrument the mercury would often stand far\nabove 31 inches in height. Boyle showed[548] that mercury could be\nmade to stand as high as 75 inches in a perfectly cleansed tube, or\nabout two and a half times as high as could be due to the pressure of\nthe atmosphere. Many theories about the pressure of imaginary fluids\nwere in consequence put forth,[549] and the subject was involved in\nmuch confusion until the adhesive or cohesive force between glass and\nmercury, when brought into perfect contact, was pointed out as the real\ninterfering cause. It seems to me, however, that the phenomenon is not\nthoroughly understood as yet.\n\n  [548] *Discourse to the Royal Society*, 28th May, 1684.\n\n  [549] Robert Hooke\u2019s *Posthumous Works*, p.\u00a0365.\n\nGay-Lussac observed that the temperature of boiling water was very\ndifferent in some kinds of vessels from what it was in others. It is\nonly when in contact with metallic surfaces or sharply broken edges\nthat the temperature is fixed at 100\u00b0\u00a0C. The suspended freezing of\nliquids is another case where the action of a law of nature appears\nto be interrupted. Spheroidal ebullition was at first sight a most\nanomalous phenomenon; it was almost incredible that water should not\nboil in a red-hot vessel, or that ice could actually be produced in a\nred-hot crucible. These paradoxical results are now fully explained as\ndue to the interposition of a non-conducting film of vapour between the\nglobule of liquid and the sides of the vessel. The feats of conjurors\nwho handle liquid metals are accounted for in the same manner. At one\ntime the *passive state* of steel was regarded as entirely anomalous.\nIt may be assumed as a general law that when pieces of electro-negative\nand electro-positive metal are placed in nitric acid, and made to touch\neach other, the electro-negative metal will undergo rapid solution. But\nwhen iron is the electro-negative and platinum the electro-positive,\nthe solution of the iron entirely and abruptly ceases. Faraday\ningeniously proved that this effect is due to a thin film of oxide of\niron, which forms upon the surface of the iron and protects it.[550]\n\n  [550] *Experimental Researches in Electricity*, vol.\u00a0ii. pp.\u00a0240\u2013245.\n\nThe law of gravity is so simple, and disconnected from the other laws\nof nature, that it never suffers any disturbance, and is in no way\ndisguised, but by the complication of its own effects. It is otherwise\nwith those secondary laws of the planetary system which have only\nan empirical basis. The fact that all the long known planets and\nsatellites have a similar motion from west to east is not necessitated\nby any principles of mechanics, but points to some common condition\nexisting in the nebulous mass from which our system has been evolved.\nThe retrograde motions of the satellites of Uranus constituted a\ndistinct breach in this law of uniform direction, which became all\nthe more interesting when the single satellite of Neptune was also\nfound to be retrograde. It now became probable, as Baden Powell well\nobserved, that the anomaly would cease to be singular, and become a\ncase of another law, pointing to some general interference which has\ntaken place on the bounds of the planetary system. Not only have the\nsatellites suffered from this perturbance, but Uranus is also anomalous\nin having an axis of rotation lying nearly in the ecliptic; and Neptune\nconstitutes a partial exception to the empirical law of Bode concerning\nthe distances of the planets, which circumstance may possibly be due to\nthe same disturbance.\n\nGeology is a science in which accidental exceptions are likely to\noccur. Only when we find strata in their original relative positions\ncan we surely infer that the order of succession is the order of\ntime. But it not uncommonly happens that strata are inverted by the\nbending and doubling action of extreme pressure. Landslips may carry\none body of rock into proximity with an unrelated series, and produce\nresults apparently inexplicable.[551] Floods, streams, icebergs, and\nother casual agents, may lodge remains in places where they would be\nwholly unexpected. Though such interfering causes have been sometimes\nwrongly supposed to explain important discoveries, the geologist must\nbear the possibility of interference in mind. Scarcely more than a\ncentury ago it was held that fossils were accidental productions of\nnature, mere forms into which minerals had been shaped by no peculiar\ncause. Voltaire appears not to have accepted such an explanation; but\nfearing that the occurrence of fossil fishes on the Alps would support\nthe Mosaic account of the deluge, he did not hesitate to attribute\nthem to the remains of fishes accidentally brought there by pilgrims.\nIn arch\u00e6ological investigations the greatest caution is requisite\nin allowing for secondary burials in ancient tombs and tumuli, for\nimitations, forgeries, casual coincidences, disturbance by subsequent\nraces or by other arch\u00e6ologists. In common life extraordinary events\nwill happen from time to time, as when a shepherdess in France was\nastonished at an iron chain falling out of the sky close to her, the\nfact being that Gay-Lussac had thrown it out of his balloon, which was\npassing over her head at the time.\n\n  [551] Murchison\u2019s *Silurian System*, vol.\u00a0ii. p.\u00a0733, &c.\n\n\n*Novel and Unexplained Exceptions.*\n\nWhen a law of nature appears to fail because some other law has\ninterfered with its action, two cases may present themselves;--the\ninterfering law may be a known one, or it may have been previously\nundetected. In the first case, which we have sufficiently considered in\nthe preceding section, we have nothing to do but calculate as exactly\nas possible the amount of interference, and make allowance for it; the\napparent failure of the law under examination should then disappear.\nBut in the second case the results may be much more important. A\nphenomenon which cannot be explained by any known laws may indicate the\ninterference of undiscovered natural forces. The ancients could not\nhelp perceiving that the general tendency of bodies downwards failed\nin the case of the loadstone, nor would the doctrine of essential\nlightness explain the exception, since the substance drawn upwards by\nthe loadstone is a heavy metal. We now see that there was no breach in\nthe perfect generality of the law of gravity, but that a new form of\nenergy manifested itself in the loadstone for the first time.\n\nOther sciences show us that laws of nature, rigorously true and exact,\nmay be developed by those who are ignorant of more complex phenomena\ninvolved in their application. Newton\u2019s comprehension of geometrical\noptics was sufficient to explain all the ordinary refractions and\nreflections of light. The simple laws of the bending of rays apply\nto all rays, whatever the character of the undulations composing\nthem. Newton suspected the existence of other classes of phenomena\nwhen he spoke of rays as *having sides*; but it remained for later\nexperimentalists to show that light is a transverse undulation, like\nthe bending of a rod or cord.\n\nDalton\u2019s atomic theory is doubtless true of all chemical compounds,\nand the essence of it is that the same compound will always be found\nto contain the same elements in the same definite proportions. Pure\ncalcium carbonate contains 48 parts by weight of oxygen to 40 of\ncalcium and 12 of carbon. But when careful analyses were made of a\ngreat many minerals, this law appeared to fail. What was unquestionably\nthe same mineral, judging by its crystalline form and physical\nproperties, would give varying proportions of its components, and\nwould sometimes contain unusual elements which yet could not be set\ndown as mere impurities. Dolomite, for instance, is a compound of the\ncarbonates of magnesia and lime, but specimens from different places\ndo not exhibit any fixed ratio between the lime and magnesia. Such\nfacts could be reconciled with the laws of Dalton only by supposing the\ninterference of a new law, that of Isomorphism.\n\nIt is now established that certain elements are related to each other,\nso that they can, as it were, step into each other\u2019s places without\napparently altering the shapes of the crystals which they constitute.\nThe carbonates of iron, calcium, and magnesium, are nearly identical\nin their crystalline forms, hence they may crystallise together in\nharmony, producing mixed minerals of considerable complexity, which\nnevertheless perfectly verify the laws of equivalent proportions. This\nprinciple of isomorphism once established, not only explains what\nwas formerly a stumbling-block, but gives valuable aid to chemists\nin deciding upon the constitution of new salts, since compounds of\nisomorphous elements which have identical crystalline forms must\npossess corresponding chemical formul\u00e6.\n\nWe may expect that from time to time extraordinary phenomena will\nbe discovered, and will lead to new views of nature. The recent\nobservation, for instance, that the resistance of a bar of selenium to\na current of electricity is affected in an extraordinary degree by rays\nof light falling upon the selenium, points to a new relation between\nlight and electricity. The allotropic changes which sulphur, selenium,\nand phosphorus undergo by an alteration in the amount of latent heat\nwhich they contain, will probably lead at some future time to important\ninferences concerning the molecular constitution of solids and liquids.\nThe curious substance ozone has perplexed many chemists, and Andrews\nand Tait thought that it afforded evidence of the decomposition of\noxygen by the electric discharge. The researches of Sir B.\u00a0C. Brodie\nnegative this notion, and afford evidence of the real constitution of\nthe substance,[552] which still, however, remains exceptional in its\nproperties and relations, and affords a hope of important discoveries\nin chemical theory.\n\n  [552] *Philosophical Transactions* (1872), vol.\u00a0clxii. No. 23.\n\n\n*Limiting Exceptions.*\n\nWe pass to cases where exceptional phenomena are actually\nirreconcilable with a law of nature previously regarded as true. Error\nmust now be allowed to have been committed, but the error may be more\nor less extensive. It may happen that a law holding rigorously true of\nthe facts actually under notice had been extended by generalisation to\nother series of facts then unexamined. Subsequent investigation may\nshow the falsity of this generalisation, and the result must be to\nlimit the law for the future to those objects of which it is really\ntrue. The contradiction to our previous opinions is partial and not\ntotal.\n\nNewton laid down as a result of experiment that every ray of\nhomogeneous light has a definite refrangibility, which it preserves\nthroughout its course until extinguished. This is one case of the\ngeneral principle of undulatory movement, which Herschel stated under\nthe title \u201cPrinciple of Forced Vibrations\u201d (p.\u00a0451), and asserted to\nbe absolutely without exception. But Herschel himself described in\nthe *Philosophical Transactions* for 1845 a curious appearance in\na solution of quinine; as viewed by transmitted light the solution\nappeared colourless, but in certain aspects it exhibited a beautiful\ncelestial blue tint. Curiously enough the colour is seen only in the\nfirst portion of liquid which the light enters. Similar phenomena in\nfluor-spar had been described by Brewster in 1838. Professor Stokes,\nhaving minutely investigated the phenomena, discovered that they were\nmore or less present in almost all vegetable infusions, and in a number\nof mineral substances. He came to the conclusion that this phenomenon,\ncalled by him Fluorescence, could only be explained by an alteration\nin the refrangibility of the rays of light; he asserts that light-rays\nof very short length of vibration in falling upon certain atoms excite\nundulations of greater length, in opposition to the principle of\nforced vibrations. No complete explanation of the mode of change is\nyet possible, because it depends upon the intimate constitution of\nthe atoms of the substances concerned; but Professor Stokes believes\nthat the principle of forced vibrations is true only so long as the\nexcursions of an atom are very small compared with the magnitude of the\ncomplex molecules.[553]\n\n  [553] *Philosophical Transactions* (1852), vol.\u00a0cxlii. pp.\u00a0465, 548,\n  &c.\n\nIt is well known that in Calorescence the refrangibility of rays is\nincreased and the wave-length diminished. Rays of obscure heat and low\nrefrangibility may be concentrated so as to heat a solid substance, and\nmake it give out rays belonging to any part of the spectrum, and it\nseems probable that this effect arises from the impact of distinct but\nconflicting atoms. Nor is it in light only that we discover limiting\nexceptions to the law of forced vibrations; for if we notice gentle\nwaves lapping upon the stones at the edge of a lake we shall see that\neach larger wave in breaking upon a stone gives rise to a series of\nsmaller waves. Thus there is constantly in progress a degradation in\nthe magnitude of water-waves. The principle of forced vibrations seems\nthen to be too generally stated by Herschel, but it must be a difficult\nquestion of mechanical theory to discriminate the circumstances in\nwhich it does and does not hold true.\n\nWe sometimes foresee the possible existence of exceptions yet unknown\nby experience, and limit the statement of our discoveries accordingly.\nExtensive inquiries have shown that all substances yet examined fall\ninto one of two classes; they are all either ferro-magnetic, that is,\nmagnetic in the same way as iron, or they are diamagnetic like bismuth.\nBut it does not follow that every substance must be ferro-magnetic or\ndiamagnetic. The magnetic properties are shown by Sir W. Thomson[554]\nto depend upon the specific inductive capacities of the substance\nin three rectangular directions. If these inductive capacities are\nall positive, we have a ferro-magnetic substance; if negative, a\ndiamagnetic substance; but if the specific inductive capacity were\npositive in one direction and negative in the others, we should have\nan exception to previous experience, and could not place the substance\nunder either of the present recognised classes.\n\n  [554] *Philosophical Magazine*, 4th Series, vol.\u00a0i. p.\u00a0182.\n\nSo many gases have been reduced to the liquid state, and so many solids\nfused, that scientific men rather hastily adopted the generalisation\nthat all substances could exist in all three states. A certain number\nof gases, such as oxygen, hydrogen, and nitrogen, have resisted all\nefforts to liquefy them, and it now seems probable from the experiments\nof Dr. Andrews that they are limiting exceptions. He finds that above\n31\u00b0\u00a0C. carbonic acid cannot be liquefied by any pressure he could\napply, whereas below this temperature liquefaction is always possible.\nBy analogy it becomes probable that even hydrogen might be liquefied if\ncooled to a very low temperature. We must modify our previous views,\nand either assert that *below a certain critical temperature* every\ngas may be liquefied, or else we must assume that a highly condensed\ngas is, when above the critical temperature, undistinguishable from\na liquid. At the same time we have an explanation of a remarkable\nexception presented by liquid carbonic acid to the general rule that\ngases expand more by heat than liquids. Liquid carbonic acid was found\nby Thilorier in 1835 to expand more than four times as much as air;\nbut by the light of Andrews\u2019 experiments we learn to regard the liquid\nas rather a highly condensed gas than an ordinary liquid, and it is\nactually possible to reduce the gas to the apparently liquid condition\nwithout any abrupt condensation.[555]\n\n  [555] Maxwell, *Theory of Heat*, p.\u00a0123.\n\nLimiting exceptions occur most frequently in the natural sciences\nof Botany, Zoology, Geology, &c., the laws of which are empirical.\nIn innumerable instances the confident belief of one generation\nhas been falsified by the wider observation of a succeeding one.\nAristotle confidently held that all swans are white,[556] and the\nproposition seemed true until not a hundred years ago black swans were\ndiscovered in Western Australia. In zoology and physiology we may\nexpect a fundamental identity to exist in the vital processes, but\ncontinual discoveries show that there is no limit to the apparently\nanomalous expedients by which life is reproduced. Alternate generation,\nfertilisation for several successive generations, hermaphroditism, are\nopposed to all we should expect from induction founded upon the higher\nanimals. But such phenomena are only limiting exceptions showing that\nwhat is true of one class is not true of another. In certain of the\ncephalopoda we meet the extraordinary fact that an arm of the male is\ncast off and lives independently until it encounters the female.\n\n  [556] *Prior Analytics*, ii. 2, 8, and elsewhere.\n\n\n*Real Exceptions to Supposed Laws.*\n\nThe exceptions which we have lastly to consider are the most important\nof all, since they lead to the entire rejection of a law or theory\nbefore accepted. No law of nature can fail; there are no such things\nas real exceptions to real laws. Where contradiction exists it must\nbe in the mind of the experimentalist. Either the law is imaginary\nor the phenomena which conflict with it; if, then, by our senses we\nsatisfy ourselves of the actual occurrence of the phenomena, the law\nmust be rejected as illusory. The followers of Aristotle held that\nnature abhors a vacuum, and thus accounted for the rise of water\nin a pump. When Torricelli pointed out the visible fact that water\nwould not rise more than 33 feet in a pump, nor mercury more than\nabout 30 inches in a glass tube, they attempted to represent these\nfacts as limiting exceptions, saying that nature abhorred a vacuum to\na certain extent and no further. But the Academicians del Cimento\ncompleted their discomfiture by showing that if we remove the pressure\nof the surrounding air, and in proportion as we remove it, nature\u2019s\nfeelings of abhorrence decrease and finally disappear altogether. Even\nAristotelian doctrines could not stand such direct contradiction.\n\nLavoisier\u2019s ideas concerning the constitution of acids received\ncomplete refutation. He named oxygen the *acid generator*, because he\nbelieved that all acids were compounds of oxygen, a generalisation\nbased on insufficient data. Berthollet, as early as 1789, proved by\nanalysis that hydrogen sulphide and prussic acid, both clearly acting\nthe part of acids, were devoid of oxygen; the former might perhaps have\nbeen interpreted as a limiting exception, but when so powerful an acid\nas hydrogen chloride (muriatic acid) was found to contain no oxygen the\ntheory had to be relinquished. Berzelius\u2019 theory of the dual formation\nof chemical compounds met a similar fate.\n\nIt is obvious that all conclusive *experimenta crucis* constitute real\nexceptions to the supposed laws of the theory which is overthrown.\nNewton\u2019s corpuscular theory of light was not rejected on account of its\nabsurdity or inconceivability, for in these respects it is, as we have\nseen, far superior to the undulatory theory. It was rejected because\ncertain small fringes of colour did not appear in the exact place\nand of the exact size in which calculation showed that they ought to\nappear according to the theory (pp.\u00a0516\u2013521). One single fact clearly\nirreconcilable with a theory involves its rejection. In the greater\nnumber of cases, what appears to be a fatal exception may be afterwards\nexplained away as a singular or disguised result of the laws with which\nit seems to conflict, or as due to the interference of extraneous\ncauses; but if we fail thus to reduce the fact to congruity, it remains\nmore powerful than any theories or any dogmas.\n\nOf late years not a few of the favourite doctrines of geologists have\nbeen rudely destroyed. It was the general belief that human remains\nwere to be found only in those deposits which are actually in progress\nat the present day, so that the creation of man appeared to have taken\nplace in this geological age. The discovery of a single worked flint\nin older strata and in connexion with the remains of extinct mammals\nwas sufficient to explode such a doctrine. Similarly, the opinions\nof geologists have been altered by the discovery of the Eozo\u00f6n in the\nLaurentian rocks of Canada; it was previously held that no remains of\nlife occurred in any older strata than those of the Cambrian system. As\nthe examination of the strata of the globe becomes more complete, our\nviews of the origin and succession of life upon the globe must undergo\nmany changes.\n\n\n*Unclassed Exceptions.*\n\nAt every period of scientific progress there will exist a multitude\nof unexplained phenomena which we know not how to regard. They are\nthe outstanding facts upon which the labours of investigators must\nbe exerted,--the ore from which the gold of future discovery is to\nbe extracted. It might be thought that, as our knowledge of the laws\nof nature increases, the number of such exceptions should decrease;\nbut, on the contrary, the more we know the more there is yet to\nexplain. This arises from several reasons; in the first place, the\nprincipal laws and forces in nature are numerous, so that he who bears\nin mind the wonderfully large numbers developed in the doctrine of\ncombinations, will anticipate the existence of immensely numerous\nrelations of one law to another. When we are once in possession of a\nlaw, we are potentially in possession of all its consequences; but\nit does not follow that the mind of man, so limited in its powers\nand capacities, can actually work them all out in detail. Just as\nthe aberration of light was discovered empirically, though it should\nhave been foreseen, so there are multitudes of unexplained facts, the\nconnexion of which with laws of nature already known to us, we should\nperceive, were we not hindered by the imperfection of our deductive\npowers. But, in the second place, as will be more fully pointed out,\nit is not to be supposed that we have approximated to an exhaustive\nknowledge of nature\u2019s powers. The most familiar facts may teem with\nindications of forces, now secrets hidden from us, because we have\nnot mind-directed eyes to discriminate them. The progress of science\nwill consist in the discovery from time to time of new exceptional\nphenomena, and their assignment by degrees to one or other of the\nheads already described. When a new fact proves to be merely a false,\napparent, singular, divergent, or accidental exception, we gain a more\nminute and accurate acquaintance with the effects of laws already known\nto exist. We have indeed no addition to what was implicitly in our\npossession, but there is much difference between knowing the laws of\nnature and perceiving all their complicated effects. Should a new fact\nprove to be a limiting or real exception, we have to alter, in part or\nin whole, our views of nature, and are saved from errors into which we\nhad fallen. Lastly, the new fact may come under the sixth class, and\nmay eventually prove to be a novel phenomenon, indicating the existence\nof new laws and forces, complicating but not otherwise interfering with\nthe effects of laws and forces previously known.\n\nThe best instance which I can find of an unresolved exceptional\nphenomenon, consists in the anomalous vapour-densities of phosphorus,\narsenic, mercury, and cadmium. It is one of the most important\nlaws of chemistry, discovered by Gay-Lussac, that equal volumes of\ngases exactly correspond to equivalent weights of the substances.\nNevertheless phosphorus and arsenic give vapours exactly twice as\ndense as they should do by analogy, and mercury and cadmium diverge\nin the other direction, giving vapours half as dense as we should\nexpect. We cannot treat these anomalies as limiting exceptions, and\nsay that the law holds true of substances generally but not of these;\nfor the properties of gases (p.\u00a0601), usually admit of the widest\ngeneralisations. Besides, the preciseness of the ratio of divergence\npoints to the real observance of the law in a modified manner. We might\nendeavour to reduce the exceptions by doubling the atomic weights of\nphosphorus and arsenic, and halving those of mercury and cadmium. But\nthis step has been maturely considered by chemists, and is found to\nconflict with all the other analogies of the substances and with the\nprinciple of isomorphism. One of the most probable explanations is,\nthat phosphorus and arsenic produce vapour in an allotropic condition,\nwhich might perhaps by intense heat be resolved into a simpler gas of\nhalf the density; but facts are wanting to support this hypothesis, and\nit cannot be applied to the other two exceptions without supposing that\ngases and vapours generally are capable of resolution into something\nsimpler. In short, chemists can at present make nothing of these\nanomalies. As Hofmann says, \u201cTheir philosophical interpretation belongs\nto the future.... They may turn out to be typical facts, round which\nmany others of the like kind may come hereafter to be grouped; and\nthey may prove to be allied with special properties, or dependent on\nparticular conditions as yet unsuspected.\u201d[557]\n\n  [557] Hofmann\u2019s *Introduction to Chemistry*, p.\u00a0198.\n\nIt would be easy to point out a great number of other unexplained\nanomalies. Physicists assert, as an absolutely universal law, that\nin liquefaction heat is absorbed;[558] yet sulphur is at least an\napparent exception. The two substances, sulphur and selenium, are, in\nfact, very anomalous in their relations to heat. Sulphur may be said\nto have two melting points, for, though liquid like water at 120\u00b0\u00a0C.,\nit becomes quite thick and tenacious between 221\u00b0 and 249\u00b0, and melts\nagain at a higher temperature. Both sulphur and selenium may be thrown\ninto several curious states, which chemists conveniently dispose of by\ncalling them *allotropic*, a term freely used when they are puzzled\nto know what has happened. The chemical and physical history of iron,\nagain, is full of anomalies; not only does it undergo inexplicable\nchanges of hardness and texture in its alloys with carbon and other\nelements, but it is almost the only substance which conveys sound with\ngreater velocity at a higher than at a lower temperature, the velocity\nincreasing from 20\u00b0 to 100\u00b0\u00a0C., and then decreasing. Silver also is\nanomalous in regard to sound. These are instances of inexplicable\nexceptions, the bearing of which must be ascertained in the future\nprogress of science.\n\n  [558] Stewart\u2019s *Elementary Treatise on Heat*, p.\u00a080.\n\nWhen the discovery of new and peculiar phenomena conflicting with\nour theories of the constitution of nature is reported to us, it\nbecomes no easy task to steer a philosophically correct course between\ncredulity and scepticism. We are not to assume, on the one hand, that\nthere is any limit to the wonders which nature can present to us.\nNothing except the contradictory is really impossible, and many things\nwhich we now regard as common-place were considered as little short\nof the miraculous when first perceived. The electric telegraph was\na visionary dream among medi\u00e6val physicists;[559] it has hardly yet\nceased to excite our wonder; to our descendants centuries hence it\nwill probably appear inferior in ingenuity to some inventions which\nthey will possess. Now every strange phenomenon may be a secret spring\nwhich, if rightly touched, will open the door to new chambers in the\npalace of nature. To refuse to believe in the occurrence of anything\nstrange would be to neglect the most precious chances of discovery.\nWe may say with Hooke, that \u201cthe believing strange things possible\nmay perhaps be an occasion of taking notice of such things as another\nwould pass by without regard as useless.\u201d We are not, therefore,\nto shut our ears even to such apparently absurd stories as those\nconcerning second-sight, clairvoyance, animal magnetism, ode force,\ntable-turning, or any of the popular delusions which from time to time\nare current. The facts recorded concerning these matters are facts in\nsome sense or other, and they demand explanation, either as new natural\nphenomena, or as the results of credulity and imposture. Most of the\nsupposed phenomena referred to have been, or by careful investigation\nwould doubtless be, referred to the latter head, and the absence of\nscientific ability in many of those who describe them is sufficient to\ncast a doubt upon their value.\n\n  [559] Jevons, *Proceedings of the Manchester Literary and\n  Philosophical Society*, 6th March, 1877, vol.\u00a0xvi. p.\u00a0164. See also\n  Mr. W.\u00a0E. A. Axon\u2019s note on the same subject, ibid. p.\u00a0166.\n\nIt is to be remembered that according to the principle of the inverse\nmethod of probability, the probability of any hypothetical explanation\nis affected by the probability of each other possible explanation. If\nno other reasonable explanation could be suggested, we should be forced\nto look upon spiritualist manifestations as indicating mysterious\ncauses. But as soon as it is shown that fraud has been committed in\nseveral important cases, and that in other cases persons in a credulous\nand excited state of mind have deceived themselves, the probability\nbecomes very considerable that similar explanations may apply to most\nlike manifestations. The performances of conjurors sufficiently prove\nthat it requires no very great skill to perform tricks the *modus\noperandi* of which shall entirely escape the notice of spectators. It\nis on these grounds of probability that we should reject the so-called\nspiritualist stories, and not simply because they are strange.\n\nCertainly in the obscure phenomena of mind, those relating to memory,\ndreams, somnambulism, and other peculiar states of the nervous\nsystem, there are many inexplicable and almost incredible facts, and\nit is equally unphilosophical to believe or to disbelieve without\nclear evidence. There are many facts, too, concerning the instincts\nof animals, and the mode in which they find their way from place to\nplace, which are at present quite inexplicable. No doubt there are many\nstrange things not dreamt of in our philosophy, but this is no reason\nwhy we should believe in every strange thing which is reported to have\nhappened.\n\n\n\n\nCHAPTER XXX.\n\nCLASSIFICATION.\n\n\nThe extensive subject of Classification has been deferred to a late\npart of this treatise, because it involves questions of difficulty, and\ndid not seem naturally to fall into an earlier place. But it must not\nbe supposed that, in now formally taking up the subject, we are for\nthe first time entertaining the notion of classification. All logical\ninference involves classification, which is indeed the necessary\naccompaniment of the action of judgment. It is impossible to detect\nsimilarity between objects without thereby joining them together in\nthought, and forming an incipient class. Nor can we bestow a common\nname upon objects without implying the existence of a class. Every\ncommon name is the name of a class, and every name of a class is a\ncommon name. It is evident also that to speak of a general notion or\nconcept is but another way of speaking of a class. Usage leads us to\nemploy the word classification in some cases and not in others. We\nare said to form the *general notion* parallelogram when we regard\nan infinite number of possible four-sided rectilinear figures as\nresembling each other in the common property of possessing parallel\nsides. We should be said to form a *class*, Trilobite, when we place\ntogether in a museum a number of specimens resembling each other in\ncertain defined characters. But the logical nature of the operation\nis the same in both cases. We form a *class* of figures called\nparallelograms and we form a *general notion* of trilobites.\n\nScience, it was said at the outset, is the detection of identify, and\nclassification is the placing together, either in thought or in actual\nproximity of space, those objects between which identity has been\ndetected. Accordingly, the value of classification is co-extensive with\nthe value of science and general reasoning. Whenever we form a class\nwe reduce multiplicity to unity, and detect, as Plato said, the one in\nthe many. The result of such classification is to yield generalised\nknowledge, as distinguished from the direct and sensuous knowledge of\nparticular facts. Of every class, so far as it is correctly formed, the\nprinciple of substitution is true, and whatever we know of one object\nin a class we know of the other objects, so far as identity has been\ndetected between them. The facilitation and abbreviation of mental\nlabour is at the bottom of all mental progress. The reasoning faculties\nof Newton were not different in nature from those of a ploughman; the\ndifference lay in the extent to which they were exerted, and the number\nof facts which could be treated. Every thinking being generalises more\nor less, but it is the depth and extent of his generalisations which\ndistinguish the philosopher. Now it is the exertion of the classifying\nand generalising powers which enables the intellect of man to cope\nin some degree with the infinite number of natural phenomena. In the\nchapters upon combinations and permutations it was made evident, that\nfrom a few elementary differences immense numbers of combinations\ncan be produced. The process of classification enables us to resolve\nthese combinations, and refer each one to its place according to one\nor other of the elementary circumstances out of which it was produced.\nWe restore nature to the simple conditions out of which its endless\nvariety was developed. As Professor Bowen has said,[560] \u201cThe first\nnecessity which is imposed upon us by the constitution of the mind\nitself, is to break up the infinite wealth of Nature into groups and\nclasses of things, with reference to their resemblances and affinities,\nand thus to enlarge the grasp of our mental faculties, even at the\nexpense of sacrificing the minuteness of information which can be\nacquired only by studying objects in detail. The first efforts in\nthe pursuit of knowledge, then, must be directed to the business\nof classification. Perhaps it will be found in the sequel, that\nclassification is not only the beginning, but the culmination and the\nend, of human knowledge.\u201d\n\n  [560] *A Treatise on Logic, or, the Laws of Pure Thought*, by Francis\n  Bowen, Professor of Moral Philosophy in Harvard College, Cambridge,\n  United States, 1866, p.\u00a0315.\n\n\n*Classification Involving Induction.*\n\nThe purpose of classification is the detection of the laws of nature.\nHowever much the process may in some cases be disguised, classification\nis not really distinct from the process of perfect induction, whereby\nwe endeavour to ascertain the connexions existing between properties of\nthe objects under treatment. There can be no use in placing an object\nin a class unless something more than the fact of being in the class is\nimplied. If we arbitrarily formed a class of metals and placed therein\na selection from the list of known metals made by ballot, we should\nhave no reason to expect that the metals in question would resemble\neach other in any points except that they are metals, and have been\nselected by the ballot. But when chemists select from the list the five\nmetals, potassium, sodium, c\u00e6sium, rubidium, and lithium and call them\nthe Alkaline metals, a great deal is implied in this classification.\nOn comparing the qualities of these substances they are all found to\ncombine very energetically with oxygen, to decompose water at all\ntemperatures, and to form strongly basic oxides, which are highly\nsoluble in water, yielding powerfully caustic and alkaline hydrates\nfrom which water cannot be expelled by heat. Their carbonates are also\nsoluble in water, and each metal forms only one chloride. It may also\nbe expected that each salt of one of the metals will correspond to a\nsalt of each other metal, there being a general analogy between the\ncompounds of these metals and their properties.\n\nNow in forming this class of alkaline metals, we have done more than\nmerely select a convenient order of statement. We have arrived at a\ndiscovery of certain empirical laws of nature, the probability being\nvery considerable that a metal which exhibits some of the properties\nof alkaline metals will also possess the others. If we discovered\nanother metal whose carbonate was soluble in water, and which\nenergetically combined with water at all temperatures, producing a\nstrongly basic oxide, we should infer that it would form only a single\nchloride, and that generally speaking, it would enter into a series\nof compounds corresponding to the salts of the other alkaline metals.\nThe formation of this class of alkaline metals then, is no mere matter\nof convenience; it is an important and successful act of inductive\ndiscovery, enabling us to register many undoubted propositions as\nresults of perfect induction, and to make a great number of inferences\ndepending upon the principles of imperfect induction.\n\nAn excellent instance as to what classification can do, is found in\nMr. Lockyer\u2019s researches on the sun.[561] Wanting some guide as to\nwhat more elements to look for in the sun\u2019s photosphere, he prepared\na classification of the elements according as they had or had not\nbeen traced in the sun, together with a detailed statement of the\nchief chemical characters of each element. He was then able to observe\nthat the elements found in the sun were for the most part those\nforming stable compounds with oxygen. He then inferred that other\nelements forming stable oxides would probably exist in the sun, and\nhe was rewarded by the discovery of five such metals. Here we have\nempirical and tentative classification leading to the detection of the\ncorrelation between existence in the sun, and the power of forming\nstable oxides and then leading by imperfect induction to the discovery\nof more coincidences between these properties.\n\n  [561] *Proceedings of the Royal Society*, November, 1873, vol.\u00a0xxi.\n  p.\u00a0512.\n\nProfessor Huxley has defined the process of classification in the\nfollowing terms.[562] \u201cBy the classification of any series of objects,\nis meant the actual or ideal arrangement together of those which are\nlike and the separation of those which are unlike; the purpose of this\narrangement being to facilitate the operations of the mind in clearly\nconceiving and retaining in the memory the characters of the objects in\nquestion.\u201d\n\n  [562] *Lectures on the Elements of Comparative Anatomy*, 1864, p.\u00a01.\n\nThis statement is doubtless correct, so far as it goes, but it does\nnot include all that Professor Huxley himself implicitly treats under\nclassification. He is fully aware that deep correlations, or in\nother terms deep uniformities or laws of nature, will be disclosed\nby any well chosen and profound system of classification. I should\ntherefore propose to modify the above statement, as follows:--\u201cBy\nthe classification of any series of objects, is meant the actual or\nideal arrangement together of those which are like and the separation\nof those which are unlike, the purpose of this arrangement being,\nprimarily, to disclose the correlations or laws of union of properties\nand circumstances, and, secondarily, to facilitate the operations\nof the mind in clearly conceiving and retaining in the memory the\ncharacters of the objects in question.\u201d\n\n\n*Multiplicity of Modes of Classification.*\n\nIn approaching the question how any given group of objects may be\nbest classified, let it be remarked that there must generally be an\nunlimited number of modes of classifying a group of objects. Misled, as\nwe shall see, by the problem of classification in the natural sciences,\nphilosophers seem to think that in each subject there must be one\nessentially natural system of classification which is to be selected,\nto the exclusion of all others. This erroneous notion probably arises\nalso in part from the limited powers of thought and the inconvenient\nmechanical conditions under which we labour. If we arrange the books\nin a library catalogue, we must arrange them in some one order; if we\ncompose a treatise on mineralogy, the minerals must be successively\ndescribed in some one arrangement; if we treat such simple things as\ngeometrical figures, they must be taken in some fixed order. We shall\nnaturally select that arrangement which appears to be most convenient\nand instructive for our principal purpose. But it does not follow\nthat this method of arrangement possesses any exclusive excellence,\nand there will be usually many other possible arrangements, each\nvaluable in its own way. A perfect intellect would not confine itself\nto one order of thought, but would simultaneously regard a group of\nobjects as classified in all the ways of which they are capable. Thus\nthe elements may be classified according to their atomicity into the\ngroups of monads, dyads, triads, tetrads, pentads, and hexads, and\nthis is probably the most instructive classification; but it does not\nprevent us from also classifying them according as they are metallic\nor non-metallic, solid, liquid or gaseous at ordinary temperatures,\nuseful or useless, abundant or scarce, ferro-magnetic or diamagnetic,\nand so on.\n\nMineralogists have spent a great deal of labour in trying to discover\nthe supposed natural system of classification for minerals. They have\nconstantly encountered the difficulty that the chemical composition\ndoes not run together with the crystallographic form, and the various\nphysical properties of the mineral. Substances identical in the\nforms of their crystals, especially those belonging to the first or\ncubical system of crystals, are often found to have no resemblance\nin chemical composition. The same substance, again, is occasionally\nfound crystallised in two essentially different crystallographic\nforms; calcium carbonate, for instance, appearing as calc-spar and\narragonite. The simple truth is that if we are unable to discover\nany correspondence, or, as we may call it, any *correlation* between\nthe properties of minerals, we cannot make any one arrangement which\nwill enable us to treat all these properties in a single system of\nclassification. We must classify minerals in as many different ways\nas there are different groups of unrelated properties of sufficient\nimportance. Even if, for the purpose of describing minerals\nsuccessively in a treatise, we select one chief system, that, for\ninstance, having regard to chemical composition, we ought mentally to\nregard the minerals as classified in all other useful modes.\n\nExactly the same may be said of the classification of plants. An\nimmense number of different modes of classifying plants have been\nproposed at one time or other, an exhaustive account of which will be\nfound in the article on classification in Rees\u2019s \u201cCyclop\u00e6dia,\u201d or in\nthe introduction to Lindley\u2019s \u201cVegetable Kingdom.\u201d There have been the\nFructists, such as C\u00e6salpinus, Morison, Hermann, Boerhaave or Gaertner,\nwho arranged plants according to the form of the fruit. The Corollists,\nRivinus, Ludwig, and Tournefort, paid attention chiefly to the number\nand arrangement of the parts of the corolla. Magnol selected the calyx\nas the critical part, while Sauvage arranged plants according to their\nleaves; nor are these instances more than a small selection from the\nactual variety of modes of classification which have been tried. Of\nsuch attempts it may be said that every system will probably yield some\ninformation concerning the relations of plants, and it is only after\ntrying many modes that it is possible to approximate to the best.\n\n\n*Natural and Artificial Systems of Classification.*\n\nIt has been usual to distinguish systems of classification as natural\nand artificial, those being called natural which seemed to express the\norder of existing things as determined by nature. Artificial methods of\nclassification, on the other hand, included those formed for the mere\nconvenience of men in remembering or treating natural objects.\n\nThe difference, as it is commonly regarded, has been well described\nby Amp\u00e9re,[563] as follows: \u201cWe can distinguish two kinds of\nclassifications, the natural and the artificial. In the latter kind,\nsome characters, arbitrarily chosen, serve to determine the place of\neach object; we abstract all other characters, and the objects are thus\nfound to be brought near to or to be separated from each other, often\nin the most bizarre manner. In natural systems of classification, on\nthe contrary, we employ concurrently all the characters essential to\nthe objects with which we are occupied, discussing the importance of\neach of them; and the results of this labour are not adopted unless\nthe objects which present the closest analogy are brought most near\ntogether, and the groups of the several orders which are formed from\nthem are also approximated in proportion as they offer more similar\ncharacters. In this way it arises that there is always a kind of\nconnexion, more or less marked, between each group and the group which\nfollows it.\u201d\n\n  [563] *Essai sur la Philosophie des Sciences*, p.\u00a09.\n\nThere is much, however, that is vague and logically false in this\nand other definitions which have been proposed by naturalists to\nexpress their notion of a natural system. We are not informed how the\n*importance* of a resemblance is to be determined, nor what is the\nmeasure of the *closeness* of analogy. Until all the words employed\nin a definition are made clear in meaning, the definition itself is\nworse than useless. Now if the views concerning classification here\nupheld are true, there can be no sharp and precise distinction between\nnatural and artificial systems. All arrangements which serve any\npurpose at all must be more or less natural, because, if closely enough\nscrutinised, they will involve more resemblances than those whereby the\nclass was defined.\n\nIt is true that in the biological sciences there would be one\narrangement of plants or animals which would be conspicuously\ninstructive, and in a certain sense natural, if it could be attained,\nand it is that after which naturalists have been in reality striving\nfor nearly two centuries, namely, that *arrangement which would display\nthe genealogical descent of every form from the original life germ*.\nThose morphological resemblances upon which the classification of\nliving beings is almost always based are inherited resemblances, and\nit is evident that descendants will usually resemble their parents and\neach other in a great many points.\n\nI have said that a natural is distinguished from an arbitrary or\nartificial system only in degree. It will be found almost impossible\nto arrange objects according to any circumstance without finding that\nsome correlation of other circumstances is thus made apparent. No\narrangement could seem more arbitrary than the common alphabetical\narrangement according to the initial letter of the name. But we cannot\nscrutinise a list of names of persons without noticing a predominance\nof Evans\u2019s and Jones\u2019s, under the letters E and J, and of names\nbeginning with Mac under the letter M. The predominance is so great\nthat we could not attribute it to chance, and inquiry would of course\nshow that it arose from important facts concerning the nationality\nof the persons. It would appear that the Evans\u2019s and Jones\u2019s were of\nWelsh descent, and those whose names bear the prefix Mac of Keltic\ndescent. With the nationality would be more or less strictly correlated\nmany peculiarities of physical constitution, language, habits, or\nmental character. In other cases I have been interested in noticing\nthe empirical inferences which are displayed in the most arbitrary\narrangements. If a large register of the names of ships be examined\nit will often be found that a number of ships bearing the same name\nwere built about the same time, a correlation due to the occurrence of\nsome striking incident shortly previous to the building of the ships.\nThe age of ships or other structures is usually correlated with their\ngeneral form, nature of materials, &c., so that ships of the same name\nwill often resemble each other in many points.\n\nIt is impossible to examine the details of some of the so-called\nartificial systems of classification of plants, without finding that\nmany of the classes are natural in character. Thus in Tournefort\u2019s\narrangement, depending almost entirely on the formation of the corolla,\nwe find the natural orders of the Labiat\u00e6, Crucifer\u00e6, Rosace\u00e6,\nUmbellifer\u00e6, Liliace\u00e6, and Papilionace\u00e6, recognised in his 4th, 5th,\n6th, 7th, 9th, and 10th classes. Many of the classes in Linn\u00e6us\u2019\ncelebrated sexual system also approximate to natural classes.\n\n\n*Correlation of Properties.*\n\nHabits and usages of language are apt to lead us into the error of\nimagining that when we employ different words we always mean different\nthings. In introducing the subject of classification nominally I was\ncareful to draw the reader\u2019s attention to the fact that all reasoning\nand all operations of scientific method really involve classification,\nthough we are accustomed to use the name in some cases and not in\nothers. The name *correlation* requires to be used with the same\nqualification. Things are correlated (*con*, *relata*) when they are\nso related or bound to each other that *where one is the other is, and\nwhere one is not the other is not*. Throughout this work we have then\nbeen dealing with correlations. In geometry the occurrence of three\nequal angles in a triangle is correlated with the existence of three\nequal sides; in physics gravity is correlated with inertia; in botany\nexogenous growth is correlated with the possession of two cotyledons,\nor the production of flowers with that of spiral vessels. Wherever a\nproposition of the form A\u00a0=\u00a0B is true there correlation exists. But it\nis in the classificatory sciences especially that the word correlation\nhas been employed.\n\nWe find it stated that in the class Mammalia the possession of two\noccipital condyles, with a well-ossified basi-occipital, is correlated\nwith the possession of mandibles, each ramus of which is composed of a\nsingle piece of bone, articulated with the squamosal element of the\nskull, and also with the possession of mamm\u00e6 and non-nucleated red\nblood-corpuscles. Professor Huxley remarks[564] that this statement of\nthe character of the class mammalia is something more than an arbitrary\ndefinition; it is a statement of a law of correlation or co-existence\nof animal structures, from which most important conclusions are\ndeducible. It involves a generalisation to the effect that in nature\nthe structures mentioned are always found associated together. This\namounts to saying that the formation of the class mammalia involves an\nact of inductive discovery, and results in the establishment of certain\nempirical laws of nature. Professor Huxley has excellently expressed\nthe mode in which discoveries of this kind enable naturalists to make\ndeductions or predictions with considerable confidence, but he has also\npointed out that such inferences are likely from time to time to prove\nmistaken. I will quote his own words:\n\n  [564] *Lectures on the Elements of Comparative Anatomy, and on the\n  Classification of Animals*, 1864, p.\u00a03.\n\n\u201cIf a fragmentary fossil be discovered, consisting of no more than\na ramus of a mandible, and that part of the skull with which it\narticulated, a knowledge of this law may enable the pal\u00e6ontologist to\naffirm, with great confidence, that the animal of which it formed a\npart suckled its young, and had non-nucleated red blood-corpuscles; and\nto predict that should the back part of that skull be discovered, it\nwill exhibit two occipital condyles and a well-ossified basi-occipital\nbone.\n\n\u201cDeductions of this kind, such as that made by Cuvier in the famous\ncase of the fossil opossum of Montmartre, have often been verified,\nand are well calculated to impress the vulgar imagination; so that\nthey have taken rank as the triumphs of the anatomist. But it should\ncarefully be borne in mind, that, like all merely empirical laws, which\nrest upon a comparatively narrow observational basis, the reasoning\nfrom them may at any time break down. If Cuvier, for example, had had\nto do with a fossil Thylacinus instead of a fossil Opossum, he would\nnot have found the marsupial bones, though the inflected angle of the\njaw would have been obvious enough. And so, though, practically,\nany one who met with a characteristically mammalian jaw would be\njustified in expecting to find the characteristically mammalian occiput\nassociated with it; yet, he would be a bold man indeed, who should\nstrictly assert the belief which is implied in this expectation, viz.,\nthat at no period of the world\u2019s history did animals exist which\ncombined a mammalian occiput with a reptilian jaw, or *vice vers\u00e2*.\u201d\n\nOne of the most distinct and remarkable instances of correlation in\nthe animal world is that which occurs in ruminating animals, and which\ncould not be better stated than in the following extract from the\nclassical work of Cuvier:[565]\n\n  [565] *Ossemens Fossiles*, 4th edit. vol.\u00a0i. p.\u00a0164. Quoted by\n  Huxley, *Lectures*, &c., p.\u00a05.\n\n\u201cI doubt if any one would have divined, if untaught by observation,\nthat all ruminants have the foot cleft, and that they alone have it. I\ndoubt if any one would have divined that there are frontal horns only\nin this class: that those among them which have sharp canines for the\nmost part lack horns.\n\n\u201cHowever, since these relations are constant, they must have some\nsufficient cause; but since we are ignorant of it, we must make good\nthe defect of the theory by means of observation: it enables us to\nestablish empirical laws which become almost as certain as rational\nlaws when they rest on sufficiently repeated observations; so that\nnow whoso sees merely the print of a cleft foot may conclude that the\nanimal which left this impression ruminated, and this conclusion is as\ncertain as any other in physics or morals. This footprint alone then,\nyields, to him who observes it, the form of the teeth, the form of the\njaws, the form of the vertebr\u00e6, the form of all the bones of the legs,\nof the thighs, of the shoulders, and of the pelvis of the animal which\nhas passed by: it is a surer mark than all those of Zadig.\u201d\n\nWe meet with a good instance of the purely empirical correlation\nof circumstances when we classify the planets according to their\ndensities and periods of axial rotation.[566] If we examine a table\nspecifying the usual astronomical elements of the solar system, we find\nthat four planets resemble each other very closely in the period of\naxial rotation, and the same four planets are all found to have high\ndensities, thus:--\n\n  [566] Chambers, *Descriptive Astronomy*, 1st edit. p.\u00a023.\n\n  Name of        Period of Axial\n  Planet.          Rotation.         Density.\n\n  Mercury      24 hours  5 minutes     7\u00b794\n  Venus        23  \"    21  \"          5\u00b733\n  Earth        23  \"    56  \"          5\u00b767\n  Mars         24  \"    37  \"          5\u00b784\n\nA similar table for the other larger planets, is as follows:--\n\n  Jupiter       9 hours 55 minutes     1\u00b736\n  Saturn       10  \"    29  \"           \u00b774\n  Uranus        9  \"    30  \"           \u00b797\n  Neptune      --  \"    --             1\u00b702\n\nIt will be observed that in neither group is the equality of the\nrotational period or the density more than rudely approximate;\nnevertheless the difference of the numbers in the first and second\ngroup is so very well marked, the periods of the first being at least\ndouble and the densities four or five times those of the second, that\nthe coincidence cannot be attributed to accident. The reader will\nalso notice that the first group consists of the planets nearest to\nthe sun; that with the exception of the earth none of them possess\nsatellites; and that they are all comparatively small. The second group\nare furthest from the sun, and all of them possess several satellites,\nand are comparatively great. Therefore, with but slight exceptions, the\nfollowing correlations hold true:--\n\nInterior planets. Long period. Small size. High Density. No satellites.\nExterior \" Short \" Great \" Low \" Many \"\n\nThese coincidences point with much probability to a difference in the\norigin of the two groups, but no further explanation of the matter is\nyet possible.\n\nThe classification of comets according to their periods by Mr.\nHind and Mr. A.\u00a0S. Davies, tends to establish the conclusion that\ndistinct groups of comets have been brought into the solar system\nby the attractive powers of Jupiter, Uranus, or other planets.[567]\nThe classification of nebul\u00e6 as commenced by the two Herschels, and\ncontinued by Lord Rosse, Mr. Huggins, and others, will probably lead\nat some future time to the discovery of important empirical laws\nconcerning the constitution of the universe. The minute examination and\nclassification of meteorites, as carried on by Mr. Sorby and others,\nseems likely to afford us an insight into the formation of the heavenly\nbodies.\n\n  [567] *Philosophical Magazine*, 4th Series, vol.\u00a0xxxix. p.\u00a0396;\n  vol.\u00a0xl. p. 183; vol.\u00a0xli. p.\u00a044. See also Proctor, *Popular Science\n  Review*, October 1874, p.\u00a0350.\n\nWe should never fail to remember the slightest and most inexplicable\ncorrelations, for they may prove of importance in the future.\nDiscoveries begin when we are least expecting them. It is a significant\nfact, for instance, that the greater number of variable stars are of\na reddish colour. Not all variable stars are red, nor all red stars\nvariable; but considering that only a small fraction of the observed\nstars are known to be variable, and only a small fraction are red, the\nnumber which fall into both classes is too great to be accidental.[568]\nIt is also remarkable that the greater number of stars possessing great\nproper motion are double stars, the star 61 Cygni being especially\nnoticeable in this respect.[569] The correlation in these cases is\nnot without exception, but the preponderance is so great as to point\nto some natural connexion, the exact nature of which must be a matter\nfor future investigation. Herschel remarked that the two double stars\n61 Cygni and \u03b1 Centauri of which the orbits were well ascertained,\nevidently belonged to the same family or genus.[570]\n\n  [568] Humboldt, *Cosmos* (Bohn), vol.\u00a0iii. p.\u00a0224.\n\n  [569] Baily, British *Association Catalogue*, p.\u00a048.\n\n  [570] *Outlines of Astronomy*, \u00a7 850, 4th edit. p.\u00a0578.\n\n\n*Classification in Crystallography.*\n\nPerhaps the most perfect and instructive instance of classification\nwhich we can find is furnished by the science of crystallography\n(p.\u00a0133). The system of arrangement now generally adopted is\nconspicuously natural, and is even mathematically perfect. A crystal\nconsists in every part of similar molecules similarly related to the\nadjoining molecules, and connected with them by forces the nature of\nwhich we can only learn by their apparent effects. But these forces\nare exerted in space of three dimensions, so that there is a limited\nnumber of suppositions which can be entertained as to the relations of\nthese forces. In one case each molecule will be similarly related to\nall those which are next to it; in a second case, it will be similarly\nrelated to those in a certain plane, but differently related to those\nnot in that plane. In the simpler cases the arrangement of molecules is\nrectangular; in the remaining cases oblique either in one or two planes.\n\nIn order to simplify the explanation and conception of the complicated\nphenomena which crystals exhibit, an hypothesis has been invented which\nis an excellent instance of the Descriptive Hypotheses before mentioned\n(p.\u00a0522). Crystallographers imagine that there are within each crystal\ncertain axes, or lines of direction, by the comparative length and the\nmutual inclination of which the nature of the crystal is determined.\nIn one class of crystals there are three such axes lying in one plane,\nand a fourth perpendicular to that plane; but in all the other classes\nthere are imagined to be only three axes. Now these axes can be varied\nin three ways as regards length: they may be (1) all equal, or (2) two\nequal and one unequal, or (3) all unequal. They may also be varied in\nfour ways as regards direction: (1) they may be all at right angles\nto each other; (2) two axes may be oblique to each other and at right\nangles to the third; (3) two axes may be at right angles to each other\nand the third oblique to both; (4) the three axes may be all oblique.\nNow, if all the variations as regards length were combined with those\nregarding direction, it would seem to be possible to have twelve\nclasses of crystals in all, the enumeration being then logically and\ngeometrically complete. But as a matter of empirical observation, many\nof these classes are not found to occur, oblique axes being seldom or\nnever equal. There remain seven recognised classes of crystals, but\neven of these one class is not positively known to be represented in\nnature.\n\nThe first class of crystals is defined by possessing three equal\nrectangular axes, and equal elasticity in all directions. The primary\nor simple form of the crystals is the cube, but by the removal of the\ncorners of the cube by planes variously inclined to the axes, we have\nthe regular octohedron, the dodecahedron, and various combinations of\nthese forms. Now it is a law of this class of crystals that as each\naxis is exactly like each other axis, every modification of any corner\nof a crystal must be repeated symmetrically with regard to the other\naxes; thus the forms produced are symmetrical or regular, and the\nclass is called the *Regular System* of crystals. It includes a great\nvariety of substances, some of them being elements, such as carbon in\nthe form of diamond, others more or less complex compounds, such as\nrock-salt, potassium iodide and bromide, the several kinds of alum,\nfluor-spar, iron bisulphide, garnet, spinelle, &c. No correlation\nthen is apparent between the form of crystallisation and the chemical\ncomposition. But what we have to notice is that the physical properties\nof the crystallised substances with regard to light, heat, electricity,\n&c., are closely similar. Light and heat undulations, wherever they\nenter a crystal of the regular system, spread with equal rapidity in\nall directions, just as they would in a uniform fluid. Crystals of the\nregular system accordingly do not in any case exhibit the phenomena\nof double refraction, unless by mechanical compression we alter the\nconditions of elasticity. These crystals, again, expand equally in all\ndirections when heated, and if we could cut a sufficiently large plate\nfrom a cubical crystal, and examine the sound vibrations of which it\nis capable, we should find that they indicated an equal elasticity\nin every direction. Thus we see that a great number of important\nproperties are correlated with that of crystallisation in the regular\nsystem, and as soon as we know that the primary form of a substance\nis the cube, we are able to infer with approximate certainty that it\npossesses all these properties. The class of regular crystals is then\nan eminently natural class, one disclosing many general laws connecting\ntogether the physical and mechanical properties of the substances\nclassified.\n\nIn the second class of crystals, called the dimetric, square prismatic,\nor pyramidal system, there are also three axes at right angles to each\nother; two of the axes are equal, but the third or principal axis is\nunequal, being either greater or less than either of the other two. In\nsuch crystals accordingly the elasticity and other properties are alike\nin all directions perpendicular to the principal axis, but vary in all\nother directions. If a point within a crystal of this system be heated,\nthe heat spreads with equal rapidity in planes perpendicular to the\nprincipal axis, but more or less rapidly in the direction of this axis,\nso that the isothermal surface is an ellipsoid of revolution round that\naxis.\n\nNearly the same statement may be made concerning the third or hexagonal\nor rhombohedral system of crystals, in which there are three axes lying\nin one plane and meeting at angles of 60\u00b0, while the fourth axis is\nperpendicular to the other three. The hexagonal prism and rhombohedron\nare the commonest forms assumed by crystals of this system, and in\nice, quartz, and calc-spar, we have abundance of beautiful specimens\nof the various shapes produced by the modification of the primitive\nform. Calc-spar alone is said to crystallise in at least 700 varieties\nof form. Now of all the crystals belonging both to this and the\ndimetric class, we know that a ray of light passing in the direction\nof the principal axis will be refracted singly as in a crystal of the\nregular system; but in every other direction the light will suffer\ndouble refraction being separated into two rays, one of which obeys\nthe ordinary law of refraction, but the other a much more complicated\nlaw. The other physical properties vary in an analogous manner. Thus\ncalc-spar expands by heat in the direction of the principal axis, but\ncontracts a little in directions perpendicular to it. So closely are\nthe physical properties correlated that Mitscherlich, having observed\nthe law of expansion in calc-spar, was enabled to predict that the\ndouble refracting power of the substance would be decreased by a rise\nof temperature, as was proved by experiment to be the case.\n\nIn the fourth system, called the trimetric, rhombic, or right prismatic\nsystem, there are three axes, at right angles, but all unequal in\nlength. It may be asserted in general terms that the mechanical\nproperties vary in such crystals in every direction, and heat spreads\nso that the isothermal surface is an ellipsoid with three unequal axes.\n\nIn the remaining three classes, called the monoclinic, diclinic, and\ntriclinic, the axes are more or less oblique, and at the same time\nunequal. The complication of phenomena is therefore greatly increased,\nand it need only be stated that there are always two directions in\nwhich a ray is singly refracted, but that in all other directions\ndouble refraction takes place. The conduction of heat is unequal in\nall directions, the isothermal surface being an ellipsoid of three\nunequal axes. The relations of such crystals to other phenomena are\noften very complicated, and hardly yet reduced to law. Some crystals,\ncalled pyro-electric, manifest vitreous electricity at some points of\ntheir surface, and resinous electricity at other points when rising in\ntemperature, the character of the electricity being changed when the\ntemperature sinks again. This production of electricity is believed to\nbe connected with the hemihedral character of the crystals exhibiting\nit. The crystalline structure of a substance again influences its\nmagnetic behaviour, the general law being that the direction in which\nthe molecules of a crystal are most approximated tends to place itself\naxially or equatorially between the poles of a magnet, respectively\nas the body is magnetic or diamagnetic. Further questions arise if we\napply pressure to crystals. Thus doubly refracting crystals with one\nprincipal axis acquire two axes when the pressure is perpendicular in\ndirection to the principal axis.\n\nAll the phenomena peculiar to crystalline bodies are thus closely\ncorrelated with the formation of the crystal, or will almost\ncertainly be found to be so as investigation proceeds. It is upon\nempirical observation indeed that the laws of connexion are in the\nfirst place founded, but the simple hypothesis that the elasticity\nand approximation of the particles vary in the directions of the\ncrystalline axes allows of the application of deductive reasoning. The\nwhole of the phenomena are gradually being proved to be consistent with\nthis hypothesis, so that we have in this subject of crystallography\na beautiful instance of successful classification, connected with\na nearly perfect physical hypothesis. Moreover this hypothesis was\nverified experimentally as regards the mechanical vibrations of sound\nby Savart, who found that the vibrations in a plate of biaxial crystal\nindicated the existence of varying elasticity in varying directions.\n\n\n*Classification an Inverse and Tentative Operation.*\n\nIf attempts at so-called natural classification are really attempts\nat perfect induction, it follows that they are subject to the remarks\nwhich were made upon the inverse character of the inductive process,\nand upon the difficulty of every inverse operation (pp.\u00a011, 12, 122,\n&c.). There will be no royal road to the discovery of the best system,\nand it will even be impossible to lay down rules of procedure to\nassist those who are in search of a good arrangement. The only logical\nrule would be as follows:--Having given certain objects, group them\nin every way in which they can be grouped, and then observe in which\nmethod of grouping the correlation of properties is most conspicuously\nmanifested. But this method of exhaustive classification will in almost\nevery case be impracticable, owing to the immensely great number of\nmodes in which a comparatively small number of objects may be grouped\ntogether. About sixty-three elements have been classified by chemists\nin six principal groups as monad, dyad, triad, &c., elements, the\nnumbers in the classes varying from three to twenty elements. Now if we\nwere to calculate the whole number of ways in which sixty-three objects\ncan be arranged in six groups, we should find the number to be so great\nthat the life of the longest lived man would be wholly inadequate\nto enable him to go through these possible groupings. The rule of\nexhaustive arrangement, then, is absolutely impracticable. It follows\nthat mere haphazard trial cannot as a general rule give any useful\nresult. If we were to write the names of the elements in succession\nupon sixty-three cards, throw them into a ballot-box, and draw them\nout haphazard in six handfuls time after time, the probability is\nexcessively small that we should take them out in a specified order,\nthat for instance at present adopted by chemists.\n\nThe usual mode in which an investigator proceeds to form a\nclassification of a new group of objects seems to consist in\ntentatively arranging them according to their most obvious\nsimilarities. Any two objects which present a close resemblance to\neach other will be joined and formed into the rudiment of a class, the\ndefinition of which will at first include all the apparent points of\nresemblance. Other objects as they come to our notice will be gradually\nassigned to those groups with which they present the greatest number\nof points of resemblance, and the definition of a class will often\nhave to be altered in order to admit them. The early chemists could\nhardly avoid classing together the common metals, gold, silver, copper,\nlead, and iron, which present such conspicuous points of similarity as\nregards density, metallic lustre, malleability, &c. With the progress\nof discovery, however, difficulties began to present themselves in such\na grouping. Antimony, bismuth, and arsenic are distinctly metallic\nas regards lustre, density, and some chemical properties, but are\nwanting in malleability. The recently discovered tellurium presents\ngreater difficulties, for it has many of the physical properties of\nmetal, and yet all its chemical properties are analogous to those of\nsulphur and selenium, which have never been regarded as metals. Great\nchemical differences again are discovered by degrees between the\nfive metals mentioned; and the class, if it is to have any chemical\nvalidity, must be made to include other elements, having none of the\noriginal properties on which the class was founded. Hydrogen is a\ntransparent colourless gas, and the least dense of all substances; yet\nin its chemical analogies it is a metal, as suggested by Faraday[571]\nin 1838, and almost proved by Graham;[572] it must be placed in the\nsame class as silver. In this way it comes to pass that almost every\nclassification which is proposed in the early stages of a science will\nbe found to break down as the deeper similarities of the objects come\nto be detected. The most obvious points of difference will have to be\nneglected. Chlorine is a gas, bromine a liquid, and iodine a solid,\nand at first sight these might have seemed formidable circumstances to\noverlook; but in chemical analogy the substances are closely united.\nThe progress of organic chemistry, again, has yielded wholly new ideas\nof the similarities of compounds. Who, for instance, would recognise\nwithout extensive research a close similarity between glycerine and\nalcohol, or between fatty substances and ether? The class of paraffins\ncontains three substances gaseous at ordinary temperatures, several\nliquids, and some crystalline solids. It required much insight to\ndetect the analogy which exists between such apparently different\nsubstances.\n\n  [571] *Life of Faraday*, vol.\u00a0ii. p.\u00a087.\n\n  [572] *Proceedings of the Royal Society*, vol.\u00a0xvii. p.\u00a0212.\n  *Chemical and Physical Researches*, reprint, by Young and Angus\n  Smith, p.\u00a0290.\n\nThe science of chemistry now depends to a great extent on a correct\nclassification of the elements, as will be learnt by consulting\nthe able article on Classification by Professor G.\u00a0C. Foster in\nWatts\u2019 *Dictionary of Chemistry*. But the present system of chemical\nclassification was not reached until at least three previous false\nsystems had been long entertained. And though there is much reason to\nbelieve that the present mode of classification according to atomicity\nis substantially correct, errors may yet be discovered in the details\nof the grouping.\n\n\n*Symbolic Statement of the Theory of Classification.*\n\nThe theory of classification can be explained in the most complete\nand general manner, by reverting for a time to the use of the Logical\nAlphabet, which was found to be of supreme importance in Formal Logic.\nThat form expresses the necessary classification of all objects and\nideas as depending on the laws of thought, and there is no point\nconcerning the purpose and methods of classification which may not\nbe stated precisely by the use of letter combinations, the only\ninconvenience being the abstract form in which the subject is thus\nrepresented.\n\nIf we pay regard only to three qualities in which things may resemble\neach other, namely, the qualities A, B, C, there are according to\nthe laws of thought eight possible classes of objects, shown in the\nfourth column of the Logical Alphabet (p.\u00a094). If there exist objects\nbelonging to all these eight classes, it follows that the qualities A,\nB, C, are subject to no conditions except the primary laws of thought\nand things (p.\u00a05). There is then no special law of nature to discover,\nand, if we arrange the objects in any one order rather than another, it\nmust be for the purpose of showing that the combinations are logically\ncomplete.\n\nSuppose, however, that there are but four kinds of objects possessing\nthe qualities A, B, C, and that these kinds are represented by the\ncombinations ABC, A*b*C, *a*B*c*, *abc*. The order of arrangement will\nnow be of importance; for if we place them in the order\n\n  {  ABC            {  A*b*C\n  { *a*B*c*         { *abc*\n\nplacing the B\u2019s first and those which are *b*\u2019s last, we shall perhaps\noverlook the law of correlation of properties involved. But if we\narrange the combinations as follows\n\n  { ABC             { *a*B*c*\n  { A*b*C           { *abc*\n\nit becomes apparent at once that where A is, and only where A is, the\nproperty C is to be found, B being indifferently present and absent.\nThe second arrangement then would be called a natural one, as rendering\nmanifest the conditions under which the combinations exist.\n\nAs a further instance, let us suppose that eight objects are presented\nto us for classification, which exhibit combinations of the five\nproperties, A, B, C, D, E, in the following manner:--\n\n  ABC*d*E          *a*BC*d*E\n  AB*cde*          *a*B*cde*\n  A*b*CDE          *ab*CDE\n  A*bc*D*e*        *abc*D*e*\n\nThey are now classified, so that those containing A stand first, and\nthose devoid of A second, but no other property seems to be correlated\nwith A. Let us alter this arrangement and group the combinations thus:--\n\n   ABC*d*E           A*b*CDE\n   AB*cde*           A*bc*D*e*\n  *a*BC*d*E         *ab*CDE\n  *a*B*cde*         *abc*D*e*\n\nIt requires little examination to discover that in the first group B is\nalways present and D absent, whereas in the second group, B is always\nabsent and D present. This is the result which follows from a law of\nthe form B\u00a0=\u00a0d (p.\u00a0136), so that in this mode of arrangement we readily\ndiscover correlation between two letters. Altering the groups again as\nfollows:--\n\n   ABC*d*E           AB*cde*\n  *a*BC*d*E         *a*B*cde*\n   A*b*CDE           A*bc*D*e*\n  *ab*CDE           *abc*D*e*,\n\nwe discover another evident correlation between C and E. Between A and\nthe other letters, or between the two pairs of letters B, D and C, E,\nthere is no logical connexion.\n\nThis example may seem tedious, but it will be found instructive in this\nway. We are classifying only eight objects or combinations, in each\nof which only five qualities are considered. There are only two laws\nof correlation between four of those five qualities, and those laws\nare of the simplest logical character. Yet the reader would hardly\ndiscover what those laws are, and confidently assign them by rapid\ncontemplation of the combinations, as given in the first group. Several\ntentative classifications must probably be made before we can resolve\nthe question. Let us now suppose that instead of eight objects and five\nqualities, we have, say, five hundred objects and fifty qualities. If\nwe were to attempt the same method of exhaustive grouping which we\nbefore employed, we should have to arrange the five hundred objects in\nfifty different ways, before we could be sure that we had discovered\neven the simpler laws of correlation. But even the successive\ngrouping of all those possessing each of the fifty properties would\nnot necessarily give us all the laws. There might exist complicated\nrelations between several properties simultaneously, for the detection\nof which no rule of procedure whatever can be given.\n\n\n*Bifurcate Classification.*\n\nEvery system of classification ought to be formed on the principles of\nthe Logical Alphabet. Each superior class should be divided into two\ninferior classes, distinguished by the possession and non-possession\nof a single specified difference. Each of these minor classes, again,\nis divisible by any other quality whatever which can be suggested,\nand thus every classification logically consists of an infinitely\nextended series of subaltern genera and species. The classifications\nwhich we form are in reality very small fragments of those which would\ncorrectly and fully represent the relations of existing things. But if\nwe take more than four or five qualities into account, the number of\nsubdivisions grows impracticably large. Our finite minds are unable to\ntreat any complex group exhaustively, and we are obliged to simplify\nand generalise scientific problems, often at the risk of overlooking\nparticular conditions and exceptions.\n\nEvery system of classes displayed in the manner of the Logical Alphabet\nmay be called *bifurcate*, because every class branches out at each\nstep into two minor classes, existent or imaginary. It would be a\ngreat mistake to regard this arrangement as in any way a peculiar or\nspecial method; it is not only a natural and important one, but it is\nthe inevitable and only system which is logically perfect, according\nto the fundamental laws of thought. All other arrangements of classes\ncorrespond to the bifurcate arrangement, with the implication that\nsome of the minor classes are not represented among existing things.\nIf we take the genus A and divide it into the species AB and AC, we\nimply two propositions, namely that in the class A, the properties of B\nand C never occur together, and that they are never both absent; these\npropositions are logically equivalent to one, namely AB\u00a0=\u00a0A*c*. Our\nclassification is then identical with the following bifurcate one:--\n\n                       A\n                       |\n            +----------+----------+\n            |                     |\n           AB                    A*b*\n            |                     |\n     +------+------+       +------+------+\n     |             |       |             |\n  ABC = 0        AB*c*   A*b*C       A*bc* = 0\n\nIf, again, we divide the genus A into three species, AB, AC, AD, we\nare either logically in error, or else we must be understood to imply\nthat, as regards the other letters, there exist only three combinations\ncontaining A, namely AB*cd*, A*b*C*d*, and A*bc*D.\n\nThe logical necessity of bifurcate classification has been clearly and\ncorrectly stated in the *Outline of a New System of Logic* by George\nBentham, the eminent botanist, a work of which the logical value has\nbeen quite overlooked until lately. Mr. Bentham points out, in p.\u00a0113,\nthat every classification must be essentially bifurcate, and takes, as\nan example, the division of vertebrate animals into four sub-classes,\nas follows:--\n\n  Mammifera--endowed with mamm\u00e6 and lungs.\n  Birds      without mamm\u00e6 but with lungs and wings.\n  Fish       deprived of lungs.\n  Reptiles   deprived of mamm\u00e6 and wings but with lungs.\n\nWe have, then, as Mr. Bentham says, three bifid divisions, thus\nrepresented:--\n\n                        Vertebrata\n                            |\n                +-----------+-----------+\n                |                       |\n       Endowed with lungs       deprived of lungs\n                |                    = Fish.\n       +--------+----------------+\n       |                         |\n  Endowed with              deprived of\n     mamm\u00e6                     mamm\u00e6\n  = Mammifera.                   |\n                          +------+------+\n                          |             |\n                      with wings   without wings\n                       = Birds.     = Reptiles.\n\nIt is quite evident that according to the laws of thought even this\narrangement is incomplete. The sub-class mammifera must either have\nwings or be deprived of them; we must either subdivide this class, or\nassume that none of the mammifera have wings, which is, as a matter of\nfact, the case, the wings of bats not being true wings in the meaning\nof the term as applied to birds. Fish, again, ought to be considered\nwith regard to the possession of mamm\u00e6 and wings; and in leaving them\nundivided we really imply that they never have mamm\u00e6 nor wings, the\nwings of the flying-fish, again, being no exception. If we resort to\nthe use of our letters and define them as follows--\n\n  A = vertebrata,\n  B = having lungs,\n  C = having mamm\u00e6,\n  D = having wings,\n\nthen there are four existent classes of vertebrata which appear to be\nthus described--\n\n  ABC     AB*c*D     AB*cd*     A*b*.\n\nBut in reality the combinations are implied to be\n\n  ABC*d* = Mammifera,\n  AB*c*D = Birds,\n  AB*cd* = Reptiles,\n  A*bcd* = Fish,\n\nand we imply at the same time that the other four conceivable\ncombinations containing B, C, or D, namely ABCD, A*b*CD, A*b*C*d*, and\nA*bc*D, do not exist in nature.\n\nMr. Bentham points out[573] that it is really this method of\nclassification which was employed by Lamarck and De Candolle in their\nso-called analytical arrangement of the French Flora. He gives as an\nexample a table of the principal classes of De Candolle\u2019s system, as\nalso a bifurcate arrangement of animals after the method proposed\nby Dum\u00e9ril in his *Zoologie Analytique*, this naturalist being\ndistinguished by his clear perception of the logical importance of the\nmethod. A bifurcate classification of the animal kingdom may also be\nfound in Professor Reay Greene\u2019s *Manual of the C\u0153lenterata*, p.\u00a018.\n\n  [573] *Essai sur la Nomenclature et la Classification*, Paris, 1823,\n  pp. 107, 108.\n\nThe bifurcate form of classification seems to be needless when the\nquality according to which we classify any group of things admits\nof numerical discrimination. It would seem absurd to arrange things\naccording as they have one degree of the quality or not one degree,\ntwo degrees or not two degrees, and so on. The elements are classified\naccording as the atom of each saturates one, two, three, or more atoms\nof a monad element, such as chlorine, and they are called accordingly\nmonad, dyad, triad, tetrad elements, and so on. It would be useless to\napply the bifid arrangement, thus:--\n\n       Element\n          |\n    +-----+-------+\n    |             |\n  Monad       not-Monad\n                  |\n        +---------+---------+\n        |                   |\n      Dyad              not-Dyad\n                            |\n                  +---------+---------+\n                  |                   |\n                Triad             not-Triad\n                                      |\n                            +---------+--------+\n                            |                  |\n                         Tetrad            not-Tetrad.\n\nThe reason of this is that, by the nature of number (p.\u00a0157) every\nnumber is logically discriminated from every other number. There can\nthus be no logical confusion in a numerical arrangement, and the series\nof numbers indefinitely extended is also exhaustive. Every thing\nadmitting of a quality expressible in numbers must find its place\nsomewhere in the series of numbers. The chords in music correspond to\nthe simpler numerical ratios and must admit of complete exhaustive\nclassification in respect to the complexity of the ratios forming\nthem. Plane rectilinear figures may be classified according to the\nnumbers of their sides, as triangles, quadrilateral figures, pentagons,\nhexagons, heptagons, &c. The bifurcate arrangement is not false when\napplied to such series of objects; it is even necessarily involved in\nthe arrangement which we do apply, so that its formal statement is\nneedless and tedious. The same may be said of the division of portions\nof space. Reid and Kames endeavoured to cast ridicule on the bifurcate\narrangement[574] by proposing to classify the parts of England into\nMiddlesex and what is not Middlesex, dividing the latter again into\nKent and what is not Kent, Sussex and what is not Sussex; and so on.\nThis is so far, however, from being an absurd proceeding that it is\nrequisite to assure us that we have made an exhaustive enumeration of\nthe parts of England.\n\n  [574] George Bentham, *Outline of a New System of Logic*, p.\u00a0115.\n\n\n*The Five Predicables.*\n\nAs a rule it is highly desirable to consign to oblivion the ancient\nlogical names and expressions, which have infested the science for\nmany centuries past. If logic is ever to be a useful and progressive\nscience, logicians must distinguish between logic and the history of\nlogic. As in the case of any other science it may be desirable to\nexamine the course of thought by which logic has, before or since the\ntime of Aristotle, been brought to its present state; the history of a\nscience is always instructive as giving instances of the mode in which\ndiscoveries take place. But at the same time we ought carefully to\ndisencumber the statement of the science itself of all names and other\nvestiges of antiquity which are not actually useful at the present day.\n\nAmong the ancient expressions which may well be excepted from such\nconsiderations and retained in use, are the \u201cFive Words\u201d or \u201cFive\nPredicables\u201d which were described by Porphyry in his introduction to\nAristotle\u2019s Organum. Two of them, *Genus* and *Species*, are the most\nvenerable names in philosophy, having probably been first employed\nin their present logical meanings by Socrates. In the present day it\nrequires some mental effort, as remarked by Grote, to see anything\nimportant in the invention of notions now so familiar as those of Genus\nand Species. But in reality the introduction of such terms showed the\nrise of the first germs of logic and scientific method; it showed that\nmen were beginning to analyse their processes of thought.\n\nThe Five Predicables are Genus, Species, Difference, Property, and\nAccident, or in the original Greek, \u03b3\u03ad\u03bd\u03bf\u03c2, \u03b5\u1f36\u03b4\u03bf\u03c2, \u03b4\u03b9\u03b1\u03c6\u03bf\u03c1\u03ac, \u1f34\u03b4\u03b9\u03bf\u03bd,\n\u03c3\u03c5\u03bc\u03b2\u03b5\u03b2\u03b7\u03ba\u03cc\u03c2. Of these, Genus may be taken to mean any class of objects\nwhich is regarded as broken up into two minor classes, which form\nSpecies of it. The genus is defined by a certain number of qualities or\ncircumstances which belong to all objects included in the class, and\nwhich are sufficient to mark out these objects from all others which\nwe do not intend to include. Interpreted as regards intension, then,\nthe genus is a group of qualities; interpreted as regards extension, it\nis a group of objects possessing those qualities. If another quality\nbe taken into account which is possessed by some of the objects and\nnot by the others, this quality becomes a difference which divides\nthe genus into two species. We may interpret the species either in\nintension or extension; in the former respect it is more than the genus\nas containing one more quality, the difference: in the latter respect\nit is less than the genus as containing only a portion of the group\nconstituting the genus. We may say, then, with Aristotle, that in one\nsense the genus is in the species, namely in intension, and in another\nsense the species is in the genus, namely in extension. The difference,\nit is evident, can be interpreted in intension only.\n\nA Property is a quality which belongs to the whole of a class, but does\nnot enter into the definition of that class. A generic property belongs\nto every individual object contained in the genus. It is a property\nof the genus parallelogram that the opposite angles are equal. If we\nregard a rectangle as a species of parallelogram, the difference being\nthat *one* angle is a right angle, it follows as a specific property\nthat all the angles are right angles. Though a property in the strict\nlogical sense must belong to each of the objects included in the class\nof which it is a property, it may or may not belong to other objects.\nThe property of having the opposite angles equal may belong to many\nfigures besides parallelograms, for instance, regular hexagons. It is a\nproperty of the circle that all triangles constructed upon the diameter\nwith the apex upon the circumference are right-angled triangles, and\n*vice vers\u00e2*, all curves of which this is true must be circles. A\nproperty which thus belongs to the whole of a class and only to that\nclass, corresponds to the \u1f34\u03b4\u03b9\u03bf\u03bd of Aristotle and Porphyry; we might\nconveniently call it *a peculiar property*. Every such property enables\nus to make a statement in the form of a simple identity (p.\u00a037). Thus\nwe know it to be a peculiar property of the circle that for a given\nlength of perimeter it encloses a greater area than any other possible\ncurve; hence we may say--\n\n  Curve of equal curvature = curve of greatest area.\n\nIt is a peculiar property of equilateral triangles that they are\nequiangular, and *vice vers\u00e2*, it is a peculiar property of equiangular\ntriangles that they are equilateral. It is a property of crystals\nof the regular system that they are devoid of the power of double\nrefraction, but this is not a property peculiar to them, because\nliquids and gases are devoid of the same property.\n\nAn Accident, the fifth and last of the Predicables, is any quality\nwhich may or may not belong to certain objects, and which has no\nconnexion with the classification adopted. The particular size of\na crystal does not in the slightest degree affect the form of the\ncrystal, nor does the manner in which it is grouped with other\ncrystals; these, then, are accidents as regards a crystallographic\nclassification. With respect to the chemical composition of a\nsubstance, again, it is an accident whether the substance be\ncrystallised or not, or whether it be organised or not. As regards\nbotanical classification the absolute size of a plant is an accident.\nThus we see that a logical accident is any quality or circumstance\nwhich is not known to be correlated with those qualities or\ncircumstances forming the definition of the species.\n\nThe meanings of the Predicables can be clearly explained by our\nsymbols. Let A be any definite group of qualities and B another quality\nor group of qualities; then A will constitute a genus, and AB, A*b*\nwill be species of it, B being the difference. Let C, D and E be other\nqualities or groups of qualities, and on examining the combinations in\nwhich A, B, C, D, E occur let them be as follows:--\n\n  ABCDE          A*b*C*d*E\n  ABCD*e*        A*b*C*de*.\n\nHere we see that wherever A is we also find C, so that C is a generic\nproperty; D occurs always with B, so that it constitutes a specific\nproperty, while E is indifferently present and absent, so as not to be\nrelated to any other letter; it represents, therefore, an accident. It\nwill now be seen that the Logical Alphabet represents an interminable\nseries of subordinate genera and species; it is but a concise symbolic\nstatement of what was involved in the ancient doctrine of the\nPredicables.\n\n\n*Summum Genus and Infima Species.*\n\nAs a genus means any class whatever which is regarded as composed\nof minor classes or species, it follows that the same class will be\na genus in one point of view and a species in another. Metal is a\ngenus as regards alkaline metal, a species as regards element, and\nany extensive system of classes consists of a series of subordinate,\nor as they are technically called, *subaltern* genera and species.\nThe question, however, arises, whether such a chain of classes\nhas a definite termination at either end. The doctrine of the old\nlogicians was to the effect that it terminated upwards in a *genus\ngeneralissimum* or *summum genus*, which was not a species of any\nwider class. Some very general notion, such as substance, object, or\nthing, was supposed to be so comprehensive as to include all thinkable\nobjects, and for all practical purposes this might be so. But as I\nhave already explained (p.\u00a074), we cannot really think of any object\nor class without thereby separating it from what is not that object or\nclass. All thinking is relative, and implies discrimination, so that\nevery class and every logical notion must have its negative. If so,\nthere is no such thing as a *summum genus*; for we cannot frame the\nrequisite notion of a class forming it without implying the existence\nof another class discriminated from it; add this new negative class to\nthe supposed *summum genus*, and we form a still higher genus, which is\nabsurd.\n\nAlthough there is no absolute summum genus, nevertheless relatively to\nany branch of knowledge or any particular argument, there is always\nsome class or notion which bounds our horizon as it were. The chemist\nrestricts his view to material substances and the forces manifested\nin them; the mathematician extends his view so as to comprehend all\nnotions capable of numerical discrimination. The biologist, on the\nother hand, has a narrower sphere containing only organised bodies, and\nof these the botanist and the zoologist take parts. In other subjects\nthere may be a still narrower summum genus, as when the lawyer regards\nonly reasoning beings of his own country together with their property.\n\nIn the description of the Logical Alphabet it was pointed out (p.\u00a093)\nthat every series of combinations is really the development of a\nsingle class, denoted by X, which letter was accordingly placed in the\nfirst column of the table on p. 94. This is the formal acknowledgment\nof the principle clearly stated by De Morgan, that all reasoning\nproceeds within an assumed summum genus. But at the same time the fact\nthat X as a logical term must have its negative *x*, shows that it\ncannot be an absolute summum genus.\n\nThere arises, again, the question whether there be any such thing as\nan *infima species*, which cannot be divided into minor species. The\nancient logicians were of opinion that there always was some assignable\nclass which could only be divided into individuals, but this doctrine\nappears to be theoretically incorrect, as Mr. George Bentham long ago\nstated.[575] We may put an arbitrary limit to the subdivision of our\nclasses at any point convenient to our purpose. The crystallographer\nwould not generally treat as different species crystalline forms which\ndiffer only in the degree of development of the faces. The naturalist\noverlooks innumerable slight differences between animals which he\nrefers to the same species. But in a strictly logical point of view\nclassification might be carried on as long as there is a difference,\nhowever minute, between two objects, and we might thus go on until we\narrive at individual objects which are numerically distinct in the\nlogical sense attributed to that expression in the chapter upon Number.\nEither, then, we must call the individual the *infima species* or allow\nthat there is no such thing at all.\n\n  [575] *Outline of a New System of Logic*, 1827, p.\u00a0117.\n\n\n*The Tree of Porphyry.*\n\nBoth Aristotle and Plato were acquainted with the value of bifurcate\nclassification, which they occasionally employed in an explicit manner.\nIt is impossible too that Aristotle should state the laws of thought,\nand employ the predicables without implicitly recognising the logical\nnecessity of that method. It is, however, in Porphyry\u2019s remarkable\nand in many respects excellent *Introduction to the Categories of\nAristotle* that we find the most distinct account of it. Porphyry not\nonly fully and accurately describes the Predicables, but incidentally\nintroduces an example for illustrating those predicables, which\nconstitutes a good specimen of bifurcate classification. Translating\nhis words[576] freely we may say that he takes Substance as the genus\nto be divided, under which are successively placed as Species--Body,\nAnimated Body, Animal, Rational Animal, and Man. Under Man, again,\ncome Socrates, Plato, and other particular men. Now of these notions\nSubstance is the genus generalissimum, and is a genus only, not a\nspecies. Man, on the other hand, is the species specialissima (infima\nspecies), and is a species only, not a genus. Body is a species of\nsubstance, but a genus of animated body, which, again, is a species of\nbody but a genus of animal. Animal is a species of animated body, but\na genus of rational animal, which, again, is a species of animal, but\na genus of man. Finally, man is a species of rational animal, but is a\nspecies merely and not a genus, being divisible only into particular\nmen.\n\n  [576] *Porphyrii Isagoge*, Caput ii. 24.\n\nPorphyry proceeds at some length to employ his example in further\nillustration of the predicables. We do not find in Porphyry\u2019s own\nwork any scheme or diagram exhibiting this curious specimen of\nclassification, but some of the earlier commentators and epitome\nwriters drew what has long been called the Tree of Porphyry. This\ndiagram, which may be found in most elementary works on Logic,[577] is\nalso called the Ramean Tree, because Ramus insisted much upon the value\nof Dichotomy. With the exception of Jeremy Bentham[578] and George\nBentham, hardly any modern logicians have shown an appreciation of the\nvalue of bifurcate classification. The latter author has treated the\nsubject, both in his *Outline of a New System of Logic* (pp.\u00a0105\u2013118),\nand in his earlier work entitled *Essai sur la Nomenclature et la\nClassification des Principales Branches d\u2019Art-et-Science* (Paris,\n1823), which consists of a free translation or improved version of his\nuncle\u2019s Essay on Classification in the *Chrestomathia*. Some interest\nattaches to the history of the Tree of Porphyry and Ramus, because\nit is the prototype of the Logical Alphabet which lies at the basis\nof logical method. Jeremy Bentham speaks truly of \u201cthe matchless\nbeauty of the Ramean Tree.\u201d After fully showing its logical value as\nan exhaustive method of classification, and refuting the objections\nof Reid and Kames, on a wrong ground, as I think, he proceeds to\ninquire to what length it may be carried. He correctly points out two\nobjections to the extensive use of bifid arrangements, (1) that they\nsoon become impracticably extensive and unwieldy, and (2) that they\nare uneconomical. In his day the recorded number of different species\nof plants was 40,000, and he leaves the reader to estimate the immense\nnumber of branches and the enormous area of a bifurcate table which\nshould exhibit all these species in one scheme. He also points out the\napparent loss of labour in making any large bifurcate classification;\nbut this he considers to be fully recompensed by the logical value of\nthe result, and the logical training acquired in its execution. Jeremy\nBentham, then, fully recognises the value of the Logical Alphabet under\nanother name, though he apprehends also the limit to its use placed by\nthe finiteness of our mental and manual powers.\n\n  [577] Jevons, *Elementary Lessons in Logic*, p.\u00a0104.\n\n  [578] *Chrestomathia; being a Collection of Papers, &c.* London,\n  1816, Appendix V.\n\n\n*Does Abstraction imply Generalisation?*\n\nBefore we can acquire a sound comprehension of the subject of\nclassification we must answer the very difficult question whether\nlogical abstraction does or does not imply generalisation. It comes to\nexactly the same thing if we ask whether a species may be coextensive\nwith its genus, or whether, on the other hand, the genus must contain\nmore than the species. To abstract logically is (p.\u00a027), to overlook or\nwithdraw our notice from some point of difference. Whenever we form a\nclass we abstract, for the time being, the differences of the objects\nso united in respect of some common quality. If we class together a\ngreat number of objects as dwelling-houses, we overlook the fact that\nsome dwelling-houses are constructed of stone, others of brick, wood,\niron, &c. Often at least the abstraction of a circumstance increases\nthe number of objects included under a class according to the law of\nthe inverse relation of the quantities of extension and intension\n(p.\u00a026). Dwelling-house is a wider term than brick-dwelling-house.\nHouse is more general than dwelling-house. But the question before\nus is, whether abstraction *always* increases the number of objects\nincluded in a class, which amounts to asking whether the law of\nthe inverse relation of logical quantities is *always* true. The\ninterest of the question partly arises from the fact, that so high\na philosophical authority as Mr. Herbert Spencer has denied that\ngeneralisation is implied in abstraction,[579] making this doctrine\nthe ground for rejecting previous methods of classifying the sciences,\nand for forming an ingenious but peculiar method of his own. The\nquestion is also a fundamental one of the highest logical importance,\nand involves subtle difficulties which have made me long hesitate in\nforming a decisive opinion.\n\n  [579] *The Classification of the Sciences*, &c., 3rd edit. p.\u00a07.\n  *Essays: Scientific, Political, and Speculative*, vol.\u00a0iii. p.\u00a013.\n\nLet us attempt to answer the question by examination of a few examples.\nCompare the two classes *gun* and *iron gun*. It is certain that there\nare many guns which are not made of iron, so that abstraction of the\ncircumstance \u201cmade of iron\u201d increases the extent of the notion. Next\ncompare *gun* and *metallic gun*. All guns made at the present day\nconsist of metal, so that the two notions seem to be coextensive;\nbut guns were at first made of pieces of wood bound together like\na tub, and as the logical term gun takes no account of time, it\nmust include all guns that have ever existed. Here again extension\nincreases as intension decreases. Compare once more \u201csteam-locomotive\nengine\u201d and \u201clocomotive engine.\u201d In the present day, as far as I am\naware, all locomotives are worked by steam, so that the omission of\nthat qualification might seem not to widen the term; but it is quite\npossible that in some future age a different motive power may be used\nin locomotives; and as there is no limitation of time in the use of\nlogical terms, we must certainly assume that there is a class of\nlocomotives not worked by steam, as well as a class that is worked by\nsteam. When the natural class of Euphorbiace\u00e6 was originally formed,\nall the plants known to belong to it were devoid of corollas; it\nwould have seemed therefore that the two classes \u201cEuphorbiace\u00e6,\u201d and\n\u201cEuphorbiace\u00e6 devoid of Corollas,\u201d were of equal extent. Subsequently\na number of plants plainly belonging to the same class were found in\ntropical countries, and they possessed bright coloured corollas.\nNaturalists believe with the utmost confidence that \u201cRuminants\u201d and\n\u201cRuminants with cleft feet\u201d are identical terms, because no ruminant\nhas yet been discovered without cleft feet. But we can see no\nimpossibility in the conjunction of rumination with uncleft feet, and\nit would be too great an assumption to say that we are certain that an\nexample of it will never be met with. Instances can be quoted, without\nend, of objects being ultimately discovered combining properties which\nhad never before been seen together. In the animal kingdom the Black\nSwan, the Ornithorhynchus Paradoxus, and more recently the singular\nfish called Ceratodus Forsteri, all discovered in Australia, have\nunited characters never previously known to coexist. At the present\ntime deep-sea dredging is bringing to light many animals of an\nunprecedented nature. Singular exceptional discoveries may certainly\noccur in other branches of science. When Davy first discovered metallic\npotassium, it was a well established empirical law that all metallic\nsubstances possessed a high specific gravity, the least dense of the\nmetals then known being zinc, of which the specific gravity is 7\u00b71. Yet\nto the surprise of chemists, potassium was found to be an undoubted\nmetal of less density than water, its specific gravity being 0\u00b7865.\n\nIt is hardly requisite to prove by further examples that our knowledge\nof nature is incomplete, so that we cannot safely assume the\nnon-existence of new combinations. Logically speaking, we ought to\nleave a place open for animals which ruminate but are without cleft\nfeet, and for every possible intermediate form of animal, plant, or\nmineral. A purely logical classification must take account not only of\nwhat certainly does exist, but of what may in after ages be found to\nexist.\n\nI will go a step further, and say that we must have places in our\nscientific classifications for purely imaginary existences. A large\nproportion of the mathematical functions which are conceivable have no\napplication to the circumstances of this world. Physicists certainly do\ninvestigate the nature and consequences of forces which nowhere exist.\nNewton\u2019s *Principia* is full of such investigations. In one chapter of\nhis *M\u00e9canique C\u00e9leste* Laplace indulges in a remarkable speculation\nas to what the laws of motion would have been if momentum, instead of\nvarying simply as the velocity, had been a more complicated function\nof it. I have already mentioned (p.\u00a0223) that Airy contemplated the\nexistence of a world in which the laws of force should be such that\na perpetual motion would be possible, and the Law of Conservation of\nEnergy would not hold true.\n\nThought is not bound down to the limits of what is materially existent,\nbut is circumscribed only by those Fundamental Laws of Identity,\nContradiction and Duality, which were laid down at the outset. This\nis the point at which I should differ from Mr. Spencer. He appears\nto suppose that a classification is complete if it has a place for\nevery existing object, and this may perhaps seem to be practically\nsufficient; but it is subject to two profound objections. Firstly, we\ndo not know all that exists, and therefore in limiting our classes we\nare erroneously omitting multitudes of objects of unknown form and\nnature which may exist either on this earth or in other parts of space.\nSecondly, as I have explained, the powers of thought are not limited by\nmaterial existences, and we may, or, for some purposes, must imagine\nobjects which probably do not exist, and if we imagine them we ought to\nfind places for them in the classifications of science.\n\nThe chief difficulty of this subject, however, consists in the fact\nthat mathematical or other certain laws may entirely forbid the\nexistence of some combinations. The circle may be defined as a plane\ncurve of equal curvature, and it is a property of the circle that it\ncontains the greatest area within the least possible perimeter. May we\nthen contemplate mentally a circle not a figure of greatest possible\narea? Or, to take a still simpler example, a parallelogram possesses\nthe property of having the opposite angles equal. May we then mentally\ndivide parallelograms into two classes according as they do or do\nnot have their opposite angles equal? It might seem absurd to do so,\nbecause we know that one of the two species of parallelogram would be\nnon-existent. But, then, unless the student had previously contemplated\nthe existence of both species as possible, what is the meaning of the\nthirty-fourth proposition of Euclid\u2019s first book? We cannot deny or\ndisprove the existence of a certain combination without thereby in a\ncertain way recognising that combination as an object of thought.\n\nThe conclusion at which I arrive is in opposition to that of Mr.\nSpencer. I think that whenever we abstract a quality or circumstance we\ndo generalise or widen the notion from which we abstract. Whatever the\nterms A, B, and C may be, I hold that in strict logic AB is mentally\na wider term than ABC, because AB includes the two species ABC and\nAB*c*. The term A is wider still, for it includes the four species\nABC, AB*c*, A*b*C, A*bc*. The Logical Alphabet, in short, is the only\nlimit of the classes of objects which we must contemplate in a purely\nlogical point of view. Whatever notions be brought before us, we must\nmentally combine them in all the ways sanctioned by the laws of thought\nand exhibited in the Logical Alphabet, and it is a matter for after\nconsideration to determine how many of these combinations exist in\noutward nature, or how many are actually forbidden by the conditions of\nspace. A classification is essentially a mental, not a material thing.\n\n\n*Discovery of Marks or Characteristics.*\n\nAlthough the chief purpose of classification is to disclose the\ndeepest and most general resemblances of the objects classified, yet\nthe practical value of a system will depend partly upon the ease with\nwhich we can refer an object to its proper class, and thus infer\nconcerning it all that is known generally of that class. This operation\nof discovering to which class of a system a certain specimen or case\nbelongs, is generally called *Diagnosis*, a technical term familiarly\nused by physicians, who constantly require to diagnose or determine\nthe nature of the disease from which a patient is suffering. Now every\nclass is defined by certain specified qualities or circumstances, the\nwhole of which are present in every object contained in the class,\nand *not all present* in any object excluded from it. These defining\ncircumstances ought to consist of the deepest and most important\ncircumstances, by which we vaguely mean those probably forming the\nconditions with which the minor circumstances are correlated. But it\nwill often happen that the so-called important points of an object\nare not those which can most readily be observed. Thus the two great\nclasses of phanerogamous plants are defined respectively by the\npossession of two cotyledons or seed-leaves, and one cotyledon. But\nwhen a plant comes to our notice and we want to refer it to the right\nclass, it will often happen that we have no seed at all to examine, in\norder to discover whether there be one seed-leaf or two in the germ.\nEven if we have a seed it will often be small, and a careful dissection\nunder the microscope will be requisite to ascertain the number of\ncotyledons. Occasionally the examination of the germ would mislead us,\nfor the cotyledons may be obsolete, as in Cuscuta, or united together,\nas in Clintonia. Botanists therefore seldom actually refer to the\nseed for such information. Certain other characters of a plant are\ncorrelated with the number of seed-leaves; thus monocotyledonous plants\nalmost always possess leaves with parallel veins like those of grass,\nwhile dicotyledonous plants have leaves with reticulated veins like\nthose of an oak leaf. In monocotyledonous plants, too, the parts of the\nflower are most often three or some multiple of three in number, while\nin dicotyledonous plants the numbers four and five and their multiples\nprevail. Botanists, therefore, by a glance at the leaves and flowers\ncan almost certainly refer a plant to its right class, and can infer\nnot only the number of cotyledons which would be found in the seed\nor young plant, but also the structure of the stem and other general\ncharacters.\n\nAny conspicuous and easily discriminated property which we thus\nselect for the purpose of deciding to which class an object belongs,\nmay be called a *characteristic*. The logical conditions of a good\ncharacteristic mark are very simple, namely, that it should be\npossessed by all objects entering into a certain class, and by none\nothers. Every characteristic should enable us to assert a simple\nidentity; if A is a characteristic, and B, viewed intensively, the\nclass of objects of which it is the mark, then A\u00a0=\u00a0B ought to be\ntrue. The characteristic may consist either of a single quality or\ncircumstance, or of a group of such, provided that they all be constant\nand easily detected. Thus in the classification of mammals the teeth\nare of the greatest assistance, not because a slight variation in the\nnumber and form of the teeth is of importance in the general economy\nof the animal, but because such variations are proved by empirical\nobservation to coincide with most important differences in the general\naffinities. It is found that the minor classes and genera of mammals\ncan be discriminated accurately by their teeth, especially by the\nforemost molars and the hindmost pre-molars. Some teeth, indeed, are\noccasionally missing, so that zoologists prefer to trust to those\ncharacteristic teeth which are most constant,[580] and to infer from\nthem not only the arrangement of the other teeth, but the whole\nconformation of the animal.\n\n  [580] Owen, *Essay on the Classification and Geographical\n  Distribution of the Mammalia*, p.\u00a020.\n\nIt is a very difficult matter to mark out a boundary-line between the\nanimal and vegetable kingdoms, and it may even be doubted whether\na rigorous boundary can be established. The most fundamental and\nimportant difference of a vegetable as compared with an animal\nsubstance probably consists in the absence of nitrogen from the\nconstituent membranes. Supposing this to be the case, the difficulty\narises that in examining minute organisms we cannot ascertain directly\nwhether they contain nitrogen or not. Some minor but easily detected\ncircumstance is therefore needed to discriminate between animals and\nvegetables, and this is furnished to some extent by the fact that the\nproduction of starch granules is restricted to the vegetable kingdom.\nThus the Desmidiace\u00e6 may be safely assigned to the vegetable kingdom,\nbecause they contain starch. But we must not employ this characteristic\nnegatively; the Diatomace\u00e6 are probably vegetables, though they do not\nproduce starch.\n\n\n*Diagnostic Systems of Classification.*\n\nWe have seen that diagnosis is the process of discovering the place in\nany system of classes, to which an object has been referred by some\nprevious investigation, the object being to avail ourselves of the\ninformation relating to such an object which has been accumulated and\nrecorded. It is obvious that this is a matter of great importance,\nfor, unless we can recognise, from time to time, objects or substances\nwhich have been investigated, recorded discoveries would lose their\nvalue. Even a single investigator must have means of recording and\nsystematising his observations of any large groups of objects like the\nvegetable and animal kingdoms.\n\nNow whenever a class has been properly formed, a definition must have\nbeen laid down, stating the qualities and circumstances possessed by\nall the objects which are intended to be included in the class, and\nnot possessed *completely* by any other objects. Diagnosis, therefore,\nconsists in comparing the qualities of a certain object with the\ndefinitions of a series of classes; the absence in the object of any\none quality stated in the definition excludes it from the class thus\ndefined; whereas, if we find every point of a definition exactly\nfulfilled in the specimen, we may at once assign it to the class in\nquestion. It is of course by no means certain that everything which has\nbeen affirmed of a class is true of all objects afterwards referred\nto the class; for this would be a case of imperfect inference, which\nis never more than matter of probability. A definition can only make\nknown a finite number of the qualities of an object, and it always\nremains possible that objects agreeing in those assigned qualities will\ndiffer in others. *An individual cannot be defined*, and can only be\nmade known by the exhibition of the individual itself, or by a material\nspecimen exactly representing it. But this and other questions relating\nto definition must be treated when I am able to take up the subject of\nlanguage in another work.\n\nDiagnostic systems of classification should, as a general rule, be\narranged on the bifurcate method explicitly. Any quality may be chosen\nwhich divides the whole group of objects into two distinct parts,\nand each part may be sub-divided successively by any prominent and\nwell-marked circumstance which is present in a large part of the genus\nand not in the other. To refer an object to its proper place in such an\narrangement we have only to note whether it does or does not possess\nthe successive critical differenti\u00e6. Dana devised a classification of\nthis kind[581] by which to refer a crystal to its place in the series\nof six or seven classes already described. If a crystal has all its\nedges modified alike or the angles replaced by three or six similar\nplanes, it belongs to the monometric system; if not, we observe\nwhether the number of similar planes at the extremity of the crystal\nis three or some multiple of three, in which case it is a crystal\nof the hexagonal system; and so we proceed with further successive\ndiscriminations. To ascertain the name of a mineral by examination with\nthe blow-pipe, an arrangement more or less evidently on the bifurcate\nplan, has been laid down by Von Kobell.[582] Minerals are divided\naccording as they possess or do not possess metallic lustre; as they\nare fusible or not fusible, according as they do or do not on charcoal\ngive a metallic bead, and so on.\n\n  [581] Dana\u2019s *Mineralogy*, vol.\u00a0i. p.\u00a0123; quoted in Watts\u2019\n  *Dictionary of Chemistry*, vol.\u00a0ii. p.\u00a0166.\n\n  [582] *Instructions for the Discrimination of Minerals by Simple\n  Chemical Experiments*, by Franz von Kobell, translated from the\n  German by R.\u00a0C. Campbell. Glasgow, 1841.\n\nPerhaps the best example to be found of an arrangement devised simply\nfor the purpose of diagnosis, is Mr. George Bentham\u2019s *Analytical Key\nto the Natural Orders and Anomalous Genera of the British Flora*, given\nin his *Handbook of the British Flora*.[583] In this scheme, the great\ncomposite family of plants, together with the closely approximate\ngenus Jasione, are first separated from all other flowering plants\nby the compound character of their flowers. The remaining plants are\nsub-divided according as the perianth is double or single. Since no\nplants are yet known in which the perianth can be said to have three\nor more distinct rings, this division becomes practically the same as\none into double and not-double. Flowers with a double perianth are\nnext discriminated according as the corolla does or does not consist\nof one piece; according as the ovary is free or not free; as it is\nsimple or not simple; as the corolla is regular or irregular; and so\non. On looking over this arrangement, it will be found that numerical\ndiscriminations often occur, the numbers of petals, stamens, capsules,\nor other parts being the criteria, in which cases, as already explained\n(p.\u00a0697), the actual exhibition of the bifid division would be tedious.\n\n  [583] Edition of 1866, p. lxiii.\n\nLinn\u00e6us appears to have been perfectly acquainted with the nature and\nuses of diagnostic classification, which he describes under the name\nof Synopsis, saying:[584]--\u201cSynopsis tradit Divisiones arbitrarias,\nlongiores aut breviores, plures aut pauciores: a Botanicis in genere\nnon agnoscenda. Synopsis est dichotomia arbitraria, qu\u00e6 instar vi\u00e6 ad\nBotanicem ducit. Limites autem non determinat.\u201d\n\n  [584] *Philosophia Botanica* (1770), \u00a7 154, p.\u00a098.\n\nThe rules and tables drawn out by chemists to facilitate the discovery\nof the nature of a substance in qualitative analysis are usually\narranged on the bifurcate method, and form excellent examples of\ndiagnostic classification, the qualities of the substances produced\nin testing being in most cases merely characteristic properties of\nlittle importance in other respects. The chemist does not detect\npotassium by reducing it to the state of metallic potassium, and\nthen observing whether it has all the principal qualities belonging\nto potassium. He selects from among the whole number of compounds of\npotassium that salt, namely the compound of platinum tetra-chloride,\nand potassium chloride, which has the most distinctive appearance,\nas it is comparatively insoluble and produces a peculiar yellow and\nhighly crystalline precipitate. Accordingly, potassium is present\nwhenever this precipitate can be produced by adding platinum chloride\nto a solution. The fine purple or violet colour which potassium\nsalts communicate to the blowpipe flame, had long been used as a\ncharacteristic mark. Some other elements were readily detected by the\ncolouring of the blowpipe flame, barium giving a pale yellowish green,\nand salts of strontium a bright red. By the use of the spectroscope\nthe coloured light given off by an incandescent vapour is made to give\nperfectly characteristic marks of the elements contained in the vapour.\n\nDiagnosis seems to be identical with the process termed by the ancient\nlogicians *abscissio infiniti*, the cutting off of the infinite or\nnegative part of a genus when we discover by observation that an\nobject possesses a particular difference. At every step in a bifurcate\ndivision, some objects possessing the difference will fall into\nthe affirmative part or species; all the remaining objects in the\nworld fall into the negative part, which will be infinite in extent.\nDiagnosis consists in the successive rejection from further notice of\nthose infinite classes with which the specimen in question does not\nagree.\n\n\n*Index Classifications.*\n\nUnder classification we may include all arrangements of objects or\nnames, which we make for saving labour in the discovery of an object.\nEven alphabetical indices are real classifications. No such arrangement\ncan be of use unless it involves some correlation of circumstances, so\nthat knowing one thing we learn another. If we merely arrange letters\nin the pigeon-holes of a secretaire we establish a correlation, for all\nletters in the first hole will be written by persons, for instance,\nwhose names begin with A, and so on. Knowing then the initial letter of\nthe writer\u2019s name, we know also the place of the letter, and the labour\nof search is thus reduced to one twenty-sixth part of what it would be\nwithout arrangement.\n\nNow the purpose of a catalogue is to discover the place in which an\nobject is to be found; but the art of cataloguing involves logical\nconsiderations of some importance. We want to establish a correlation\nbetween the place of an object and some circumstance about the object\nwhich shall enable us readily to refer to it; this circumstance\ntherefore should be that which will most readily dwell in the memory\nof the searcher. A piece of poetry will be best remembered by the\nfirst line of the piece, and the name of the author will be the next\nmost definite circumstance; a catalogue of poetry should therefore be\narranged alphabetically according to the first word of the piece, or\nthe name of the author, or, still better, in both ways. It would be\nimpossible to arrange poems according to their subjects, so vague and\nmixed are these found to be when the attempt is made.\n\nIt is a matter of considerable literary importance to decide upon the\nbest mode of cataloguing books, so that any required book in a library\nshall be most readily found. Books may be classified in a great number\nof ways, according to subject, language, date, or place of publication,\nsize, the initial words of the text or title-page, or colophon, the\nauthor\u2019s name, the publisher\u2019s name, the printer\u2019s name, the character\nof the type, and so on. Every one of these modes of arrangement may be\nuseful, for we may happen to remember one circumstance about a book\nwhen we have forgotten all others; but as we cannot usually go to the\nexpense of forming more than two or three indices, we must select\nthose circumstances which will lead to the discovery of a book most\nfrequently. Many of the criteria mentioned are evidently inapplicable.\n\nThe language in which a book is written is definite enough, provided\nthat the whole book is written in the same language; but it is obvious\nthat language gives no means for the subdivision and arrangement of\nthe literature of any one people. Classification by subjects would be\nan exceedingly useful method if it were practicable, but experience\nshows it to be a logical absurdity. It is a very difficult matter to\nclassify the sciences, so complicated are the relations between them.\nBut with books the complication is vastly greater, since the same book\nmay treat of different sciences, or it may discuss a problem involving\nmany branches of knowledge. A good account of the steam-engine will be\nantiquarian, so far as it traces out the earliest efforts at discovery;\npurely scientific, as regards the principles of thermodynamics\ninvolved; technical, as regards the mechanical means of applying\nthose principles; economical, as regards the industrial results of\nthe invention; biographical, as regards the lives of the inventors.\nA history of Westminster Abbey might belong either to the history of\narchitecture, the history of the Church, or the history of England.\nIf we abandon the attempt to carry out an arrangement according to\nthe natural classification of the sciences, and form comprehensive\npractical groups, we shall be continually perplexed by the occurrence\nof intermediate cases, and opinions will differ *ad infinitum* as to\nthe details. If, to avoid the difficulty about Westminster Abbey, we\nform a class of books devoted to the History of Buildings, the question\nwill then arise whether Stonehenge is a building, and if so, whether\ncromlechs, mounds, and monoliths are so. We shall be uncertain whether\nto include lighthouses, monuments, bridges, &c. In regard to literary\nworks, rigorous classification is still less possible. The same work\nmay partake of the nature of poetry, biography, history, philosophy,\nor if we form a comprehensive class of Belles-lettres, nobody can say\nexactly what does or does not come under the term.\n\nMy own experience entirely bears out the opinion of De Morgan, that\nclassification according to the name of the author is the only one\npracticable in a large library, and this method has been admirably\ncarried out in the great catalogue of the British Museum. The name\nof the author is the most precise circumstance concerning a book,\nwhich usually dwells in the memory. It is a better characteristic of\nthe book than anything else. In an alphabetical arrangement we have\nan exhaustive classification, including a place for every name. The\nfollowing remarks[585] of De Morgan seem therefore to be entirely\ncorrect. \u201cFrom much, almost daily use, of catalogues for many years,\nI am perfectly satisfied that a classed catalogue is more difficult\nto use than to make. It is one man\u2019s theory of the subdivision of\nknowledge, and the chances are against its suiting any other man. Even\nif all doubtful works were entered under several different heads, the\nfrontier of the dubious region would itself be a mere matter of doubt.\nI never turn from a classed catalogue to an alphabetical one without\na feeling of relief and security. With the latter I can always, by\ntaking proper pains, make a library yield its utmost; with the former\nI can never be satisfied that I have taken proper pains, until I have\nmade it, in fact, as many different catalogues as there are different\nheadings, with separate trouble for each. Those to whom bibliographical\nresearch is familiar, know that they have much more frequently to\nhunt an author than a subject: they know also that in searching for a\nsubject, it is never safe to take another person\u2019s view, however good,\nof the limits of that subject with reference to their own particular\npurposes.\u201d\n\n  [585] *Philosophical Magazine*, 3rd Series (1845), vol.\u00a0xxvi. p.\u00a0522.\n  See also De Morgan\u2019s evidence before the Royal Commission on the\n  British Museum in 1849, Report (1850), Questions, 5704*-5815*,\n  6481\u20136513. This evidence should be studied by every person who wishes\n  to understand the elements of Bibliography.\n\nIt is often desirable, however, that a name catalogue should be\naccompanied by a subordinate subject catalogue, but in this case\nno attempt should be made to devise a theoretically complete\nclassification. Every principal subject treated in a book should\nbe entered separately in an alphabetical list, under the name most\nlikely to occur to the searcher, or under several names. This method\nwas partially carried out in Watts\u2019 *Bibliotheca Britannica*, but\nit was excellently applied in the admirable subject index to the\n*British Catalogue of Books*, and equally well in the *Catalogue\nof the Manchester Free Library* at Campfield, drawn up under the\ndirection of Mr. Crestadoro, this latter being the most perfect model\nof a printed catalogue with which I am acquainted. The Catalogue of\nthe London Library is also in the right form, and has a useful index\nof subjects, though it is too much condensed and abbreviated. The\npublic catalogue of the British Museum is arranged as far as possible\naccording to the alphabetical order of the authors\u2019 names, but in\nwriting the titles for this catalogue several copies are simultaneously\nproduced by a manifold writer, so that a catalogue according to the\norder of the books on the shelves, and another according to the first\nwords of the title-page, are created by a mere rearrangement of the\nspare copies. In the *English Cyclop\u00e6dia* it is suggested that twenty\ncopies of the book titles might readily have been utilised in forming\nadditional catalogues, arranged according to the place of publication,\nthe language of the book, the general nature of the subject, and so\nforth.[586] An excellent suggestion has also been made to the effect\nthat each book when published should have a fly-leaf containing half\na dozen printed copies of the title, drawn up in a form suitable for\ninsertion in catalogues. Every owner of a library could then easily\nmake accurate printed catalogues to suit his own purposes, by merely\ncutting out these titles and pasting them in books in any desirable\norder.\n\n  [586] *English Cyclop\u00e6dia, Arts and Sciences*, vol.\u00a0v. p.\u00a0233.\n\nIt will hardly be a digression to point out the enormous saving of\nlabour, or, what comes to the same thing, the enormous increase in our\navailable knowledge, both literary and scientific, which arises from\nthe formation of extensive indices. The \u201cState Papers,\u201d containing\nthe whole history of the nation, were practically sealed to literary\ninquirers until the Government undertook the task of calendaring and\nindexing them. The British Museum Catalogue is another national work,\nof which the importance in advancing knowledge cannot be overrated.\nThe Royal Society is doing great service in publishing a complete\ncatalogue of memoirs upon physical science. The time will perhaps\ncome when our views upon this subject will be extended, and either\nGovernment or some public society will undertake the systematic\ncataloguing and indexing of masses of historical and scientific\ninformation which are now almost closed against inquiry.\n\n\n*Classification in the Biological Sciences.*\n\nThe great generalisations established in the works of Herbert Spencer\nand Charles Darwin have thrown much light upon other sciences, and\nhave removed several difficulties out of the way of the logician. The\nsubject of classification has long been studied in almost exclusive\nreference to the arrangement of animals and plants. Systematic botany\nand zoology have been commonly known as the Classificatory Sciences,\nand scientific men seemed to suppose that the methods of arrangement,\nwhich were suitable for living creatures, must be the best for all\nother classes of objects. Several mineralogists, especially Mohs, have\nattempted to arrange minerals in genera and species, just as if they\nhad been animals capable of reproducing their kind with variations.\nThis confusion of ideas between the relationship of living forms and\nthe logical relationship of things in general prevailed from the\nearliest times, as manifested in the etymology of words. We familiarly\nspeak of a *kind* of things meaning a class of things, and the kind\nconsists of those things which are *akin*, or come of the same race.\nWhen Socrates and his followers wanted a name for a class regarded in a\nphilosophical light, they adopted the analogy in question, and called\nit a \u03b3\u03ad\u03bd\u03bf\u03c2, or race, the root \u03b3\u03b5\u03bd- being connected with the notion of\ngeneration.\n\nSo long as species of plants and animals were believed to proceed from\ndistinct acts of Creation, there was no apparent reason why methods of\nclassification suitable to them should not be treated as a guide to\nthe classification of other objects generally. But when once we regard\nthese resemblances as hereditary in their origin, we see that the\nsciences of systematic botany and zoology have a special character of\ntheir own. There is no reason to suppose that the same kind of natural\nclassification which is best in biology will apply also in mineralogy,\nin chemistry, or in astronomy. The logical principles which underlie\nall classification are of course the same in natural history as in the\nsciences of lifeless matter, but the special resemblances which arise\nfrom the relation of parent and offspring will not be found to prevail\nbetween different kinds of crystals or mineral bodies.\n\nThe genealogical view of the relations of animals and plants leads us\nto discard all notions of a regular progression of living forms, or\nany theory as to their symmetrical relations. It was at one time a\nquestion whether the ultimate scheme of natural classification would\nlead to arrangement in a simple line, or a circle, or a combination\nof circles. Macleay\u2019s once celebrated system was a circular one, and\neach class-circle was composed of five order-circles, each of which was\ncomposed again of five tribe-circles, and so on, the subdivision being\nat each step into five minor circles. Macleay held that in the animal\nkingdom there are five sub-kingdoms--the Vertebrata, Annulosa, Radiata,\nAcrita, and Mollusca. Each of these was again divided into five--the\nVertebrata, consisting of Mammalia, Reptilia, Pisces, Amphibia, and\nAves.[587] It is evident that in such a symmetrical system the animals\nwere made to suit themselves to the classes instead of the classes\nbeing suited to the animals.\n\n  [587] Swainson, \u201cTreatise on the Geography and Classification of\n  Animals,\u201d *Cabinet Cyclop\u00e6dia*, p.\u00a0201.\n\nWe now perceive that the ultimate system will have the form of an\nimmensely extended genealogical tree, which will be capable of\nrepresentation by lines on a plane surface of sufficient extent.\nStrictly speaking, this genealogical tree ought to represent the\ndescent of each individual living form now existing or which has\nexisted. It should be as personal and minute in its detail of\nrelations, as the Stemma of the Kings of England. We must not assume\nthat any two forms are exactly alike, and in any case they are\nnumerically distinct. Every parent then must be represented at the apex\nof a series of divergent lines, representing the generation of so many\nchildren. Any complete system of classification must regard individuals\nas the infim\u00e6 species. But as in the lower races of animals and\nplants the differences between individuals are slight and apparently\nunimportant, while the numbers of such individuals are immensely\ngreat, beyond all possibility of separate treatment, scientific men\nhave always stopped at some convenient but arbitrary point, and have\nassumed that forms so closely resembling each other as to present no\nconstant difference were all of one kind. They have, in short, fixed\ntheir attention entirely upon the main features of family difference.\nIn the genealogical tree which they have been unconsciously aiming to\nconstruct, diverging lines meant races diverging in character, and the\npurpose of all efforts at so-called natural classification was to trace\nout the descents between existing groups of plants or animals.\n\nNow it is evident that hereditary descent may have in different\ncases produced very different results as regards the problem of\nclassification. In some cases the differentiation of characters may\nhave been very frequent, and specimens of all the characters produced\nmay have been transmitted to the present time. A living form will then\nhave, as it were, an almost infinite number of cousins of various\ndegrees, and there will be an immense number of forms finely graduated\nin their resemblances. Exact and distinct classification will then\nbe almost impossible, and the wisest course will be not to attempt\narbitrarily to distinguish forms closely related in nature, but to\nallow that there exist transitional forms of every degree, to mark\nout if possible the extreme limits of the family relationship, and\nperhaps to select the most generalised form, or that which presents the\ngreatest number of close resemblances to others of the family, as the\n*type* of the whole.\n\nMr. Darwin, in his most interesting work upon Orchids, points out\nthat the tribe of Malaxe\u00e6 are distinguished from Epidendre\u00e6 by the\nabsence of a caudicle to the pollinia; but as some of the Malaxe\u00e6\nhave a minute caudicle, the division really breaks down in the most\nessential point. \u201cThis is a misfortune,\u201d he remarks,[588] \u201cwhich every\nnaturalist encounters in attempting to classify a largely developed\nor so-called natural group, in which, relatively to other groups,\nthere has been little extinction. In order that the naturalist may be\nenabled to give precise and clear definitions of his divisions, whole\nranks of intermediate or gradational forms must have been utterly swept\naway: if here and there a member of the intermediate ranks has escaped\nannihilation, it puts an effectual bar to any absolutely distinct\ndefinition.\u201d\n\n  [588] Darwin, *Fertilisation of Orchids*, p.\u00a0159.\n\nIn other cases a particular plant or animal may perhaps have\ntransmitted its form from generation to generation almost unchanged,\nor, what comes to the same result, those forms which diverged in\ncharacter from the parent stock may have proved unsuitable to their\ncircumstances, and perished. We shall then find a particular form\nstanding apart from all others, and marked by many distinct characters.\nOccasionally we may meet with specimens of a race which was formerly\nfar more common but is now undergoing extinction, and is nearly the\nlast of its kind. Thus we explain the occurrence of exceptional forms\nsuch as are found in the Amphioxus. The Equisetace\u00e6 perplex botanists\nby their want of affinity to other orders of Acrogenous plants. This\ndoubtless indicates that their genealogical connection with other\nplants must be sought for in the most distant ages of geological\ndevelopment.\n\nConstancy of character, as Mr. Darwin has said,[589] is what is chiefly\nvalued and sought after by naturalists; that is to say, naturalists\nwish to find some distinct family mark, or group of characters, by\nwhich they may clearly recognise the relationship of descent between a\nlarge group of living forms. It is accordingly a great relief to the\nmind of the naturalist when he comes upon a definitely marked group,\nsuch as the Diatomace\u00e6, which are clearly separated from their nearest\nneighbours the Desmidiace\u00e6 by their siliceous framework and the absence\nof chlorophyll. But we must no longer think that because we fail in\ndetecting constancy of character the fault is in our classificatory\nsciences. Where gradation of character really exists, we must devote\nourselves to defining and registering the degrees and limits of that\ngradation. The ultimate natural arrangement will often be devoid of\nstrong lines of demarcation.\n\n  [589] *Descent of Man*, vol.\u00a0i. p.\u00a0214.\n\nLet naturalists, too, form their systems of natural classification\nwith all care they can, yet it will certainly happen from time to\ntime that new and exceptional forms of animals or vegetables will be\ndiscovered and will require the modification of the system. A natural\nsystem is directed, as we have seen, to the discovery of empirical laws\nof correlation, but these laws being purely empirical will frequently\nbe falsified by more extensive investigation. From time to time the\nnotions of naturalists have been greatly widened, especially in the\ncase of Australian animals and plants, by the discovery of unexpected\ncombinations of organs, and such events must often happen in the\nfuture. If indeed the time shall come when all the forms of plants are\ndiscovered and accurately described, the science of Systematic Botany\nwill then be placed in a new and more favourable position, as remarked\nby Alphonse Decandolle.[590]\n\n  [590] *Laws of Botanical Nomenclature*, p.\u00a016.\n\nIt ought to be remembered that though the genealogical classification\nof plants or animals is doubtless the most instructive of all, it is\nnot necessarily the best for all purposes. There may be correlations\nof properties important for medicinal, or other practical purposes,\nwhich do not correspond to the correlations of descent. We must regard\nthe bamboo as a tree rather than a grass, although it is botanically\na grass. For legal purposes we may continue with advantage to treat\nthe whale, seal, and other cetace\u00e6, as fish. We must also class plants\naccording as they belong to arctic, alpine, temperate, sub-tropical or\ntropical regions. There are causes of likeness apart from hereditary\nrelationship, and *we must not attribute exclusive excellence to any\none method of classification*.\n\n\n*Classification by Types.*\n\nPerplexed by the difficulties arising in natural history from the\ndiscovery of intermediate forms, naturalists have resorted to what they\ncall classification by types. Instead of forming one distinct class\ndefined by the invariable possession of certain assigned properties,\nand rigidly including or excluding objects according as they do or\ndo not possess all these properties, naturalists select a typical\nspecimen, and they group around it all other specimens which resemble\nthis type more than any other selected type. \u201cThe type of each genus,\u201d\nwe are told,[591] \u201cshould be that species in which the characters\nof its group are best exhibited and most evenly balanced.\u201d It would\nusually consist of those descendants of a form which had undergone\nlittle alteration, while other descendants had suffered slight\ndifferentiation in various directions.\n\n  [591] Waterhouse, quoted by Woodward in his *Rudimentary Treatise of\n  Recent and Fossil Shells*, p.\u00a061.\n\nIt would be a great mistake to suppose that this classification by\ntypes is a logically distinct method. It is either not a real method\nof classification at all, or it is merely an abbreviated mode of\nrepresenting a complicated system of arrangement. A class must be\ndefined by the invariable presence of certain common properties. If,\nthen, we include an individual in which one of these properties does\nnot appear, we either fall into logical contradiction, or else we form\na new class with a new definition. Even a single exception constitutes\na new class by itself, and by calling it an exception we merely imply\nthat this new class closely resembles that from which it diverges in\none or two points only. Thus in the definition of the natural order\nof Rosace\u00e6, we find that the seeds are one or two in each carpel, but\nthat in the genus Spir\u00e6a there are three or four; this must mean either\nthat the number of seeds is not a part of the fixed definition of the\nclass, or else that Spir\u00e6a does not belong to that class, though it\nmay closely approximate to it. Naturalists continually find themselves\nbetween two horns of a dilemma; if they restrict the number of marks\nspecified in a definition so that every form intended to come within\nthe class shall possess all those marks, it will then be usually found\nto include too many forms; if the definition be made more particular,\nthe result is to produce so-called anomalous genera, which, while they\nare held to belong to the class, do not in all respects conform to its\ndefinition. The practice has hence arisen of allowing considerable\nlatitude in the definition of natural orders. The family of Crucifer\u00e6,\nfor instance, forms an exceedingly well-marked natural order, and among\nits characters we find it specified that the fruit is a pod, divided\ninto two cells by a thin partition, from which the valves generally\nseparate at maturity; but we are also informed that, in a few genera,\nthe pod is one-celled, or indehiscent, or separates transversely into\nseveral joints.[592] Now this must either mean that the formation of\nthe pod is not an essential point in the definition of the family, or\nthat there are several closely associated families.\n\n  [592] Bentham\u2019s *Handbook of the British Flora* (1866), p.\u00a025.\n\nThe same holds true of typical classification. The type itself is\nan individual, not a class, and no other object can be exactly like\nthe type. But as soon as we abstract the individual peculiarities\nof the type and thus specify a finite number of qualities in which\nother objects may resemble the type, we immediately constitute\na class. If some objects resemble the type in some points, and\nothers in other points, then each definite collection of points of\nresemblance constitutes intensively a separate class. The very notion\nof classification by types is in fact erroneous in a logical point of\nview. The naturalist is constantly occupied in endeavouring to mark\nout definite groups of living forms, where the forms themselves do not\nin many cases admit of such rigorous lines of demarcation. A certain\nlaxity of logical method is thus apt to creep in, the only remedy for\nwhich will be the frank recognition of the fact, that, according to the\ntheory of hereditary descent, gradation of characters is probably the\nrule, and precise demarcation between groups the exception.\n\n\n*Natural Genera and Species.*\n\nOne important result of the establishment of the theory of evolution\nis to explode all notions about natural groups constituting separate\ncreations. Naturalists long held that every plant belongs to some\nspecies, marked out by invariable characters, which do not change by\ndifference of soil, climate, cross-breeding, or other circumstances.\nThey were unable to deny the existence of such things as sub-species,\nvarieties, and hybrids, so that a species of plants was often\nsubdivided and classified within itself. But then the differences upon\nwhich this sub-classification depended were supposed to be variable,\nand thus distinguished from the invariable characters imposed upon the\nwhole species at its creation. Similarly a natural genus was a group of\nspecies, and was marked out from other genera by eternal differences of\nstill greater importance.\n\nWe now, however, perceive that the existence of any such groups as\ngenera and species is an arbitrary creation of the naturalist\u2019s\nmind. All resemblances of plants are natural so far as they express\nhereditary affinities; but this applies as well to the variations\nwithin the species as to the species itself, or to the larger groups.\nAll is a matter of degree. The deeper differences between plants have\nbeen produced by the differentiating action of circumstances during\nmillions of years, so that it would naturally require millions of\nyears to undo this result, and prove experimentally that the forms can\nbe approximated again. Sub-species may sometimes have arisen within\nhistorical times, and varieties approaching to sub-species may often\nbe produced by the horticulturist in a few years. Such varieties can\neasily be brought back to their original forms, or, if placed in the\noriginal circumstances, will themselves revert to those forms; but\naccording to Darwin\u2019s views all forms are capable of unlimited change,\nand it might possibly be, unlimited reversion if suitable circumstances\nand sufficient time be granted.\n\nMany fruitless attempts have been made to establish a rigorous\ncriterion of specific and generic difference, so that these classes\nmight have a definite value and rank in all branches of biology.\nLinn\u00e6us adopted the view that the species was to be defined as a\ndistinct creation, saying,[593] \u201cSpecies tot numeramus, quot divers\u00e6\nform\u00e6 in principio sunt creat\u00e6;\u201d or again, \u201cSpecies tot sunt, quot\ndiversas formas ab initio produxit Infinitum Ens; qu\u00e6 form\u00e6, secundum\ngenerationis inditas leges, produxere plures, at sibi semper similes.\u201d\nOf genera he also says,[594] \u201cGenus omne est naturale, in primordio\ntale creatum.\u201d It was a common doctrine added to and essential to that\nof distinct creation that these species could not produce intermediate\nand variable forms, so that we find Linn\u00e6us obliged by the ascertained\nexistence of hybrids to take a different view in another work; he\nsays,[595] \u201cNovas species immo et genera ex copula diversarum specierum\nin regno vegetabilium oriri primo intuitu paradoxum videtur; interim\nobservationes sic fieri non ita dissuadent.\u201d Even supposing in the\npresent day that we could assent to the notion of a certain number of\ndistinct creational acts, this notion would not help us in the theory\nof classification. Naturalists have never pointed out any method of\ndeciding what are the results of distinct creations, and what are\nnot. As Darwin says,[596] \u201cthe definition must not include an element\nwhich cannot possibly be ascertained, such as an act of creation.\u201d\nIt is, in fact, by investigation of forms and classification that we\nshould ascertain what were distinct creations and what were not; this\ninformation would be a result and not a means of classification.\n\n  [593] *Philosophia Botanica* (1770), \u00a7 157, p.\u00a099.\n\n  [594] *Ibid.* \u00a7 159, p.\u00a0100.\n\n  [595] *Am\u0153nitates Academic\u00e6* (1744), vol.\u00a0i. p.\u00a070. Quoted in\n  *Edinburgh Review*, October 1868, vol.\u00a0cxxviii. pp.\u00a0416, 417.\n\n  [596] *Descent of Man*, vol.\u00a0i. p.\u00a0228.\n\nAgassiz seemed to consider that he had discovered an important\nprinciple, to the effect that general plan or structure is the true\nground for the discrimination of the great classes of animals, which\nmay be called branches of the animal kingdom.[597] He also thought that\ngenera are definite and natural groups. \u201cGenera,\u201d he says,[598] \u201care\nmost closely allied groups of animals, differing neither in form, nor\nin complication of structure, but simply in the ultimate structural\npeculiarities of some of their parts; and this is, I believe, the best\ndefinition which can be given of genera.\u201d But it is surely apparent\nthat there are endless degrees both of structural peculiarity and of\ncomplication of structure. It is impossible to define the amount of\nstructural peculiarity which constitutes the genus as distinguished\nfrom the species.\n\n  [597] Agassiz, *Essay on Classification*, p.\u00a0219.\n\n  [598] *Ibid.* p.\u00a0249.\n\nThe form which any classification of plants or animals tends to take is\nthat of an unlimited series of subaltern classes. Originally botanists\nconfined themselves for the most part to a small number of such\nclasses. Linn\u00e6us adopted Class, Order, Genus, Species, and Variety, and\neven seemed to think that there was something essentially natural in a\nfive-fold arrangement of groups.[599]\n\n  [599] *Philosophia Botanica*, \u00a7 155, p.\u00a098.\n\nWith the progress of botany intermediate and additional groups\nhave gradually been introduced. According to the Laws of Botanical\nNomenclature adopted by the International Botanical Congress, held at\nParis[600] in August 1867, no less than twenty-one names of classes\nare recognised--namely, Kingdom, Division, Sub-division, Class,\nSub-class, Cohort, Sub-cohort, Order, Sub-order, Tribe, Sub-tribe,\nGenus, Sub-genus, Section, Sub-section, Species, Sub-species, Variety,\nSub-variety, Variation, Sub-variation. It is allowed by the authors of\nthis scheme, that the rank or degree of importance to be attributed\nto any of these divisions may vary in a certain degree according to\nindividual opinion. The only point on which botanists are not allowed\ndiscretion is as to the order of the successive sub-divisions; any\ninversion of the arrangement, such as division of a genus into tribes,\nor of a tribe into orders, is quite inadmissible. There is no reason\nto suppose that even the above list is complete and inextensible. The\nBotanical Congress itself recognised the distinction between variations\naccording as they are Seedlings, Half-breeds, or *Lusus Natur\u00e6*.\nThe complication of the inferior classes is increased again by the\nexistence of *hybrids*, arising from the fertilisation of one species\nby another deemed a distinct species, nor can we place any limit to the\nminuteness of discrimination of degrees of breeding short of an actual\npedigree of individuals.\n\n  [600] *Laws of Botanical Nomenclature*, by Alphonse Decandolle,\n  translated from the French, 1868, p.\u00a019.\n\nIt will be evident to the reader that in the remarks upon\nclassification as applied to the Natural Sciences, given in this\nand the preceding sections, I have not in the least attempted to\ntreat the subject in a manner adequate to its extent and importance.\nA volume would be insufficient for tracing out the principles of\nscientific method specially applicable to these branches of science.\nWhat more I may be able to say upon the subject will be better said,\nif ever, when I am able to take up the closely-connected subjects of\nScientific Nomenclature, Terminology, and Descriptive Representation.\nIn the meantime, I have wished to show, in a negative point of\nview, that natural classification in the animal and vegetable\nkingdoms is a special problem, and that the particular methods and\ndifficulties to which it gives rise are not those common to all cases\nof classification, as so many physicists have supposed. Genealogical\nresemblances are only a special case of resemblances in general.\n\n\n*Unique or Exceptional Objects.*\n\nIn framing a system of classification in almost any branch of science,\nwe must expect to meet with unique or peculiar objects, which stand\nalone, having comparatively few analogies with other objects. They may\nalso be said to be *sui generis*, each unique object forming, as it\nwere, a genus by itself; or they are called *nondescript*, because from\nthus standing apart it is difficult to find terms in which to describe\ntheir properties. The rings of Saturn, for instance, form a unique\nobject among the celestial bodies. We have indeed considered this and\nmany other instances of unique objects in the preceding chapter on\nExceptional Phenomena. Apparent, Singular, and Divergent Exceptions\nespecially, are analogous to unique objects.\n\nIn the classification of the elements, Carbon stands apart as a\nsubstance entirely unique in its powers of producing compounds. It is\nconsidered to be a quadrivalent element, and it obeys all the ordinary\nlaws of chemical combination. Yet it manifests powers of affinity in\nsuch an exalted degree that the substances in which it appears are\nmore numerous than all the other compounds known to chemists. Almost\nthe whole of the substances which have been called organic contain\ncarbon, and are probably held together by the carbon atoms, so that\nmany chemists are now inclined to abandon the name Organic Chemistry,\nand substitute the name Chemistry of the Carbon Compounds. It used to\nbe believed that the production of organic compounds could be effected\nonly by the action of vital force, or of some inexplicable cause\ninvolved in the phenomena of life; but it is now found that chemists\nare able to commence with the elementary materials, pure carbon,\nhydrogen, and oxygen, and by strictly chemical operations to combine\nthese so as to form complicated organic compounds. So many substances\nhave already been formed that we might be inclined to generalise and\ninfer that all organic compounds might ultimately be produced without\nthe agency of living beings. Thus the distinction between the organic\nand the inorganic kingdoms seems to be breaking down, but our wonder at\nthe peculiar powers of carbon must increase at the same time.\n\nIn considering generalisation, the law of continuity was applied\nchiefly to physical properties capable of mathematical treatment. But\nin the classificatory sciences, also, the same important principle\nis often beautifully exemplified. Many objects or events seem to be\nentirely exceptional and abnormal, and in regard to degree or magnitude\nthey may be so termed; but it is often easy to show that they are\nconnected by intermediate links with ordinary cases. In the organic\nkingdoms there is a common groundwork of similarity running through\nall classes, but particular actions and processes present themselves\nconspicuously in particular families and classes. Tenacity of life\nis most marked in the Rotifera, and some other kinds of microscopic\norganisms, which can be dried and boiled without loss of life. Reptiles\nare distinguished by torpidity, and the length of time they can live\nwithout food. Birds, on the contrary, exhibit ceaseless activity and\nhigh muscular power. The ant is as conspicuous for intelligence and\nsize of brain among insects as the quadrumana and man among vertebrata.\nAmong plants the Leguminos\u00e6 are distinguished by a tendency to sleep,\nfolding their leaves at the approach of night. In the genus Mimosa,\nespecially the Mimosa pudica, commonly called the sensitive plant,\nthe same tendency is magnified into an extreme irritability, almost\nresembling voluntary motion. More or less of the same irritability\nprobably belongs to vegetable forms of every kind, but it is of course\nto be investigated with special ease in such an extreme case. In the\nGymnotus and Torpedo, we find that organic structures can act like\ngalvanic batteries. Are we to suppose that such animals are entirely\nanomalous exceptions; or may we not justly expect to find less intense\nmanifestations of electric action in all animals?\n\nSome extraordinary differences between the modes of reproduction\nof animals have been shown to be far less than was at first sight\napparent. The lower animals seem to differ entirely from the higher\nones in the power of reproducing lost limbs. A kind of crab has the\nhabit of casting portions of its claws when much frightened, but\nthey soon grow again. There are multitudes of smaller animals which,\nlike the Hydra, may be cut in two and yet live and develop into new\ncomplete individuals. No mammalian animal can reproduce a limb, and\nin appearance there is no analogy. But it was suggested by Blumenbach\nthat the healing of a wound in the higher animals really represents in\na lower degree the power of reproducing a limb. That this is true may\nbe shown by adducing a multitude of intermediate cases, each adjoining\npair of which are clearly analogous, so that we pass gradually from one\nextreme to the other. Darwin holds, moreover, that any such restoration\nof parts is closely connected with that perpetual replacement of\nthe particles which causes every organised body to be after a time\nentirely new as regards its constituent substance. In short, we\napproach to a great generalisation under which all the phenomena of\ngrowth, restoration, and maintenance of organs are effects of one and\nthe same power.[601] It is perhaps still more surprising to find that\nthe complicated process of reproduction in the higher animals may be\ngradually traced down to a simpler and simpler form, which at last\nbecomes undistinguishable from the budding out of one plant from the\nstem of another. By a great generalisation we may regard all the modes\nof reproduction of organic life as alike in their nature, and varying\nonly in complexity of development.[602]\n\n  [601] Darwin, *The Variation of Animals and Plants*, vol.\u00a0ii.\n  pp.\u00a0293, 359, &c.; quoting Paget, *Lectures on Pathology*, 1853,\n  pp.\u00a0152, 164.\n\n  [602] *Ibid.* vol.\u00a0ii. p.\u00a0372.\n\n\n*Limits of Classification.*\n\nScience can extend only so far as the power of accurate classification\nextends. If we cannot detect resemblances, and assign their exact\ncharacter and amount, we cannot have that generalised knowledge which\nconstitutes science; we cannot infer from case to case. Classification\nis the opposite process to discrimination. If we feel that two tastes\ndiffer, the tastes of two kinds of wine for instance, the mere fact of\ndifference existing prevents inference. The detection of the difference\nsaves us, indeed, from false inference, because so far as difference\nexists, inference is impossible. But classification consists in\ndetecting resemblances of all degrees of generality, and ascertaining\nexactly how far such resemblances extend, while assigning precisely the\npoints at which difference begins. It enables us, then, to generalise,\nand make inferences where it is possible, and it saves us at the same\ntime from going too far. A full classification constitutes a complete\nrecord of all our knowledge of the objects or events classified,\nand the limits of exact knowledge are identical with the limits of\nclassification.\n\nIt must by no means be supposed that every group of natural objects\nwill be found capable of rigorous classification. There may be\nsubstances which vary by insensible degrees, consisting, for instance,\nin varying mixtures of simpler substances. Granite is a mixture of\nquartz, felspar, and mica, but there are hardly two specimens in which\nthe proportions of these three constituents are alike, and it would\nbe impossible to lay down definitions of distinct species of granite\nwithout finding an infinite variety of intermediate species. The\nonly true classification of granites, then, would be founded on the\nproportions of the constituents present, and a chemical or microscopic\nanalysis would be requisite, in order that we might assign a specimen\nto its true position in the series. Granites vary, again, by insensible\ndegrees, as regards the magnitude of the crystals of felspar and mica.\nPrecisely similar remarks might be made concerning the classification\nof other plutonic rocks, such as syenite, basalt, pumice-stone, lava.\n\nThe nature of a ray of homogeneous light is strictly defined, either\nby its place in the spectrum or by the corresponding wave-length, but\na ray of mixed light admits of no simple classification; any of the\ninfinitely numerous rays of the continuous spectrum may be present or\nabsent, or present in various intensities, so that we can only class\nand define a mixed colour by defining the intensity and wave-length\nof each ray of homogeneous light which is present in it. Complete\nspectroscopic analysis and the determination of the intensity of\nevery part of the spectrum yielded by a mixed ray is requisite for\nits accurate classification. Nearly the same may be said of complex\nsounds. A simple sound undulation, if we could meet with such a sound,\nwould admit of precise and exhaustive classification as regards pitch,\nthe length of wave, or the number of waves reaching the ear per\nsecond being a sufficient criterion. But almost all ordinary sounds,\neven those of musical instruments, consist of complex aggregates of\nundulations of different pitches, and in order to classify the sound\nwe should have to measure the intensities of each of the constituent\nsounds, a work which has been partially accomplished by Helmholtz, as\nregards the vowel sounds. The different tones of voice distinctive\nof different individuals must also be due to the intermixture of\nminute waves of various pitch, which are yet quite beyond the range\nof experimental investigation. We cannot, then, at present attempt to\nclassify the different kinds or *timbres* of sound.\n\nThe difficulties of classification are still greater when a varying\nphenomenon cannot be shown to be a mixture of simpler phenomena. If\nwe attempt to classify tastes, we may rudely group them according as\nthey are sweet, bitter, saline, alkaline, acid, astringent or fiery;\nbut it is evident that these groups are bounded by no sharp lines\nof definition. Tastes of mixed or intermediate character may exist\nalmost *ad infinitum*, and what is still more troublesome, the tastes\nclearly united within one class may differ more or less from each\nother, without our being able to arrange them in subordinate genera and\nspecies. The same remarks may be made concerning the classification of\nodours, which may be roughly grouped according to the arrangement of\nLinn\u00e6us as, aromatic, fragrant, ambrosiac, alliaceous, fetid, virulent,\nnauseous. Within each of these vague classes, however, there would be\ninfinite shades of variety, and each class would graduate into other\nclasses. The odours which can be discriminated by an acute nose are\ninfinite; every rock, stone, plant, or animal has some slight smell,\nand it is well known that dogs, or even blind men, can discriminate\npersons by a slight distinctive odour which usually passes unnoticed.\n\nSimilar remarks may be made concerning the feelings of the human mind,\ncalled emotions. We know what is anger, grief, fear, hatred, love;\nand many systems for classifying these feelings have been proposed.\nThey may be roughly distinguished according as they are pleasurable\nor painful, prospective or retrospective, selfish or sympathetic,\nactive or passive, and possibly in many other ways; but each mode of\narrangement will be indefinite and unsatisfactory when followed into\ndetails. As a general rule, the emotional state of the mind at any\nmoment will be neither pure anger nor pure fear, nor any one pure\nfeeling, but an indefinite and complex aggregate of feelings. It may\nbe that the state of mind is really a sum of several distinct modes\nof agitation, just as a mixed colour is the sum of the several rays\nof the spectrum. In this case there may be more hope of some method\nof analysis being successfully applied at a future time. But it may\nbe found that states of mind really graduate into each other so that\nrigorous classification would be hopeless.\n\nA little reflection will show that there are whole worlds of\nexistences which in like manner are incapable of logical analysis and\nclassification. One friend may be able to single out and identify\nanother friend by his countenance among a million other countenances.\nFaces are capable of infinite discrimination, but who shall classify\nand define them, or say by what particular shades of feature he does\njudge? There are of course certain distinct types of face, but each\ntype is connected with each other type by infinite intermediate\nspecimens. We may classify melodies according to the major or minor\nkey, the character of the time, and some other distinct points;\nbut every melody has, independently of such circumstances, its\nown distinctive character and effect upon the mind. We can detect\ndifferences between the styles of literary, musical, or artistic\ncompositions. We can even in some cases assign a picture to its\npainter, or a symphony to its composer, by a subtle feeling of\nresemblances or differences which may be felt, but cannot be described.\n\nFinally, it is apparent that in human character there is unfathomable\nand inexhaustible diversity. Every mind is more or less like every\nother mind; there is always a basis of similarity, but there is a\nsuperstructure of feelings, impulses, and motives which is distinctive\nfor each person. We can sometimes predict the general character of the\nfeelings and actions which will be produced by a given external event\nin an individual well known to us; but we also know that we are often\ninexplicably at fault in our inferences. No one can safely generalise\nupon the subtle variations of temper and emotion which may arise even\nin a person of ordinary character. As human knowledge and civilisation\nprogress, these characteristic differences tend to develop and multiply\nthemselves, rather than decrease. Character grows more many-sided. Two\nwell educated Englishmen are far better distinguished from each other\nthan two common labourers, and these are better distinguished than two\nAustralian aborigines. The complexities of existing phenomena probably\ndevelop themselves more rapidly than scientific method can overtake\nthem. In spite of all the boasted powers of science, we cannot really\napply scientific method to our own minds and characters, which are more\nimportant to us than all the stars and nebul\u00e6.\n\n\n\n\nBOOK VI.\n\n\n\n\nCHAPTER XXXI.\n\nREFLECTIONS ON THE RESULTS AND LIMITS OF SCIENTIFIC METHOD.\n\n\nBefore concluding a work on the Principles of Science, it will not be\ninappropriate to add some remarks upon the limits and ultimate bearings\nof the knowledge which we may acquire by the employment of scientific\nmethod. All science consists, it has several times been stated, in the\ndetection of identities in the action of natural agents. The purpose of\ninductive inquiry is to ascertain the apparent existence of necessary\nconnection between causes and effects, expressed in the form of natural\nlaws. Now so far as we thus learn the invariable course of nature, the\nfuture becomes the necessary sequel of the present, and we are brought\nbeneath the sway of powers with which nothing can interfere.\n\nBy degrees it is found, too, that the chemistry of organised substances\nis not entirely separated from, but is continuous with, that of earth\nand stones. Life seems to be nothing but a special form of energy\nwhich is manifested in heat and electricity and mechanical force. The\ntime may come, it almost seems, when the tender mechanism of the brain\nwill be traced out, and every thought reduced to the expenditure of\na determinate weight of nitrogen and phosphorus. No apparent limit\nexists to the success of scientific method in weighing and measuring,\nand reducing beneath the sway of law, the phenomena both of matter\nand of mind. And if mental phenomena be thus capable of treatment by\nthe balance and the micrometer, can we any longer hold that mind is\ndistinct from matter? Must not the same inexorable reign of law which\nis apparent in the motions of brute matter be extended to the subtle\nfeelings of the human heart? Are not plants and animals, and ultimately\nman himself, merely crystals, as it were, of a complicated form? If\nso, our boasted free will becomes a delusion, moral responsibility a\nfiction, spirit a mere name for the more curious manifestations of\nmaterial energy. All that happens, whether right or wrong, pleasurable\nor painful, is but the outcome of the necessary relations of time and\nspace and force.\n\nMaterialism seems, then, to be the coming religion, and resignation\nto the nonentity of human will the only duty. Such may not generally\nbe the reflections of men of science, but I believe that we may thus\ndescribe the secret feelings of fear which the constant advance of\nscientific investigation excites in the minds of many. Is science,\nthen, essentially atheistic and materialistic in its tendency?\nDoes the uniform action of material causes, which we learn with an\never-increasing approximation to certainty, preclude the hypothesis of\na benevolent Creator, who has not only designed the existing universe,\nbut who still retains the power to alter its course from time to time?\n\nTo enter upon actual theological discussions would be evidently beyond\nthe scope of this work. It is with the scientific method common to all\nthe sciences, and not with any of the separate sciences, that we are\nconcerned. Theology therefore would be at least as much beyond my scope\nas chemistry or geology. But I believe that grave misapprehensions\nexist as regards the very nature of scientific method. There are\nscientific men who assert that the interposition of Providence is\nimpossible, and prayer an absurdity, because the laws of nature are\ninductively proved to be invariable. Inferences are drawn not so\nmuch from particular sciences as from the logical nature of science\nitself, to negative the impulses and hopes of men. Now I may state\nthat my own studies in logic lead me to call in question such negative\ninferences. Laws of nature are uniformities observed to exist in the\naction of certain material agents, but it is logically impossible to\nshow that all other agents must behave as these do. The too exclusive\nstudy of particular branches of physical science seems to generate an\nover-confident and dogmatic spirit. Rejoicing in the success with which\na few groups of facts are brought beneath the apparent sway of laws,\nthe investigator hastily assumes that he is close upon the ultimate\nsprings of being. A particle of gelatinous matter is found to obey\nthe ordinary laws of chemistry; yet it moves and lives. The world is\ntherefore asked to believe that chemistry can resolve the mysteries of\nexistence.\n\n\n*The Meaning of Natural Law.*\n\nPindar speaks of Law as the Ruler of the Mortals and the Immortals, and\nit seems to be commonly supposed that the so-called Laws of Nature, in\nlike manner, rule man and his Creator. The course of nature is regarded\nas being determined by invariable principles of mechanics which have\nacted since the world began, and will act for evermore. Even if the\norigin of all things is attributed to an intelligent creative mind,\nthat Being is regarded as having yielded up arbitrary power, and as\nbeing subject like a human legislator to the laws which he has himself\nenacted. Such notions I should describe as superficial and erroneous,\nbeing derived, as I think, from false views of the nature of scientific\ninference, and the degree of certainty of the knowledge which we\nacquire by inductive investigation.\n\nA law of nature, as I regard the meaning of the expression, is not a\nuniformity which must be obeyed by all objects, but merely a uniformity\nwhich is as a matter of fact obeyed by those objects which have come\nbeneath our observation. There is nothing whatever incompatible with\nlogic in the discovery of objects which should prove exceptions to any\nlaw of nature. Perhaps the best established law is that which asserts\nan invariable correlation to exist between gravity and inertia, so that\nall gravitating bodies are found to possess inertia, and all bodies\npossessing inertia are found to gravitate. But it would be no reproach\nto our scientific method, if something were ultimately discovered\nto possess gravity without inertia. Strictly defined and correctly\ninterpreted, the law itself would acknowledge the possibility; for with\nthe statement of every law we ought properly to join an estimate of the\nnumber of instances in which it has been observed to hold true, and the\nprobability thence calculated, that it will hold true in the next case.\nNow, as we found (p.\u00a0259), no finite number of instances can warrant\nus in expecting with certainty that the next instance will be of like\nnature; in the formulas yielded by the inverse method of probabilities\na unit always appears to represent the probability that our inference\nwill be mistaken. I demur to the assumption that there is any necessary\ntruth even in such fundamental laws of nature as the Indestructibility\nof Matter, the Conservation of Energy, or the Laws of Motion. Certain\nit is that men of science have recognised the conceivability of\nother laws, and even investigated their mathematical consequences.\nAiry investigated the mathematical conditions of a perpetual motion\n(p.\u00a0223), and Laplace and Newton discussed imaginary laws of forces\ninconsistent with those observed to operate in the universe (pp.\u00a0642,\n706).\n\nThe laws of nature, as I venture to regard them, are simply general\npropositions concerning the correlation of properties which have been\nobserved to hold true of bodies hitherto observed. On the assumption\nthat our experience is of adequate extent, and that no arbitrary\ninterference takes place, we are then able to assign the probability,\nalways less than certainty, that the next object of the same apparent\nnature will conform to the same laws.\n\n\n*Infiniteness of the Universe.*\n\nWe may safely accept as a satisfactory scientific hypothesis the\ndoctrine so grandly put forth by Laplace, who asserted that a perfect\nknowledge of the universe, as it existed at any given moment, would\ngive a perfect knowledge of what was to happen thenceforth and for\never after. Scientific inference is impossible, unless we may regard\nthe present as the outcome of what is past, and the cause of what is\nto come. To the view of perfect intelligence nothing is uncertain. The\nastronomer can calculate the positions of the heavenly bodies when\nthousands of generations of men shall have passed away, and in this\nfact we have some illustration, as Laplace remarks, of the power which\nscientific prescience may attain. Doubtless, too, all efforts in the\ninvestigation of nature tend to bring us nearer to the possession of\nthat ideally perfect power of intelligence. Nevertheless, as Laplace\nwith profound wisdom adds,[603] we must ever remain at an infinite\ndistance from the goal of our aspirations.\n\n  [603] *Th\u00e9orie Analytique des Probabilit\u00e9s*, quoted by Babbage,\n  *Ninth Bridgewater Treatise*, p.\u00a0173.\n\nLet us assume, for a time at least, as a highly probable hypothesis,\nthat whatever is to happen must be the outcome of what is; there then\narises the question, What is? Now our knowledge of what exists must\never remain imperfect and fallible in two respects. Firstly, we do\nnot know all the matter that has been created, nor the exact manner\nin which it has been distributed through space. Secondly, assuming\nthat we had that knowledge, we should still be wanting in a perfect\nknowledge of the way in which the particles of matter will act upon\neach other. The power of scientific prediction extends at the most\nto the limits of the data employed. Every conclusion is purely\nhypothetical and conditional upon the non-interference of agencies\npreviously undetected. The law of gravity asserts that every body tends\nto approach towards every other body, with a certain determinate force;\nbut, even supposing the law to hold true, it does not assert that the\nbody *will* approach. No single law of nature can warrant us in making\nan absolute prediction. We must know all the laws of nature and all\nthe existing agents acting according to those laws before we can say\nwhat will happen. To assume, then, that scientific method can take\neverything within its cold embrace of uniformity, is to imply that the\nCreator cannot outstrip the intelligence of his creatures, and that\nthe existing Universe is not infinite in extent and complexity, an\nassumption for which I see no logical basis whatever.\n\n\n*The Indeterminate Problem of Creation.*\n\nA second and very serious misapprehension concerning the import of a\nlaw of nature may now be pointed out. It is not uncommonly supposed\nthat a law determines the character of the results which shall take\nplace, as, for instance, that the law of gravity determines what force\nof gravity shall act upon a given particle. Surely a little reflection\nmust render it plain that a law by itself determines nothing. It is\n*law plus agents obeying law which has results*, and it is no function\nof law to govern or define the number and place of its own agents.\nWhether a particle of matter shall gravitate, depends not only upon the\nlaw of Newton, but also upon the distribution of surrounding particles.\nThe theory of gravitation may perhaps be true throughout all time\nand in all parts of space, and the Creator may never find occasion\nto create those possible exceptions to it which I have asserted to\nbe conceivable. Let this be as it may; our science cannot certainly\ndetermine the question. Certain it is, that the law of gravity does not\nalone determine the forces which may be brought to bear at any point of\nspace. The force of gravitation acting upon any particle depends upon\nthe mass, distance, and relative position of all the other particles\nof matter within the bounds of space at the instant in question.\nEven assuming that all matter when once distributed through space at\nthe Creation was thenceforth to act in an invariable manner without\nsubsequent interference, yet the actual configuration of matter at any\nmoment, and the consequent results of the law of gravitation, must have\nbeen entirely a matter of free choice.\n\nChalmers has most distinctly pointed out that the existing\n*collocations* of the material world are as important as the laws\nwhich the objects obey. He remarks that a certain class of writers\nentirely overlook the distinction, and forget that mere laws without\ncollocations would have afforded no security against a turbid and\ndisorderly chaos.[604] Mill has recognised[605] the truth of Chalmers\u2019\nstatement, without drawing the proper inferences from it. He says[606]\nof the distribution of matter through space, \u201cWe can discover nothing\nregular in the distribution itself; we can reduce it to no uniformity,\nto no law.\u201d More lately the Duke of Argyll in his well-known work on\nthe *Reign of Law* has drawn attention to the profound distinction\nbetween laws and collocations of causes.\n\n  [604] *First Bridgewater Treatise* (1834), pp.\u00a016\u201324.\n\n  [605] *System of Logic*, 5th edit. bk. III. chap. V. \u00a7 7; chap. XVI.\n  \u00a7 3.\n\n  [606] *System of Logic*, vol.\u00a0i. p.\u00a0384.\n\nThe original conformation of the material universe, as far as we can\ntell, was free from all restriction. There was unlimited space in\nwhich to frame it, and an unlimited number of material particles,\neach of which could be placed in any one of an infinite number of\ndifferent positions. It should be added, that each particle might be\nendowed with any one of an infinite number of quantities of *vis viva*\nacting in any one of an infinite number of different directions. The\nproblem of Creation was, then, what a mathematician would call *an\nindeterminate problem*, and it was indeterminate in a great number of\nways. Infinitely numerous and various universes might then have been\nfashioned by the various distribution of the original nebulous matter,\nalthough all the particles of matter should obey the law of gravity.\n\nLucretius tells us how in the original rain of atoms some of these\nlittle bodies diverged from the rectilinear direction, and coming into\ncontact with other atoms gave rise to the various combinations of\nsubstances which exist. He omitted to tell us whence the atoms came,\nor by what force some of them were caused to diverge; but surely these\nomissions involve the whole question. I accept the Lucretian conception\nof creation when properly supplemented. Every atom which existed in\nany point of space must have existed there previously, or must have\nbeen created there by a previously existing Power. When placed there\nit must have had a definite mass and a definite energy. Now, as before\nremarked, an unlimited number of atoms can be placed in unlimited space\nin an unlimited number of modes of distribution. Out of infinitely\ninfinite choices which were open to the Creator, that one choice must\nhave been made which has yielded the Universe as it now exists.\n\nIt would be a mistake, indeed, to suppose that the law of gravity,\nwhen it holds true, is no restriction on the distribution of\nforce. That law is a geometrical law, and it would in many cases\nbe mathematically impossible, as far as we can see, that the force\nof gravity acting on one particle should be small while that on a\nneighbouring particle is great. We cannot conceive that even Omnipotent\nPower should make the angles of a triangle greater than two right\nangles. The primary laws of thought and the fundamental notions of the\nmathematical sciences do not seem to admit of error or alteration. Into\nthe metaphysical origin and meaning of the apparent necessity attaching\nto such laws I have not attempted to inquire in this work, and it is\nnot requisite for my present purpose. If the law of gravity were the\nonly law of nature and the Creator had chosen to render all matter\nobedient to that law, there would doubtless be restrictions upon the\neffects derivable from any one distribution of matter.\n\n\n*Hierarchy of Natural Laws.*\n\nA further consideration presents itself. A natural law like that\nof gravity expresses a certain uniformity in the action of agents\nsubmitted to it, and this produces, as we have seen, certain\ngeometrical restrictions upon the effects which those agents may\nproduce. But there are other forces and laws besides gravity. One\nforce may override another, and two laws may each be obeyed and may\neach disguise the action of the other. In the intimate constitution of\nmatter there may be hidden springs which, while acting in accordance\nwith their own fixed laws, may lead to sudden and unexpected changes.\nSo at least it has been found from time to time in the past, and so\nthere is every reason to believe it will be found in the future. To\nthe ancients it seemed incredible that one lifeless stone could make\nanother leap towards it. A piece of iron while it obeys the magnetic\nforce of the loadstone does not the less obey the law of gravity.\nA plant gravitates downwards as regards every constituent cell or\nfibre, and yet it persists in growing upwards. Life is altogether an\nexception to the simpler phenomena of mineral substances, not in the\nsense of disproving those laws, but in superadding forces of new and\ninexplicable character. Doubtless no law of chemistry is broken by the\naction of the nervous cells, and no law of physics by the pulses of the\nnervous fibres, but something requires to be added to our sciences in\norder that we may explain these subtle phenomena.\n\nNow there is absolutely nothing in science or in scientific method to\nwarrant us in assigning a limit to this hierarchy of laws. When in\nmany undoubted cases we find law overriding law, and at certain points\nin our experience producing unexpected results, we cannot venture to\naffirm that we have exhausted the strange phenomena which may have been\nprovided for in the original constitution of matter. The Universe might\nhave been so designed that it should go for long intervals through the\nsame round of unvaried existence, and yet that events of exceptional\ncharacter should be produced from time to time. Babbage showed in that\nmost profound and eloquent work, *The Ninth Bridgewater Treatise*, that\nit was theoretically possible for human artists to design a machine,\nconsisting of metallic wheels and levers, which should work invariably\naccording to a simple law of action during any finite number of steps,\nand yet at a fixed moment, however distant, should manifest a single\nbreach of law. Such an engine might go on counting, for instance, the\nnatural numbers until they would reach a number requiring for its\nexpression a hundred million digits. \u201cIf every letter in the volume\nnow before the reader\u2019s eyes,\u201d says Babbage,[607] \u201cwere changed into\na figure, and if all the figures contained in a thousand such volumes\nwere arranged in order, the whole together would yet fall far short\nof the vast induction the observer would have had in favour of the\ntruth of the law of natural numbers.... Yet shall the engine, true to\nthe prediction of its director, after the lapse of myriads of ages,\nfulfil its task, and give that one, the first and only exception to\nthat time-sanctioned law. What would have been the chances against the\nappearance of the excepted case, immediately prior to its occurrence?\u201d\n\n  [607] *Ninth Bridgewater Treatise*, p.\u00a0140.\n\nAs Babbage further showed,[608] a calculating engine, after proceeding\nthrough any required number of motions according to a first law, may\nbe made suddenly to suffer a change, so that it shall then commence\nto calculate according to a wholly new law. After giving the natural\nnumbers for a finite time, it might suddenly begin to give triangular,\nor square, or cube numbers, and these changes might be conceived\ntheoretically as occurring time after time. Now if such occurrences can\nbe designed and foreseen by a human artist, it is surely within the\ncapacity of the Divine Artist to provide for analogous changes of law\nin the mechanism of the atom, or the construction of the heavens.\n\n  [608] *Ibid.* pp.\u00a034\u201343.\n\nPhysical science, so far as its highest speculations can be trusted,\ngives some indication of a change of law in the past history of the\nUniverse. According to Sir W. Thomson\u2019s deductions from Fourier\u2019s\n*Theory of Heat*, we can trace down the dissipation of heat by\nconduction and radiation to an infinitely distant time when all\nthings will be uniformly cold. But we cannot similarly trace the\nheat-history of the Universe to an infinite distance in the past.\nFor a certain negative value of the time the formul\u00e6 give impossible\nvalues, indicating that there was some initial distribution of heat\nwhich could not have resulted, according to known laws of nature,[609]\nfrom any previous distribution.[610] There are other cases in which a\nconsideration of the dissipation of energy leads to the conception of\na limit to the antiquity of the present order of things.[611] Human\nscience, of course, is fallible, and some oversight or erroneous\nsimplification in these theoretical calculations may afterwards be\ndiscovered; but as the present state of scientific knowledge is the\nonly ground on which erroneous inferences from the uniformity of nature\nand the supposed reign of law are founded, I am right in appealing to\nthe present state of science in opposition to these inferences. Now\nthe theory of heat places us in the dilemma either of believing in\nCreation at an assignable date in the past, or else of supposing that\nsome inexplicable change in the working of natural laws then took\nplace. Physical science gives no countenance to the notion of infinite\nduration of matter in one continuous course of existence. And if in\ntime past there has been a discontinuity of law, why may there not be\na similar event awaiting the world in the future? Infinite ingenuity\ncould have implanted some agency in matter so that it might never yet\nhave made its tremendous powers manifest. We have a very good theory\nof the conservation of energy, but the foremost physicists do not\ndeny that there may possibly be forms of energy, neither kinetic nor\npotential, and therefore of unknown nature.[612]\n\n  [609] Professor Clifford, in his most interesting lecture on \u201cThe\n  First and Last Catastrophe\u201d (*Fortnightly Review*, April 1875,\n  p.\u00a0480, reprint by the Sunday Lecture Society, p.\u00a024), objects that I\n  have erroneously substituted \u201cknown laws of nature\u201d for \u201cknown laws\n  of conduction of heat.\u201d I quite admit the error, without admitting\n  all the conclusions which Professor Clifford proceeds to draw; but I\n  maintain the paragraph unchanged, in order that it may be discussed\n  in the Preface.\n\n  [610] Tait\u2019s *Thermodynamics*, p.\u00a038. *Cambridge Mathematical\n  Journal*, vol.\u00a0iii. p.\u00a0174.\n\n  [611] Clerk Maxwell\u2019s *Theory of Heat*, p.\u00a0245.\n\n  [612] Maxwell\u2019s *Theory of Heat*, p.\u00a092.\n\nWe can imagine reasoning creatures dwelling in a world where the\natmosphere was a mixture of oxygen and inflammable gas like the\nfire-damp of coal-mines. If devoid of fire, they might have lived\nthrough long ages unconscious of the tremendous forces which a single\nspark would call into play. In the twinkling of an eye new laws might\ncome into action, and the poor reasoning creatures, so confident about\ntheir knowledge of the reign of law in their world, would have no time\nto speculate upon the overthrow of all their theories. Can we with our\nfinite knowledge be sure that such an overthrow of our theories is\nimpossible?\n\n\n*The Ambiguous Expression, \u201cUniformity of Nature.\u201d*\n\nI have asserted that serious misconception arises from an erroneous\ninterpretation of the expression Uniformity of Nature. Every law of\nnature is the statement of a certain uniformity observed to exist among\nphenomena, and since the laws of nature are invariably obeyed, it seems\nto follow that the course of nature itself is uniform, so that we can\nsafely judge of the future by the present. This inference is supported\nby some of the results of physical astronomy. Laplace proved that\nthe planetary system is stable, so that no perturbation which planet\nproduces upon planet can become so great as to cause disruption and\npermanent alteration of the planetary orbits. A full comprehension of\nthe law of gravity shows that all such disturbances are essentially\nperiodic, so that after the lapse of millions of years the planets will\nreturn to the same relative positions, and a new cycle of disturbances\nwill then commence.\n\nAs other branches of science progress, we seem to gain assurance\nthat no great alteration of the world\u2019s condition is to be expected.\nConflict with a comet has long been the cause of fear, but now it is\ncredibly asserted that we have passed through a comet\u2019s tail without\nthe fact being known at the time, or manifested by any more serious a\nphenomenon than a slight luminosity of the sky. More recently still\nthe earth is said to have touched the comet Biela, and the only result\nwas a beautiful and perfectly harmless display of meteors. A decrease\nin the heating power of the sun seems to be the next most probable\ncircumstance from which we might fear the extinction of life on the\nearth. But calculations founded on reasonable physical data show\nthat no appreciable change can be going on, and experimental data to\nindicate a change are wholly wanting. Geological investigations show\nindeed that there have been extensive variations of climate in past\ntimes; vast glaciers and icebergs have swept over the temperate regions\nat one time, and tropical vegetation has flourished near the poles\nat another time. But here again the vicissitudes of climate assume a\nperiodic character, so that the stability of the earth\u2019s condition does\nnot seem to be threatened.\n\nAll these statements may be reasonable, but they do not establish the\nUniformity of Nature in the sense that extensive alterations or sudden\ncatastrophes are impossible. In the first place, Laplace\u2019s theory of\nthe stability of the planetary system is of an abstract character, as\npaying regard to nothing but the mutual gravitation of the planetary\nbodies and the sun. It overlooks several physical causes of change\nand decay in the system which were not so well known in his day as at\npresent, and it also presupposes the absence of any interruption of the\ncourse of things by conflict with foreign astronomical bodies.\n\nIt is now acknowledged by astronomers that there are at least two ways\nin which the *vis viva* of the planets and satellites may suffer loss.\nThe friction of the tides upon the earth produces a small quantity\nof heat which is radiated into space, and this loss of energy must\nresult in a decrease of the rotational velocity, so that ultimately\nthe terrestrial day will become identical with the year, just as the\nperiods of revolution of the moon upon its axis and around the earth\nhave already become equal. Secondly, there can be little doubt that\ncertain manifestations of electricity upon the earth\u2019s surface depend\nupon the relative motions of the planets and the sun, which give\nrise to periods of increased intensity. Such electrical phenomena\nmust result in the production and dissipation of heat, the energy\nof which must be drawn, partially at least, from the moving bodies.\nThis effect is probably identical (p.\u00a0570) with the loss of energy of\ncomets attributed to the so-called resisting medium. But whatever be\nthe theoretical explanation of these phenomena, it is almost certain\nthat there exists a tendency to the dissipation of the energy of the\nplanetary system, which will, in the indefinite course of time, result\nin the fall of the planets into the sun.\n\nIt is hardly probable, however, that the planetary system will be left\nundisturbed throughout the enormous interval of time required for the\ndissipation of its energy in this way. Conflict with other bodies is\nso far from being improbable, that it becomes approximately certain\nwhen we take very long intervals of time into account. As regards\ncometary conflicts, I am by no means satisfied with the negative\nconclusions drawn from the remarkable display on the evening of the\n27th of November, 1872. We may often have passed through the tail of a\ncomet, the light of which is probably an electrical manifestation no\nmore substantial than the aurora borealis. Every remarkable shower of\nshooting stars may also be considered as proceeding from a cometary\nbody, so that we may be said to have passed through the thinner parts\nof innumerable comets. But the earth has probably never passed, in\ntimes of which we have any record, through the nucleus of a comet,\nwhich consists perhaps of a dense swarm of small meteorites. We can\nonly speculate upon the effects which might be produced by such a\nconflict, but it would probably be a much more serious event than any\nyet registered in history. The probability of its occurrence, too,\ncannot be assigned; for though the probability of conflict with any one\ncometary nucleus is almost infinitesimal, yet the number of comets is\nimmensely great (p.\u00a0408).\n\nIt is far from impossible, again, that the planetary system may be\ninvaded by bodies of greater mass than comets. The sun seems to be\nplaced in so extensive a portion of empty space that its own proper\nmotion would not bring it to the nearest known star (\u03b1 Centauri) in\nless than 139,200 years. But in order to be sure that this interval of\nundisturbed life is granted to our globe, we must prove that there are\nno stars moving so as to meet us, and no dark bodies of considerable\nsize flying through intervening space unknown to us. The intrusion\nof comets into our system, and the fact that many of them have\nhyperbolic paths, is sufficient to show that the surrounding parts of\nspace are occupied by multitudes of dark bodies of some size. It is\nquite probable that small suns may have cooled sufficiently to become\nnon-luminous; for even if we discredit the theory that the variation of\nbrightness of periodic stars is due to the revolution of dark companion\nstars, yet there is in our own globe an unquestionable example of a\nsmaller body which has cooled below the luminous point.\n\nAltogether, then, it is a mere assumption that the uniformity of nature\ninvolves the unaltered existence of our own globe. There is no kind\nof catastrophe which is too great or too sudden to be theoretically\nconsistent with the reign of law. For all that our science can tell,\nhuman history may be closed in the next instant of time. The world may\nbe dashed to pieces against a wandering star; it may be involved in a\nnebulous atmosphere of hydrogen to be exploded a second afterwards; it\nmay be scorched up or dissipated into vapour by some great explosion in\nthe sun; there might even be within the globe itself some secret cause\nof disruption, which only needs time for its manifestation.\n\nThere are some indications, as already noticed (p.\u00a0660), that violent\ndisturbances have actually occurred in the history of the solar system.\nOlbers sought for the minor planets on the supposition that they were\nfragments of an exploded planet, and he was rewarded with the discovery\nof some of them. The retrograde motion of the satellites of the more\ndistant planets, the abnormal position of the poles of Uranus and the\nexcessive distance of Neptune, are other indications of some violent\nevent, of which we have no other evidence. I adduce all these facts\nand arguments, not to show that there is any considerable probability,\nas far as we can judge, of interruption within the scope of human\nhistory, but to prove that the Uniformity of Nature is theoretically\nconsistent with the most unexpected events of which we can form a\nconception.\n\n\n*Possible States of the Universe.*\n\nWhen we give the rein to scientific imagination, it becomes apparent\nthat conflict of body with body must not be regarded as the rare\nexception, but as the general rule and the inevitable fate of each\nstar system. So far as we can trace out the results of the law of\ngravitation, and of the dissipation of energy, the universe must be\nregarded as undergoing gradual condensation into a single cold solid\nbody of gigantic dimensions. Those who so frequently use the expression\nUniformity of Nature seem to forget that the Universe might exist\nconsistently with the laws of nature in the most diverse conditions. It\nmight consist, on the one hand, of a glowing nebulous mass of gaseous\nsubstances. The heat might be so intense that all elements, even carbon\nand silicon, would be in the state of gas, and all atoms, of whatever\nnature, would be flying about in chemical independence, diffusing\nthemselves almost uniformly in the neighbouring parts of space. There\nwould then be no life, unless we can apply that name to the passage\nthrough each part of space of similar average trains of atoms, the\nparticular succession of atoms being governed only by the theory of\nprobability, and the law of divergence from a mean exhibited in the\nArithmetical Triangle. Such a universe would correspond partially to\nthe Lucretian rain of atoms, and to that nebular hypothesis out of\nwhich Laplace proposed philosophically to explain the evolution of the\nplanetary system.\n\nAccording to another extreme supposition, the intense heat-energy of\nthis nebulous mass might be radiated away into the unknown regions\nof outer space. The attraction of gravity would exert itself between\neach two particles, and the energy of motion thence arising would,\nby incessant conflicts, be resolved into heat and dissipated.\nInconceivable ages might be required for the completion of this\nprocess, but the dissipation of energy thus proceeding could end only\nin the production of a cold and motionless universe. The relation of\ncause and effect, as we see it manifested in life and growth, would\ndegenerate into the constant existence of every particle in a fixed\nposition relative to every other particle. Logical and geometrical\nresemblances would still exist between atoms, and between groups of\natoms crystallised in their appropriate forms for evermore. But time,\nthe great variable, would bring no variation, and as to human hopes and\ntroubles, they would have gone to eternal rest.\n\nScience is not really adequate to proving that such is the inevitable\nfate of the universe, for we can seldom trust our best-established\ntheories far from their data. Nevertheless, the most probable\nspeculations which we can form as to the history, especially of our own\nplanetary system, is that it originated in a heated revolving nebulous\nmass of gas, and is in a state of excessively slow progress towards\nthe cold and stony condition. Other speculative hypotheses might\ndoubtless be entertained. Every hypothesis is pressed by difficulties.\nIf the whole universe be cooling, whither does the heat go? If we are\nto get rid of it entirely, outer space must be infinite in extent, so\nthat it shall never be stopped and reflected back. But not to speak\nof metaphysical difficulties, if the medium of heat undulations be\ninfinite in extent, why should not the material bodies placed in\nit be infinite also in number and aggregate mass? It is apparent\nthat we are venturing into speculations which surpass our powers of\nscientific inference. But then I am arguing negatively; I wish to show\nthat those who speak of the uniformity of nature, and the reign of\nlaw, misinterpret the meaning involved in those expressions. Law is\nnot inconsistent with extreme diversity, and, so far as we can read\nthe history of this planetary system, it did probably originate in\nheated nebulous matter, and man\u2019s history forms but a brief span in\nits progress towards the cold and stony condition. It is by doubtful\nand speculative hypotheses alone that we can avoid such a conclusion,\nand I depart least from undoubted facts and well-established laws\nwhen I assert that, whatever uniformities may underlie the phenomena\nof nature, constant variety and ever-progressing change is the real\noutcome.\n\n\n*Speculations on the Reconcentration of Energy.*\n\nThere are unequivocal indications, as I have said, that the material\nuniverse, as we at present see it, is progressing from some act of\ncreation, or some discontinuity of existence of which the date may be\napproximately fixed by scientific inference. It is progressing towards\na state in which the available energy of matter will be dissipated\nthrough infinite surrounding space, and all matter will become cold\nand lifeless. This constitutes, as it were, the historical period of\nphysical science, that over which our scientific foresight may more\nor less extend. But in this, as in other cases, we have no right to\ninterpret our experience negatively, so as to infer that because the\npresent state of things began at a particular time, there was no\nprevious existence. It may be that the present period of material\nexistence is but one of an indefinite series of like periods. All that\nwe can see, and feel, and infer, and reason about may be, as it were,\nbut a part of one single pulsation in the existence of the universe.\n\nAfter Sir W. Thomson had pointed out the preponderating tendency\nwhich now seems to exist towards the conversion of all energy into\nheat-energy, and its equal diffusion by radiation throughout space,\nthe late Professor Rankine put forth a remarkable speculation.[613] He\nsuggested that the ethereal, or, as I have called it, the *adamantine*\nmedium in which all the stars exist, and all radiation takes place, may\nhave bounds, beyond which only empty space exists. All heat undulations\nreaching this boundary will be totally reflected, according to the\ntheory of undulations, and will be reconcentrated into foci situated in\nvarious parts of the medium. Whenever a cold and extinct star happens\nto pass through one of these foci, it will be instantly ignited and\nresolved by intense heat into its constituent elements. Discontinuity\nwill occur in the history of that portion of matter, and the star will\nbegin its history afresh with a renewed store of energy.\n\n  [613] *Report of the British Association* (1852), Report of Sections,\n  p.\u00a012.\n\nThis is doubtless a mere speculation, practically incapable of\nverification by observation, and almost free from restrictions afforded\nby present knowledge. We might attribute various shapes to the\nadamantine medium, and the consequences would be various. But there\nis this value in such speculations, that they draw attention to the\nfiniteness of our knowledge. We cannot deny the possible truth of such\nan hypothesis, nor can we place a limit to the scientific imagination\nin the framing of other like hypotheses. It is impossible, indeed, to\nfollow out our scientific inferences without falling into speculation.\nIf heat be radiated into outward space, it must either proceed *ad\ninfinitum*, or it must be stopped somewhere. In the latter case we fall\nupon Rankine\u2019s hypothesis. But if the material universe consist of a\nfinite collection of heated matter situated in a finite portion of an\ninfinite adamantine medium, then either this universe must have existed\nfor a finite time, or else it must have cooled down during the infinity\nof past time indefinitely near to the absolute zero of temperature. I\nobjected to Lucretius\u2019 argument against the destructibility of matter,\nthat we have no knowledge whatever of the laws according to which\nit would undergo destruction. But we do know the laws according to\nwhich the dissipation of heat appears to proceed, and the conclusion\ninevitably is that a finite heated material body placed in a perfectly\ncold infinitely extended medium would in an infinite time sink to\nzero of temperature. Now our own world is not yet cooled down near to\nzero, so that physical science seems to place us in the dilemma of\nadmitting either the finiteness of past duration of the world, or else\nthe finiteness of the portion of medium in which we exist. In either\ncase we become involved in metaphysical and mechanical difficulties\nsurpassing our mental powers.\n\n\n*The Divergent Scope for New Discovery.*\n\nIn the writings of some recent philosophers, especially of Auguste\nComte, and in some degree John Stuart Mill, there is an erroneous\nand hurtful tendency to represent our knowledge as assuming an\napproximately complete character. At least these and many other\nwriters fail to impress upon their readers a truth which cannot be\ntoo constantly borne in mind, namely, that the utmost successes which\nour scientific method can accomplish will not enable us to comprehend\nmore than an infinitesimal fraction of what there doubtless is to\ncomprehend.[614] Professor Tyndall seems to me open to the same charge\nin a less degree. He remarks[615] that we can probably never bring\nnatural phenomena completely under mathematical laws, because the\napproach of our sciences towards completeness may be asymptotic, so\nthat however far we may go, there may still remain some facts not\nsubject to scientific explanation. He thus likens the supply of novel\nphenomena to a convergent series, the earlier and larger terms of which\nhave been successfully disposed of, so that comparatively minor groups\nof phenomena alone remain for future investigators to occupy themselves\nupon.\n\n  [614] Mr. C.\u00a0J. Monroe objects that in this statement I do injustice\n  to Comte, who, he thinks, did impress upon his readers the inadequacy\n  of our mental powers compared with the vastness of the subject matter\n  of science. The error of Comte, he holds, was in maintaining that\n  science had been carried about as far as it is worth while to carry\n  it, which is a different matter. In either case, Comte\u2019s position is\n  so untenable that I am content to leave the question undecided.\n\n  [615] *Fragments of Science*, p.\u00a0362.\n\nOn the contrary, as it appears to me, the supply of new and unexplained\nfacts is divergent in extent, so that the more we have explained, the\nmore there is to explain. The further we advance in any generalisation,\nthe more numerous and intricate are the exceptional cases still\ndemanding further treatment. The experiments of Boyle, Mariotte,\nDalton, Gay-Lussac, and others, upon the physical properties of gases,\nmight seem to have exhausted that subject by showing that all gases\nobey the same laws as regards temperature, pressure, and volume. But\nin reality these laws are only approximately true, and the divergences\nafford a wide and quite unexhausted field for further generalisation.\nThe recent discoveries of Professor Andrews have summed up some of\nthese exceptional facts under a wider generalisation, but in reality\nthey have opened to us vast new regions of interesting inquiry,\nand they leave wholly untouched the question why one gas behaves\ndifferently from another.\n\nThe science of crystallography is that perhaps in which the most\nprecise and general laws have been detected, but it would be untrue\nto assert that it has lessened the area of future discovery. We can\nshow that each one of the seven or eight hundred forms of calcite is\nderivable by geometrical modifications from an hexagonal prism; but\nwho has attempted to explain the molecular forces producing these\nmodifications, or the chemical conditions in which they arise? The law\nof isomorphism is an important generalisation, for it establishes a\ngeneral resemblance between the forms of crystallisation of natural\nclasses of elements. But if we examine a little more closely we find\nthat these forms are only approximately alike, and the divergence\npeculiar to each substance is an unexplained exception.\n\nBy many similar illustrations it might readily be shown that in\nwhatever direction we extend our investigations and successfully\nharmonise a few facts, the result is only to raise up a host of other\nunexplained facts. Can any scientific man venture to state that there\nis less opening now for new discoveries than there was three centuries\nago? Is it not rather true that we have but to open a scientific book\nand read a page or two, and we shall come to some recorded phenomenon\nof which no explanation can yet be given? In every such fact there is a\npossible opening for new discoveries, and it can only be the fault of\nthe investigator\u2019s mind if he can look around him and find no scope for\nthe exercise of his faculties.\n\n\n*Infinite Incompleteness of the Mathematical Sciences.*\n\nThere is one privilege which a certain amount of knowledge should\nconfer; it is that of becoming aware of the weakness of our powers\ncompared with the tasks which they might undertake if stronger. To the\npoor savage who cannot count twenty the arithmetical accomplishments of\nthe schoolboy are miraculously great. The schoolboy cannot comprehend\nthe vastly greater powers of the student, who has acquired facility in\nalgebraic processes. The student can but look with feelings of surprise\nand reverence at the powers of a Newton or a Laplace. But the question\nat once suggests itself, Do the powers of the highest human intellect\nbear a finite ratio to the things which are to be understood and\ncalculated? How many further steps must we take in the rise of mental\nability and the extension of mathematical methods before we begin to\nexhaust the knowable?\n\nI am inclined to find fault with mathematical writers because they\noften exult in what they can accomplish, and omit to point out that\nwhat they do is but an infinitely small part of what might be done.\nThey exhibit a general inclination, with few exceptions, not to do\nso much as mention the existence of problems of an impracticable\ncharacter. This may be excusable as far as the immediate practical\nresult of their researches is in question, but the custom has the\neffect of misleading the general public into the fallacious notion\nthat mathematics is a *perfect* science, which accomplishes what it\nundertakes in a complete manner. On the contrary, it may be said that\nif a mathematical problem were selected by chance out of the whole\nnumber which might be proposed, the probability is infinitely slight\nthat a human mathematician could solve it. Just as the numbers we can\ncount are nothing compared with the numbers which might exist, so the\naccomplishments of a Laplace or a Lagrange are, as it were, the little\ncorner of the multiplication-table, which has really an infinite extent.\n\nI have pointed out that the rude character of our observations prevents\nus from being aware of the greater number of effects and actions\nin nature. It must be added that, if we perceive them, we should\nusually be incapable of including them in our theories from want of\nmathematical power. Some persons may be surprised that though nearly\ntwo centuries have elapsed since the time of Newton\u2019s discoveries, we\nhave yet no general theory of molecular action. Some approximations\nhave been made towards such a theory. Joule and Clausius have measured\nthe velocity of gaseous atoms, or even determined the average distance\nbetween the collisions of atom and atom. Thomson has approximated to\nthe number of atoms in a given bulk of substance. Rankine has formed\nsome reasonable hypotheses as to the actual constitution of atoms. It\nwould be a mistake to suppose that these ingenious results of theory\nand experiment form any appreciable approach to a complete solution\nof molecular motions. There is every reason to believe, judging from\nthe spectra of the elements, their atomic weights and other data, that\nchemical atoms are very complicated structures. An atom of pure iron\nis probably a far more complicated system than that of the planets\nand their satellites. A compound atom may perhaps be compared with\na stellar system, each star a minor system in itself. The smallest\nparticle of solid substance will consist of a great number of such\nstellar systems united in regular order, each bounded by the other,\ncommunicating with it in some manner yet wholly incomprehensible. What\nare our mathematical powers in comparison with this problem?\n\nAfter two centuries of continuous labour, the most gifted men have\nsucceeded in calculating the mutual effects of three bodies each\nupon the other, under the simple hypothesis of the law of gravity.\nConcerning these calculations we must further remember that they are\npurely approximate, and that the methods would not apply where four or\nmore bodies are acting, and all produce considerable effects upon each\nother. There is reason to believe that each constituent of a chemical\natom goes through an orbit in the millionth part of the twinkling of\nan eye. In each revolution it is successively or simultaneously under\nthe influence of many other constituents, or possibly comes into\ncollision with them. It is no exaggeration to say that mathematicians\nhave the least notion of the way in which they could successfully\nattack so difficult a problem of forces and motions. As Herschel has\nremarked,[616] each of these particles is for ever solving differential\nequations, which, if written out in full, might belt the earth.\n\n  [616] *Familiar Lectures on Scientific Subjects*, p.\u00a0458.\n\nSome of the most extensive calculations ever made were those required\nfor the reduction of the measurements executed in the course of the\nTrigonometrical Survey of Great Britain. The calculations arising out\nof the principal triangulation occupied twenty calculators during\nthree or four years, in the course of which the computers had to solve\nsimultaneous equations involving seventy-seven unknown quantities.\nThe reduction of the levellings required the solution of a system of\nninety-one equations. But these vast calculations present no approach\nwhatever to what would be requisite for the complete treatment of\nany one physical problem. The motion of glaciers is supposed to\nbe moderately well understood in the present day. A glacier is a\nviscid, slowly yielding mass, neither absolutely solid nor absolutely\nrigid, but it is expressly remarked by Forbes,[617] that not even an\napproximate solution of the mathematical conditions of such a moving\nmass can yet be possible. \u201cEvery one knows,\u201d he says, \u201cthat such\nproblems are beyond the compass of exact mathematics;\u201d but though\nmathematicians may know this, they do not often enough impress that\nknowledge on other people.\n\n  [617] *Philosophical Magazine*, 3rd Series, vol.\u00a0xxvi. p.\u00a0406.\n\nThe problems which are solved in our mathematical books consist of\na small selection of those which happen from peculiar conditions to\nbe solvable. But the very simplest problem in appearance will often\ngive rise to impracticable calculations. Mr. Todhunter[618] seems to\nblame Condorcet, because in one of his memoirs he mentions a problem\nto solve which would require a great and impracticable number of\nsuccessive integrations. Now, if our mathematical sciences are to cope\nwith the problems which await solution, we must be prepared to effect\nan unlimited number of successive integrations; yet at present, and\nalmost beyond doubt for ever, the probability that an integration taken\nhaphazard will come within our powers is exceedingly small.\n\n  [618] *History of the Theory of Probability*, p.\u00a0398.\n\nIn some passages of that remarkable work, the *Ninth Bridgewater\nTreatise* (pp.\u00a0113\u2013115), Babbage has pointed out that if we had power\nto follow and detect the minutest effects of any disturbance, each\nparticle of existing matter would furnish a register of all that has\nhappened. \u201cThe track of every canoe--of every vessel that has yet\ndisturbed the surface of the ocean, whether impelled by manual force or\nelemental power, remains for ever registered in the future movement of\nall succeeding particles which may occupy its place. The furrow which\nit left is, indeed, instantly filled up by the closing waters; but they\ndraw after them other and larger portions of the surrounding element,\nand these again, once moved, communicate motion to others in endless\nsuccession.\u201d We may even say that \u201cThe air itself is one vast library,\non whose pages are for ever written all that man has ever said or\neven whispered. There, in their mutable but unerring characters, mixed\nwith the earliest as well as the latest sighs of mortality, stand for\never recorded, vows unredeemed, promises unfulfilled, perpetuating in\nthe united movements of each particle the testimony of man\u2019s changeful\nwill.\u201d\n\nWhen we read reflections such as these, we may congratulate ourselves\nthat we have been endowed with minds which, rightly employed, can\nform some estimate of their incapacity to trace out and account for\nall that proceeds in the simpler actions of material nature. It ought\nto be added that, wonderful as is the extent of physical phenomena\nopen to our investigation, intellectual phenomena are yet vastly more\nextensive. Of this I might present one satisfactory proof were space\navailable by pointing out that the mathematical functions employed\nin the calculations of physical science form an infinitely small\nfraction of the functions which might be invented. Common trigonometry\nconsists of a great series of useful formul\u00e6, all of which arise out\nof the relation of the sine and cosine expressed in one equation,\nsin\u00a0^{2}*x*\u00a0+\u00a0cos\u00a0^{2}*x*\u00a0=\u00a01. But this is not the only trigonometry\nwhich may exist; mathematicians also recognise hyperbolic trigonometry,\nof which the fundamental equation is cos\u00a0^{2}*x*\u00a0-\u00a0sin\u00a0^{2}*x*\u00a0=\u00a01. De\nMorgan has pointed out that the symbols of ordinary algebra form but\nthree of an interminable series of conceivable systems.[619] As the\nlogarithmic operation is to addition or addition to multiplication, so\nis the latter to a higher operation, and so on without limit.\n\n  [619] *Trigonometry and Double Algebra*, chap. ix.\n\nWe may rely upon it that immense, and to us inconceivable, advances\nwill be made by the human intellect, in the absence of any catastrophe\nto the species or the globe. Within historical periods we can trace\nthe rise of mathematical science from its simplest germs. We can\nprove our descent from ancestors who counted only on their fingers.\nHow infinitely is a Newton or a Laplace above those simple savages.\nPythagoras is said to have sacrificed a hecatomb when he discovered the\nforty-seventh proposition of Euclid, and the occasion was worthy of\nthe sacrifice. Archimedes was beside himself when he first perceived\nhis beautiful mode of determining specific gravities. Yet these great\ndiscoveries are the commonplaces of our school books. Step by step we\ncan trace upwards the acquirement of new mental powers. What could be\nmore wonderful than Napier\u2019s discovery of logarithms, a new mode of\ncalculation which has multiplied perhaps a hundredfold the working\npowers of every computer, and has rendered easy calculations which were\nbefore impracticable? Since the time of Newton and Leibnitz worlds of\nproblems have been solved which before were hardly conceived as matters\nof inquiry. In our own day extended methods of mathematical reasoning,\nsuch as the system of quaternions, have been brought into existence.\nWhat intelligent man will doubt that the recondite speculations of a\nCayley, a Sylvester, or a Clifford may lead to some new development of\nnew mathematical power, at the simplicity of which a future age will\nwonder, and yet wonder more that to us they were so dark and difficult.\nMay we not repeat the words of Seneca: \u201cVeniet tempus, quo ista qu\u00e6\nnunc latent, in lucem dies extrahat, et longioris \u00e6vi diligentia:\nad inquisitionem tantorum \u00e6tas una non sufficit. Veniet tempus, quo\nposteri nostri tam aperta nos nescisse mirentur.\u201d\n\n\n*The Reign of Law in Mental and Social Phenomena.*\n\nAfter we pass from the so-called physical sciences to those which\nattempt to investigate mental and social phenomena, the same general\nconclusions will hold true. No one will be found to deny that there\nare certain uniformities of thinking and acting which can be detected\nin reasoning beings, and so far as we detect such laws we successfully\napply scientific method. But those who attempt to establish social or\nmoral sciences soon become aware that they are dealing with subjects\nof enormous perplexity. Take as an instance the science of political\neconomy. If a science at all, it must be a mathematical science,\nbecause it deals with quantities of commodities. But as soon as we\nattempt to draw out the equations expressing the laws of demand and\nsupply, we discover that they have a complexity entirely surpassing\nour powers of mathematical treatment. We may lay down the general form\nof the equations, expressing the demand and supply for two or three\ncommodities among two or three trading bodies, but all the functions\ninvolved are so complicated in character that there is not much fear\nof scientific method making rapid progress in this direction. If such\nbe the prospects of a comparatively formal science, like political\neconomy, what shall we say of moral science? Any complete theory of\nmorals must deal with quantities of pleasure and pain, as Bentham\npointed out, and must sum up the general tendency of each kind of\naction upon the good of the community. If we are to apply scientific\nmethod to morals, we must have a calculus of moral effects, a kind\nof physical astronomy investigating the mutual perturbations of\nindividuals. But as astronomers have not yet fully solved the problem\nof three gravitating bodies, when shall we have a solution of the\nproblem of three moral bodies?\n\nThe sciences of political economy and morality are comparatively\nabstract and general, treating mankind from simple points of view,\nand attempting to detect general principles of action. They are to\nsocial phenomena what the abstract sciences of chemistry, heat, and\nelectricity are to the concrete science of meteorology. Before we\ncan investigate the actions of any aggregate of men, we must have\nfairly mastered all the more abstract sciences applying to them,\nsomewhat in the way that we have acquired a fair comprehension of\nthe simpler truths of chemistry and physics. But all our physical\nsciences do not enable us to predict the weather two days hence with\nany great probability, and the general problem of meteorology is almost\nunattempted as yet. What shall we say then of the general problem of\nsocial science, which shall enable us to predict the course of events\nin a nation?\n\nSeveral writers have proposed to lay the foundations of the science\nof history. Buckle undertook to write the *History of Civilisation\nin England*, and to show how the character of a nation could be\nexplained by the nature of the climate and the fertility of the soil.\nHe omitted to explain the contrast between the ancient Greek nation\nand the present one; there must have been an extraordinary revolution\nin the climate or the soil. Auguste Comte detected the simple laws\nof the course of development through which nations pass. There are\nalways three phases of intellectual condition,--the theological, the\nmetaphysical, and the positive; applying this general law of progress\nto concrete cases, Comte was enabled to predict that in the hierarchy\nof European nations, Spain would necessarily hold the highest place.\nSuch are the parodies of science offered to us by the *positive*\nphilosophers.\n\nA science of history in the true sense of the term is an absurd notion.\nA nation is not a mere sum of individuals whom we can treat by the\nmethod of averages; it is an organic whole, held together by ties of\ninfinite complexity. Each individual acts and re-acts upon his smaller\nor greater circle of friends, and those who acquire a public position\nexert an influence on much larger sections of the nation. There will\nalways be a few great leaders of exceptional genius or opportunities,\nthe unaccountable phases of whose opinions and inclinations sway the\nwhole body. From time to time arise critical situations, battles,\ndelicate negotiations, internal disturbances, in which the slightest\nincidents may change the course of history. A rainy day may hinder a\nforced march, and change the course of a campaign; a few injudicious\nwords in a despatch may irritate the national pride; the accidental\ndischarge of a gun may precipitate a collision the effects of which\nwill last for centuries. It is said that the history of Europe depended\nat one moment upon the question whether the look-out man upon Nelson\u2019s\nvessel would or would not descry a ship of Napoleon\u2019s expedition\nto Egypt which was passing not far off. In human affairs, then,\nthe smallest causes may produce the greatest effects, and the real\napplication of scientific method is out of the question.\n\n\n*The Theory of Evolution.*\n\nProfound philosophers have lately generalised concerning the production\nof living forms and the mental and moral phenomena regarded as their\nhighest development. Herbert Spencer\u2019s theory of evolution purports\nto explain the origin of all specific differences, so that not even\nthe rise of a Homer or a Beethoven would escape from his broad\ntheories. The homogeneous is unstable and must differentiate itself,\nsays Spencer, and hence comes the variety of human institutions\nand characters. In order that a living form shall continue to exist\nand propagate its kind, says Darwin, it must be suitable to its\ncircumstances, and the most suitable forms will prevail over and\nextirpate those which are less suitable. From these fruitful ideas\nare developed theories of evolution and natural selection which go\nfar towards accounting for the existence of immense numbers of living\ncreatures--plants, and animals. Apparent adaptations of organs to\nuseful purposes, which Paley regarded as distinct products of creative\nintelligence, are now seen to follow as natural effects of a constantly\nacting tendency. Even man, according to these theories, is no distinct\ncreation, but rather an extreme case of brain development. His nearest\ncousins are the apes, and his pedigree extends backwards until it joins\nthat of the lowliest zoophytes.\n\nThe theories of Darwin and Spencer are doubtless not demonstrated; they\nare to some extent hypothetical, just as all the theories of physical\nscience are to some extent hypothetical, and open to doubt. Judging\nfrom the immense numbers of diverse facts which they harmonise and\nexplain, I venture to look upon the theories of evolution and natural\nselection in their main features as two of the most probable hypotheses\never proposed. I question whether any scientific works which have\nappeared since the *Principia* of Newton are comparable in importance\nwith those of Darwin and Spencer, revolutionising as they do all our\nviews of the origin of bodily, mental, moral, and social phenomena.\n\nGranting all this, I cannot for a moment admit that the theory of\nevolution will destroy theology. That theory embraces several laws\nor uniformities which are observed to be true in the production of\nliving forms; but these laws do not determine the size and figure of\nliving creatures, any more than the law of gravitation determines the\nmagnitudes and distances of the planets. Suppose that Darwin is correct\nin saying that man is descended from the Ascidians: yet the precise\nform of the human body must have been influenced by an infinite train\nof circumstances affecting the reproduction, growth, and health of the\nwhole chain of intermediate beings. No doubt, the circumstances being\nwhat they were, man could not be otherwise than he is, and if in any\nother part of the universe an exactly similar earth, furnished with\nexactly similar germs of life, existed, a race must have grown up there\nexactly similar to the human race.\n\nBy a different distribution of atoms in the primeval world a different\nseries of living forms on this earth would have been produced. From the\nsame causes acting according to the same laws, the same results will\nfollow; but from different causes acting according to the same laws,\ndifferent results will follow. So far as we can see, then, infinitely\ndiverse living creatures might have been created consistently with the\ntheory of evolution, and the precise reason why we have a backbone,\ntwo hands with opposable thumbs, an erect stature, a complex brain,\nabout 223 bones, and many other peculiarities, is only to be found in\nthe original act of creation. I do not, any less than Paley, believe\nthat the eye of man manifests design. I believe that the eye was\ngradually developed, and we can in fact trace its gradual development\nfrom the first germ of a nerve affected by light-rays in some simple\nzoophyte. In proportion as the eye became a more accurate instrument\nof vision, it enabled its possessor the better to escape destruction,\nbut the ultimate result must have been contained in the aggregate of\nthe causes, and these causes, as far as we can see, were subject to the\narbitrary choice of the Creator.\n\nAlthough Agassiz was clearly wrong in holding that every species of\nliving creature appeared on earth by the immediate intervention of\nthe Creator, which would amount to saying that no laws of connection\nbetween forms are discoverable, yet he seems to be right in asserting\nthat living forms are distinct from those produced by purely physical\ncauses. \u201cThe products of what are commonly called physical agents,\u201d\nhe says,[620] \u201care everywhere the same (*i.e.* upon the whole surface\nof the earth), and have always been the same (*i.e.* during all\ngeological periods); while organised beings are everywhere different\nand have differed in all ages. Between two such series of phenomena\nthere can be no causal or genetic connection.\u201d Living forms as we now\nregard them are essentially variable, but from constant mechanical\ncauses constant effects would ensue. If vegetable cells are formed\non geometrical principles being first spherical, and then by mutual\ncompression dodecahedral, then all cells should have similar forms. In\nthe Foraminifera and some other lowly organisms, we seem to observe the\nproduction of complex forms on geometrical principles. But from similar\ncauses acting according to similar laws only similar results could\nbe produced. If the original life germ of each creature is a simple\nparticle of protoplasm, unendowed with any distinctive forces, then the\nwhole of the complex phenomena of animal and vegetable life are effects\nwithout causes. Protoplasm may be chemically the same substance, and\nthe germ-cell of a man and of a fish may be apparently the same, so far\nas the microscope can decide; but if certain cells produce men, and\nothers as uniformly produce a species of fish, there must be a hidden\nconstitution determining the extremely different results. If this were\nnot so, the generation of every living creature from the uniform germ\nwould have to be regarded as a distinct act of creation.\n\n  [620] Agassiz, *Essay on Classification*, p.\u00a075.\n\nTheologians have dreaded the establishment of the theories of Darwin\nand Huxley and Spencer, as if they thought that those theories could\nexplain everything upon the purest mechanical and material principles,\nand exclude all notions of design. They do not see that those theories\nhave opened up more questions than they have closed. The doctrine of\nevolution gives a complete explanation of no single living form. While\nshowing the general principles which prevail in the variation of living\ncreatures, it only points out the infinite complexity of the causes\nand circumstances which have led to the present state of things. Any\none of Mr. Darwin\u2019s books, admirable though they all are, consists\nbut in the setting forth of a multitude of indeterminate problems. He\nproves in the most beautiful manner that each flower of an orchid is\nadapted to some insect which frequents and fertilises it, and these\nadaptations are but a few cases of those immensely numerous ones which\nhave occurred in the lives of plants and animals. But why orchids\nshould have been formed so differently from other plants, why anything,\nindeed, should be as it is, rather than in some of the other infinitely\nnumerous possible modes of existence, he can never show. The origin\nof everything that exists is wrapped up in the past history of the\nuniverse. At some one or more points in past time there must have been\narbitrary determinations which led to the production of things as they\nare.\n\n\n*Possibility of Divine Interference.*\n\nI will now draw the reader\u2019s attention to pages 149 to 152. I there\npointed out that all inductive inference involves the assumption that\nour knowledge of what exists is complete, and that the conditions of\nthings remain unaltered between the time of our experience and the\ntime to which our inferences refer. Recurring to the illustration\nof a ballot-box, employed in the chapter on the inverse method of\nprobabilities, we assume when predicting the probable nature of\nthe next drawing, firstly, that our previous drawings have been\nsufficiently numerous to give us knowledge of the contents of the\nbox; and, secondly, that no interference with the ballot-box takes\nplace between the previous and the next drawings. The results yielded\nby the theory of probability are quite plain. No finite number of\ncasual drawings can give us sure knowledge of the contents of the\nbox, so that, even in the absence of all disturbance, our inferences\nare merely the best which can be made, and do not approach to\ninfallibility. If, however, interference be possible, even the theory\nof probability ceases to be applicable, for, the amount and nature\nof that interference being arbitrary and unknown, there ceases to\nbe any connection between premises and conclusion. Many years of\nreflection have not enabled me to see the way of avoiding this hiatus\nin scientific certainty. The conclusions of scientific inference appear\nto be always of a hypothetical and provisional nature. Given certain\nexperience, the theory of probability yields us the true interpretation\nof that experience and is the surest guide open to us. But the best\ncalculated results which it can give are never absolute probabilities;\nthey are purely relative to the extent of our information. It seems to\nbe impossible for us to judge how far our experience gives us adequate\ninformation of the universe as a whole, and of all the forces and\nphenomena which can have place therein.\n\nI feel that I cannot in the space remaining at my command in the\npresent volume, sufficiently follow out the lines of thought suggested,\nor define with precision my own conclusions. This chapter contains\nmerely *Reflections* upon subjects of so weighty a character that I\nshould myself wish for many years--nay for more than a lifetime of\nfurther reflection. My purpose, as I have repeatedly said, is the\npurely negative one of showing that atheism and materialism are no\nnecessary results of scientific method. From the preceding reviews of\nthe value of our scientific knowledge, I draw one distinct conclusion,\nthat we cannot disprove the possibility of Divine interference in\nthe course of nature. Such interference might arise, so far as our\nknowledge extends, in two ways. It might consist in the disclosure\nof the existence of some agent or spring of energy previously\nunknown, but which effects a given purpose at a given moment. Like\nthe pre-arranged change of law in Babbage\u2019s imaginary calculating\nmachine, there may exist pre-arranged surprises in the order of\nnature, as it presents itself to us. Secondly, the same Power, which\ncreated material nature, might, so far as I can see, create additions\nto it, or annihilate portions which do exist. Such events are in a\ncertain sense inconceivable to us; yet they are no more inconceivable\nthan the existence of the world as it is. The indestructibility of\nmatter, and the conservation of energy, are very probable scientific\nhypotheses, which accord satisfactorily with experiments of scientific\nmen during a few years past, but it would be gross misconception of\nscientific inference to suppose that they are certain in the sense\nthat a proposition in geometry is certain. Philosophers no doubt hold\nthat *de nihilo nihil fit*, that is to say, their senses give them no\nmeans of imagining to the mind how creation can take place. But we are\non the horns of a trilemma; we must either deny that anything exists,\nor we must allow that it was created out of nothing at some moment of\npast time, or that it existed from eternity. The first alternative is\nabsurd; the other two seem to me equally conceivable.\n\n\n*Conclusion.*\n\nIt may seem that there is one point where our speculations must end,\nnamely where contradiction begins. The laws of Identity and Difference\nand Duality were the foundations from which we started, and they are,\nso far as I can see, the foundations which we can never quit without\ntottering. Scientific Method must begin and end with the laws of\nthought, but it does not follow that it will save us from encountering\ninexplicable, and at least apparently contradictory results. The nature\nof continuous quantity leads us into extreme difficulties. Any finite\nspace is composed of an infinite number of infinitely small spaces,\neach of which, again, is composed of an infinite number of spaces of\na second order of smallness; these spaces of the second order are\ncomposed, again, of infinitely small spaces of the third order. Even\nthese spaces of the third order are not absolute geometrical points\nanswering to Euclid\u2019s definition of a point, as position without\nmagnitude. Go on as far as we will, in the subdivision of continuous\nquantity, yet we never get down to the absolute point. Thus scientific\nmethod leads us to the inevitable conception of an infinite series\nof successive orders of infinitely small quantities. If so, there is\nnothing impossible in the existence of a myriad universes within the\ncompass of a needle\u2019s point, each with its stellar systems, and its\nsuns and planets, in number and variety unlimited. Science does nothing\nto reduce the number of strange things that we may believe. When fairly\npursued it makes absurd drafts upon our powers of comprehension and\nbelief.\n\nSome of the most precise and beautiful theorems in mathematical science\nseem to me to involve apparent contradiction. Can we imagine that a\npoint moving along a perfectly straight line towards the west would\never get round to the east and come back again, having performed, as\nit were, a circuit through infinite space, yet without ever diverging\nfrom a perfectly straight direction? Yet this is what happens to the\nintersecting point of two straight lines in the same plane, when one\nline revolves. The same paradox is exhibited in the hyperbola regarded\nas an infinite ellipse, one extremity of which has passed to an\ninfinite distance and come back in the opposite direction. A varying\nquantity may change its sign by passing either through zero or through\ninfinity. In the latter case there must be one intermediate value of\nthe variable for which the variant is indifferently negative infinity\nand positive infinity. Professor Clifford tells me that he has found\na mathematical function which approaches infinity as the variable\napproaches a certain limit; yet at the limit the function is finite!\nMathematicians may shirk difficulties, but they cannot make such\nresults of mathematical principles appear otherwise than contradictory\nto our common notions of space.\n\nThe hypothesis that there is a Creator at once all-powerful\nand all-benevolent is pressed, as it must seem to every candid\ninvestigator, with difficulties verging closely upon logical\ncontradiction. The existence of the smallest amount of pain and evil\nwould seem to show that He is either not perfectly benevolent, or\nnot all-powerful. No one can have lived long without experiencing\nsorrowful events of which the significance is inexplicable. But if we\ncannot succeed in avoiding contradiction in our notions of elementary\ngeometry, can we expect that the ultimate purposes of existence shall\npresent themselves to us with perfect clearness? I can see nothing to\nforbid the notion that in a higher state of intelligence much that is\nnow obscure may become clear. We perpetually find ourselves in the\nposition of finite minds attempting infinite problems, and can we be\nsure that where we see contradiction, an infinite intelligence might\nnot discover perfect logical harmony?\n\nFrom science, modestly pursued, with a due consciousness of the extreme\nfinitude of our intellectual powers, there can arise only nobler and\nwider notions of the purpose of Creation. Our philosophy will be an\naffirmative one, not the false and negative dogmas of Auguste Comte,\nwhich have usurped the name, and misrepresented the tendencies of a\ntrue *positive philosophy*. True science will not deny the existence of\nthings because they cannot be weighed and measured. It will rather lead\nus to believe that the wonders and subtleties of possible existence\nsurpass all that our mental powers allow us clearly to perceive. The\nstudy of logical and mathematical forms has convinced me that even\nspace itself is no requisite condition of conceivable existence.\nEverything, we are told by materialists, must be here or there,\nnearer or further, before or after. I deny this, and point to logical\nrelations as my proof.\n\nThere formerly seemed to me to be something mysterious in the\ndenominators of the binomial expansion (p.\u00a0190), which are reproduced\nin the natural constant \u03b5, or\n\n  1\u00a0+\u00a01/1\u00a0+\u00a01/(1\u00a0.\u00a02)\u00a0+\u00a01/(1\u00a0.\u00a02\u00a0.\u00a03)\u00a0+\u00a0...\n\nand in many results of mathematical analysis. I now perceive, as\nalready explained (pp.\u00a033, 160, 383), that they arise out of the fact\nthat the relations of space do not apply to the logical conditions\ngoverning the numbers of combinations as contrasted to those of\npermutations. So far am I from accepting Kant\u2019s doctrine that space is\na necessary form of thought, that I regard it as an accident, and an\nimpediment to pure logical reasoning. Material existences must exist in\nspace, no doubt, but intellectual existences may be neither in space\nnor out of space; they may have no relation to space at all, just as\nspace itself has no relation to time. For all that I can see, then,\nthere may be intellectual existences to which both time and space are\nnullities.\n\nNow among the most unquestionable rules of scientific method is\nthat first law that *whatever phenomenon is, is*. We must ignore no\nexistence whatever; we may variously interpret or explain its meaning\nand origin, but, if a phenomenon does exist, it demands some kind\nof explanation. If then there is to be competition for scientific\nrecognition, the world without us must yield to the undoubted existence\nof the spirit within. Our own hopes and wishes and determinations are\nthe most undoubted phenomena within the sphere of consciousness. If men\ndo act, feel, and live as if they were not merely the brief products of\na casual conjunction of atoms, but the instruments of a far-reaching\npurpose, are we to record all other phenomena and pass over these? We\ninvestigate the instincts of the ant and the bee and the beaver, and\ndiscover that they are led by an inscrutable agency to work towards\na distant purpose. Let us be faithful to our scientific method, and\ninvestigate also those instincts of the human mind by which man is led\nto work as if the approval of a Higher Being were the aim of life.\n\n\n\n\nINDEX.\n\n\n  Abacus, logical, 104;\n    arithmetical, 107;\n    Panchrestus, 182.\n\n  Aberration of light, 561;\n    systematic, 547.\n\n  Abscissio infiniti, 79, 713.\n\n  Abstract terms, 27;\n    number, 159.\n\n  Abstraction, 704;\n    logical, 25;\n    numerical, 158;\n    of indifferent circumstances, 97.\n\n  Accademia del Cimento, 427, 432, 436, 527.\n\n  Accident, logical, 700.\n\n  Accidental discovery, 529.\n\n  Achromatic lenses, 432.\n\n  Actinometer, 337.\n\n  Adamantine medium, 605, 751.\n\n  Adjectives, 14, 30, 31, 35;\n    indeterminate, 41.\n\n  Adrain, of New Brunswick, 375.\n\n  Affirmation, 44.\n\n  Agassiz, on genera, 726;\n    on creation of species, 763.\n\n  Agreement, 44.\n\n  Airy, Sir George Biddell, on perpetual motion, 223;\n    new property of sphere, 232;\n    pendulum experiments, 291, 304, 348, 567;\n    standard clock, 353;\n    book on *Errors of Observation*, 395;\n    tides, 488;\n    extra-polation, 495;\n    Thales\u2019 eclipse, 537;\n    interference of light, 539;\n    density of earth, 291.\n\n  Alchemists, 505;\n    how misled, 428.\n\n  Algebra, 123, 155, 164;\n    Diophantine, 631.\n\n  Algebraic, equations, 123;\n    geometry, 633.\n\n  Allotropic state, 663, 670.\n\n  Alloys, possible number, 191;\n    properties, 528.\n\n  Alphabet, the Logical, 93, 104, 125;\n    Morse, 193.\n\n  Alphabet, permutations of letters of the, 174, 179.\n\n  Alphabetic indexes, 714.\n\n  Alternative relations, 67;\n    exclusive and unexclusive, 205.\n\n  Amp\u00e8re, electricity, 547;\n    classification, 679.\n\n  Anagrams, 128.\n\n  Analogy, 627;\n    of logical and numerical terms, 160;\n    and generalisation, 596;\n    in mathematical sciences, 631;\n    in theory of undulations, 635;\n    in astronomy, 638;\n    failure of, 641.\n\n  Analysis, logical, 122.\n\n  Andrews, Prof. Thomas, experiments on gaseous state, 71, 613, 665, 753.\n\n  Angstr\u00f6m, on spectrum, 424.\n\n  Angular magnitude, 305, 306, 326.\n\n  Antecedent defined, 225.\n\n  Anticipation of Nature, 509.\n\n  Anticipations, of Principle of Substitution, 21;\n    of electric telegraph, 671.\n\n  Apparent, equality, 275;\n    sequence of events, 409.\n\n  Approximation, theory of, 456;\n    to exact laws, 462;\n    mathematical principles of, 471;\n    arithmetic of, 481.\n\n  Aqueous vapour, 500.\n\n  Aquinas, on disjunctive propositions, 69.\n\n  Arago, photometer, 288;\n    rotating disc, 535;\n    his philosophic character, 592.\n\n  Archimedes, *De Aren\u00e6 Numero*, 195;\n    centre of gravity, 363.\n\n  Arcual unit, 306, 330.\n\n  Argyll, Duke of, 741.\n\n  Aristarchus on sun\u2019s and moon\u2019s distances, 294.\n\n  Aristotelian doctrines, 666.\n\n  Aristotle, dictum, 21;\n    singular terms, 39;\n    overlooked simple identities, 40;\n    order of premises, 114;\n    logical error, 117;\n    definition of time, 307;\n    on science, 595;\n    on white swans, 666.\n\n  Arithmetic, reasoning in, 167;\n    of approximate quantities, 481.\n\n  Arithmetical triangle, 93, 143, 182, 202, 378, 383;\n    diagram of, 184;\n    connection with Logical Alphabet, 189;\n    in probability, 208.\n\n  Asteroids, discovery of, 412, 748.\n\n  Astronomy, physical, 459.\n\n  Atmospheric tides, 553.\n\n  Atomic theory, 662.\n\n  Atomic weights, 563.\n\n  Atoms, size of, 195;\n    impossibility of observing, 406.\n\n  Augustin on time, 307.\n\n  Average, 359, 360;\n    divergence from, 188;\n    etymology of, 363.\n\n  Axes of crystals, 686.\n\n  Axioms of algebra, 164.\n\n\n  Babbage, Charles, calculating machine, 107, 231, 743;\n    lighthouse signals, 194;\n    natural constants, 329;\n    Mosaic history, 412;\n    universal and general truths, 646;\n    change of law, 230;\n    persistence of effects, 757.\n\n  Bacon, Francis Lord, *Novum Organum*, 107;\n    on induction, 121;\n    biliteral cipher, 193;\n    First Aphorism, 219;\n    on causes, 221;\n    Copernican system, 249, 638;\n    deficient powers of senses, 278;\n    observation, 402;\n    Natural History, 403;\n    use of hypothesis, 506;\n    his method, 507;\n    *experimentum crucis*, 519;\n    error of his method, 576;\n    ostensive, clandestine instances, &c., 608, 610;\n    *latens precessus*, 619.\n\n  Bacon, Roger, on the rainbow, 526, 598.\n\n  Baily, Francis, 272;\n    density of earth, 342, 566;\n    experiments with torsion balance, 370, 397, 432, 567\u20138;\n    motions of stars, 572.\n\n  Bain,  Alexander, on powers of mind, 4;\n    Mill\u2019s reform of logic, 227.\n\n  Baker\u2019s poem, *The Universe*, 621.\n\n  Balance, use of the chemical, 292, 351, 354, 369;\n    delicacy of, 304;\n    vibrations of, 369.\n\n  Ballot, Buys, experiment on sound, 541.\n\n  Ballot-box, simile of, 150, 251\u20136, 765.\n\n  Barbara, 55, 57, 88, 105, 141.\n\n  Baroko, 85.\n\n  Barometer, 659;\n    Gay Lussac\u2019s standard, 346;\n    variations, 337, 346, 349.\n\n  Bartholinus on double refraction, 585.\n\n  Base-line, measurement of, 304.\n\n  Bauhusius, verses of, 175.\n\n  Baxendell, Joseph, 552.\n\n  Beneke, on substitution, 21.\n\n  Bennet, momentum of light, 435.\n\n  Bentham, George, 15;\n    bifurcate classification, 695;\n    infima species, 702;\n    works on classification, 703;\n    analytical key to flora, 712.\n\n  Bentham, Jeremy, on analogy, 629;\n    bifurcate classification, 703.\n\n  Benzenberg\u2019s experiment, 388.\n\n  Bernoulli, Daniel, planetary orbits, 250;\n    resisting media and projectiles, 467;\n    vibrations, 476.\n\n  Bernoulli, James, 154;\n    numbers of, 124;\n    Protean verses, 175;\n    *De Arte Conjectandi* quoted, 176, 183;\n    on figurate numbers, 183;\n    theorem of, 209;\n    false solution in probability, 213;\n    solution of inverse problem, 261.\n\n  Bessel, F.\u00a0W., 375;\n    law of error, 384;\n    formula for periodic variations, 488;\n    use of hypothesis, 506;\n    solar parallax, 560\u20132;\n    ellipticity of earth, 565;\n    pendulum experiments, 604.\n\n  Bias, 393, 402.\n\n  Biela\u2019s comet, 746.\n\n  Bifurcate classification, 694.\n\n  Binomial theorem, 190;\n    discovery of, 231.\n\n  Biot, on tension of vapour, 500.\n\n  Blind experiments, 433.\n\n  Bode\u2019s law, 147, 257, 660.\n\n  Boethius, quoted, 33;\n    on kinds of mean, 360.\n\n  Boiling point, 442, 659.\n\n  Bonnet\u2019s theory of reproduction, 621.\n\n  Boole, George, on sign of equality, 15;\n    his calculus of logic, 23, 113, 634;\n    on logical terms, 33;\n    law of commutativeness, 35;\n    use of *some*, 41\u20132;\n    disjunctive propositions, 70;\n    Venn on his method, 90;\n    *Laws of Thought*, 155;\n    statistical conditions, 168;\n    propositions numerically definite, 172;\n    on probability, 199;\n    general method in probabilities, 206;\n    Laplace\u2019s solution of inverse problem, 256;\n    law of error, 377.\n\n  Borda, his repeating circle, 290.\n\n  Boscovich\u2019s hypothesis, 512.\n\n  Botany, 666, 678, 681;\n    modes of classification, 678;\n    systematic, 722;\n    nomenclature of, 727.\n\n  Bowen, Prof. Francis, on inference, 118;\n    classification, 674.\n\n  Boyle\u2019s, Robert, law of gaseous pressure, 468, 470, 619;\n    on hypothesis, 510;\n    barometer, 659.\n\n  Bradley, his observations, 384;\n    accuracy of, 271;\n    aberration of light, 535.\n\n  Bravais, on law of error, 375.\n\n  Brewer, W.\u00a0H., 142.\n\n  Brewster, Sir David, iridescent colours, 419;\n    spectrum, 429;\n    Newton\u2019s theory of colours, 518;\n    refractive indices, 10, 527;\n    optic axes, 446.\n\n  British Museum, catalogue of, 717.\n\n  Brodie, Sir B.\u00a0C., on errors of experiment, 388, 464;\n    ozone, 663.\n\n  Brown, Thomas, on cause, 224.\n\n  Buckle, Thomas, on constancy of average, 656;\n    science of history, 760.\n\n  Buffon, on probability, 215;\n    definition of genius, 576.\n\n  Bunsen, Robert, spectrum, 244;\n    photometrical researches, 273, 324, 441;\n    calorimeter, 343.\n\n  Butler, Bishop, on probability, 197.\n\n\n  Calorescence, 664.\n\n  Camestres, 84.\n\n  Canton, on compressibility of water, 338.\n\n  Carbon, 640, 728;\n    conductibility of, 442.\n\n  Cardan, on inclined plane, 501.\n\n  Cards, combinations of, 190.\n\n  Carlini, pendulum experiments, 567.\n\n  Carnot\u2019s law, 606.\n\n  Carpenter, Dr. W.\u00a0B., 412.\n\n  Catalogues, art of making, 714.\n\n  Cauchy, undulatory theory, 468.\n\n  Cause, 220;\n    definition of, 224.\n\n  Cavendish\u2019s experiment, 272, 566.\n\n  Cayley, Professor, 145;\n    on mathematical tables, 331;\n    numbers of chemical compounds, 544.\n\n  Celarent, 55.\n\n  Centre of gravity, 363, 524;\n    of oscillation, gyration, &c., 364.\n\n  Centrobaric bodies, 364.\n\n  Certainty, 235, 266.\n\n  Cesare, 85.\n\n  Chalmers, on collocations, 740.\n\n  Chance, 198.\n\n  Character, human, 733.\n\n  Characteristics, 708.\n\n  Chauvenet, Professor W., on treatment of observations, 391.\n\n  Chemical affinity, 614;\n    analysis, 713.\n\n  Chladni, 446.\n\n  Chloroform, discovery of, 531.\n\n  Chronoscope, 616.\n\n  Cipher, 32;\n    Bacon\u2019s, 193.\n\n  Circle, circumference of, 389.\n\n  Circumstances, indifferent, 419.\n\n  Circumstantial evidence, 264.\n\n  Clairaut, 650, 651;\n    on gravity, 463.\n\n  Classes, 25;\n    problem of common part of three, 170.\n\n  Classification, 673;\n    involving induction, 675;\n    multiplicity of modes, 677;\n    natural and artificial systems, 679;\n    in crystallography, 685;\n    symbolic statement of, 692;\n    bifurcate, 694;\n    an inverse and tentative operation, 689;\n    diagnostic, 710;\n    by indexes, 714;\n    of books, 715;\n    in biological sciences, 718;\n    genealogical, 719;\n    by types, 722;\n    limits of, 730.\n\n  Clifford, Professor, on types of compound statements, 143, 529;\n    first and last catastrophe, 744;\n    mathematical function, 768.\n\n  Clocks, astronomical, 340, 353.\n\n  Clouds, 447;\n    cirrous, 411.\n\n  Coincidences, 128;\n    fortuitous, 261;\n    measurement by, 292;\n    method of, 291.\n\n  Collective terms, 29, 39.\n\n  Collocations of matter, 740.\n\n  Colours, iridescent, 419;\n    natural, 518;\n    perception of, 437;\n    of spectrum, 584.\n\n  Combinations, 135, 142;\n    doctrine of, 173;\n    of letters of alphabet, 174;\n    calculations of, 180;\n    higher orders of, 194.\n\n  Combinatorial analysis, 176.\n\n  Comets, 449;\n    number of, 408;\n    hyperbolic, 407;\n    classification of, 684;\n    conflict with, 746\u20137;\n    Halley\u2019s comet, 537;\n    Lexell\u2019s comet, 651.\n\n  Commutativeness, law of, 35, 72, 177.\n\n  Comparative use of instruments, 299.\n\n  Compass, variations of, 281.\n\n  Complementary statements, 144.\n\n  Compossible alternatives, 69.\n\n  Compound statements, 144;\n    events, 204.\n\n  Compounds, chemical, 192.\n\n  Comte, Auguste, on probability, 200, 214;\n    on prevision, 536;\n    his positive philosophy, 752, 760, 768.\n\n  Concrete number, 159.\n\n  Conditions, of logical symbols, 32;\n    removal of usual, 426;\n    interference of unsuspected, 428;\n    maintenance of similar, 443;\n    approximation to natural, 465.\n\n  Condorcet, 2;\n    his problem, 253.\n\n  Confusion of elements, 237.\n\n  Conical refraction, 653.\n\n  Conjunction  of  planets,  293, 657.\n\n  Consequent, definition of, 225.\n\n  Conservation of energy, 738.\n\n  Constant numbers of nature, 328;\n    mathematical, 330;\n    physical, 331;\n    astronomical, 332;\n    terrestrial, 333;\n    organic, 333;\n    social, 334.\n\n  Continuity, law of, 615, 729;\n    sense of, 493;\n    detection of, 610;\n    failure of, 619.\n\n  Continuous quantity, 274, 485.\n\n  Contradiction, law of, 31, 74.\n\n  Contrapositive,  proposition,  84, 136;\n    conversion, 83.\n\n  Conversion of propositions, 46, 118.\n\n  Copernican theory, 522, 625, 638, 647.\n\n  Copula, 16.\n\n  Cornu, velocity of light, 561.\n\n  Corpuscular theory, 520, 538, 667.\n\n  Correction, method of, 346.\n\n  Correlation, 678, 681.\n\n  Cotes, Roger, use of mean, 359;\n    method of least squares, 377.\n\n  Coulomb, 272.\n\n  Couple, mechanical, 653.\n\n  Creation, problem of, 740.\n\n  Crookes\u2019 radiometer, 435.\n\n  Cross divisions, 144.\n\n  Crystallography, 648, 654, 658, 678, 754;\n    systems of, 133;\n    classification in, 685.\n\n  Crystals, 602;\n    Dana\u2019s classification of, 711;\n    pseudomorphic, 658.\n\n  Curves, use of, 392, 491, 496;\n    of various degrees, 473.\n\n  Cuvier, on experiment, 423;\n    on inferences, 682.\n\n  Cyanite, 609.\n\n  Cycloid, 633.\n\n  Cycloidal pendulum, 461.\n\n  Cypher, 124.\n\n\n  D\u2019Alembert, blunders in probability, 213, 214;\n    on gravity, 463.\n\n  Dalton, laws of, 464, 471;\n    atomic theory, 662.\n\n  Darapti, 59.\n\n  Darii, 56.\n\n  Darwin, Charles, his works, 131;\n    negative results of observation, 413;\n    arguments against his theory, 437;\n    cultivated plants, 531;\n    his influence, 575;\n    classification, 718;\n    constancy of character in classification, 720\u20131;\n    on definition, 726;\n    restoration of limbs, 730;\n    tendency of his theory, 762, 764.\n\n  Davy, Sir H., on new instruments, 270;\n    nature of heat, 343, 417;\n    detection of salt in electrolysis, 428.\n\n  Day, sidereal, 310;\n    length of, 289.\n\n  Decandolle, on classification, 696.\n\n  Decyphering, 124.\n\n  Deduction, 11, 49.\n\n  Deductive reasoning, 534;\n    miscellaneous forms of, 60;\n    probable, 209.\n\n  Definition, 39, 62, 711, 723;\n    purpose of, 54;\n    of cause and power, 224.\n\n  De Morgan, Augustus, negative terms, 14;\n    Aristotle\u2019s logic, 18;\n    relatives, 23;\n    logical universe, 43;\n    complex propositions, 75;\n    contraposition, 83;\n    formal logic quoted, 101;\n    error of his system, 117;\n    anagram of his name, 128;\n    numerically definite reasoning, 168\u2013172;\n    probability, 198;\n    belief, 199;\n    experiments in probability, 207;\n    probable deductive arguments, 209\u2013210;\n    trisection of angle, 233;\n    probability of inference, 259;\n    arcual unit, 306;\n    mathematical tables, 331;\n    personal error, 348;\n    average, 363;\n    his works on probability, 394\u2013395;\n    apparent sequence, 409;\n    sub-equality, 480;\n    rule of approximation, 481;\n    negative areas, 529;\n    generalisation, 600;\n    double algebra, 634;\n    bibliography, 716;\n    catalogues, 716;\n    extensions of algebra, 758.\n\n  Density, unit of, 316;\n    of earth, 387;\n    negative, 642.\n\n  Descartes, vortices, 517;\n    geometry, 632.\n\n  Description, 62.\n\n  Design, 762\u2013763.\n\n  Determinants, inference by, 50.\n\n  Development, logical, 89, 97.\n\n  Diagnosis, 708.\n\n  Dichotomy, 703.\n\n  Difference, 44;\n    law of, 5;\n    sign of, 17;\n    representation of, 45;\n    inference with, 52, 166;\n    form of, 158.\n\n  Differences of numbers, 185.\n\n  Differential calculus, 477.\n\n  Differential thermometer, 345.\n\n  Diffraction of light, 420.\n\n  Dimensions, theory of, 325.\n\n  Dip-needle, observation of, 355.\n\n  Direct deduction, 49.\n\n  Direction of motion, 47.\n\n  Discontinuity, 620.\n\n  Discordance, of theory and experiment, 558;\n    of theories, 587.\n\n  Discoveries, accidental, 529;\n    predicted, 536;\n    scope for, 752.\n\n  Discrimination, 24;\n    power of, 4.\n\n  Disjunctive, terms, 66;\n    conjunction, 67;\n    propositions, 66;\n    syllogism, 77;\n    argument, 106.\n\n  Dissipation of energy, 310.\n\n  Distance of statements, 144.\n\n  Divergence from average, 188.\n\n  Diversity, 156.\n\n  Divine interference, 765.\n\n  Dollond, achromatic lenses, 608.\n\n  Donkin, Professor, 375;\n    on probability, 199, 216;\n    principle of inverse method, 244.\n\n  Double refraction, 426.\n\n  Dove\u2019s law of winds, 534.\n\n  Draper\u2019s law, 606.\n\n  Drobitsch, 15.\n\n  Duality, 73, 81;\n    law of, 5, 45, 92, 97.\n\n  Dulong and Petit, 341, 471.\n\n  Duration, 308.\n\n\n  \u03b5, 330, 769.\n\n  Earth, density of, 387;\n    ellipticity, 565.\n\n  Eclipses, 656;\n    Egyptian records of, 246;\n    of Jupiter\u2019s satellites, 294, 372;\n    solar, 486.\n\n  Electric, sense, 405;\n    acid, 428;\n    fluid, 523.\n\n  Electric telegraph, anticipations of, 671.\n\n  Electricity, theories of, 522;\n    duality of, 590.\n\n  Electrolysis, 428, 530.\n\n  Electro-magnet, use of, 423.\n\n  Elements, confusion of, 237;\n    definition, 427;\n    classification, 676, 677, 690.\n\n  Elimination, 58.\n\n  Ellicott, observation on clocks, 455.\n\n  Ellipsis, 41;\n    of terms, 57.\n\n  Elliptic variation, 474.\n\n  Ellipticity of earth, 565.\n\n  Ellis, A.\u00a0J., contributions to formal logic, 172.\n\n  Ellie, Leslie, 23, 375.\n\n  Ellis, W., on moon\u2019s influence, 410.\n\n  Emanation, law of, 463.\n\n  Emotions, 732.\n\n  Empirical, knowledge, 505, 525\u2013526;\n    measurement, 552.\n\n  Encke, on mean, 386, 389;\n    his comet, 570, 605;\n    on resisting medium, 523;\n    solar parallax, 562.\n\n  Energy, unit of, 322;\n    conservation of, 465;\n    reconcentration of, 751.\n\n  English language, words in, 175.\n\n  E\u00f6zoon canadense, 412, 668.\n\n  Equality, sign of, 14;\n    axiom, 163;\n    four meanings of, 479.\n\n  Equations, 46, 53, 160;\n    solution of, 123.\n\n  Equilibrium, unstable, 276, 654.\n\n  Equisetace\u00e6, 721.\n\n  Equivalence of propositions, 115, 120, 132;\n    remarkable case of, 529, 657.\n\n  Eratosthenes, sieve of, 82, 123, 139;\n    measurement of degree, 293.\n\n  Error, function, 330, 376, 381;\n    elimination of, 339, 353;\n    personal, 347;\n    law of, 374;\n    origin of law, 383;\n    verification of law, 383;\n    probable, 386;\n    mean, 387;\n    constant, 396;\n    variation of small errors, 479.\n\n  Ether, luminiferous, 512, 514, 605.\n\n  Euclid, axioms, 51, 163;\n    indirect proof, 84;\n    10th book, 117th proposition, 275;\n    on analogy, 631.\n\n  Euler, on certainty of inference, 238;\n    corpuscular theory, 435;\n    gravity, 463;\n    on ether, 514.\n\n  Everett, Professor, unit of angle, 306;\n    metric system, 328.\n\n  Evolution, theory of, 761.\n\n  Exact science, 456.\n\n  Exceptions, 132, 644, 728;\n    classification of, 645;\n    imaginary, 647;\n    apparent, 649;\n    singular, 652;\n    divergent, 655;\n    accidental, 658;\n    novel, 661;\n    limiting, 663;\n    real, 666;\n    unclassed, 668.\n\n  Excluded middle, law of, 6.\n\n  Exclusive alternatives, 68.\n\n  Exhaustive investigation, 418.\n\n  Expansion, of bodies, 478;\n    of liquids, 488.\n\n  Experiment, 400, 416;\n    in probability, 208;\n    test or blind, 433;\n    negative results of, 434;\n    limits of, 437;\n    collective, 445;\n    simplification of, 422;\n    failure in simplification, 424.\n\n  Experimentalist, character of, 574, 592.\n\n  Experimentum crucis, 518, 667.\n\n  Explanation, 532.\n\n  Extent of meaning, 26;\n    of terms, 48.\n\n  Extrapolation, 495.\n\n\n  Factorials, 179.\n\n  Facts, importance of false, 414;\n    conformity with, 516.\n\n  Fallacies, 62;\n    analysed by indirect method, 102;\n    of observation, 408.\n\n  Faraday, Michael, measurement of gold-leaf, 296;\n    on gravity, 342, 589;\n    magnetism of gases, 352;\n    vibrating plate, 419;\n    electric poles, 421;\n    circularly polarised light, 424, 588, 630;\n    freezing mixtures, 427;\n    magnetic experiments, 431, 434;\n    lines of magnetic force, 446, 580;\n    errors of experiment, 465;\n    electrolysis, 502;\n    velocity of light, 520;\n    prediction, 543;\n    relations of physical forces, 547;\n    character of, 578, 587;\n    ray vibrations, 579;\n    mathematical power, 580;\n    philosophic reservation of opinion, 592;\n    use of heavy glass, 609;\n    electricity, 612;\n    radiant matter, 642;\n    hydrogen, 691.\n\n  Fatality, belief in, 264.\n\n  Ferio, 56.\n\n  Figurate numbers, 183, 186.\n\n  Figure of earth, 459, 565.\n\n  Fizeau, use of Newton\u2019s rings, 297, 582;\n    fixity of properties, 313;\n    velocity of light, 441, 561.\n\n  Flamsteed, use of wells, 294;\n    standard stars, 301;\n    parallax of pole-star, 338;\n    selection of observations, 358;\n    astronomical instruments, 391;\n    solar eclipses, 486.\n\n  Fluorescence, 664.\n\n  Fontenelle on the senses, 405.\n\n  Forbes, J.\u00a0D., 248.\n\n  Force, unit of, 322, 326;\n    emanating, 464;\n    representation of, 633.\n\n  Formul\u00e6, empirical, 487;\n    rational, 489.\n\n  Fortia, *Trait\u00e9 des Progressions*, 183.\n\n  Fortuitous coincidences, 261.\n\n  Fossils, 661.\n\n  Foster, G.\u00a0C., on classification, 691.\n\n  Foucault, rotating mirror, 299;\n    pendulum, 342, 431, 522;\n    on velocity of light, 441, 521, 561.\n\n  Fourier, Joseph, theory of dimensions, 325;\n    theory of heat, 469, 744.\n\n  Fowler, Thomas, on method of difference, 439;\n    reasoning from case to case, 227.\n\n  Frankland, Professor Edward, on spectrum of gases, 606.\n\n  Franklin\u2019s experiments on heat, 424.\n\n  Fraunhofer, dark lines of spectrum, 429.\n\n  Freezing-point, 546.\n\n  Freezing mixtures, 546.\n\n  Fresnel, inflexion of light, 420;\n    corpuscular theory, 521;\n    on use of hypothesis, 538;\n    double refraction, 539.\n\n  Friction, 417;\n    determination of, 347.\n\n  Function, definitions of, 489.\n\n  Functions, discovery of, 496.\n\n\n  Galileo, 626;\n    on cycloid, 232, 235;\n    differential method of observation, 344;\n    projectiles, 447, 466;\n    use of telescope, 522;\n    gravity, 604;\n    principle of continuity, 617.\n\n  Gallon, definition of, 318.\n\n  Galton, Francis, divergence from mean, 188;\n    works by, 188, 655;\n    on hereditary genius, 385, 655.\n\n  Galvanometer, 351.\n\n  Gani\u00e8res, de, 182.\n\n  Gases, 613;\n    properties of, 601, 602;\n    perfect, 470;\n    liquefiable, 665.\n\n  Gauss, pendulum experiments, 316;\n    law of error, 375\u20136;\n    detection of error, 396;\n    on gravity, 463.\n\n  Gay Lussac, on boiling point, 659;\n    law of, 669.\n\n  Genealogical classification, 680, 719.\n\n  General, terms, 29;\n    truths, 647;\n    notions, 673.\n\n  Generalisation, 2, 594, 704;\n    mathematical, 168;\n    two meanings of, 597;\n    value of, 599;\n    hasty, 623.\n\n  Genius, nature of, 575.\n\n  Genus, 433, 698;\n    generalissimum, 701;\n    natural, 724.\n\n  Geology, 667;\n    records in, 408;\n    slowness of changes, 438;\n    exceptions in, 660.\n\n  Geometric mean, 361.\n\n  Geometric reasoning, 458;\n    certainty of, 267.\n\n  Giffard\u2019s injector, 536.\n\n  Gilbert, on rotation of earth, 249;\n    magnetism of silver, 431;\n    experimentation, 443.\n\n  Gladstone, J.\u00a0H., 445.\n\n  Glaisher, J.\u00a0W.\u00a0L., on mathematical tables, 331;\n    law of error, 375, 395.\n\n  Gold, discovery of, 413.\n\n  Gold-assay process, 434.\n\n  Gold-leaf, thickness of, 296.\n\n  Graham, Professor Thomas, on chemical affinity, 614;\n    continuity, 616;\n    nature of hydrogen, 691.\n\n  Grammar, 39;\n    rules of, 31.\n\n  Grammatical, change, 119;\n    equivalence, 120.\n\n  Gramme, 317.\n\n  Graphical method, 492.\n\n  Gravesande, on inflection of light, 420.\n\n  Gravity, 422, 512, 514, 604, 740;\n    determination of, 302;\n    elimination of, 427;\n    law of, 458, 462, 474;\n    inconceivability of, 510;\n    Newton\u2019s theory, 555;\n    variation of, 565;\n    discovery of law, 581;\n    Faraday on, 589;\n    discontinuity in, 620;\n    Aristotle on, 649;\n    Hooke\u2019s experiment, 436.\n\n  Grimaldi on the spectrum, 584.\n\n  Grove, Mr. Justice, on ether, 514;\n    electricity, 615.\n\n  Guericke, Otto von, 432.\n\n\n  Habit, formation of, 618.\n\n  Halley, trade-winds, 534.\n\n  Halley\u2019s comet, 537, 570.\n\n  Hamilton, Sir William, disjunctive propositions, 69;\n    inference, 118;\n    free-will, 223.\n\n  Hamilton, Sir W. Rowan, on conical refraction, 540;\n    quaternions, 634.\n\n  Harley, Rev. Robert, on Boole\u2019s logic, 23, 155.\n\n  Harris, standards of length, 312.\n\n  Hartley, on logic, 7.\n\n  Hatchett, on alloys, 191.\n\n  Haughton, Professor, on tides, 450;\n    muscular exertion, 490.\n\n  Ha\u00fcy, on crystallography, 529.\n\n  Hayward, R.\u00a0B., 142.\n\n  Heat, unit of, 324;\n    measurement of, 349;\n    experiments on, 444;\n    mechanical equivalent of, 568.\n\n  Heavy glass, 588, 609.\n\n  Helmholtz, on microscopy, 406;\n    undulations, 414;\n    sound, 476.\n\n  Hemihedral crystals, 649.\n\n  Herschel, Sir John, on rotation of plane of polarisation of light, 129, 630;\n    quartz crystals, 246;\n    numerical precision, 273;\n    photometry, 273;\n    light of stars, 302;\n    actinometer, 337;\n    mean and average, 363;\n    eclipses of Jupiter\u2019s satellites, 372;\n    law of error, 377;\n    error in observations, 392;\n    on observation, 400;\n    moon\u2019s influence on clouds, 410;\n    comets, 411;\n    spectrum analysis, 429;\n    collective instances, 447;\n    principle of forced vibrations, 451, 663;\n    meteorological variations, 489;\n    double stars, 499, 685;\n    direct action, 502;\n    use of theory, 508;\n    ether, 515;\n    *experimentum crucis*, 519;\n    interference of light, 539;\n    interference of sound, 540;\n    density of earth, 567;\n    residual phenomena, 569;\n    helicoidal dissymmetry, 630;\n    fluorescence, 664.\n\n  Hindenburg, on combinatorial analysis, 176.\n\n  Hipparchus, used method of repetition, 289;\n    longitudes of stars, 294.\n\n  Hippocrates, area of lunule, 480.\n\n  History, science of, 760.\n\n  Hobbes, Thomas, definition of cause, 224;\n    definition of time, 307;\n    on hypothesis, 510.\n\n  Hofmann, unit called crith, 321;\n    on prediction, 544;\n    on anomalies, 670.\n\n  Homogeneity, law of, 159, 327.\n\n  Hooke, on gravitation, 436, 581;\n    philosophical method, 507;\n    on strange things, 671.\n\n  Hopkinson, John, 194;\n    method of interpolation, 497.\n\n  Horrocks, use of mean, 358;\n    use of hypothesis, 507.\n\n  Hume on perception, 34.\n\n  Hutton, density of earth, 566.\n\n  Huxley, Professor Thomas, 764;\n    on hypothesis, 509;\n    classification, 676;\n    mammalia, 682;\n    pal\u00e6ontology, 682.\n\n  Huyghens, theory of pendulum, 302;\n    pendulum standard, 315;\n    cycloidal pendulum, 341;\n    differential method, 344;\n    distant stars, 405;\n    use of hypothesis, 508;\n    philosophical method of, 585;\n    on analogy, 639.\n\n  Hybrids, 727.\n\n  Hydrogen, expansion of, 471;\n    refractive power, 527;\n    metallic nature of, 691.\n\n  Hygrometry, 563.\n\n  Hypotheses, use of, 265, 504;\n    substitution of simple hypotheses, 458;\n    working hypotheses, 509;\n    requisites of, 510;\n    descriptive, 522, 686;\n    representative, 524;\n    probability of, 559.\n\n\n  Identical propositions, 119.\n\n  Identities, simple, 37;\n    partial, 40;\n    limited, 42;\n    simple and partial, 111;\n    inference from, 51, 55.\n\n  Identity, law of, 5, 6, 74;\n    expression of, 14;\n    propagating power, 20;\n    reciprocal, 46.\n\n  Illicit process, of major term, 65, 103;\n    of minor term, 65.\n\n  Immediate inference, 50, 61.\n\n  Imperfect induction, 146, 149.\n\n  Inclusion, relation of, 40.\n\n  Incommensurable quantities, 275.\n\n  Incompossible events, 205.\n\n  Independence of small effects, 475.\n\n  Independent events, 204.\n\n  Indestructibility of matter, 465.\n\n  Indexes, classification by, 714;\n    formation of, 717.\n\n  India-rubber, properties of, 545.\n\n  Indirect method of deduction, 49, 81;\n    illustrations of, 98;\n    fallacies analysed by, 102;\n    the test of equivalence, 115.\n\n  Induction, 11, 121;\n    symbolic statement of, 131;\n    perfect, 146;\n    imperfect, 149;\n    philosophy of, 218;\n    grounds of, 228;\n    illustrations of, 229;\n    quantitative, 483;\n    problem of two classes, 134;\n    problem of three classes, 137.\n\n  Inductive truths, classes of, 219.\n\n  Inequalities, reasoning by, 47, 163, 165\u2013166.\n\n  Inference, 9;\n    general formula of, 17;\n    immediate, 50;\n    with two simple identities, 51;\n    from simple and partial identity, 53;\n    with partial identities, 55;\n    by sum of predicates, 61;\n    by disjunctive propositions, 76;\n    indirect method of, 81;\n    nature of, 118;\n    principle of mathematical, 162;\n    certainty of, 236.\n\n  Infima species, 701, 702.\n\n  Infiniteness of universe, 738.\n\n  Inflection of light, 420.\n\n  Instanti\u00e6, citantes, evocantes, radii, curriculi, 270;\n    monodic\u00e6, irregulares, heteroclit\u00e6, 608;\n    clandestin\u00e6, 610.\n\n  Instruments of measurement, 284.\n\n  Insufficient enumeration, 176.\n\n  Integration, 123.\n\n  Intellect, etymology of, 5.\n\n  Intension of logical terms, 26, 48;\n    of propositions, 47.\n\n  Interchangeable system, 20.\n\n  Interpolation, 495;\n    in meteorology, 497.\n\n  Inverse, process, 12;\n    operation, 122, 689;\n    problem of two classes, 134;\n    problem of three classes, 137;\n    problem of probability, 240, 251;\n    rules of inverse method, 257;\n    simple illustrations, 253;\n    general solution, 255.\n\n  Iodine, the substance X, 523.\n\n  Iron, properties of, 528, 670.\n\n  *Is*, ambiguity of verb, 16, 41.\n\n  Isomorphism, 662.\n\n  Ivory, 375.\n\n\n  James, Sir H., on density of earth, 567.\n\n  Jenkin, Professor Fleming, 328.\n\n  Jevons, W.\u00a0S., on use of mean, 361;\n    on pedesis or molecular movement of microscopic particles, 406, 549;\n    cirrous clouds, 411;\n    spectrum analysis, 429;\n    elevated rain-gauges, 430;\n    experiments on clouds, 447;\n    on muscular exertion, 490;\n    resisting medium, 570;\n    anticipations of the electric telegraph, 671.\n\n  Jones, Dr. Bence, Life of Faraday, 578.\n\n  Jordanus, on the mean, 360.\n\n  Joule, 545;\n    on thermopile, 299, 300;\n    mechanical equivalent of heat, 325, 347, 568;\n    temperature of air, 343;\n    rarefaction, 444;\n    on Thomson\u2019s prediction, 543;\n    molecular theory of gases, 548;\n    friction, 549;\n    thermal phenomena of fluids, 557.\n\n  Jupiter, satellites of, 372, 458, 638, 656;\n    long inequality of, 455;\n    figure of, 556.\n\n\n  Kames, Lord, on bifurcate classification, 697.\n\n  Kant, disjunctive propositions, 69;\n    analogy, 597;\n    doctrine of space, 769.\n\n  Kater\u2019s pendulum, 316.\n\n  Keill, law of emanating forces, 464;\n    axiom of simplicity, 625.\n\n  Kepler, on star-discs, 390;\n    comets, 408;\n    laws of, 456;\n    refraction, 501;\n    character of, 578.\n\n  Kinds of things, 718.\n\n  King Charles and the Royal Society, 647.\n\n  Kirchhoff, on lines of spectrum, 245.\n\n  Kohlrausch, rules of approximate calculation, 479.\n\n\n  Lagrange, formula for interpolation, 497;\n    accidental discovery, 531;\n    union of algebra and geometry, 633.\n\n  Lambert, 15.\n\n  Lamont, 452.\n\n  Language, 8, 628, 643.\n\n  Laplace, on probability, 200, 216;\n    principles of inverse method, 242;\n    solution of inverse problem, 256;\n    planetary motions, 249, 250;\n    conjunctions of planets, 293;\n    observation of tides, 372;\n    atmospheric tides, 367;\n    law of errors, 378;\n    dark stars, 404;\n    hyperbolic comets, 407;\n    his works on probability, 395;\n    velocity of gravity, 435;\n    stability of planetary system, 448, 746;\n    form of Jupiter, 556;\n    corpuscular theory, 521;\n    ellipticity of earth, 565;\n    velocity of sound, 571;\n    analogy, 597;\n    law of gravity, 615;\n    inhabitants of planets, 640;\n    laws of motion, 706;\n    power of science, 739.\n\n  Lavoisier, mistaken inference of, 238;\n    pyrometer, 287;\n    on experiments, 423;\n    prediction of, 544;\n    theory, 611;\n    on acids, 667\n\n  Law, 3;\n    of simplicity, 33, 72, 161;\n    commutativeness, 35, 160;\n    disjunctive relation, 71;\n    unity, 72, 157, 162;\n    identity, 74;\n    contradiction, 74, 82;\n    duality, 73, 74, 81, 97, 169;\n    homogeneity, 159;\n    error, 374;\n    continuity, 615;\n    of Boyle, 619;\n    natural, 737.\n\n  Laws, of thought, 6;\n    empirical mathematical, 487;\n    of motion, 617;\n    of botanical nomenclature, 727;\n    natural hierarchy of, 742.\n\n  Least squares, method of, 386, 393.\n\n  Legendre, on geometry, 275;\n    rejection of observations, 391;\n    method of least squares, 377.\n\n  Leibnitz, 154, 163;\n    on substitution, 21;\n    propositions, 42;\n    blunder in probability, 213;\n    on Newton, 515;\n    continuity, 618.\n\n  Leslie, differential thermometer, 345;\n    radiating power, 425;\n    on affectation of accuracy, 482.\n\n  Letters, combinations of, 193.\n\n  Leverrier, on solar parallax, 562.\n\n  Lewis, Sir G.\u00a0C., on time, 307.\n\n  Life is change, 173.\n\n  Light, intensity of, 296;\n    unit, 324;\n    velocity, 535, 560, 561;\n    science of, 538;\n    total reflection, 650;\n    waves of, 637;\n    classification of, 731.\n\n  Lighthouses, Babbage on, 194.\n\n  Limited identities, 42;\n    inference of 59.\n\n  Lindsay, Prof. T.\u00a0M., 6, 21.\n\n  Linear variation, 474.\n\n  Linn\u00e6us on synopsis, 712;\n    genera and species, 725.\n\n  Liquid state, 601, 614.\n\n  Locke, John, on induction, 121;\n    origin of number, 157;\n    on probability, 215;\n    the word power, 221.\n\n  Lockyer, J. Norman, classification of elements, 676.\n\n  Logarithms, 148;\n    errors in tables, 242.\n\n  Logic, etymology of name, 5.\n\n  Logical abacus, 104.\n\n  Logical alphabet, 93, 116, 173, 417, 701;\n    table of, 94;\n    connection with arithmetical triangle, 189;\n    in probability, 205.\n\n  Logical conditions, numerical meaning of, 171.\n\n  Logical machine, 107.\n\n  Logical relations, number of, 142.\n\n  Logical slate, 95.\n\n  Logical truths, certainty of, 153.\n\n  Lottery, the infinite, 2.\n\n  Lovering, Prof., on ether, 606.\n\n  Lubbock and Drinkwater-Bethune, 386, 395.\n\n  Lucretius, rain of atoms, 223, 741;\n    indestructibility of matter, 622.\n\n\n  Machine, logical, 107.\n\n  Macleay, system of classification, 719.\n\n  Magnetism of gases, 352.\n\n  Mallet, on earthquakes, 314.\n\n  Malus, polarised light, 530.\n\n  Mammalia, characters of, 681.\n\n  Manchester Literary and Philosophical Society, papers quoted, 137, 143, 168.\n\n  Mansel, on disjunctive propositions, 69.\n\n  Mars, white spots of, 596.\n\n  Maskelyne, on personal error, 347;\n    deviation of plumbline, 369;\n    density of earth, 566.\n\n  Mass, unit of, 317, 325.\n\n  Mathematical science, 767;\n    incompleteness of, 754.\n\n  Matter, uniform properties of, 603;\n    variable properties, 606.\n\n  Matthiessen, 528.\n\n  Maximum points, 371.\n\n  Maxwell, Professor Clerk, on the balance, 304;\n    natural system of standards, 311, 319;\n    velocity of electricity, 442;\n    on Faraday, 580;\n    his book on *Matter and Motion*, 634.\n\n  Mayer, proposed repeating circle, 290;\n    on mechanical equivalent of heat, 568, 572.\n\n  Mean, etymology of, 359\u2013360;\n    geometric, 362;\n    fictitious, 363;\n    precise, 365;\n    probable, 385;\n    rejection of, 389;\n    method of, 357, 554.\n\n  Mean error, 387.\n\n  Meaning, of names, 25;\n    of propositions, 47.\n\n  Measurement, of phenomena, 270;\n    methods of, 282;\n    instruments, 284;\n    indirect, 296;\n    accuracy of, 303;\n    units and standards of, 305;\n    explained results of, 554;\n    agreement of modes of, 564.\n\n  Mediate statements, 144.\n\n  Melodies, possible number of, 191.\n\n  Melvill, Thomas, on the spectrum, 429.\n\n  *Membra dividentia*, 68.\n\n  Metals, probable character of new, 258;\n    transparency, 548;\n    classification, 675;\n    density, 706.\n\n  Method, indirect, 98;\n    of avoidance of error, 340;\n    differential, 344;\n    correction, 346;\n    compensation, 350;\n    reversal, 354;\n    means, 357;\n    least squares, 377, 386, 393;\n    variations, 439;\n    graphical, 492;\n    Baconian, 507.\n\n  Meteoric streams, 372.\n\n  Meteoric cycle, 537.\n\n  Metre, 349;\n    error of, 314.\n\n  Metric system, 318, 323.\n\n  Michell, speculations, 212;\n    on double stars, 247;\n    Pleiades, 248;\n    torsion balance, 566.\n\n  Middle term undistributed, 64.\n\n  Mill, John Stuart, disjunctive propositions, 69;\n    induction, 121, 594;\n    music, 191;\n    probability, 200\u2013201, 222;\n    supposed reform of logic, 227;\n    deductive method, 265, 508;\n    elimination of chance, 385;\n    joint method of agreement and difference, 425;\n    method of variations, 484;\n    on collocations, 740;\n    erroneous tendency of his philosophy, 752.\n\n  Miller, Prof. W.\u00a0H., kilogram, 318.\n\n  Mind, powers of, 4;\n    phenomena of, 672.\n\n  Minerals, classification of, 678.\n\n  Minor term, illicit process of, 65.\n\n  Mistakes, 7.\n\n  *Modus, tolendo ponens*, 77;\n    *ponendo tollens*, 78.\n\n  Molecular movement, or pedesis, 406.\n\n  Molecules, number of, 195.\n\n  Momentum, 322, 326.\n\n  Monro, C.\u00a0J., correction by, 172;\n    on Comte, 753.\n\n  Monstrous productions, 657.\n\n  Moon, supposed influence on clouds, 410;\n    atmosphere of, 434;\n    motions, 485;\n    fall towards earth, 555.\n\n  Morse alphabet, 193.\n\n  Mother of pearl, 419.\n\n  M\u00fcller, Max, on etymology of intellect, 5.\n\n  Multiplication in logic, 161.\n\n  Murphy, J.\u00a0J., on disjunctive relation, 71.\n\n  Murray, introduced use of ice, 343.\n\n  Muscular susurrus, 298.\n\n  Music, possible combinations of, 191.\n\n\n  Names, 25;\n    of persons, ships, &c., 680.\n\n  Nature, 1;\n    laws of, 737;\n    uniformity of, 745.\n\n  Nebular theory, 427.\n\n  Negation, 44.\n\n  Negative arguments, 621.\n\n  Negative density, 642.\n\n  Negative premises, 63, 103.\n\n  Negative propositions, 43.\n\n  Negative results of experiment, 434.\n\n  Negative terms, 14, 45, 54, 74.\n\n  Neil on use of hypothesis, 509.\n\n  Neptune, discovery of, 537, 660.\n\n  Newton, Sir Isaac, binomial theorem, 231;\n    spectrum, 262, 418, 420, 424, 583;\n    rings of, 288, 470;\n    velocity of sound, 295;\n    wave-lengths, 297;\n    use of pendulum, 303;\n    on time, 308;\n    definition of matter, 316;\n    pendulum experiment, 348, 443, 604;\n    centrobaric bodies, 365;\n    on weight, 422;\n    achromatic lenses, 432;\n    resistance of space, 435;\n    absorption of light, 445;\n    planetary motions, 249, 457, 463, 466, 467;\n    infinitesimal calculus, 477;\n    as an alchemist, 505;\n    his knowledge of Bacon\u2019s works, 507;\n    *hypotheses non fingo*, 515;\n    on vortices, 517;\n    theory of colours, 518;\n    corpuscular theory of light, 520;\n    fits of easy reflection, &c., 523;\n    combustible substances, 527;\n    gravity, 555, 650;\n    density of earth, 566;\n    velocity of sound, 571;\n    third law of motion, 622;\n    his rules of philosophising, 625;\n    fluxions, 633;\n    theory of sound, 636;\n    negative density, 642;\n    rays of light having sides, 662.\n\n  Newtonian Method, 581.\n\n  Nicholson, discovery of electrolysis, 530.\n\n  *Ninth Bridgewater Treatise* quoted, 743, 757.\n\n  Nipher, Professor, on muscular exertion, 490.\n\n  Noble, Captain, chronoscope, 308, 616.\n\n  Nomenclature, laws of botanical, 727.\n\n  Non-observation, arguments from, 411.\n\n  Norwood\u2019s measurement of a degree, 272.\n\n  Nothing, 32.\n\n  Number, nature of, 153, 156;\n    concrete and abstract, 159, 305.\n\n  Numbers, prime, 123;\n    of Bernoulli, 124;\n    figurate, 183;\n    triangular, &c., 185.\n\n  Numerical abstraction, 158.\n\n\n  Observation, 399;\n    mental conditions, 402;\n    instrumental and sensual conditions, 404;\n    external conditions, 407.\n\n  Obverse statements, 144.\n\n  Ocean, depth of, 297.\n\n  Odours, 732.\n\n  Oersted, on electro-magnetism, 530, 535.\n\n  *Or*, meaning of, 70.\n\n  Order, of premises, 114;\n    of terms, 33.\n\n  Orders of combinations, 194.\n\n  Original research, 574.\n\n  Oscillation, centre of, 364.\n\n  Ostensive instances, 608.\n\n  Ozone, 663.\n\n\n  \u03c0, value of, 234, 529.\n\n  Pack of cards, arrangement of, 241.\n\n  Paley on design, 762, 763.\n\n  Parallax, of stars, 344;\n    of sun, 560.\n\n  Parallel forces, 652.\n\n  Paralogism, 62.\n\n  Parity of reasoning, 268.\n\n  Partial identities, 40, 55, 57, 111;\n    induction of, 130.\n\n  Particular quantity, 56.\n\n  Particulars, reasoning from, 227.\n\n  Partition, 29.\n\n  Pascal, 176;\n    arithmetical machine, 107;\n    arithmetical triangle, 182;\n    binomial formula, 182;\n    error in probabilities, 213;\n    barometer, 519.\n\n  Passive state of steel, 659.\n\n  Pedesis, or molecular movement of microscopic particles, 406, 612.\n\n  Peirce, Professor, 23;\n    on rejection of observations, 391.\n\n  Pendulum, 290, 302, 315;\n    faults of, 311;\n    vibrations, 453, 454;\n    cycloidal, 461.\n\n  Perfect induction, 146, 149.\n\n  Perigon, 306.\n\n  Permutations, 173, 178;\n    distinction from combinations, 177.\n\n  Personal error, 347.\n\n  Photometry, 288.\n\n  Physiology, exceptions in, 666.\n\n  Planets, conjunctions of, 181, 187, 657;\n    discovery of, 412;\n    motions, 457;\n    perturbations of, 657;\n    classification, 683;\n    system of, 748.\n\n  Plants, classification of, 678.\n\n  Plateau\u2019s experiments, 427.\n\n  Plato on science, 595.\n\n  Plattes, Gabriel, 434, 438.\n\n  Pliny on tides, 451.\n\n  Plumb-line, divergence of, 461.\n\n  Plurality, 29, 156.\n\n  Poinsot, on probability, 214.\n\n  Poisson, on principle of the inverse method, 244;\n    work on Probability, 395;\n    Newton\u2019s rings, 470;\n    simile of ballot box, 524.\n\n  Polarisation, 653;\n    discovery of, 530.\n\n  Pole-star, 652;\n    observations of, 366.\n\n  Poles, of magnets, 365;\n    of battery, 421.\n\n  Political economy, 760.\n\n  Porphyry, on the Predicables, 698;\n    tree of, 702.\n\n  Port Royal logic, 22.\n\n  Positive philosophy, 760, 768.\n\n  Pouillet\u2019s pyrheliometer, 337.\n\n  Powell, Baden, 623;\n    on planetary motions, 660.\n\n  Power, definition of, 224.\n\n  Predicables, 698.\n\n  Prediction, 536, 739;\n    in science of light, 538;\n    theory of undulations, 540;\n    other sciences, 542;\n    by inversion of cause and effect, 545.\n\n  Premises, order of, 114.\n\n  Prime numbers, 123, 139;\n    formula for, 230.\n\n  *Principia*, Newton\u2019s, 581, 583.\n\n  Principle, of probability, 200;\n    inverse method, 242;\n    forced vibrations, 451;\n    approximation, 471;\n    co-existence of small vibrations, 476;\n    superposition of small effects, 476.\n\n  Probable error, 555.\n\n  Probability, etymology of, 197;\n    theory of, 197;\n    principles, 200;\n    calculations, 203;\n    difficulties of theory, 213;\n    application of theory, 215;\n    in induction, 219;\n    in judicial proceedings, 216;\n    works on, 394;\n    results of law, 656.\n\n  Problems, to be worked by reader, 126;\n    inverse problem of two classes, 135;\n    of three classes, 137.\n\n  Proclus, commentaries of, 232.\n\n  Proctor, R.\u00a0A., star-drifts, 248.\n\n  Projectiles, theory of, 466.\n\n  Proper names, 27.\n\n  Properties, generality of, 600;\n    uniform, 603;\n    extreme instances, 607;\n    correlation, 681.\n\n  Property, logical, 699;\n    peculiar, 699.\n\n  Proportion, simple, 501.\n\n  Propositions, 36;\n    negative, 43;\n    conversion of, 46;\n    twofold meaning, 47;\n    disjunctive, 66;\n    equivalence of, 115;\n    identical, 119;\n    tautologous, 119.\n\n  Protean verses, 175.\n\n  Protoplasm, 524, 764.\n\n  Prout\u2019s law, 263, 464.\n\n  Provisional units, 323.\n\n  Proximate statements, 144.\n\n  Pyramidal numbers, 185.\n\n  Pythagoras, on duality, 95;\n    on the number seven, 262, 624.\n\n\n  Quadric variation, 474.\n\n  Qualitative, reasoning, 48;\n    propositions, 119.\n\n  Quantification of predicate, 41.\n\n  Quantitative, reasoning, 48;\n    propositions, 119;\n    questions, 278;\n    induction, 483.\n\n  Quantities, continuous, 274;\n    incommensurable, 275.\n\n  Quaternions, 160, 634.\n\n  Quetelet, 188;\n    experiment on probability, 208;\n    on mean and average, 363;\n    law of error, 378, 380;\n    verification of law of error, 385.\n\n\n  Radian, 306.\n\n  Radiant matter, 642.\n\n  Radiation of heat, 430.\n\n  Radiometer, 435.\n\n  Rainbow, theory of, 526, 533.\n\n  Rainfall, variation of, 430.\n\n  Ramean tree, 703, 704.\n\n  Ramsden\u2019s balance, 304.\n\n  Rankine, on specific heat of air, 557;\n    reconcentration of energy, 751.\n\n  Rational formul\u00e6, 489.\n\n  Rayleigh, Lord, on graphical method, 495.\n\n  Reasoning, arithmetical, 167;\n    numerically definite, 168;\n    geometrical, 458.\n\n  Recorde, Robert, 15.\n\n  Reduction, of syllogisms, 85;\n    *ad absurdum*, 415;\n    of observations, 552, 572.\n\n  Reflection, total, 650.\n\n  Refraction, atmospheric, 340, 356, 500;\n    law of, 501;\n    conical, 540;\n    double, 585.\n\n  Regnault, dilatation of mercury, 342;\n    measurement of heat, 350;\n    exact experiment, 397;\n    on Boyle\u2019s law, 468, 471;\n    latent heat of steam, 487;\n    graphical method, 494;\n    specific heat of air, 557.\n\n  Reid, on bifurcate classification, 697.\n\n  Reign of law, 741, 759.\n\n  Rejection of observations, 390.\n\n  Relation, sign of, 17;\n    logic of, 22;\n    logical, 35;\n    axiom of, 164.\n\n  Repetition, method of, 287, 288.\n\n  Representative hypotheses, 524.\n\n  Reproduction, modes of, 730.\n\n  Reservation of judgment, 592.\n\n  Residual effects, 558;\n    phenomena, 560, 569.\n\n  Resisting medium, 310, 523, 570.\n\n  Resonance, 453.\n\n  Reusch, on substitution, 21.\n\n  Reversal, method of, 354.\n\n  Revolution, quantity of, 306.\n\n  Robertson, Prof. Croom, 27, 101.\n\n  Robison, electric curves, 446.\n\n  Rock-salt, 609.\n\n  R\u0153mer, divided circle, 355;\n    velocity of light, 535.\n\n  Roscoe, Prof., photometrical researches, 273;\n    solubility of salts, 280;\n    constant flame, 441;\n    absorption of gases, 499;\n    vanadium, 528;\n    atomic weight of vanadium, 392, 649.\n\n  Rousseau on geometry, 233.\n\n  Rules, of inference, 9, 17;\n    indirect method of inference, 89;\n    for calculation of combinations, 180;\n    of probabilities, 203;\n    of inverse method, 257;\n    for elimination of error, 353.\n\n  Rumford, Count, experiments on heat, 343, 350, 467.\n\n  Ruminants, Cuvier on, 683.\n\n  Russell, Scott, on sound, 541.\n\n\n  Sample, use of, 9.\n\n  Sandeman, on perigon, 306;\n    approximate arithmetic, 481.\n\n  Saturn, motions of satellites, 293;\n    rings, 293.\n\n  Schehallien, attraction of, 369, 566.\n\n  Schottus, on combinations, 179.\n\n  Schwabe, on sun-spots, 452.\n\n  Science, nature of, 1, 673.\n\n  Selenium, 663, 670.\n\n  Self-contradiction, 32.\n\n  Senior\u2019s definition of wealth, 75.\n\n  Senses, fallacious indications of, 276.\n\n  Seven, coincidences of number, 262;\n    fallacies of, 624.\n\n  Sextus, fatality of name, 264.\n\n  Sieve of Eratosthenes, 82, 123, 139.\n\n  Similars, substitution of, 17.\n\n  Simple identity, 37, 111;\n    inference of, 58;\n    contrapositive, 86;\n    induction of, 127.\n\n  Simple statement, 143.\n\n  Simplicity, law of, 33, 58, 72.\n\n  Simpson, discovery of property of chloroform, 531.\n\n  Simultaneity of knowledge, 34.\n\n  Singular names, 27;\n    terms, 129.\n\n  Siren, 10, 298, 421.\n\n  Slate, the logical, 95.\n\n  Smeaton\u2019s experiments, on water-wheels, 347;\n    windmills, 401, 441.\n\n  Smee, Alfred, logical machines, 107.\n\n  Smell, delicacy of, 437.\n\n  Smithsonian Institution, 329.\n\n  Smyth, Prof. Piazzi, 452.\n\n  Socrates, on the sun, 611.\n\n  Solids, 602.\n\n  Solubility of salts, 279.\n\n  *Some*, the adjective, 41, 56.\n\n  Sorites, 60.\n\n  Sound, observations on, 356;\n    undulations, 405, 421;\n    velocity of, 571;\n    classification of sounds, 732.\n\n  Space, relations of, 220.\n\n  Species, 698;\n    infima, 701;\n    natural, 724.\n\n  Specific gravities, 301;\n    heat of air, 557.\n\n  Spence, on boiling point, 546.\n\n  Spencer, Herbert, nature of logic, 4, 7;\n    sign of equality, 15;\n    rhythmical motion, 448;\n    abstraction, 705;\n    philosophy of, 718, 761, 762.\n\n  Spectroscope, 437.\n\n  Spectrum, 583.\n\n  Spiritualism, 671.\n\n  Spontaneous generation, 432.\n\n  Standards of measurement, 305;\n    the bar, 312;\n    terrestrial, 314;\n    pendulum, 315;\n    provisional, 318;\n    natural system, 319.\n\n  Stars, discs of, 277;\n    motions of, 280, 474;\n    variations of, 281;\n    approach or recess, 298;\n    standard stars, 301;\n    apparent diameter, 390;\n    variable, 450;\n    proper motions, 572;\n    Bruno on, 639;\n    new, 644;\n    pole-star, 652;\n    conflict with wandering stars, 748.\n\n  Stas, M., his balance, 304;\n    on atomic weights, 464.\n\n  Statements, kinds of, 144.\n\n  Statistical conditions, 168.\n\n  Stevinus, on inclined plane, 622.\n\n  Stewart, Professor Balfour, on resisting medium, 570;\n    theory of exchanges, 571.\n\n  Stifels, arithmetical triangle, 182.\n\n  Stokes, Professor, on resistance, 475;\n    fluorescence, 664.\n\n  Stone, E.\u00a0J., heat of the stars, 370;\n    temperature of earth\u2019s surface, 452;\n    transit of Venus, 562.\n\n  Struve on double stars, 247.\n\n  Substantial terms, 28.\n\n  Substantives, 14.\n\n  Substitution of similars, 17, 45, 49, 104, 106;\n    anticipations of, 21.\n\n  Substitutive weighing, 345.\n\n  *Sui generis*, 629, 728.\n\n  Sulphur, 670.\n\n  Summum genus, 93, 701.\n\n  Sun, distance, 560;\n    variations of spots, 452.\n\n  Superposition, of small effects, 450;\n    small motions, 476.\n\n  Swan, W., on sodium light, 430.\n\n  Syllogism, 140;\n    moods of, 55, 84, 85, 88, 105, 141;\n    numerically definite, 168.\n\n  Symbols, use of, 13, 31, 32;\n    of quantity, 33.\n\n  Synthesis, 122;\n    of terms, 30.\n\n\n  Table-turning, 671.\n\n  Tacit knowledge, 43.\n\n  Tacquet on combinations, 179.\n\n  Tait, P.\u00a0G., 375;\n    theory of comets, 571.\n\n  Talbot on the spectrum, 429.\n\n  Tartaglia on projectiles, 466.\n\n  Tastes, classification of, 732.\n\n  Tautologous propositions, 119.\n\n  Teeth, use in classification, 710.\n\n  Temperature, variations of, 453.\n\n  Tension of aqueous vapour, 500.\n\n  Terms, 24;\n    abstract, 27;\n    substantial, 28;\n    collective, 29;\n    synthesis of, 30;\n    negative, 45.\n\n  Terrot, Bishop, on probability, 212.\n\n  Test experiments, 347, 433.\n\n  Tetractys, 95.\n\n  Thales, predicted eclipse, 537.\n\n  Theory, results of, 534;\n    facts known by, 547;\n    quantitative, 551;\n    of exchanges, 571;\n    freedom of forming, 577;\n    of evolution, 761.\n\n  Thermometer, differential, 345;\n    reading of, 390;\n    change of zero, 390.\n\n  Thermopile, 300.\n\n  Thomas, arithmetical machine, 107.\n\n  Thomson, Archbishop, 50, 61.\n\n  Thomson, James, prediction by, 542;\n    on gaseous state, 654.\n\n  Thomson, Sir W., lighthouse signals, 194;\n    size of atoms, 195;\n    tides, 450;\n    capillary attraction, 614;\n    magnetism, 665;\n    dissipation of energy, 744.\n\n  Thomson and Tait, chronometry, 311;\n    standards of length, 315;\n    the crowbar, 460;\n    polarised light, 653.\n\n  Thomson, Sir Wyville, 412.\n\n  Thunder-cloud, 612.\n\n  Tides, 366, 450, 476, 541;\n    velocity of, 298;\n    gauge, 368;\n    atmospheric, 367, 553.\n\n  Time, 220;\n    definition of, 307.\n\n  Todhunter, Isaac, *History of the Theory of Probability*, 256, 375, 395;\n    on insoluble problems, 757.\n\n  Tooke, Horne, on cause, 226.\n\n  Torricelli, cycloid, 235;\n    his theorem, 605;\n    on barometer, 666.\n\n  Torsion balance, 272, 287.\n\n  Transit of Venus, 294, 348, 562.\n\n  Transit-circle, 355.\n\n  Tree of Porphyry, 702;\n    of Ramus, 703.\n\n  Triangle, arithmetical, 93, 182.\n\n  Triangular numbers, 185.\n\n  Trigonometrical survey, 301;\n    calculations of, 756.\n\n  Trisection of angles, 414.\n\n  Tuning-fork, 541.\n\n  Tycho Brahe, 271;\n    on star discs, 277;\n    obliquity of earth\u2019s axis, 289;\n    circumpolar stars, 366;\n    Sirius, 390.\n\n  Tyndall, Professor, on natural constants, 328;\n    magnetism of gases, 352;\n    precaution in experiments, 431;\n    use of imagination, 509;\n    on Faraday, 547;\n    magnetism, 549, 607;\n    scope for discovery, 753.\n\n  Types, of logical conditions, 140, 144;\n    of statements, 145;\n    classification by, 722.\n\n\n  Ueberweg\u2019s logic, 6.\n\n  Ultimate statements, 144.\n\n  Undistributed, attribute, 40;\n    middle term, 64, 103.\n\n  Undulations, of light, 558;\n    analogy in theory of, 635.\n\n  Undulatory theory, 468, 520, 538, 540;\n    inconceivability of, 510.\n\n  Unique objects, 728.\n\n  Unit, definition of, 157;\n    groups, 167;\n    of measurement, 305;\n    arcual, 306;\n    of time, 307;\n    space, 312;\n    density, 316;\n    mass, 317;\n    subsidiary, 320;\n    derived, 321;\n    provisional, 323;\n    of heat, 325;\n    magnetical and electrical units, 326, 327.\n\n  Unity, law of, 72.\n\n  Universe, logical, 43;\n    infiniteness of, 738;\n    heat-history of, 744, 749;\n    possible states of, 749.\n\n  Uranus, anomalies of, 660.\n\n\n  Vacuum, Nature\u2019s abhorrence of, 513.\n\n  Vapour densities, 548.\n\n  Variable, variant, 440, 441, 483.\n\n  Variation, linear, elliptic, &c., 474;\n    method of, 439.\n\n  Variations, logical, 140;\n    periodic, 447;\n    combined, 450;\n    integrated, 452;\n    simple proportional, 501.\n\n  Variety, of nature, 173;\n    of nature and art, 190;\n    higher orders of, 192.\n\n  Velocity, unit of, 321.\n\n  Venn, Rev. John, logical problem by, 90;\n    on Boole, 155;\n    his work on *Logic of Chance*, 394.\n\n  Venus, 449;\n    transits of, 294.\n\n  Verses, Protean, 175.\n\n  Vibrations, law of, 295;\n    principle of forced, 451;\n    co-existence of small, 476.\n\n  Vital force, 523.\n\n  Voltaire on fossils, 661.\n\n  Vortices, theory of, 513, 517.\n\n  Vulcan, supposed planet, 414.\n\n\n  Wallis, 124, 175.\n\n  Water, compressibility of, 338;\n    properties of, 610.\n\n  Watt\u2019s parallel motion, 462.\n\n  Waves, 599, 635;\n    nature of, 468;\n    in canals, 535;\n    earthquake, 297.\n\n  Weak arguments, effect of, 211.\n\n  Wells, on dew, 425.\n\n  Wenzel, on neutral salts, 295.\n\n  Whately, disjunctive propositions, 69;\n    probable arguments, 210.\n\n  Wheatstone, cipher, 124;\n    galvanometer, 286;\n    revolving mirror, 299, 308;\n    kaleidophone, 445;\n    velocity of electricity, 543.\n\n  Whewell, on tides, 371, 542;\n    method of least squares, 386.\n\n  Whitworth, Sir Joseph, 304, 436.\n\n  Whitworth, Rev. W.\u00a0A., on *Choice and Chance*, 395.\n\n  Wilbraham, on Boole, 206.\n\n  Williamson, Professor A.\u00a0W., chemical unit, 321;\n    prediction by, 544.\n\n  Wollaston, the goniometer, 287;\n    light of moon, 302;\n    spectrum, 429.\n\n  Wren, Sir C., on gravity, 581.\n\n\n  X, the substance, 523.\n\n\n  Yard, standard, 397.\n\n  Young, Dr. Thomas, tension of aqueous vapour, 500;\n    use of hypotheses, 508;\n    ethereal medium, 515.\n\n\n  Zero point, 368.\n\n  Zodiacal light, 276.\n\n  Zoology, 666.\n\n\nLONDON: R. CLAY, SONS, AND TAYLOR, PRINTERS,\n\n\n\n\nBY THE SAME AUTHOR.\n\n\n  ELEMENTARY LESSONS IN LOGIC: DEDUCTIVE AND INDUCTIVE. With Copious\n  Questions and Examples, and a Vocabulary of Logical Terms. Ninth\n  Edition. Fcap. 8vo. 3*s.* 6*d.*\n\n  PRIMER OF LOGIC. With Illustrations and Questions. New Edition. 18mo.\n  1*s.*\n\n  STUDIES IN DEDUCTIVE LOGIC. A Manual for Students. Crown 8vo. 6*s.*\n\n  THE SUBSTITUTION OF SIMILARS THE TRUE PRINCIPLE OF REASONING. 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KING AND CO.\n\n\n\n\nSPELLING CORRECTIONS\n\n  acording \u2192 according\n  aklaline \u2192 alkaline\n  an an \u2192 an\n  aws \u2192 laws\n  beween \u2192 between\n  BOOK III \u2192 BOOK IV\n  errror \u2192 error\n  incapadle \u2192 incapable\n  interpretion \u2192 interpretation\n  justifed \u2192 justified\n  longtitude \u2192 longitude\n  Marriotte \u2192 Mariotte\n  melecules \u2192 molecules\n  Meropolitana \u2192 Metropolitana\n  necesssarily \u2192 necessarily\n  nnmber \u2192 number\n  or \u2192 of\n  probabilty \u2192 probability\n  quantites \u2192 quantities\n  secresy \u2192 secrecy\n  sucession \u2192 succession\n  suficiently \u2192 sufficiently\n  telecope \u2192 telescope\n  verifiy \u2192 verify","title":"The principles of science : $b a treatise on logic and scientific method","tradition":"philosophy-ethics","translated_text":null,"word_count":292115}