{"collection":"philosophy-ethics","full_text_available":true,"graph_url":"https://unchartedknowledge.5gfusion.net/v2/api/page/gutenberg_43791/graph","html_url":"https://unchartedknowledge.5gfusion.net/v2/page/gutenberg_43791","id":"gutenberg_43791","license":"Public Domain (US)","rights_basis":"gutenberg_metadata_copyright_false","source":"gutenberg","source_url":"https://www.gutenberg.org/ebooks/43791","summary":null,"tags":["Category: Philosophy & Ethics","Category: Science - Biology","Category: Science - Physics","Science -- Philosophy"],"text":"NATURAL PHILOSOPHY\n\n\n  BY\n  WILHELM OSTWALD\n\n  TRANSLATED\n  BY\n  THOMAS SELTZER\n\n  _With the author's special revision for the American edition_\n\n\n\n  NEW YORK\n  HENRY HOLT AND COMPANY\n  1910\n\n\n\n\n  COPYRIGHT, 1910,\n  BY\n  HENRY HOLT AND COMPANY\n\n  _Published November_, 1910\n\n  THE QUINN & BODEN CO. PRESS\n  RAHWAY N. J.\n\n\n\n\n  The original of this book was published\n  as volume I in Reclam's B\u00dcCHER DER\n  NATURWISSENSCHAFT.\n\n\n\n\nPREFACE\n\n\nThe beginning of the twentieth century is marked by a sudden rise of\ninterest in philosophy. This is especially manifest in the vast growth\nof philosophic literature. The present movement, it is noteworthy, is by\nno means a revival proceeding from the academic philosophy traditionally\nrepresented at the universities, but has rather the original character\nof _natural philosophy_. It owes its origin to the fact that after the\nspecialization of the last half century, the synthetic factors of\nscience are again vigorously asserting themselves. The need finally to\nconsider all the numerous separate sciences from a general point of view\nand to find the connection between one's own activity and the work of\nmankind in its totality, must be regarded as the most prolific source of\nthe present philosophic movement, just as it was the source of the\nnatural philosophic endeavors a hundred years ago.\n\nBut while that old natural philosophy soon ended in a boundless sea of\nspeculation, the present movement gives promise of permanent results,\nbecause it is built upon an extremely broad basis of experience. The\nlaws of energy in the inorganic world and the laws of evolution in the\norganic world furnish mental instruments for a conceptual elaboration\nof the material provided by science, instruments capable not only of\nunifying present knowledge, but also of evoking the knowledge of the\nfuture. If it is not permissible to regard this unification as\nexhaustive and sufficient for all time, yet there is still so much left\nfor us to do in working over the material we have on hand from the\ngeneral points of view just mentioned, that the need for systematizing\nmust be satisfied before we can turn our gaze upon things more remote.\n\nThe present work is meant to serve as the first aid and guide in the\nacquisition of these comprehensive notions of the external world and the\ninner life. It is not meant to develop or uphold a \"system of\nphilosophy.\" Through long experience as a teacher the writer has learned\nthat those are the best pupils who soon go their own way. However, it\n_is_ meant to uphold a certain method, that is, the scientific (or, if\nyou will, the _natural_ scientific), which takes its problems, and\nendeavors to solve its problems, from experience and for experience. If,\nas a result, several points of view arise that differ from those of the\npresent day, and consequently demand a different attitude toward\nimportant matters in the immediate future, this very fact affords proof\nthat our present natural philosophy does not lead away from life, but\naims to form a part of our life, and has a right to.\n\n\n\n\nCONTENTS\n\n\n                                                         PAGE\n\n  INTRODUCTION                                              1\n\n\n  PART I\n\n  GENERAL THEORY OF KNOWLEDGE                              11\n\n  1.  The Formation of Concepts                            11\n\n  2.  Science                                              13\n\n  3.  The Aim of Science                                   13\n\n  4.  Concrete and Abstract                                16\n\n  5.  The Subjective Part                                  17\n\n  6.  Empirical Concepts                                   18\n\n  7.  Simple and Complex Concepts                          19\n\n  8.  The Conclusion                                       24\n\n  9.  The Natural Laws                                     28\n\n  10. The Law of Causation                                 31\n\n  11. The Purification of the Causal Relation              34\n\n  12. Induction                                            38\n\n  13. Deduction                                            40\n\n  14. Ideal Cases                                          44\n\n  15. The Determinateness of Things                        47\n\n  16. The Freedom of the Will                              50\n\n  17. The Classification of the Sciences                   53\n\n  18. The Applied Sciences                                 57\n\n\n  PART II\n\n  LOGIC, THE SCIENCE OF THE MANIFOLD, AND MATHEMATICS      61\n\n  19. The Most General Concept                             61\n\n  20. Association                                          63\n\n  21. The Group                                            65\n\n  22. Negation                                             68\n\n  23. Artificial and Natural Groups                        69\n\n  24. Arrangement of the Members                           75\n\n  25. Numbers                                              78\n\n  26. Arithmetic, Algebra, and the Theory of Numbers       79\n\n  27. Co-ordination                                        80\n\n  28. Comparison                                           82\n\n  29. Numbers                                              85\n\n  30. Signs and Names                                      86\n\n  31. The Written Language                                 89\n\n  32. Pasigraphy and Sound Writing                         92\n\n  33. Sound Writing                                        96\n\n  34. The Science of Language                              97\n\n  35. Continuity                                          101\n\n  36. Measurement                                         107\n\n  37. The Function                                        109\n\n  38. The Application of the Functional Relation          112\n\n  39. The Law of Continuity                               113\n\n  40. Time and Space                                      118\n\n  41. Recapitulation                                      124\n\n\n  PART III\n\n  THE PHYSICAL SCIENCES                                   127\n\n  42. General                                             127\n\n  43. Mechanics                                           128\n\n  44. Kinetic Energy                                      132\n\n  45. Mass and Matter                                     136\n\n  46. Energetic Mechanics                                 138\n\n  47. The Mechanistic Theories                            140\n\n  48. Complementary Branches of Mechanics                 144\n\n  49. The Theory of Heat                                  147\n\n  50. The Second Fundamental Principle                    150\n\n  51. Electricity and Magnetism                           154\n\n  52. Light                                               156\n\n  53. Chemical Energy                                     159\n\n\n  PART IV\n\n  THE BIOLOGIC SCIENCES                                   163\n\n  54. Life                                                163\n\n  55. The Storehouse of Free Energy                       168\n\n  56. The Soul                                            171\n\n  57. Feeling, Thinking, Acting                           174\n\n  58. Society                                             179\n\n  59. Language and Intercourse                            182\n\n  60. Civilization                                        184\n\n\n  INDEX                                                   187\n\n\n\n\nINTRODUCTION\n\n\nNatural science and natural philosophy are not two provinces mutually\nexclusive of each other. They belong together. They are like two roads\nleading to the same goal. This goal is the domination of nature by man,\nwhich the various natural sciences reach by collecting all the\nindividual actual relations between the natural phenomena, placing them\nin juxtaposition, and seeking to discover their interdependence, upon\nthe basis of which one phenomenon may be foretold from another with more\nor less certainty. Natural philosophy accompanies these specialized\nlabors and generalizations with similar labors and generalizations, only\nof a more universal nature. For instance, while the science of\nelectricity, as a branch of physics, deals with the relation of\nelectrical phenomena to one another and to phenomena in other branches\nof physics, natural philosophy is not only concerned with the question\nof the mutual connection of _all_ physical relations, but also endeavors\nto include in the sphere of its study chemical, biological,\nastronomical, in short, all the known phenomena. In other words,\n_natural philosophy is the most general branch of natural science_.\n\nHere two questions are usually asked. First, how can we define the\nboundary line between natural philosophy and the special sciences,\nsince, obviously, sharp lines of demarcation are out of the question?\nSecondly, how can we investigate and teach natural philosophy, when it\nis impossible for any one person to master all the sciences completely,\nand so obtain a bird's-eye view of the general relations between all the\nbranches of knowledge? To the beginner especially, who must first learn\nthe various sciences, it seems quite hopeless to devote himself to a\nstudy that presupposes a command of them.\n\nSince a discussion of the two questions will afford an excellent\npreliminary survey of the work in hand, it will be well to consider them\nin detail. In the first place, _the lack of complete and precise\nboundary lines is a general characteristic of all natural things_, and\nscience is a natural thing. If, for instance, we try to differentiate\nsharply between physics and chemistry, we are met with the same\ndifficulty. So also in biology if we try to settle beyond the shadow of\na doubt the line of separation between the animal and the vegetable\nkingdoms.\n\nIf, despite this well-known impossibility, we consider the division of\nnatural things into classes and orders as by no means useless and do not\ndiscard it, but regard it as an important scientific work, this is\npractical proof that such classification preserves its essential\nusefulness, even if it does not attain ideal definiteness. For, this\nimperfection notwithstanding, classification reaches its end, which is\na comprehensive view, and thus a mastery, of the manifoldness of\nphenomena. For example, with the overwhelming majority of organic beings\nthere is no doubt whether they are animals or plants. Similarly, most\nphenomena of inorganic nature can readily be designated as physical or\nchemical. For all such cases, therefore, the existing classification is\ngood and useful. The few cases presenting difficulty may very well be\nconsidered by themselves wherever they occur, and we need merely take\ncognizance of them here. It follows from this, to be sure, that\nclassification will be all the _better fitted to its purpose the less\nfrequently_ such doubtful cases arise, and that we have an interest in\nrepeatedly testing existing classifications with a view to finding out\nif they cannot be supplanted by more suitable ones.\n\nIn these matters it is much the same as when we look upon the waves on\nthe surface of a large body of water. Our first glance tells us that a\nnumber of waves are rolling there; and from a point giving us a\nsufficiently wide outlook, we can count them and gauge their width. But\nwhere is the line of division between one wave and the next? We\nundoubtedly see one wave following another, yet it is impossible for us\nto indicate precisely the end of one and the beginning of the next. Are\nwe then to deduce that it is superfluous or unfeasible to designate the\nwaves as different? By no means. On the contrary, in strictly\nscientific work we will endeavor to find some suitable definition of the\nboundary line between two consecutive waves. It may then be called an\narbitrary line, and in a degree arbitrary it will certainly be. But to\nthe investigator this does not matter. What concerns him is, if, with\nthe help of this definition, wave lengths can be unequivocally\ndetermined, and if this is possible, he will use the definition as\nsuitable to the purposes of science, without dismissing from his mind\nthe idea that possibly some other definition may provide an even easier\nor sharper determination. Such an one he would instantly prefer to the\nold one.\n\nThus we see that these questions of classification are not questions of\nthe so-called \"essence\" of the thing, _but pertain merely to purely\npractical arrangements for an easier and more successful mastery of\nscientific problems_. This is an extremely important point of view, much\nmore far-reaching than is apparent here at its first application.\n\nAs to the second objection, I will admit its validity. But here, too, we\nhave a phenomenon appearing in all branches and forms of science.\nTherefore we must familiarize ourselves with it in advance. Science was\ncreated by man for man's purposes, and, consequently, like all human\nachievements, possesses the indestructible quality of imperfection. But\nthe mere fact that a successful working science exists, with the help of\nwhich human life has been fundamentally modified, signifies that _the\nquality of incompleteness in human learning is no hindrance to its\nefficiency_. For what science has once worked out always contains a\nportion of truth, hence a portion of efficiency. The old corpuscular\ntheory of light, which now seems so childishly incomplete to us, was\nadequate, none the less, for satisfactorily explaining the phenomena of\nreflection and refraction, and the finest telescopes have been built\nwith its help. This is due to the _true elements_ in it, which taught us\ncorrectly to calculate the direction of rays of light in reflection and\nrefraction. The rest was merely an arbitrary accessory which had to fall\nwhen new, contradictory facts were discovered. These facts could not\nhave been taken into consideration when the theory was propounded,\nbecause they were not yet known. But when the corpuscular theory of\nlight was replaced by the theory of waves of an elastic ether, geometric\noptics at first remained quite unchanged, because the theory of straight\nlines of rays could be deduced from the new views also, though not so\neasily and smoothly. And geometric optics was then concerned with\nnothing but these straight lines, in no wise with the question of their\npropagation. It did not become clear until recently that this conception\nof straight lines of rays is incomplete, though, it is true, it made a\nfirst approach toward the presentation of actual phenomena. It fails\nwhen it comes to characterize the behavior of a pencil of rays of large\naperture. The old idea of a straight line of rays was to be replaced by\na more complex concept with more varied characteristics, namely, the\nwave-surface. The greater variety of this concept renders possible the\npresentation of the greater variety of the optical phenomena just\nmentioned. And from it proceed the very considerable advances that have\nbeen made, since the new theory was propounded, in optical instruments,\nespecially the microscope and the photographic objective, for the\npurposes of which pencils of rays of large aperture are required. The\nastronomic objective with its small angle of aperture has not undergone\nparticularly important improvements.\n\nExperience in every province of science is the same as in this. Science\nis not like a chain which snaps when only a single link proves to be\nweak. It is like a tree, or, better still, like a forest, in which all\nsorts of changes or ravages go on without causing the whole to pass out\nof existence or cease to be active. The relations between the various\nphenomena, once they become known, continue to exist as indestructible\ncomponents of all future science. It may come to pass, in fact, does\ncome to pass very frequently, that the form in which those relations\nwere first expressed prove to be imperfect, and that the relations\ncannot be maintained quite generally. It turns out that they are\nsubjected to other influences which change them because they had been\nunknown, and which could not have been taken into consideration at the\ndiscovery and first formulation of these relations. But no matter what\nchanges science may undergo, a certain residue of that first knowledge\nwill remain and never be lost. In this sense, a truth that science has\nonce gained has life eternal, that is, it will exist as long as human\nscience exists.\n\nApplying this general notion to our case, we have the following. How far\nand how generally at any given time the relations of the various\nphenomena are summed up in fixed forms, that is, in natural laws, will\ndepend upon the stage attained by each of the special sciences. But\nsince science has been in existence it has yielded a certain number of\nsuch general laws, and these, though they have been filed down a good\ndeal in form and expression, and have undergone many corrections as to\nthe limits of their application, nevertheless have preserved their\nessence, since they began their existence in the brains of human\ninvestigators. The net of the relations of phenomena grows ever wider\nand more diversified, but its chief features persist.\n\nThe same is true of an individual. No matter how limited the circle of\nhis knowledge, _it is a part of the great net, and therefore possesses\nthe quality by virtue of which the other parts readily join it as soon\nas they reach the consciousness and knowledge of the individual_. The\nman who thus enters the realm of science acquires advantages which may\nbe compared to those of a telephone in his residence. If he wishes to,\nhe may be connected with everybody else, though he will make extremely\nlimited use of his privilege, since he will try to reach only those with\nwhom he has personal relations. But once such relations have been\nestablished, the possibility of telephone communication is\nsimultaneously and automatically established. Similarly, every bit of\nknowledge that the individual appropriates will prove to be a regular\npart of the central organization, the entire extent of which he can\nnever cover, though each individual part has been made accessible to\nhim, provided he wants to take cognizance of it.\n\nThe mere beginner in learning, therefore, when receiving the most\nelementary instruction in school, or from his parents, or even from his\npersonal experiences in his surroundings, is grasping one or more\nthreads of the mighty net, and can grope his way farther along it in\norder to draw an increasing area of it into his life and the field of\nhis activity. _And this net has the valuable, even precious quality of\nbeing the same that joins the greatest and most comprehensive intellects\nin mankind to one another._ The truths a man has once grasped he need\nnever learn afresh so far as their _actual content_ is concerned, though\nnot infrequently--especially in newer sciences--he may have to see the\n_form_ of their presentation and generalization change. For this reason\nit is of such especial importance for each individual from the first to\nperceive these unalterable facts and realize that they are unalterable\nand learn to distinguish them from the alterable forms of their\npresentation. It is in this very regard that the incompleteness of human\nknowledge is most clearly revealed. Time and again in the history of\nscience form has been taken for content, and necessary changes of\nform--a merely practical question--have been confused with revolutionary\nmodifications of the content.\n\nThus, each presentation of a science has its natural philosophic\nportion. In text-books, whether elementary or advanced, the chapter on\nnatural philosophy is found usually at the beginning of the book,\nsometimes at the end, in the form of a \"general introduction,\" or\n\"general summary.\" In the special works in which the latest advances of\nscience are made known by the investigators, the natural philosophic\nportions are usually to be found in the form of theses, of principles,\nwhich are not discussed, often not even explicitly stated, but upon the\nacceptance of which depend all the special conclusions that are drawn,\nin the case in hand, from the new facts or thoughts imparted. Whether at\nthe beginning or at the end of the book, these most general principles\ndo not quite occupy the place that befits them. If at the introduction\nof the text-book, they are practically devoid of content, since the\nfacts they are meant to summarize are yet to be unfolded in the course\nof the presentation. If at the end, they come too late, since they have\nalready been applied in numerous instances, though without reference to\ntheir general nature. The best method is--and a good teacher always\nemploys this method, whether in the spoken or the written word--to let\nthe generalizations come whenever the individual facts imparted require\nand justify them.\n\nThus, all instruction in natural sciences is necessarily interspersed\nwith natural philosophy, good or bad, according to the clearheadedness\nof the teacher. If we wish to obtain a perfect survey of a complex\nstructure, as, for instance, the confusion of streets in a large city,\nwe had better not try to know each street, but study a general plan,\nfrom which we learn the comparative situation of the streets. So it is\nwell for us in studying a special science to look at our general plan,\nif for no other reason than to keep from losing our way when it may\nchance to lead through a quarter hitherto unknown. This is the purpose\nof the present work.\n\n\n\n\nPART I\n\nGENERAL THEORY OF KNOWLEDGE\n\n\n=1. The Formation of Concepts.= To the human mind, as it slowly awakens\nin every child, the world at first seems a chaos consisting of mere\nindividual experiences. The only connection between them is that they\nfollow each other consecutively. Of these experiences, all of which at\nfirst are different from one another, certain parts come to be\ndistinguished by the fact that they are repeated more frequently, and\ntherefore receive a special character, that of _being familiar_. The\nfamiliarity is due to our _recalling_ a former similar experience; in\nother words, to our feeling that there is a relation between the present\nexperience and certain former experiences. The cause of this phenomenon,\nwhich is at the basis of all mental life, is a quality common to all\nliving things, and manifesting itself in all their functions, while\nappearing but rarely or accidentally in inorganic nature. It is the\nquality by virtue of which _the oftener any process has taken place in a\nliving organism the more easily it is repeated_. Here is not yet the\nplace to show how almost all the characteristic qualities of living\nbeings, from the preservation of the species to the highest intellectual\naccomplishments, are conditioned by this special peculiarity. Suffice it\nto say that because of this quality all those processes which are\nrepeated frequently in any given living organism, assume spontaneously,\nthat is, from physiologic reasons, a character distinguishing them\nessentially from those which appear only in isolated instances, or\nsporadically.\n\nIf a living being is equipped with consciousness and thought, like man,\nthen the conscious recollections of such uniform experiences form the\nenduring or permanent part in the sum-total of his experiences. Each\ntime a complex event, like the change of seasons, for example, which we\nknow from experience repeats itself--each time a part of such an event\nreaches our consciousness, we are prepared also for the other parts that\nexperience teaches are connected with it. This makes it possible for us\nto foresee future events. What significance the foreseeing of future\nevents has for the preservation and the development of the individual as\nwell as the species can only be indicated here. To give one instance, it\nis our ability to foretell the coming of winter with the impossibility\nof obtaining food directly during the winter that causes us to refrain\nfrom at once using up all the food we have and to preserve it for the\nday of need. The ability to foretell, therefore, becomes the foundation\nof the whole structure of economic life.\n\n\n=2. Science.= The prophecy of future events based upon the knowledge of\nthe details of recurring events is called _science_ in its most general\nsense. Here, as in most cases in which language became fixed long before\nmen had a clear knowledge of the things designated, the name of the\nthing is easily associated with false ideas arising either from errors\nthat had been overcome or from other, still more accidental, causes.\nThus, the mere knowledge of _past_ events is also called science without\nany thought of its use for prophesying future events. Yet a moment's\nreflection teaches that mere knowledge of the past which is not meant\nto, or cannot, serve as a basis for shaping the future is utterly\naimless knowledge, and must take its place with other aimless activities\ncalled _play_. There are all sorts of plays requiring great acumen and\npatient application, as for example the game of chess; and no one has\nthe right to prevent any individual from pursuing such games. But the\nplayer for his part must not demand special regard for his activity. By\nusing his energies for his personal pleasure and not for a social\npurpose, that is, for a general human purpose, he loses every claim to\nthe social encouragement of his activity, and must be content if only\nhis individual rights are respected; and that, too, only so long as the\nsocial interests do not suffer by it.\n\n\n=3. The Aim of Science.=These views are deliberately opposed to a very\nwidespread idea that science should be cultivated \"for its own sake,\"\nand not for the sake of the benefits it actually brings or may be made\nto bring. We reply that there is nothing at all which is done merely\n\"for its own sake.\" Everything, without exception, is done for human\npurposes. These purposes range from momentary personal satisfaction to\nthe most comprehensive social services involving disregard of one's own\nperson. But in all our actions we never get beyond the sphere of the\nhuman. If, therefore, the phrase \"for its own sake\" means anything, it\nmeans that science should be followed for the sake of the immediate\npleasure it affords, that is to say, as _play_ (as we have just\ncharacterized it), and in the \"for-its-own-sake\" demand there is hidden\na misunderstood idealism, which, on closer inspection, resolves itself\ninto its very opposite, the degradation of science.\n\nThe element of truth hidden in that misunderstood phrase is, that in a\nhigher state of culture it is found better to disregard the _immediate_\ntechnical application in the pursuit of science, and to aim only for the\ngreatest possible perfection and depth in the solution of its individual\nproblems. Whether this is the correct method of procedure and when it is\nso, is solely a question of the general state of culture. In the early\nstages of human civilization such a demand is utterly meaningless, and\nall science is necessarily and naturally confined to immediate life. But\nthe wider and more complex human relations become, the wider and surer\nmust the ability to predict future events become. Then it is the\nfunction of prophesying science to have answers ready for questions\nwhich as yet have not become pressing, but which with further\ndevelopment may sooner or later become so.\n\nIn the net-like interlacing of the sciences, that is, of the various\nfields of knowledge, described in the introduction, we must always\nreckon with the fact that our anticipation of what kind of knowledge we\nshall next need must always remain very incomplete. It is possible to\nforesee future needs in general outline with more or less certainty, but\nit is impossible to be prepared for particular individual cases which\nlie on the _border line_ of such anticipation, and which may sometimes\nbecome of the utmost importance and urgency. Therefore it is one of the\nmost important functions of science to achieve as _perfect_ an\nelaboration as possible of _all_ the relations conceivable, and in this\npractical necessity lies the foundation of the general or _theoretical_\nelaboration of science.\n\n=The Science of Concepts.= Here the question immediately arises: how can\nwe secure such perfection? The answer to this general preliminary\nquestion of all the sciences belongs to the sphere of the first or the\nmost general of all the sciences, a knowledge of which is presupposed\nfor the pursuit of the other sciences. Since its foundation by the Greek\nphilosopher Aristotle it has borne the name of _logic_, which name,\netymologically speaking, hints suspiciously at the _word_, and the word,\nas is known, steps in where ideas are wanting. Here, however, we have to\ndeal with the very science of ideas, to which language bears the\nrelation only of a means--and often an inadequate means--to an end. We\nhave already seen how, through the physiologic fact of _memory_,\nexperiences are found in our consciousness which are similar, that is,\npartially coinciding with one another. These coinciding parts are those\nconcerning which we can make predictions, for the very reason that they\ncoincide in every single instance, and they alone, therefore, constitute\nthat part of our experience which bears results and hence has\nsignificance.\n\n\n=4. Concrete and Abstract.= Such coinciding or repeated parts of similar\nexperiences we call, as already stated, _concepts_. But here, too,\nattention must immediately be drawn to a linguistic imperfection, which\nconsists in the fact that in such a group of coinciding experiences we\ndesignate by the same name both the isolated experience or the object of\na special experience and the totality of _all_ the coinciding\nexperiences; in other words, all the similar experiences. Thus, _horse_\nmeans, on the one hand, quite a definite thing which for the moment\nforms an object of our experience, and, on the other, the totality of\nall possible similar objects which have been present in our former\nexperiences, and which we shall meet in our future experiences. It is\ntrue that these two sorts of contents of consciousness bearing the same\nname are distinguished also as _concrete_ and _abstract_, and there is\nan inclination to attribute \"reality\" only to the first, while the\nother, as \"mere entities in thought,\" are relegated to a lesser degree\nof reality. As a matter of fact, the difference, though important, is of\nquite another kind. It is the difference between the _momentary\nexperience_, as opposed to the totality of the corresponding _memories_\nand _expectations_. Hence not so much a difference in _reality_ as in\n_presence_. However, our observations have already made it apparent that\npresence alone never yields knowledge. A necessary part of knowledge is\nthe memory of former similar experiences. For without such memory and\nthe corresponding comparison, it is quite impossible for us to get at\nthose things which agree and which, therefore, may be predicted; and we\nshould stand before every one of our experiences with the helplessness\nof a new-born babe.[A]\n\n[A] Sometimes on suddenly awaking from a profound sleep a person finds\nhimself for the moment deprived of his personal stock of memories,\nunable to recall where and in what circumstances he is. No one who has\nexperienced such a condition can ever forget the terrifying sense of\nhelplessness it brings.\n\n\n=5. The Subjective Part.= We shall therefore have to recognize realities\nin abstract ideas in so far as they must rest upon some experiences to\nbe at all intelligible to us. Since the formation of concepts depends\nupon memories, and these may refer, according to the individual, to very\ndifferent parts of the same experience of different individuals,\nconcepts always possess an element dependent upon the individual, or a\n_subjective_ element. This, however, does not consist in the _addition_\nby the individual of new parts not found in the experience, but, on the\ncontrary, in the different _choice_ out of what is found in the\nexperience. If every individual absorbed all parts of the experience,\nthe individual, or subjective, differences would disappear. And since\nscientific experience endeavors to make the absorption of experiences as\ncomplete as possible, it aims nearer and nearer to this ideal by seeking\nto equalize the subjective deficiency of the individual memory through\nthe collocation of as many and as various memories as possible, thus\nfilling in the subjective gaps in experience as far as possible and\nrendering them harmless.\n\n\n=6. Empirical Concepts.= First and unconditionally those concepts\npossess reality which always and without exception are based on\n_experienced_ facts. But we can easily make manifold arbitrary\ncombinations of concepts from different experiences, since our memory\nfreely places them at our disposal, and from such a combination we can\nform a new concept. Of course it is not necessary that our arbitrary\ncombination should also be found in our past or future experiences. On\nthe contrary, we may rather expect that there could be many more\narbitrary combinations not to be found in experience than combinations\nlater \"confirmed\" by experience. The former are purposeless because\nunreal, the latter, on the contrary, are of the utmost consequence\nbecause upon them is based the real aim of knowledge, prediction. The\nformer are those which have brought the very \"reality\" of the concepts\ninto ill repute, while the latter show that the formation and the mutual\nreaction of the concepts practically constitute the entire content of\nall science. It is of the greatest importance, therefore, to distinguish\nbetween the two kinds of concept combinations, and the study of this\ndifferentiation forms the content of that most general of all the\nsciences which we have characterized as logic, or, better, the science\nof concepts.\n\n\n=7. Simple and Complex Concepts.= The formation of concepts consists, as\nwe have seen, in the selection of those parts of different but similar\nexperiences which coincide with one another and in the elimination of\nthose that are different in kind. The results of such a procedure may\nvary greatly according to the number and the difference of the\nexperiences placed in relation with one another. If, for example, we\ncompare only a few experiences, and if, moreover, these experiences are\nvery similar to one another, then the resulting concepts will contain\nvery many parts that agree. But at the same time they will have the\npeculiarity of not being applicable to other experiences, since these\nare without some of the coinciding parts of that narrower circle. Thus,\nfor example, the concept which a rustic chained to the soil all his life\nhas of human work does not apply to the work of the city man. A concept\nwill embrace a larger number of individual cases in proportion as it\ncontains fewer different parts. And by systematically following out this\nthought we arrive at the conclusion that the concepts that are simple\nand have no different parts at all find the widest application or are\nthe most general.\n\nThe elimination of the non-coinciding parts from the concept-forming\nexperience is called _abstraction_. Obviously abstraction must be\ncarried the farther the more numerous and the more varied the\nexperiences from which the concepts are abstracted, and the simplest\nconcepts are the most abstract.\n\nBy looking back over the ground just traversed, the less abstract ideas\nmay also be regarded as the _more complex_ in contradistinction to the\nsimpler ones. Only we must guard against the error of literal\ninterpretation and not suppose that the less simple concepts have really\nbeen compounded of the simpler ones. In point of origin they actually\nexisted first, since the experience contains the ensemble of all the\nparts, those which have been retained as well as those which have been\neliminated. It is only later, by a characteristic mental operation,\nafter we have analyzed the more complex concept, that is, after we have\ndisclosed the simpler concepts existing in it, that we can compound it\nagain; in other words, execute its synthesis.\n\nThese relations bear a striking resemblance to the relations known from\nchemistry to exist between substances, namely, between elements and\ncompounds. From the chaos of all objects of experimentation (chemistry\npurposely limits itself to ponderable bodies) the _pure_ substances are\nsifted out--an operation corresponding to the formation of concepts. The\npure substances prove to be either _simple_ or _compound_, and the\ncompounds are so constituted that they can each be reduced to a limited\nnumber of simple substances. The simple substances, or _elements_,\nretain this quality of simplicity only until they are recalled; that is,\nuntil it has been proved that they, too, can be resolved into still\nsimpler elements. The same is true of the simple concepts. They can\nclaim simplicity only until their complex nature is demonstrated.\n\nWith all these similarities we must be extremely careful never to forget\nthe differences existing alongside the agreements. So hereafter we shall\nmake no further use of the chemical simile. It was brought into\nrequisition merely in order to acquaint the beginner the more readily\nwith the entire method of investigation by means of a more familiar\nfield of thought and study. It is quite certain, however, that side by\nside with the given similarities there are also radical differences.\nMoreover, the notion of simple and complex concepts or \"ideas\" had been\nelaborated by John Locke long before chemistry reached its present state\nof clearness concerning the concept of the elements.\n\nNevertheless since then the relation has been completely reversed. While\nthe study of the chemical elements has in the meantime undergone great\ndevelopment, so that not only have the elements of all the substances\ncoming under the observation of the chemist been discovered, but,\ninversely, many compound substances have been constructed from their\nelements, not even an approach to such a development is apparent in the\nstudy of concepts. On the contrary, the whole matter has remained at\nabout the same point as that to which John Locke had brought it in the\nsecond half of the seventeenth century. This is due above all to the\nopinion of the most influential philosophers, that Aristotle's logic, or\nscience of concepts, is absolutely true as well as exhaustive and\ncomplete, so that, at the utmost, what is left for later generations to\ndo is only to make a change in the form in which the matter is\npresented. It is true that in more recent times the grave error of this\nview is beginning to be recognized. We realize that Aristotle's logic\nembraces but a very small part of the entire field, though in this part\nhe displays the greatest genius. But beyond this general recognition no\ngreat step forward has been made. Not even a provisional table of the\nelementary concepts has been propounded and applied since Locke.\n\nHence in the following investigation we shall have to speak of the\nelements or the simpler parts of a complex concept only in the sense\nthat these concept elements are simpler as compared with the complex\nconcepts, but not in the sense that the simplest or truly elementary\nconcepts have already been worked out. It must be left to later\ninvestigators to find these, and it may be expected that the reduction\nof some concepts until then considered elementary into still simpler\nones will take place chiefly in times of great intellectual progress.\n\n_Complex concepts_ can, in the first place, be formed from experience,\nfor in an empirical concept we meet with several conceptual component\nparts which can be separated from one another by a process of\nabstraction, but are always found together in the given experiences. For\nexample, the concept _horse_ has originated from a very frequent,\nsimilarly repeated experience. On analysis it is found to contain a vast\nnumber of other concepts, such as quadruped, vertebrate animal,\nwarm-blooded, hairiness, and so on. Horse, then, is obviously a _complex\nempirical concept_.\n\nOn the other hand, we can combine as many simple concepts as we please,\neven if we did not find them combined in experience, for in reality\nthere is nothing to hinder us from uniting all the concepts provided by\nmemory into any combinations we please. In this way we obtain _complex\narbitrary concepts_.\n\nThe task of science can now be even more sharply defined than before by\nthe fact that it _permits the construction of arbitrary concepts which\nin circumstances to be foreseen become empirical concepts_. This is\nanother expression for _prediction_, which we recognized as the\ncharacteristic of science. It goes deeper than the previous definition,\nbecause here the means for its realization are given.\n\n\n=8. The Conclusion.= First let us consider the scientific import of the\ncomplex empirical concepts. It consists in the fact that they accustom\nus to the coexistence of the corresponding elements of a concept. So\nthat when, in a new experience, we meet with some of these elements\ntogether, we immediately suppose that we shall find in the same\nexperience the other elements also which have not yet been ascertained.\nSuch a supposition is called a _conclusion_. A conclusion always exceeds\nthe present experience by predicting an expected experience. Therefore,\nthe form of a conclusion is the universal form of scientific\npredication.\n\nA conclusion must contain at least two concepts, the one which is\nexperienced, and the one which, on the basis of this experience, is\nexpected. Every complex empirical concept makes such a conclusion\npossible after it has been separated into simpler concepts. And the\nsimplest case is naturally the one in which there are only _two_ parts,\nor in which only two parts are taken into consideration.\n\nTo what extent such a conclusion is valid, that is to say, to what\nextent the experience produces the anticipated concept, obviously\ndepends upon the reply to a very definite fundamental question. If in\nexperience the union of the two parts of the concept occurs\n_invariably_, so that one part of the concept is never experienced\nunless the other part is also experienced, then there is the _greatest_\nprobability that the expected experience will also have the same\ncharacter, and that the conclusion will prove valid or true. To be sure,\nthere is no way of making certain that the coincident occurrence of the\ntwo concepts, which experience has shown to be _without exception_\nhitherto, will continue to be so also in the future. For our only means\nof penetrating into the future consists in applying that conclusion from\nprevious experiences to future experiences, and it can therefore by no\nmeans claim absolute validity. There are, however, different _degrees of\ncertainty_, or, rather, _probability_, attaching to such a conclusion.\nIn experiences that occur but rarely the probability is that so far we\nhave experienced only certain combinations of simple concepts, while\nothers, though occurring, have not yet entered within the limited circle\nof our experience. In such a case a conclusion of the kind mentioned\nabove may be right, but there is also some probability of its being\nfalse. On the other hand, in experiences which happen extremely\nfrequently and in the most diverse circumstances, and in which we always\nfind the constant and unexceptional combination, the probability is very\nstrong that we shall find the combination in future experiences also,\nand the probability of the conclusion approaches practical certainty. Of\ncourse, we can never quite exclude the possibility that new relations\nnever as yet experienced might enter, by which the conclusion which\nhitherto has always been true would now become false, whether because\nthe expectation entertained prove invalid in single instances or in all\ncases.\n\nIt follows from this that in general, our conclusions will have the\ngreater probability the more generally and the oftener the corresponding\nexperiences have occurred and are occurring. Such concepts as are found\nconsistently in many experiences otherwise different are called\n_general_ concepts, and therefore the probability of the conclusions\ndescribed will be the greater the more _general_ the concepts to which\nthey refer. This obtains to such a degree that we feel that certain very\ngeneral conclusions must be true always and without exception, and it is\n\"unthinkable\" to us that they can ever in any circumstances prove not\nvalid. Such a statement, however, is never anything else than a hidden\nappeal to experience. For the mere putting of the question, whether the\nconclusion can also be false, demonstrates that the opposite of what has\nproved to be the experience so far can be conceived, and the assertion\nof its \"unthinkability\" only signifies that such an experience cannot be\nevoked in the mind by the _memory_ for the very reason that, as has been\npremised, there are no such memories because the experiences did not\nexist. But since, on the other hand, there is no hindrance to thinking\nany combinations of concepts at will, we have not the least difficulty,\nas everybody knows, in thinking any sort of \"nonsense\" whatsoever. Only\nit is impossible to reproduce such combinations from memory.\n\nThe scientific conclusion, therefore, first takes the form: if A is,\nthen B is also. Here A and B represent the two simple concepts which are\nknown from experience to be found together in the more complex concept\nC. The word \"is\" signifies here some empirical reality corresponding to\nthe concepts. The conclusion may therefore also be expressed, somewhat\nmore circumstantially and more precisely, in this form: if A is\nexperienced, the experience of B is also expected. The evoking of this\nexpectation, which implies its justification, is due to the recollection\nof the coincidence of the two concepts in former experiences, and the\nprobability depends, in the manner described above, upon the number of\nvalid cases. Here it must be observed that even individual cases in\nwhich our expectations have been deceived do not for the most part lead\nus to regard the conclusion as generally untrue, that is, to abandon the\nexpectation of B from A. For we know that our experience is always\n_incomplete_, that in certain circumstances we fail to notice existing\nfactors, and that, therefore, our failure to find that relation valid\nwhich, on other occasions, has been found to be valid, may be attributed\nto subjective causes. In case, however, of the repeated occurrence of\nsuch disappointments, we will look elsewhere for relations between these\nand other elements of experience, in order that thereafter we may\nforesee such cases also and include them in our anticipations.\n\n\n=9. The Natural Laws.= The facts just described have very frequently\nfound expression in the doctrine of the _laws of nature_, in connection\nwith which we have often, as in the man-made social or political laws,\nconceived of a lawmaker, who, for some reasons, or perhaps arbitrarily,\nhas ordained that things should be as they are and not otherwise. But\nthe intellectual history of the origin of the laws of nature shows that\nhere the process is quite a different one. The laws of nature do not\ndecree what shall happen, but _inform us what has happened and what is\nwont to happen_. The knowledge of these laws, therefore, makes it\npossible for us, as I have emphasized again and again, to foresee the\nfuture in a certain degree and, in some measure, also to determine it.\nWe determine the future by constructing those relations in which the\ndesired results appear. If we cannot do so either because of ignorance\nor because of inaccessibility to the required relations, then we have no\nprospect of fashioning the future according to our desires. The wider\nour knowledge of the natural laws, that is, of the actual behavior of\nthings, the more likely and more numerous the possibilities for\nfashioning the future according to our desires. In this way science can\nbe conceived of as the study of how to become happy. For he is happy\nwhose desires are fulfilled.\n\nIn this conception the natural laws indicate what simpler concepts are\nfound in complex concepts. The complex concept _water_ contains the\nsimpler ones _liquid_, a certain _density_, _transparency_,\n_colorlessness_,[B] and many others. The sentences, water is a liquid,\nwater has a density of one, water is transparent, water is colorless,\nor, pale blue, etc., are so many natural laws.\n\n[B] More precisely, a very pale blue.\n\nNow what predictions do those natural laws enable us to make?\n\nThey enable us to predict that when we have recognized a given body as\nwater by virtue of the above properties, we are justified in expecting\nto find in the same body all the other known properties of water. And so\nfar experience has invariably confirmed such expectations.\n\nFurthermore, we may expect that if in a given specimen of water we\ndiscover a relation which up to that time was unknown, we shall find\nthis relation also in all the other specimens of water even though they\nwere not tested for that particular relation. It is obvious how\nenormously this facilitates the progress of science. For it is only\nnecessary to determine this new relation in some one case accessible to\nthe investigator to enable us to predict the same relation in all the\nother cases without subjecting them to a new test. As a matter of fact,\nthis is the general method that science pursues. It is this that makes\nit possible for science to make regular and generally valid progress\nthrough the labors of the most various investigators who work\nindependently of one another, and often know nothing of one another.\n\nOf course, it must not be forgotten that such conclusions are always\nobtained in accordance with the following formula: _things have been so\nuntil now, therefore we expect that they will be so in the future_. In\nevery such case, therefore, there is the possibility of error. Thus far,\nwhenever an expectation was not realized, it was almost always possible\nto find an \"explanation\" for the error. Either the inclusion of the\nspecial case in the general concept proved to be inadmissible because\nsome of its other characteristics were absent, or the accepted\ncharacterization of the concept required an improvement (limitation or\nextension). In other words, one way or another, there was a discrepancy\nbetween the concept and the experience, and, as a rule, sooner or later\nit becomes possible for us to arrive at a better adjustment between\nthem.\n\nThis general truth has often been interpreted to mean that in the end\nsuch an adjustment must of necessity always be possible to reach,\nwithout exception; in other words, that absolutely every part of an\nexperience can be demonstrated as conditioned by natural law. Evidently\nsuch an assertion far exceeds the demonstrable. And even the usual\nconclusion cannot be applied here, that because it has happened so in\nthe past it will happen so in the future also. For the part of our\nexperiences that we can grasp by natural laws is infinitesimally small\nin comparison with that in which our knowledge still fails us entirely.\nI will mention only the uncertainty in predicting the weather for only\none day ahead. Moreover, when we consider that until now only the\n_easiest_ problems had been solved, and naturally so, because they were\nmost accessible to the means at hand, then we can readily see that\nexperience offers no basis whatever for such a conclusion. We must not\nsay, therefore, that because we have been able so far to explain all\nexperiences by natural laws it will be so in the future likewise. For we\nare far from being able to explain all experiences. In fact, it is only\na very small part that we have begun to investigate. We are as little\njustified in saying that we have explained all the problems of our\nexperience that have been subjected to scientific investigation. We have\nby no means explained all of them. Every science, even mathematics,\nteems with unsolved problems. So we must resign ourselves to the present\nstatus of human knowledge and ability, and may at best express the\n_hope_ founded upon previous experience, that we shall be able to solve\nmore and more of the incalculable number of problems of our experience\nwithout indulging in any illusions as to the perfection of this work.\n\n\n=10. The Law of Causation.= By reason of its frequency and importance\nthe mental process above described has been subjected to the most\ndiverse investigations, and that most general form of the scientific\nconclusion (which we apply in ordinary life even much more frequently\nthan in science) has been raised, under the name of the law of\ncausation, to a principle anteceding all experience and to the very\ncondition making experience possible. Of this so much is true, that\nthrough the peculiar physiological organization of man, _memory in the\nmost general sense_--the easier execution of such processes as have\nalready repeatedly taken place in the organism, as against entirely new\nkinds of processes--the formation of concepts (of the recurring parts in\nthe constantly changing variety of processes), is especially stimulated\nand facilitated. By it the recurring parts of experience step into the\nforeground, and on account of their paramount practical importance for\nthe security of life, it may well be said in the sense of the theory of\nevolution and adaptation, that the entire structure and mode of life of\nthe organism, especially of the human organism, nay, perhaps life\nitself, is indissolubly bound up with that foresight and, therefore,\nwith the law of causation also. Of course, there is nothing in the way\nof calling such a relation an _a priori_ relation, if it is so desired.\nAs far as the individual is concerned it no doubt antedates all his\nexperience, since the entire organization which he inherits from his\nparents had already been formed under such an influence. But that there\ncan be forms or existence _without_ such an attribute is shown by the\nwhole world of the _inorganic_, in which, as far as our knowledge goes,\nthere is no evidence of either memory or foresight, but only of an\nimmediate passive participation in the processes of the world around\nthem.[C]\n\n[C] It cannot be objected that inorganic nature also is known to be\nsubject to the law of causation. The causal mode of regarding inorganic\nphenomena is a distinctly human one, and nothing justifies the assertion\nthat the same phenomena cannot be viewed in an entirely different\nmanner.\n\nFurther, the circumstance that the causal relation is brought about by\nthe peculiar manner in which we react upon our experiences, has\nsometimes been expressed in this way--the relation of cause and effect\ndoes not exist in nature at all, but has been introduced by men. The\nelement of truth in this is, that a quite differently organized being,\nit is to be supposed, would be able to, or would have to, arrange its\nexperiences according to quite different mutual relations. But since we\nhave no experience of such a being, we have no possibility of forming a\nvalid opinion of its behavior. On the other hand, we must recognize that\nit is possible, at least formally, to conceive also of kinds of\nexperiences with no coinciding parts, or a world in which there are no\nexperiences at all with coinciding parts. In such, therefore, prediction\nis impossible. Such a world will not call forth, even in a being endowed\nwith memory, a conception and generalization of the various experiences\nin the shape of natural laws. Consequently we must recognize that in\naddition to the _subjective_ factor in the formation of our knowledge of\nthe world, or that factor which is dependent upon our physico-psychical\nstructure, there is also the _objective_ character of the world with\nwhich we must decidedly reckon, or that character which is independent\nof us; and that in so far the natural laws contain also objective parts.\nTo represent the relation clearly to our minds by a figure, we may\ncompare the world to a heap of gravel and man to a pair of sieves, one\ncoarser than the other. As gravel passes through the double sieve\npebbles of apparently equal size accumulate between the sieves, the\nlarger ones being excluded by the first sieve and the smaller ones\nallowed to pass by the second. It would be an error to assert that all\nthe gravel consisted of such pebbles of equal size. But it would be\nequally false to assert that it was the sieves that _made_ the pebbles\nequal.\n\n\n=11. The Purification of the Causal Relation.= If by experience we have\nfound a proposition of the content, If A is, then B is also, the two\nconcepts A and B generally consist of several elements which we will\ndesignate as a, a\u00b4, a\u00b4\u00b4, a\u00b4\u00b4\u00b4, etc., and as b, b\u00b4, b\u00b4\u00b4, b\u00b4\u00b4\u00b4. Now the\nquestion arises, whether or not all these elements are essential for the\nrelation in question. It is quite possible, in fact, even highly\nprobable, that at first only a special instance of the existing\nphenomena was found, that is, that the concept A, which has been found\nto be connected with the concept B, contains other determining parts\nwhich are not at all requisite to the appearance of B.\n\nThe general method of convincing oneself of this is by eliminating one\nby one the component parts of the concept A, namely, a, a\u00b4, a\u00b4\u00b4, etc.,\nand then seeing whether B still appears. It is not always easy to carry\nout this process of elimination. Our greater or less ability to conduct\nsuch investigations depends upon whether we deal with things that are\nmerely the objects of our _observation_, and which we ourselves have not\nthe power to change (as, for example, astronomical phenomena), or with\nthings which are the objects of our _experimentation_, and which we can\ninfluence. In the latter case one or another factor is usually found\nwhich can be eliminated without the disappearance of B, and then we must\nproceed in such a way as to form a corresponding new concept A\u00b4 from the\nfactors recognized as necessary (which new concept will be more general\nthan the former A), and to express the given proposition in the improved\nform: If A\u00b4 is, then B is also.\n\nQuite similar is the case with the other member of this relation. It\noften happens that when a, or a\u00b4\u00b4, a\u00b4\u00b4\u00b4 is found, somewhat different\nthings appear, which do not fit the concept as first constructed. Then\nwe must multiply the experiences as much as possible in order to\ndetermine what constant elements are found in the concept B, and to form\nfrom these constant elements the corresponding concept B\u00b4. The improved\nproposition will then read: if A\u00b4 is, then B\u00b4 is also.\n\nThis entire process may be called the purification of the causal\nrelation. By this term we express the general fact that in first forming\nsuch a regular connection, the proper concepts are very seldom brought\ninto relation with one another at once. The cause of it is that at first\nwe make use of _existing_ concepts which had been formed for quite a\ndifferent purpose. It must therefore be regarded as a special piece of\ngood fortune if these old concepts should at once prove suited to the\nnew purpose. Furthermore, the existing concepts are as a rule so vaguely\ncharacterized by their names, which we must employ to express the new\nrelation, that for this reason also it is often necessary to determine\nempirically in what way the concept is to be definitely established.\n\nThe various sciences are constantly occupied with this work of the\nmutual adaptation of the concepts that enter into a causal relation. By\nway of example, we may take the \"self-understood\" proposition which we\nuse when we call out to a careless child when it sticks its finger into\nthe flame of a candle, \"Fire burns!\" We discover that there are\nself-luminous bodies which produce no increase of temperature, and\ntherefore no sensation of pain. We discover that there are processes of\ncombustion that develop no light, but heat enough to burn one's\nfingers. And, finally, the scientific investigation of this proposition\narrives at the general expression that, as a rule, chemical processes\nare accompanied by the development of heat, but that, conversely, such\nprocesses may also be accompanied by the absorption of heat. In this way\nthat casual sentence which we call out to the child develops into the\nextensive science of thermo-chemistry when it is subjected to the\ncontinuous purification of the causal relation, which is the general\ntask of science.\n\nIt remains to be added that in this process of adapting concepts it is\nnecessary also sometimes to follow the opposite course. This is the case\nwhen _exceptions_ are noticed in a relation as expressed for the time\nbeing; when, therefore, the proposition if A is present, then B is\npresent also, is in a great many instances valid, but occasionally\nfails. This is an indication that in the concept A an element is still\nlacking. This element, however, is present in the instances that tally,\nbut absent in the negative cases, and its absence is not noticed because\nit is not contained in A. Then it is necessary to seek this part, and\nafter it has been found, to embody it in the concept A, which thus\npasses into the new concept A\u00b4.\n\nThis case is the obverse of the former one. Here the more suitable\nconcept proves to be less general than the concept accepted temporarily,\nwhile in the first case the improved concept is more general. Hence we\nformulate the rule: exceptions to the temporary rule require a\nlimitation, while an unforeseen freedom requires an extension, of the\naccepted concept.\n\n\n=12. Induction.= The form of conclusion previously discussed, _because\nit has been so, I expect it will continue to be so in the future_, is\nthe form through which each science has arisen and has won its real\ncontent, that is, its value for the judgment of the future. It is called\n_inference by induction_, and the sciences in which it is\npreponderatingly applied are called _inductive sciences_. They are also\ncalled experiential or empirical sciences. At the basis of this\nnomenclature is the notion that there are other sciences, the deductive\nor rational sciences, in which a reverse logical procedure is applied,\nwhereby from general principles admitted to be valid in advance,\naccording to an absolutely sure logical process, conclusions of like\nabsolute validity are drawn. At the present time people are beginning to\nrecognize the fact that the deductive sciences must give up these claims\none by one, and that they already have given them up to a certain\nextent; partly because on closer study they prove to be inductive\nsciences, and partly because they must forego the title and rank of a\nscience altogether. The latter alternative applies especially to those\nprovinces of knowledge which have not been used in prophesying the\nfuture or cannot be so used.\n\nTo return to the inductive method--it is to be noted that _Aristotle_,\nwho was the first to describe it, proposed two kinds of induction, the\n_complete_ and the _incomplete_. The first has this form: since _all_\nthings of a certain kind are so, each _individual thing_ is so. While\nthe incomplete induction merely says: since _many_ things of a certain\nkind are so, _presumably_ all things of this kind are so. One instantly\nperceives that the two conclusions are essentially different. The first\nlays claim to afford an absolutely certain result. But it rests upon the\nassumption that _all_ the things of the kind in question are known and\nhave been tested as to their behavior. This hypothesis is generally\nimpossible of fulfilment, since we can never prove that there are not\nmore things of the same kind other than those known to us or tested by\nus. Moreover, the conclusion is _superfluous_, as it merely repeats\nknowledge that we have already directly acquired, since we have tested\n_all_ the things of the one kind, hence the special thing to which the\npredication refers.\n\nOn the other hand, the _incomplete_ induction affirms something that has\nnot yet been tested, and therefore involves as a condition an\n_extension_ of our knowledge, sometimes an extremely important\nextension. To be sure, it must give up the claim to unqualified or\nabsolute validity, but, to compensate, it acquires the irreplaceable\nadvantage of lending itself to practical application. Indeed, in\naccordance with the scientific practice justified by experience,\ndescribed on p. 29, the scientific inductive conclusion assumes the\nform: because it has _once_ been found to be so, it will _always_ be so.\nFrom this appears the significance of this method for the enlargement of\nscience, which, without it, would have had to proceed at an incomparably\nslower pace.\n\n\n=13. Deduction.= In addition to the inductive method, science has (p.\n38) another method, which, in a sense, should be the reverse of the\ninductive and is claimed to provide absolutely correct results. It is\ncalled the _deductive_ method, and it is described as the method that\nleads from premises of general validity by means of logical methods of\ngeneral validity to results of general validity.\n\nAs a matter of fact, there is no science that does or could work in such\na way. In the first place, we ask in vain, how can we arrive at such\ngeneral, or absolutely valid, premises, since all knowledge is of\nempiric origin and is therefore equipped with the possibility of error\nas ineradicable evidence of this origin. In the next place, we cannot\nsee how from principles at hand conclusions can be drawn the content of\nwhich exceeds that of these principles (and of the other means\nemployed). In the third place, the absolute correctness of such results\nis doubtful from the fact that blunders in the process of reasoning\ncannot be excluded even where the premises and methods are absolutely\ncorrect. In practice it has actually come to pass that in the so-called\ndeductive sciences doubts and contradictions on the part of the various\ninvestigators of the same question are by no means excluded. To wit,\nthe discussion that has been carried on for centuries, and is not yet\nended, over Euclid's parallel theorem in geometry.\n\nIf we ask whether, in the sense of the observations we have just made of\nthe formation of scientific principles, there is anything at all like\ndeduction, we can find a procedure which bears a certain resemblance\nwith that impossible procedure and which, as a matter of fact, is\nfrequently and to very good purpose applied in science. It consists in\nthe fact that general principles which have been acquired through the\nordinary incomplete induction are _applied to special instances which,\nat the proposition of the principle, had not been taken into\nconsideration_, and whose connection with the general concept had not\nbecome directly evident. Through such application of general principles\nto cases that have not been regarded before, specific natural laws are\nobtained which had not been foreseen either, but which, according to the\nprobability of the thesis and the correctness of the application are\nalso probably correct. However, the investigator, bearing in mind the\nfactor of uncertainty in these ratiocinations feels in each such\ninstance the need for testing the results by experience, and he does not\nconsider the _deduction_ complete until he had found _confirmation_ in\nexperience.\n\nDeduction, therefore, actually consists in the searching out of\nparticular instances of a principle established by induction and in its\nconfirmation by experience. This conduces to the growth of science, not\nin breadth, but in profundity. I again resort to the comparison I have\nfrequently made of science with a very complex network. At first glance\nwe cannot obtain a complete picture of all the meshes. So, at the first\nproposition of a natural law an immediate survey of the entire range of\nthe possible experiences to which it may apply is inachievable. It is a\nregular, important, and necessary part of all scientific work to learn\nthe extent of this range and investigate the specific forms which the\nlaw assumes in the remoter instances. Now, if an especially gifted and\nfar-seeing investigator has succeeded in stating in advance an\nespecially general formulation of an inductive law, it is everywhere\nconfirmed in the course of the trial applications, and the impression\neasily arises that confirmation is superfluous, since it results simply\nin what had already been \"deduced.\" In point of fact, however, the\nreverse is not infrequently the case, that the principle is _not_\nconfirmed, and conditions quite different from those anticipated are\nfound. Such discoveries, then, as a rule, constitute the starting-point\nof important and far-reaching modifications of the original formulation\nof the law in question.\n\nAs we see, deduction is a necessary complement of, in fact, a part of,\nthe inductive process. The history of the origin of a natural law is in\ngeneral as follows. The investigator notices certain agreements in\nindividual instances under his observation. He assumes that these\nagreements are general, and propounds a temporary natural law\ncorresponding to them. Then he proceeds by further experimentation to\ntest the law in order to see whether he can find full confirmation of it\nby a number of other instances. If not, he tries other formulations of\nthe law applicable to the contradictory instances, or exclusive of them,\nas not allied. Through such a process of adjustment he finally arrives\nat a principle that possesses a certain range of validity. He informs\nother scientists of the principle. These in their turn are impelled to\ntest other instances known to them to which the principle can be\napplied. Any doubts or contradictions arising from this again impel the\nauthor of the principle to carry out whatever readjustments may have\nbecome necessary. Upon the scientific imagination of the discoverer\ndepends the range of instances sufficing for the formulation of the\ngeneral inductive principle. It also frequently depends upon conscious\noperations of the mind dubbed \"scientific instinct.\" But as soon as the\nprinciple has been propounded, even if only in the consciousness of the\ndiscoverer, the deductive part of the work begins, and the consequent\ntest of the proposition has the most essential influence on the value of\nthe result.\n\nIt is immediately evident that this _deductive_ part is of all the more\nweight, the more _general_ the concepts in question are. If, in\naddition, the inductive laws posited soon prove to be of a comparatively\nhigh degree of perfection, we obtain the impression described above,\nthat an unlimited number of independent results can be deduced from a\npremise. _Kant_ was keenly alive to the peculiarity of such a view,\nwhich had been widely spread pre-eminently by _Euclid's_ presentation of\ngeometry, and he gave expression to his opinion of it in the famous\nquestion: _How are a priori judgments possible?_ We have seen that it is\nnot always a question of _a priori_ judgments, but also of inductive\nconclusions applied and tested according to deductive methods.\n\n\n=14. Ideal Cases.= Each experience may generally be considered under an\nindefinite number of various concepts, all of which may be abstracted\nfrom that experience by corresponding observations. Accordingly an\nindefinite number of natural laws would be required for prophesying that\nexperience in all its parts. Likewise the indefinite number of premises\nmust be known through the application of which those natural laws\nacquire a certain content. Thus it seems as if it were altogether\nimpossible to apply natural laws for the determination of a single\nexperience to come, and in a certain sense this is true (p. 30). For\nexample, when a child is born, we are quite incapable of foretelling the\npeculiar events that will occur in its life. Beyond the statement that\nit will live a while and then die, we can make only the broadest\nassertions qualified by numerous \"ifs\" and \"buts.\"\n\nIf, in spite of this, we arrange a very great part of our life and\nactivity according to the prophecies we make in regard to numerous\ndetails in life, basing them upon natural laws, the question arises, how\nwe get over the difficulty, or, rather, the impossibility just referred\nto.\n\nThe answer is, that we repeatedly so find or can form our experiences\nthat certain natural relations _preponderatingly_ determine the\nexperience, while the other parts that remain undetermined fall into the\nbackground. _The prophecy will cover so considerable a part of the\nexperience that we can forego previous knowledge of the rest._ We can\nforetell enough to render a practical construction of life possible, and\nincreasing experience, whether the personal experience of the individual\nor the general experience of science, constantly enlarges this\ncontrollable part of future experiences.\n\nThe procedure of science is similar to that of practical life, though\nfreer. Whenever an investigator seeks to test a natural law of the form:\nif A is so, then B is so, he endeavors to choose or formulate the\nexperiences in such a way that the fewest possible extraneous elements\nare present, and that those that are unavoidable should exert the least\npossible influence upon the relation in question. He never succeeds\ncompletely. In order, nevertheless, to reach a conclusion as to the form\nthe relation will take without extraneous influences, the following\ngeneral method is applied.\n\nA series of instances are investigated which are so adjusted that the\ninfluence of the extraneous elements grows less and less. Then the\nrelation investigated approaches a limit which is never quite reached,\nbut to which it draws nearer and nearer, the less the influence of the\nextraneous elements. And the conclusion is drawn that if it were\npossible to exclude the extraneous elements entirely, the limit of the\nrelation would be reached.\n\nA case in which none of the extraneous elements of experience operate is\ncalled an _ideal case_, and the inference from a series of values\nleading to the limit-value is an _extrapolation_. _Such extrapolations\nto the ideal case_ are a quite natural procedure in science, and a very\nlarge part of natural laws, especially all quantitative laws, that is,\nsuch as express a relation between measurable values, have precise\nvalidity only in ideal cases.\n\nWe here confront the fact that many natural laws, and among them the\nmost important, are expressed as, and taken to be, conditions _which\nnever occur in reality_. This seemingly absurd procedure is, as a matter\nof fact, the best fitted for scientific purposes, since ideal cases are\nto be distinguished by this, _that with them the natural laws take on\nthe simplest forms_. This is the result of the fact that in ideal cases\nwe intentionally and arbitrarily overlook every complication of the\ndetermining factors, and in describing ideal cases we describe the\nsimplest conceivable form of the class of experiences in question. The\nreal cases are then constructed from the ideal cases by representing\nthem as the sum of all the elements that have an influence on the\nexperience or the result. Just as we can represent the unlimited\nmultitude of finite numbers by the figures up to ten, so we can\nrepresent an unlimited quantity of complicated events by a finite number\nof natural laws, and so reach a highly serviceable approximation to\nreality.\n\nThus geometry deals with absolutely straight lines, absolutely flat\nsurfaces, and perfect spheres, though such have never been observed, and\nthe results of geometry come the closer to truth, the more nearly the\nreal lines, surfaces, and spheres correspond to the ideal demands.\nSimilarly, in physics, there are no ideal gases or mirrors, or in\nchemistry ideally pure substances, though the expressed simple laws in\nthese sciences are valid for only such bodies. The non-ideal bodies of\nthese sciences, which reality presents in various degrees of\napproximation, correspond the more closely to these laws, the slighter\nthe deviation of the real from the ideal. And the same method is applied\nin the so-called mental sciences, psychology and sociology, in which the\n\"normal eye\" and a \"state with an entirely closed door\" are examples of\nsuch idealized limit-concepts.\n\n\n=15. The Determinateness of Things.= A very widespread view and a very\ngrave one, because of its erroneous results, is _that by the natural\nlaws things are unequivocally and unalterably determined down to the\nvery minutest detail_. This is called _determinism_, and is regarded as\nan inevitable consequence of every natural scientific generalization.\nBut an accurate investigation of actual relations produces something\nrather different.\n\nThe most general formulation of the natural law: _if A is experienced,\nthen we expect B_, necessarily refers in the first place only to certain\n_parts_ of the thing experienced. For perfect similarity in two\nexperiences is excluded by the mere fact that we ourselves change\nunceasingly and one-sidedly. Consequently, no matter how accurate the\nrepetition of a former experience may be, our very participation in it,\nan element bound to enter, causes it to be different. Therefore we deal\nwith only a _partial_ repetition of any experience, and the common part\nis all the smaller a fraction of the entire experience, the more\n_general_ the concept corresponding to this part. But the most general\nand most important natural laws apply to such very general ideas, and\naccordingly they determine only a small part of the whole result. Other\nparts are determined by other laws, but we can never point out an\nexperience that has been determined completely and unequivocally by\nnatural laws known to us. For example, we know that when we throw a\nstone, it will describe an approximate parabolic curve in falling to the\nground. But if we should attempt to determine its course accurately, we\nshould have to take into consideration the resistance of the air, the\nrotatory motion of the stone upon being thrown, the movement of the\nearth, and numerous other circumstances, the exact determination of\nwhich is a matter beyond the power of all sciences. Nothing but an\n_approximate_ determination of the stone's course is possible, and every\nstep forward toward accuracy and absoluteness would require scientific\nadvances which it would probably take centuries to accomplish.\n\nScience, therefore, can by no means determine the exact linear course\nthat the stone will take in its fall. It can merely establish a certain\nbroader path within which the stone's movement will remain. And the path\nis the wider the smaller the progress science has made in the branch in\nquestion. The same conditions prevail in the case of every other\nprediction based upon natural laws. Natural laws merely provide a\ncertain frame within which the thing will remain. But which of the\ninfinitely numerous possibilities within this frame will become reality\ncan never be absolutely determined by human powers.\n\nThe belief that it is possible has been evoked merely by a far-reaching\nmethod of abstraction on the part of science. By assuming in place of\nthe stone \"a non-extended point of mass\" and by disregarding all the\nother factors which in some way (whether known or unknown) exercise an\ninfluence on the stone's movement, we can effect an apparently perfect\nsolution of the problem. But the solution is not valid for real\nexperience, merely for an ideal case, which bears only a more or less\nprofound similarity to the real. It is only such an ideal world, that\nis, a world arbitrarily removed from its actual complexity, that has the\nquality of absolute determinateness which we are wont to ascribe to the\nreal world.\n\nWe might point to the method of abstraction generally adopted in science\nand to the extrapolation to ideal cases which has just been explained,\nand regard the assertion of the absolute determinateness of events in\nthe world as a justified extrapolation to the ideal case. In other\nwords, we might say that we know all the natural laws and how to apply\nthem perfectly to the individual instances. In controversion of this it\nmust be said that the ulterior justification of such ideal extrapolation\nis not yet feasible. The justification lies in the demonstration that\nthe real cases approximate the ideal the more closely the more we\nactualize our presumptions. But in this case this is not feasible,\nsince, for the greater part of our experiences, we do not even know the\napproximate or ideal natural laws by the help of which we can construct\nsuch ideal cases. For instance, the whole province of organic life is at\npresent essentially like an unknown land, in which there are only a few\nwidely separated paths ending in _culs-de-sac_.\n\n\n=16. The Freedom of the Will.= This relation explains why, on the one\nhand, we assume a far-reaching determinateness for many things, that is,\nfor all those accessible to scientific treatment and regulation, and\nwhy, on the other hand, we have the consciousness of acting _freely_,\nthat is, of being able to control future events according to the\nrelations they bear to our wishes. Essentially there is no objection to\nbe found to a fundamental determinism which explains that this feeling\nof freedom is only a different way of saying _that a part of the causal\nchain lies within our consciousness_, and that we feel these processes\n(in themselves determined) as if we ourselves determined their course.\nNor can we prove this idea to be false, that, since the number of\nfactors which influence each experience is indefinitely great and their\nnature indefinitely complex, each event would appear to be determined in\nthe eyes of an all-comprehensive intellect. But to our finite minds an\nundetermined residue necessarily remains in each experience, and to that\nextent the world must always remain in part practically undetermined to\nhuman beings. Thus, both views, that the world is not completely\ndetermined, and that it really is, though we can never recognize that it\nis, lead practically to the same result: _that we can and must assume in\nour practical attitude to the world that it is only partially\ndetermined_.\n\nBut if two different lines of thought in the whole world of experience\neverywhere lead to the same result, they cannot be materially, but\nmerely formally or superficially, different. For those things are alike\nwhich cannot be distinguished. There is no other definition of\nalikeness. Thus, if we see that the age-long dispute between these two\nviews always breaks out afresh without seeming to be able to reach an\nend, this is readily understood, from what has been said, since the very\nsame essential arguments which can be adduced of _one_ view can be used\nas a prop for the _other_ view, because in their essential results the\ntwo are the same. I have discussed this matter because it presents a\nvery telling example of a method to be applied in all the sciences when\ndealing with the solution of old and ever recurrent moot questions. Each\ntime we encounter such problems, we must ask ourselves: what would be\nthe difference empirically if the one or the other view were correct? In\nother words, we first assume the one to be correct, and develop the\nconsequences accordingly. Then we assume the second to be correct and\ndevelop the consequences accordingly. If in the two cases the\nconsequences differ in a certain definite point, we at least have the\npossibility of ascertaining the false view by investigating in favor of\nwhich case experience decides on this point. However, we may not\nconclude that by this the other view has been proved to be entirely\ncorrect. It likewise may be false, only with the peculiar quality that\nin the case in question it leads to the correct conclusions. That such\na thing is possible, every one knows who has attentively observed his\nown experiences. How often we act correctly in actual practice, though\nwe have started out on false premises! The explanation of this\npossibility resides in the highly composite nature of each experience\nand each assumption. It is quite possible--and, in fact, it is the\ngeneral rule--that a certain view contains true elements, but _along\nwith them false elements also_. In applications of the view where the\ntrue elements are the decisive factors, true results are obtained,\ndespite the errors present. Likewise, false results will be achieved\nwhere the false elements are decisive, despite the true results that can\nbe had, or have been had, elsewhere, by means of the true elements.\nHence, in case of the \"confirmation,\" we can only conclude that that\nportion of the view essential for the instance in question is correct.\n\nOne readily perceives that these observations find application in all\nprovinces of science and life. There are no absolutely correct\nassertions, and even the falsest may in some respect be true. There are\nonly greater and lesser probabilities, and every advance made by the\nhuman intellect tends to increase the degree of probability of\nexperiential relations, or natural laws.\n\n\n=17. The Classification of the Sciences.= From the preceding\nobservations the means may be drawn for outlining a complete table of\nthe sciences. However, we must not regard it complete in the sense that\nit gives every possible ramification and turn of each science, but that\nit sets up a frame inside of which at given points each science finds\nits place, so that, in the course of progressive enlargement, the frame\nneed not be exceeded.\n\nThe basic thought upon which this classification rests is that of graded\nabstraction. We have seen (p. 19) that a concept is all the more\ngeneral, that is, is applicable to all the more experiences, the fewer\nparts or elementary concepts it contains. So we shall begin the system\nof the sciences with the most general concepts, that is, the elementary\nconcepts (or with what for the time being we shall have to consider\nelementary concepts), and, in grading the concept complexes according to\ntheir increasing diversity, set up a corresponding graded series of\nsciences. One thing more is to be noted here, that this graded series,\non account of the very large number of new concepts entering, must\nproduce a correspondingly great number of diverse sciences. For\npractical reasons groups of such grades have been combined temporarily.\nThereby a rougher classification, though one easier to obtain a survey\nof, has been made. The most suitable and lasting scheme of this sort was\noriginated by the French philosopher, _Auguste Comte_, since whom it has\nundergone a few changes.\n\nBelow is the table of the sciences, which I shall then proceed to\nexplain:\n\n    I. _Formal Sciences._ Main concept: order\n         Logic, or the science of the Manifold\n         Mathematics, or the science of Quantity\n         Geometry, or the science of Space\n         Phoronomy, or the science of Motion\n\n   II. _Physical Sciences._ Main concept: energy\n         Mechanics\n         Physics\n         Chemistry\n\n  III. _Biological Sciences._ Main concept: life\n         Physiology\n         Psychology\n         Sociology\n\nAs is evident, we first have to deal with the three great groups of the\nformal, the physical, and the biological sciences. The formal sciences\ntreat of characteristics belonging to all experiences, characteristics,\nconsequently, that enter into every known phase of life, and so affect\nscience in the broadest sense. In order immediately to overcome a\nwidespread error, I emphasize the fact that these sciences are to be\nconsidered just as experiential or empirical as the sciences of the\nother two groups, as to which there is no doubt that they are empirical.\nBut because the concepts dealt with by the first group are so extremely\nwide, and the experiences corresponding to them, therefore, are the most\ngeneral of all experiences, we easily forget that we are dealing with\nexperiences at all; and our very firmly rooted consciousness of the\nunqualified similarity of these experiences causes them to seem native\nqualities of the mind, or _a priori_ judgments. Nevertheless,\nmathematics has been proved to be an empirical science by the fact that\nin certain of its branches (the theory of numbers) laws are known which\nhave been found empirically and the \"deductive\" proof of which we have\nas yet not succeeded in obtaining. The most general concept expressed\nand operative in these sciences is the concept of order, of _conjugacy_\nor _function_, the content and significance of which will become clear\nlater in a more thorough study of the special sciences.\n\nIn the second group, the physical sciences, the arbitrariness of the\nclassification becomes very apparent, since these sciences are among the\nbest known. We are perfectly justified in regarding mechanics as a part\nof physics; and in our day physical chemistry, which in the last twenty\nyears suddenly developed into an extended and important special science,\nthrust itself between physics and chemistry.\n\nThe most general concept of the physical sciences is that of _energy_,\nwhich does not appear in the formal sciences. To be sure it is not a\nfundamental concept. On the contrary, its characteristic is undoubtedly\nthat of compositeness, or, rather, complexity.\n\nThe third group comprehends all the relations of living beings. Their\nmost general concept, accordingly, is that of _life_. By physiology is\nunderstood the entire science dealing with non-psychic life phenomena.\nIt therefore embraces what is called, in the present often chance\narrangement of scientific activities, botany, zoology, and physiology of\nthe plants, animals, and man. Psychology is the science of mental\nphenomena. As such, it is not limited to man, even though for many\nreasons he claims by far the preponderating part of it for himself.\nSociology is the science which deals with the peculiarities of the human\nrace. It may therefore be called anthropology, but in a far wider sense\nthan the word is now applied.\n\n\n=18. The Applied Sciences.= It will be remarked that the grouping of the\ntable gives no place at all in its scheme to certain branches of\nlearning taught in the universities and equally good technical\ninstitutions. We look in vain not only for theology and jurisprudence,\nbut also for astronomy, medicine, etc.\n\nThe explanation and justification of this is, that for purposes of\nsystematization we must distinguish between _pure_ and _applied_\nsciences. By virtue of their strictly conceptual exclusiveness the pure\nsciences constitute a regular hierarchy or graded series, so that all\nthe concepts that have been used and dealt with in the preceding\nsciences are repeated in the following sciences, while certain\ncharacteristic new concepts enter in addition. Thus logic, the science\nof the manifold, exercises its dominion over all the other sciences,\nwhile the specific concepts of physics and chemistry have nothing to do\nwith it, though they are of importance to all the biologic sciences.\nThrough this graded addition of new (naturally empiric) concepts, the\nconstruction of the pure sciences proceeds in strict regularity, and\ntheir problems arise exclusively from the application of new concepts to\nall the earlier ones. In other words, their problems do not reach them\naccidentally from without, but result from the action and reaction of\ntheir concepts upon one another.\n\nAt the same time there are problems that each day sets before us without\nregard to system. These come from our endeavor to improve life and avert\nevil. In the problems of life we are confronted by the whole variety of\npossible concepts, and under the day's immediate compulsion we cannot\nwait, if we are sowing crops or helping a sick man, until physiology and\nall the other appropriate sciences have solved all the problems of plant\ngrowth and the changes of the human body and human energy. When other\nsigns fail, we use the position of the stars for finding our way on the\nhigh seas. In this manner we turn the teaching of the stars, or\nastronomy, into an applied science, in which at first mechanics alone\nseemed to have a part. Later physics took a share in it, then optics\ntook a particularly prominent share, and in recent times not only did\nchemistry find its way into astronomy, but the specifically biologic\nconcept of evolution was applied in astronomy with success.\n\nThus, side by side with the pure sciences are the applied, which are to\nbe distinguished from the pure sciences by the fact that they do not\nunfold their problems systematically, but are assigned them by the\nexternal circumstances of man's life. The pure sciences, therefore,\nalmost always have a larger or smaller share in the tasks of the applied\nsciences. For instance, in building a bridge or railroad, physical\nproblems have to be taken into consideration as well as sociologic\nproblems (problems of trade), and a good physician should be a\npsychologist as well as a chemist.\n\nBut since all the individual questions arising in the applied sciences\nmay be considered essentially as problems of one or other pure science,\nthey need not be explicitly enumerated along with the pure sciences,\nespecially since their development is greatly dependent upon temporary\nconditions and is therefore incapable of simple systematization.\n\n\n\n\nPART II\n\nLOGIC, THE SCIENCE OF THE MANIFOLD, AND MATHEMATICS\n\n\n=19. The Most General Concept.= If we try to conceive the whole\nstructure of science according to the principle of the increasing\ncomplexity of concepts, the first question which confronts us is, What\nconcept is the _most general_ of all possible concepts, so general that\nit enters into every concept formation and acts as a decisive factor? In\norder to find this concept let us go back to the psycho-physical basis\nof concept formation, namely, _memory_, and let us investigate what is\nthe general characteristic determining memory. We soon perceive that if\na being were to lead an absolutely uniform existence, _no_ memories\ncould be evoked. There would be nothing by which the past could be\ndistinguished from the present, hence nothing by which to compare them.\nSo the \"primal phenomenon\" of conscious thought is the realization of a\n_difference_, a difference between memory and the present, or, to put\nthe same idea still more generally, between two memories.\n\nOur experiences, therefore, are divided into two parts, distinguished\nfrom each other. In order to predicate something of a perfectly general\nnature concerning those parts, without regard to their particular\ncontent, we must, in accordance with the means employed in human\nintercourse, designate them by a _name_. Now in all human languages\nthere is a great deal of arbitrariness and indefiniteness in the\nrelations between the concepts and the names applied to them, which\nrender all accurate work in the study of concepts extremely difficult.\nIt is necessary, therefore, to state definitely in each particular\ninstance with what conceptual content a given name is to be connected.\nEvery experience in so far as it is differentiated from other\nexperiences we shall call simply an _experience_ without making a\ndistinction between a so-called inner or outer experience.\n\nMany of the experiences remain isolated, because they are not repeated\nin a similar form, and so do not remain in our memory. They depart from\nour psychic life once for all and leave no further consequences or\nassociations. But some experiences recur with greater or less\nuniformity, and become permanent parts of psychic life. Their duration\nis by no means unlimited. For even memories fade and disappear. However,\nthey extend through a considerable part of life, and that suffices to\ngive them their character.\n\nThe aggregate of similar experiences, hence of experiences conceptually\ngeneralized, we shall call _things_. _A thing, therefore, is an\nexperience which has been repeated_, and is \"recognized\" by us. That\nis, it is felt as repeated and conceptually comprehended. In other\nwords, all experiences of which we have formed concepts are things, and\n_the concept of thing itself is the most general concept_, since,\naccording to its definition, it includes all possible concepts. Its\n\"essence,\" or determining characteristic, lies in the possibility of\ndifferentiating any one thing from another. Things we do not\ndifferentiate we call _the same_, or _identical_. Here we shall leave\nundecided the question whether this lack of differentiation occurs\nbecause we _cannot_, or because we _would not_, differentiate. All\nexperiences generalized into one concept are therefore felt or regarded\nas the same in reference to this concept. Now, since concepts arise\nunconsciously as well as consciously, the first is a case of identities\nwhich had been directly felt as such. On the other hand, in the second\ncase, the process is that of consciously disregarding or abstracting the\nexisting differences in order to form a concept into which these do not\nenter. This last process is applied in the highest degree possible in\nobtaining the concept _thing_.\n\n\n=20. Association.= The experience of the _connection_ or _relation_\nbetween various things is also derived from the nature of our\nexperiences in the most general sense. When we recall a thing A, another\nthing B comes to our mind, the memory of which is called forth by A, and\n_vice versa_. The cause of this invariably lies in some experiences in\nwhich A and B occur together. In fact, A and B must have occurred\ntogether a number of times. Otherwise they would have disappeared from\nmemory. In other words, it is the fact of the _complex concept_ which\nappears in such connections between various things. Two things, A and B,\nwhich are connected with each other in such a way, are said to be\nassociated. Association in the most general sense means nothing more\nthan that when we think of B we also have A in our consciousness, and\n_vice versa_. However, we can at will make the association more\ndefinite, so that quite definite thoughts or actions will be connected\nwith the association of B. These thoughts and actions are then the same\nfor all the individual cases occurring under the concept A and B.\n\nIf we associate with the thing B another thing C, we obtain a relation\nof the same nature as that obtained by the association of A and B. But\nat the same time a new relation arises which was not directly sought,\nnamely, the association of A to C. If A recalls B, and B recalls C, A\nmust inevitably recall C also. This psychologic law of nature is\nproductive of numberless special results. For we can apply it directly\nto still another case, the association of a fourth thing D to the thing\nC, whereby new relations are necessarily established also between A and\nD as well as between B and D. By positing the _one_ relation C : D there\narise two new relations not immediately given, namely, A : D and B : D.\nThe reason the other relations arise is because C was not taken free\nfrom all relations, but had already attached to it the relations to A\nand B. These relations of C, therefore, brought A and B into the new\nrelation with D.\n\nBy this simplest and most general example we recognize the type of the\ndeductive process (p. 41), namely, the discovery of relations which, it\nis true, have already been established by the accepted premises, but\nwhich do not directly appear in undertaking the corresponding\noperations. In the present case, to be sure, the deduction is so\napparent that the recognition of the relations in question offers not\nthe slightest difficulty. But we can easily imagine more complicated\ncases in which it is much more difficult to find the actually existing\nrelations, and so in certain circumstances we may search for them a long\ntime in vain.\n\n\n=21. The Group.= The aggregate of all individual things occurring in a\ndefinite concept, or the common characteristics of which make up this\nconcept, is called a group. Such a group may consist of a limited or\nfinite number of members, or may be unlimited, according to the nature\nof the concepts that characterize it. Thus, all the integers form an\nunlimited or infinite group, while the integers between ten and one\nhundred (or the two-digit numbers) form a limited or finite group.\n\nFrom the definition of the group concept follows the so-called classic\n_process of argumentation_ of the syllogism. Its form is: _Group A is\ndistinguished by the characteristic of B_. _The thing C belongs to group\nA. Therefore C has the characteristic of B._ The prominent part ascribed\nby _Aristotle_ and his successors to this process is based upon the\n_certainty_ which its results possess. Nevertheless, it has been pointed\nout, especially by _Kant_, that judgments or conclusions of such a\nnature (which he called analytic) have no significance at all for the\nprogress of science, since they express only what is already known. For\nin order to enable us to say that the thing C belongs to group A, we\nmust already have recognized or proved the presence of the group\ncharacteristic B in C, and in that case the conclusion only repeats what\nis already contained in the second or minor premise.\n\nThis is evident in the classic example: All men are mortal. Caius is a\nman. Therefore Caius is mortal. For if Caius's mortality were not known\n(here we are not concerned how this knowledge was obtained), we should\nhave no right to call him a man.\n\nAt the same time the character of the really scientific conclusion based\nupon the incomplete induction becomes clear. It proceeds according to\nthe following form. The attributes of the group A are the\ncharacteristics of a, b, c, d. We find in the thing C the\ncharacteristics a, b, c. Therefore we presume that the characteristic d\nwill also be found in C. The ground for this presumption is that we\nhave learned by experience that the characteristics mentioned have\nalways been found together. It is for this reason, and for this reason\nonly, that we may assume from the presence of a, b, c the presence of d.\nIn the case of an arbitrary combination, in which it is possible to\ncombine other characteristics, the conclusion is unfounded. But if, on\nthe other hand, the formation of the concept A with the characteristics\nof a, b, c, d has been caused by repeated and habitual experience, then\nthe conclusion is well founded; that is, it is probable.\n\nAs a matter of fact, however, that classic example which is supposed to\nprove the absolute certainty of the regular syllogism turns out to be a\nhidden inductive conclusion of the incomplete kind. The premise, Caius\nis a man, is based on the attributes a, b, c (for example, erect\nbearing, figure, language), while the attribute d (mortality) cannot be\nbrought under observation so long as Caius remains alive. In the sense\nof the classic logic, therefore, we are not justified in the minor\npremise, Caius is a man, while Caius is alive. The utter futility of the\nsyllogism is apparent, since, according to it, it is only of dead men\nthat we can assert that they are mortal.\n\nFrom these observations it becomes further apparent that logic, whether\nit is the superfluous classic logic or modern effective inductive logic,\nis nothing but a part of the group theory, or science of manifoldness,\nwhich appears as the first, because it is the most general member of\nthe mathematical sciences (this word taken in its widest significance).\nBut according to the hierarchic system in harmony with which the scheme\nof all the sciences had been consciously projected, we cannot expect\nanything else than that those sciences which are needful for the pursuit\nof all other sciences (and logic has always been regarded as such an\nindispensable science, or, at least, art) should be found collected and\nclassified in the first science.\n\n\n=22. Negation.= When the characteristics a, b, c, d of a group have been\ndetermined, then the aggregate of all things existing can be divided\ninto two parts, namely, the things which belong to the group A and those\nwhich do not belong to it. This second aggregate may then be regarded as\na group by itself. If we call this group \"not-A,\" it follows from the\ndefinition of this group that the two groups, A and not-A, together form\nthe aggregate of all things.\n\nThis is the meaning and the significance of the linguistic form of\n_negation_. It excludes the thing negated from any group given in a\nproposition, and this relegates it to the second or complementary group.\n\nThe characteristic of such a group is the common absence of the\ncharacteristics of the positive group. We must note here that the\nabsence of even _one_ of the characteristics a, b, c, d excludes the\nincorporation of the thing into the group A, while the mere absence of\nthis characteristic suffices to include it in the group not-A. We can\ntherefore by no means predicate of group not-A that each one of its\nmembers must lack _all_ the characteristics a, b, c, d. We can only say\nthat each of its members lacks at least one of the characteristics, but\nthat one or some may be present, and several or all may be absent. From\nthis follows a certain asymmetry of the two groups, which we must bear\nin mind.\n\nThe consideration of this subject is especially important in the\ntreatment of negation in the conclusions of formal logic. As we shall\nmake no special use of formal logic, we need not enter into it in\ndetail.\n\n\n=23. Artificial and Natural Groups.= The combination of the\ncharacteristics which are to serve for the definition of a group is at\nfirst purely arbitrary. Thus, when we have chosen such an arbitrary\ncombination, a, b, c, d, we can eliminate one of the characteristics,\nas, for example, c, and form a group with the characteristics a, b, d.\nSuch a group, which is _poorer in characteristics_, will, in general, be\n_richer in members_, for to it belong, in the first place, all the\nthings with the characteristics a, b, c, d, of which the first group\nconsisted, and in addition all the things which, though not possessing\nc, possess a, b, and d.\n\nIf we call such groups related as contain common characteristics, though\ncontaining them in different members and combinations, so that the\ndefinition of the one group can be derived from the other by the\nelimination or incorporation of individual characteristics, then we can\npostulate the general thesis _that in related groups those must be\nricher in members which are poorer in characteristics, and inversely_.\nThis is the precise statement of the proposition of the less definite\nthesis stated above.\n\nFor the purposes of systematization we have assumed that we can\narbitrarily eliminate one or another characteristic of a group. In\nexperience, however, this often proves inadmissible. As a rule we find\nthat the things which lack one of the characteristics of a group will\nalso lack a number of other characteristics; in other words, that the\ncharacteristics are not all independent of one another, but that a\ncertain number of them go together, so that they are present in a thing\neither in common or not at all.\n\nThis case, however, can be referred to the general one first described,\nby treating the characteristics belonging together as being _one_\ncharacteristic, so that the group is defined solely by the independent\ncharacteristics. Then, according to the definition, we can, without\nlosing our connection with experience, carry out that formal\nmanifoldness of all possible related groups which yields what is called\na _classification_ of the corresponding things.\n\nIf for the determination of a group a definite number of independent\ncharacteristics is taken, say, a, b, c, d, and e, then we have at first\nthe narrowest or poorest group abcde. By the elimination of one\ncharacteristic we obtain the five groups, bcde, acde, abde, abce, and\nabcd. If we omit one other characteristic we get ten different groups\nabc, abd, abe, acd, ace, ade, bcd, bce, bde, cde. Likewise, there are\nten groups with two characteristics each, and finally five groups with\none characteristic each. All these groups are related. There is a\nscience, the Theory of Combinations, which gives the rules by which, in\ngiven elements or characteristics, the kind and number of the possible\ngroups can be found. The theory of combinations enables us to obtain a\ncomplete table and survey of all possible complex concepts which can be\nformed from given simple ones (whether they be really elementary\nconcepts, or only relatively so). When in any field of science the\nfundamental concepts have been combined in this manner, a complete\nsurvey can be had of all the possible parts of this science by means of\nthe theory of combinations.\n\nIn order to present this process vividly to our minds, let us take as an\nexample the science of the chemical combination of substances which form\nan important part of chemistry. There are about eighty elements in\nchemistry, and this science has to treat of\n\n  a) each of the eighty elements by itself\n  b) all substances containing two elements and no more\n  c) all substances containing three elements\n  d, e, f, etc.) the substances containing four, five, and six, etc.,\n    elements,\n\nuntil finally we reach a group (not existing in experience) embracing\nsubstances formed of _all_ the elements. That there is no such substance\nin the present scope of human knowledge has, of course, no significance\nfor the structure of the scheme. What is significant is the fact that\nthe scheme really embraces and arranges all possible substances in such\na way that we cannot conceive of any case in which a newly discovered\nsubstance cannot after examination immediately be classed with one of\nthe existing groups.\n\nTo cite an example from another science. Physics, it will be recalled,\nmay be considered to be the science of the different kinds of energy.\nThis science, accordingly, is divided first into the study of the\nproperties of each energy, and then into the study of the relations of\ntwo energies, of three energies, of four energies, etc. Here, too, we\nmay say that in the end there can be no physical phenomenon which cannot\nbe placed in one of the groups so obtained.\n\nOf course, neither in chemistry nor in physics does this mean that each\n_new_ case will fall within the scheme obtained by the exhaustive\ncombination of elementary concepts (whether chemical elements or kinds\nof energy) _known_ at the time. It is quite possible that a new thing\nunder investigation contains a _new_ elementary concept, so that on\naccount of it the scheme must be enlarged through the embodiment of\nthis new element. But simultaneously a corresponding number of new\ngroups appear in the scheme, and the investigator's attention is\ndirected to the fact that he still has a reasonable prospect, in\nfavorable circumstances, of discovering these new things also. Thus\ncombinatory schematization serves not only to bring the existing content\nof science into such order that each single thing has its assigned\nplace, but the groups which have thereby been found to be vacant, to\nwhich as yet nothing of experience corresponds, also point to the places\nin which science can be completed by new discoveries.\n\nFrom the above presentation it is apparent how from the two concepts\n\"thing\" and \"association\" alone a great manifoldness of various and\nregular forms can be developed. They are purely empirical relations, for\nthe fact that several things can be combined in the graded series\ndescribed above according to a fixed rule does not follow merely from\nthe two concepts, but must be _experienced_. But, on the other hand,\nboth concepts are so general that the experiences obtained in some cases\ncan be applied to all possible experiences and may serve the purpose of\nclassifying and making a general survey of them.\n\nThe above statements, however, have by no means exhausted the\npossibilities. For it has been tacitly assumed that in the combination\nof several things the _sequence_ according to which this combination\ntakes place should not condition a difference of the result. This is\ntrue of a number of things, but not of all. In order, therefore, to\nexhaust the possibilities the theory of combinations must be extended\nalso to cases in which the sequence is to be taken account of, so that\nthe form ab is regarded as different from ba.\n\nWe will not undertake to work out the results of this assumption. It is\nobvious that the manifoldness of the various cases is much greater than\nif we neglect the sequence. On this point we have one more observation\nto make, that further causes for diversity exist. It is true that a\nchemical combination is not influenced by the sequence in which its\nelements enter the combination, but there do occur with the same\nelements differences in their _quantitative relations_, and thereby a\nnew complexity is introduced into the system, so that two or more\nsimilar elements can form different combinations according to the\ndifference in the quantitative relations. Still, even with this, the\nactual manifoldness is not exhausted, for from the same elements and\nwith the same quantitative relations there can arise different\nsubstances called _isomeric_, which, for all their similarity, possess\ndifferent energy contents. But the first scheme is not demolished, nor\ndoes it become impracticable because of this increase of manifoldness.\nWhat simply happens is that _several_ different things instead of one\nappear in the same group of the original scheme, the systematic\nclassification of which necessitates a further schematization by the use\nof other characteristics.\n\n\n=24. Arrangement of the Members.= Since we have started from the\nproposition that all members of a group are different from one another,\nwe have perfect liberty to arrange them. The most obvious arrangement\naccording to which some _one_ definite member is followed by a _single_\nother member and so forth (as, for example, the arrangement of the\nletters of the alphabet) is by no means the only mode of arrangement,\nthough it is the simplest. Besides this _linear_ arrangement, there is\nalso, for instance, the one in which two new members follow\nsimultaneously upon each previous one, or the members may be disposed\nlike a number of balls heaped up in a pyramid. However, we shall not\nhave much occasion to occupy ourselves with these complex types of\narrangement, and can therefore limit our considerations at first to the\nsimplest, that is, to the linear arrangement.\n\nThis simplest of all possible forms expresses itself in the fact _that\nthe immediately experienced things of our consciousness are arranged in\nthis way_. In point of fact, the contents of our consciousness proceed\nin linear order, one single new member always attaching itself to an\nexisting member. This law, however, is not strictly and invariably\nadhered to. It sometimes happens that our consciousness continues for a\nwhile to pursue the direction of thought it has once taken, although a\nbranching off had already taken place at a former point, at which a new\nchain of thought had begun. Nevertheless, one of these chains usually\nbreaks off very soon, and the linear character of the inner experience\nis immediately restored. Of certain specially powerful intellects it is\nrecorded that they could keep up several lines of thought for a\nconsiderable length of time--Julius C\u00e6sar, for instance.\n\nThe biologic peculiarity here mentioned of the linear juxtaposition of\nthe contents of our consciousness has led to the concept of _time_,\nwhich has been appropriately called a _form of inner life_. That all our\nexperiences succeed each other in time is equivalent to saying that our\nthought processes represent a group in linear arrangement. As appears\nfrom the above observations, this is by no means an absolute form,\nunalterable for all times. On the contrary, a few highly developed\nindividuals have already begun to emancipate themselves from it. But the\nexisting form is so firmly fixed through heredity and habit that it\nstill seems impracticable for most men to imagine the succession of the\ninner experiences in a different way than by a line or by _one\ndimension_. Since, on the other hand, we have all learned to feel space\nas _tri-dimensional_, although optically it appears to possess only two\ndimensions (we see length and breadth, and only infer thickness from\nsecondary characteristics), we come to recognize that the linear form by\nwhich we represent the succession of our experiences is a matter of\nadaptation, and that because the change has been extremely slight in the\ncourse of centuries it produces the impression of being unalterable.[D]\n\n[D] Mathematicians who busy themselves a great deal with the formal\ntheory of four-dimensional space, seem to acquire a capacity for\nimagining this form as easily as the three-dimensional form with which\nwe are all familiar. Therefore, despite the oft-repeated statements to\nthe contrary, it is not impossible to imagine four-dimensional space.\nOnly, we must not attempt to represent to ourselves four-dimensional\nspace in three-dimensional space, especially not without a knowledge of\nits properties.\n\nThese discussions lead to a further difference that can exist in groups\nof linear arrangement. While in the first example we chose, the\nalphabet, the sequence was quite _arbitrary_, since any other sequence\nis just as possible, the same cannot be said of experiences into which\nthe element of time enters. These are not arbitrary, but are arranged by\nspecial circumstances depending upon the aggregate of things which\nco-operate in the given experiences.\n\nWhile, therefore, a group with free members, that is, members not\ndetermined in their arrangement by special circumstances, can be brought\ninto linear order in very different ways, there are groups in which only\none of those orders actually occurs. We see at once that in free groups\nthe number of different orders possible is the greater, the greater the\ngroup itself. The theory of combinations teaches how to calculate these\nnumbers which play a very important r\u00f4le in the various provinces of\nmathematics. The naturally ordered groups always represent a single\ninstance out of these possibilities, the source of which always lies\noutside the group concept, that is, it proceeds from the things\nthemselves which are united into a group.\n\n\n=25. Numbers.= An especially important group in the linear order is that\nof the _integral numbers_. Its origin is as follows:\n\nFirst we abstract the difference of the things found in the group, that\nis, we determine, although they are different, to disregard their\ndifferences. Then we begin with some member of the group and form it\ninto a group by itself. It does not matter which member is chosen, since\nall are regarded as equivalent. Then another member is added, and the\ngroup thus obtained is again characterized as a special type. Then one\nmore member is added, and the corresponding type formed, and so on.\nExperience teaches that never has a hindrance arisen to the formation of\nnew types of this kind by the addition of a single member at a time, so\nthat the operation of this peculiar group formation may be regarded as\n_unlimited_ or _infinite_.\n\nThe groups or types thus obtained are called the _integral numbers_.\nFrom the description of the process it follows that every number has two\nneighbors, the one the number from which it arose by the addition of a\nmember, and the other the number which arose from it by the addition of\na member. In the case of the number one with which the series begins,\nthis characteristic is present in a peculiar form, the preceding group\nbeing _group zero_, that is, a group without content. This number in\nconsequence reveals certain peculiarities into which we cannot enter\nhere.\n\nNow, according to a previous observation (p. 64), not only does the\norder bring every number into relation with the preceding one, but since\nthis last for its part already possesses a great number of relations to\nall preceding, these relations exert their influence also upon the new\nrelation. This fact gives rise to extraordinarily manifold relations\nbetween the various numbers and to manifold laws governing these\nrelations. The elucidation of them forms the subject of an extensive\nscience.\n\n\n=26. Arithmetic, Algebra, and the Theory of Numbers.= From this regular\nform of the number series numerous special characteristics can be\nestablished. The investigations leading to the discovery of these\ncharacteristics are purely scientific, that is, they have no special\ntechnical aim. But they have the uncommonly great practical significance\nthat they provide for all possible arrangements and divisions of\nnumbered things, and so have instruments at hand ready for application\nto each special case as it arises. I have already pointed out that in\nthis lies the positive importance of the theoretical sciences. For\n_practical_ reasons the study of them must be as _general_ as possible.\nThis science is called _arithmetic_.\n\nArithmetic undergoes an important generalization if the individual\nnumbers in a calculation are disregarded and _abstract signs_ standing\nfor any number at all are used in their place. At first glance this\nseems superfluous, since in every real numerical calculation the numbers\nmust be reintroduced. The advantage lies in this, that in calculations\nof the same form, the required steps are formally disposed of once for\nall, so that the numerical values need be introduced only at the\nconclusion and need not be calculated at each step. Moreover, the\ngeneral laws of numerical combination appear much more clearly if the\nsigns are kept, since the result is immediately seen to be composed of\nthe participating members. Thus, _algebra_, that is, calculation with\nabstract or general quantities, has developed as an extensive and\nimportant field of general mathematics.\n\nBy the theory of numbers we understand the most general part of\narithmetic which treats of the properties of the \"numerical bodies\"\nformed in some regular way.\n\n\n=27. Co-ordination.= So far our discussion has confined itself to the\n_individual_ groups and to the properties which each one of them\nexhibits _by itself_. We shall now investigate the relations which exist\n_between two or more groups_, both with regard to their several members\nand to their aggregate.\n\nIf at first we have two groups the members of which are all\ndifferentiated from one another, then any one member of the one group\ncan be co-ordinated with any one member of the other group. This means\nthat we determine that the same should be done with every member of the\nsecond group as is done with the corresponding member of the first\ngroup. That such a rule may be carried out we must be able to do with\nthe members of all the groups whatever we do with the members of one\ngroup. In other words, no properties peculiar to individual members may\nbe utilized, but only the properties that each member possesses as a\nmember of a group. As we have seen, these are the properties of\n_association_.\n\nFirst, the co-ordination is _mutual_, that is, it is immaterial to which\nof the two groups the processes are applied. The relation of the two\ngroups is reciprocal or symmetrical.\n\nFurther, the process of co-ordination can be extended to a third and a\nfourth group and so on, with the result that what has been done in one\nof the co-ordinated groups must happen in all. If hereby the third group\nis co-ordinated with the second, the effects are quite the same as if it\nwere co-ordinated directly with the first instead of indirectly through\nthe second. And the same is true for the fourth and the fifth groups,\netc. Thus, co-ordination can be extended to any number of groups we\nplease, and each single group proves to be co-ordinated with every\nother.\n\nFinally, a group can be co-ordinated with itself, each of its members\ncorresponding to a certain definite other member. It is not impossible\nthat individual members should correspond to themselves, in which case\nthe group has _double members_, or _double points_. The limit-case is\n_identity_, in which every member corresponds to _itself_. This last\ncase cannot supply any special knowledge in itself, but may be applied\nprofitably to throw light on those observations for which it represents\nthe extreme possibility.\n\n\n=28. Comparison.= If we have two groups A and B, and if we co-ordinate\ntheir members severally, three cases may arise. Either group A is\nexhausted while there are members remaining in B, or B is exhausted\nbefore A, or, finally, both groups allow of a mutual co-ordination of\n_all_ their members. In the first case A is called, in the broader sense\nof the word, _smaller_ than B, in the second B is called smaller than A,\nin the third the two groups are said to be of _equal magnitude_. The\nexpression, \"B is greater than A,\" is equivalent to the expression, \"A\nis smaller than B,\" and inversely.\n\nIt is to be noted that the relations mentioned above are true, whether\nthe members are considered as individually different from one another or\nwhether the difference of the members is disregarded, and they are\ntreated as alike. This comes from the fact that every definite\nco-ordination of a group can be translated into every other possible\nco-ordination by exchanging two members at a time in pairs. Since in\nthis process one member is each time substituted for another, and a gap\ntherefore can never occur in its place, the group in the new arrangement\ncan be co-ordinated with the other group as successfully as in the old\narrangement. At the same time we learn from this that in every\nco-ordination of a group with itself, independently of the arrangement\nof its members, it must prove equal to itself.\n\nBy carrying out the co-ordination proof is further supplied of the\nfollowing propositions:\n\n                  { greater than }\n    If group A is { equal to     } group B\n                  { smaller than }\n\n                  { greater than }\n   and group B is { equal to     } group C\n                  { smaller than }\n\n                  { greater than }\n  then group A is { equal to     } group C\n                  { smaller than }\n\nFrom this it follows that any collection of finite groups whatsoever, of\nwhich no one is equal to the other, can always be so arranged that the\nseries should begin with the smallest and end with the greatest, and\nthat a larger should always follow a smaller. _This order would be\nunequivocal_, that is, there is only one series of the given groups\nwhich has this peculiarity. As we shall soon see, the series of integers\nis the purest type of a series so arranged.\n\nIn comparing two infinitely large groups by co-ordination, it may be\nsaid on the one hand that never will one group be exhausted while the\nother still contains members. Accordingly, it is possible to designate\ntwo unlimited or infinite groups (or as many such groups as we please)\nas _equal_ to each other. On the other hand, the statement that in both\ngroups each member of the one is co-ordinated with a member of the other\nhas no definite meaning on account of the infinitely large number of\nmembers. _The definition of equality is therefore not completely\nfulfilled_, and we must not loosely apply a principle valid for finite\ngroups to infinite groups. This consideration, which may assume very\ndifferent forms according to circumstances, explains the \"paradoxes of\nthe infinite,\" that is, the contradictions which arise when concepts of\na definite content are applied to cases possessing in part a different\ncontent. If we wish to attempt such an application, we must in each\ninstance make a special investigation as to the manner in which the\nrelations on their part change by the change of those contents (or\npremises). As a general rule we must expect that the former relations\nwill not remain valid in these circumstances without any change at all.\n\nIn the course of these observations we have learned how co-ordination\ncan be used for obtaining a number of fundamental and multifariously\napplied principles. From this alone the great importance of\nco-ordination is evident, and later we shall see that its significance\nis even more far-reaching. _The entire methodology of all the sciences\nis based upon the most manifold and many-sided application of the\nprocess of co-ordination_, and we shall have occasion to make use of it\nrepeatedly. Its significance may be briefly characterized by stating\nthat it is the most general means of bringing connection into the\naggregate of our experiences.\n\n\n=29. Counting.= The group of integral numbers, because of its\nfundamental simplicity and regularity, is by far the best basis of\nco-ordination. For while arithmetic and the theory of numbers give us a\nmost thorough acquaintance with the peculiarities of this group, we\nsecure by the process of co-ordination the right to presuppose these\npeculiarities and the possibility of finding them again in every other\ngroup which we have co-ordinated with the numerical group. The carrying\nout of such co-ordination is called _counting_, and from the premises\nmade it follows _that we can count all things in so far as we disregard\ntheir differences_.\n\nWe count when we co-ordinate in turn one member of a group after another\nwith the members of the number series that succeed one another until\nthe group to be counted is exhausted. The last number required for the\nco-ordination is called the _sum_ of the members of the counted group.\nSince the number series continues indefinitely, every given group can be\ncounted.\n\nNumerals have been co-ordinated with _names_ as well as with _signs_.\nThe former are different in the different languages, the latter are\ninternational, that is, they have the same form in all languages. From\nthis proceeds the remarkable fact that the written numbers are\nunderstood by all educated men, while the spoken numbers are\nintelligible only within the various languages.\n\nThe purpose of counting is extremely manifold. Its most frequent and\nmost important application lies in the fact that the amount affords a\n_measure for the effectiveness or the value_ of the corresponding group,\nboth increasing and decreasing simultaneously. A further number serves\nas a basis for divisions and arrangements of all kinds to be carried out\nwithin the group, whereby liberal use is made of the principle that\neverything that can be effected in the given number group can also be\neffected in the co-ordinated counted group.\n\n\n=30. Signs and Names.= The co-ordination of names and signs with numbers\ncalls for a few general remarks on co-ordination of this nature.\n\nThe possibility of carrying out the formal operations effected in one of\nthe groups upon the co-ordinated group itself facilitates to an\nextraordinary extent the practical shaping of the reality for definite\npurposes. If by counting we have ascertained that a group of people\nnumbers sixty, we can infer without actually executing the steps that it\nis possible to form these men in six rows of ten, or in five rows of\ntwelve, or in four rows of fifteen, but that we cannot obtain complete\nrows if we try to arrange them in sevens or elevens. These and\nnumberless other peculiarities we can learn of the group of men from its\namount, that is, from its co-ordination with the numerical group of\nsixty. In co-ordination, therefore, we have a means of acquainting\nourselves with facts without having to deal directly with the\ncorresponding realities.\n\nIt is clear that men will very soon notice and avail themselves of so\nenormous an advantage for the mastery and shaping of life. Thus, we see\nthe process of co-ordination in general use among the most primitive\nmen. Even the higher animals know how to utilize co-ordination\nconsciously. When the dog learns to answer to his name, when the horse\nresponds to the \"Whoa\" and the \"Gee\" of his driver there is in each case\na co-ordination of a definite action or series of actions, that is, of a\nconcept with a sign, or, in other words, of a concept with a member of\nanother group; and in this there need not be the least similarity\nbetween the things co-ordinated with each other. The only requirement is\nthat on the one hand the co-ordinated sign should be easily and\ndefinitely expressed and be to the point, and that, on the other hand,\nit should be easily \"understood,\" that is, _comprehended_ by the senses\nand unmistakably _differentiated_ from other signs co-ordinated with\nother things.\n\nThus, we find that the most frequent concepts of co-ordinated sound\nsigns form the beginnings of _language_ in the narrower sense. It is\nvery difficult to ascertain for what reasons the particular forms of\nsound signs have been chosen, nor is it a matter of great importance. In\nthe course of time the original causes have disappeared from our\nconsciousness and the present connection is purely external. This is\nevident from the enormous difference of languages in which hundreds of\ndifferent signs are employed for the same concept.\n\nNow it would be quite possible to solve the problem of co-ordinating\nwith each group of concepts a corresponding group of sounds, so that\neach concept should have its own sound, or, in other words, that the\n_co-ordination should be unambiguous_. It would not by any means be\nbeyond human power to accomplish this, if it were not for the fact that\nthe concepts themselves are still in so chaotic a state as they are at\npresent. We have seen that the attempts of Leibnitz and Locke to draw up\na system of concepts, if only in broad outline, have undergone no\nfurther development since. Even the most regulated concepts as well as\nthe familiar concepts of daily life are in ceaseless flux, while the\nco-ordinated signs are comparatively more stable. But they, too,\nundergo a slow change, as the history of languages shows, and in\naccordance with quite different laws from those which govern the change\nof concepts. The consequence is that in language the co-ordination of\nconcepts and words is far from being unambiguous. The science of\nlanguage designates the presence of several names for the same concept\nand of several concepts for the same name by the words synonym and\nhomonym. These forms, which have arisen accidentally, signify so many\n_fundamental defects_ of language, since they destroy the _principle of\nunambiguity_ upon which language is based. In consequence of the false\nconception of its nature we have until now positively shrunk from\nconsciously developing language in such a way that it should more and\nmore approach the ideal of unambiguity. Such an ideal is in fact\nscarcely known, much less recognized.\n\n\n=31. The Written Language.= Sound signs, to be sure, possess the\nadvantage of being produced easily and without any apparatus, and of\nbeing communicable over a not inconsiderable distance. But they suffer\nunder the disadvantage of transitoriness. They suffice for the purpose\nof temporary understanding and are constantly being used for that. If,\non the other hand, it is necessary to make communications over greater\ndistances or longer periods of time, sound signs must be replaced by\nmore permanent forms.\n\nFor this we turn to another sense, the sense of sight. Since optic\nsigns can travel much greater distances than sound signs without\nbecoming indistinguishable, we first have the optical telegraphs, which\nfind application, though rather limited application, in very varying\nforms, the most efficient being the heliotrope. The other sort of optic\nsigns is much more generally used. These are objectively put on\nappropriate solid bodies, and last and are understood as long as the\nobject in question lasts. Such signs form the _written language_ in the\nwidest sense, and here, too, it is a question of co-ordinating signs and\nconcepts.\n\nWhat I have said concerning the very imperfect state of our present\nconcept system is true also of these two groups. On the other hand, the\nwritten signs are not subject to such great change as the sound signs,\nbecause the sound signs must be produced anew each time, whereas the\nwritten signs inscribed on the right material may survive hundreds, even\nthousands of years. Hence it is that the written languages are, upon the\nwhole, much better developed than the spoken languages. In fact, there\nare isolated instances in which it may be said that the ideal has\nwell-nigh been reached.\n\nAs we have already pointed out, such a case is furnished by the _written\nsigns_ of numbers. By a systematic manipulation of the ten signs 0 1 2 3\n4 5 6 7 8 9 it is not only possible to co-ordinate a written sign with\nany number whatsoever, but this co-ordination is strictly unambiguous,\nthat is, each number can be written in only one way, and each numerical\nsign has only one numerical significance. This has been obtained in the\nfollowing manner:\n\nFirst, a special sign is co-ordinated to each of the group of numbers\nfrom zero to nine. The same signs are co-ordinated with the next group,\nten to nineteen, containing as many numbers as the first. To distinguish\nthe second from the first group, the sign one is used as a prefix. The\nthird group is marked by the prefixed sign two, and so on, until we\nreach group nine. The following group, in accordance with the principle\nadopted, has as its prefix the sign ten, which contains two digits. All\nthe succeeding numbers are indicated accordingly. From this the\nfollowing result is assured: First, no number in its sequence escapes\ndesignation; second, never is an aggregate sign used for two or more\ndifferent numbers. Both these circumstances suffice to secure\nunambiguity of co-ordination.\n\nIt is known that the system of rotation just described is by no means\nthe only possible one. But of all systems hitherto tried it is the\nsimplest and most logical, so that it has never had a serious rival, and\nthe clumsy notation with which the Greeks and Romans had to plague\nthemselves in their day was immediately crowded out, never to return\nagain upon the introduction of the Indo-Arabic notation, which has made\nits way in the same form among all the civilized nations and constitutes\na uniform part of all their written languages.\n\nThe comparison of the spoken and the written languages offers a very\nilluminating proof of the much greater imperfection of the language of\n_words_. The number 18654 is expressed in the English language by\neighteen thousand six hundred and fifty-four, that is, the second figure\nis named first, then the first, the third, the fourth, and the fifth. In\naddition, four different designations are used to indicate the place of\nthe figures, -teen, -thousand, -hundred, and -ty. A more aimless\nconfusion can scarcely be conceived. It would be much clearer to name\nthe figures simply in their sequence, as one-eight-six-five-four.\nBesides, this would be unambiguous. If we should desire to indicate the\n_place value_ in advance, we could do so in some conventional way, for\nexample, by stating the number of digits in advance. This, however,\nwould be superfluous, and ordinarily should be omitted.[E]\n\n[E] The usual designation of the larger groups, ten, hundred, thousand,\nmillion, billion, etc., is also quite irrational. If it is our object to\nsecure expressions for place values in as few words as possible, we find\nthat the numbers of the form 10^{2n}, in which n is a whole number, must\nreceive their own names, that is, 10, 100, 10,000, 100,000,000 etc. In\nthis way the problem of designating as many numbers as possible by as\nfew words as possible is solved.\n\n\n=32. Pasigraphy and Sound Writing.= There are two possibilities for\nco-ordination between concepts and written signs. Either the\nco-ordination is _direct_, so that it is only a matter of providing\nevery concept with a corresponding sign, or it is indirect, the signs\nserving only the purpose of expressing the _language sound_. In the\nlatter case the written language is based entirely upon the sound\nlanguage, and the only problem, comparatively easy to solve, is to\nconstruct _an unambiguous co-ordination between sound and sign_. The\nChinese script follows the direct process, but all the scripts of the\nEuropean-American civilized peoples are based on the indirect process.\n\nThis, it is true, is the case only in ordinary, non-scientific language,\nwhile for science the European nations also have to a large extent built\nup a direct concept writing. One example of this we have seen in the\nnumber signs. Musical notation furnishes another instance, though by far\nnot so perfect. The use of the different keys destroys the unambiguous\nconnection between the pitch and the note sign, and the signatures\nplaced at the beginning of a whole staff have the defect of removing the\nsign from the place where it is applied. Despite this imperfection\nmusical notation is quite international, and every one who understands\nEuropean music also understands its signs.[F]\n\n[F] It is not difficult to perfect musical notation with a view to\nunambiguity, a thing which would greatly facilitate the study of music.\n\nFundamentally we need not hesitate to recognize in _concept writing_ or\n_pasigraphy_ a more complete solution of the problem of sign\narrangement. Even the very incomplete Chinese pasigraphy renders\npossible written intercourse, especially for mercantile purposes,\nbetween the various East-Asiatic peoples who speak some dozens of\ndifferent languages. But each language community translates the common\nsigns into its own words, just as we do in the case of the number signs.\nBut in order that such a system of representation should be complete it\nmust fulfil a whole series of conditions for which scarcely a remote\npossibility is to be discerned at present.\n\nAt first the concepts could simply be taken as found in the words and\ngrammatical forms of the various languages, and each one provided with\nan arbitrary sign. Such approximately is the Chinese system. But a\nsystem of that sort entails an extreme burdening of the memory, which\nresults both from the great number of words and from the necessity of\nkeeping the signs within certain bounds of simplicity. If we consider\nthat the complex concepts are formed according to laws, to a large\nextent still unknown, from a relatively small number of _elementary_\nconcepts, we may attempt to build up the signs of the complex concepts\nby the combination of those of the elementary concepts according to\ncorresponding rules. Then it would only be necessary to learn the signs\nfor the elementary concepts and the rules of combination in order for us\nto be able to represent all the possible concepts. This would provide\neven for the natural enlargement of the concept world, since every new\nelementary concept would receive its sign and would then serve as the\nbasis from which to deduce all the complex concepts dependent upon it.\nIn fact, even should a concept hitherto regarded as elementary prove to\nbe complex, it would not be difficult to declare that its sign, like the\nname of an extinct race, is dead, and after the lapse of sufficient time\nto use it for other purposes.\n\nThe numerical signs offer an excellent example for the elucidation of\nthis subject, and at the same time serve as a proof that in limited\nprovinces the ideal has already been attained. Another very instructive\nexample is furnished by the chemical formulas, which, though they use\nthe letters of the European languages, do not associate with them sound\nconcepts, but chemical concepts. Since the chemical concepts are\nco-ordinated with certain letters, it is possible, in the first place,\nto denote the composition of all combinations qualitatively by the\ncombination of the corresponding letters. But since quantitative\ncomposition proceeds according to definite relations which are\ndetermined by a variety of specific numbers peculiar to each element and\ncalled its combining weight, we need only add to the sign of the element\nthe concept of the combining weight in order to represent in the second\nplace the quantitative composition. Further, the multiples mentioned can\nalso be given. Since, moreover, there are various substances which,\ndespite equal composition, possess different properties, the attempt\nhas been made to express this new manifoldness by the position of the\nelement signs on the paper, and in more recent times also by space\nrepresentation. And here, too, rules have been worked out in which the\nscheme affords a close approach to experience. This example shows how,\nby the constant increase of the complexity of a concept (here the\nchemical composition), ever greater and more manifold demands are made\nupon the co-ordinated scheme. The form of expression first chosen is not\nalways adequate to keep pace with the progress of science. In this case\nit must be radically changed and formed anew to meet the new demands.\n\n\n=33. Sound Writing.= In point of unambiguity of co-ordination _phonetic\nwriting_ is far more imperfect than concept writing. It is obvious that\nin phonetic writing all the faults already present in the co-ordination\nbetween concept and sound are transferred to the written language. To\nthese are added the defects as regards unambiguity occurring in\nco-ordination between sound and sign from which no language is free. In\nsome languages, in fact, notably in English, these defects amount to a\ncrying calamity. The principle of unambiguity would require that there\nshould never be a doubt as to the way in which a spoken word is written,\nand as little doubt as to the way in which a written word is spoken. It\nneeds no proof to show how often the principle is violated in every\nlanguage. In the German language the same sound is represented by f, v,\nand ph; in the English by f and ph. And in both German and English quite\ndifferent sounds are associated with c, g, s, and other letters. _The\nfact that orthographic mistakes can be made in the writing of any\nlanguage is direct proof of its imperfection_, and the oftener this\npossibility occurs the more imperfect is the language in this respect.\nWe know that the spelling reforms begun in Germany more than ten years\nago and recently in America and England, have for their object\nunambiguity in the co-ordination between sign and sound. Still it must\nbe admitted that this tendency has not always been pursued\nundeviatingly. A few innovations, in fact, undoubtedly represent a step\nbackward.\n\n\n=34. The Science of Language.= A comparison of our investigations--which\nwe cannot present in detail but only indicate--with the science of\nlanguage or philology as taught in the universities and in a great\nnumber of books, reveals a great difference between them. This academic\nphilology makes a most exhaustive study of relations, which from the\npoint of view of the purpose of language are of no consequence whatever,\nsuch as most of the rules and usages of grammar. A study of this sort\nmust naturally confine itself to a mere determination of whether certain\nindividuals or groups of individuals have or have not conformed to these\nrules. Even the chief subject of modern comparative philology, the study\nof the relations of the word forms to one another and their changes in\nthe course of history, both within the language communities and when\ntransferred to other localities, appear to be quite useless from the\npoint of view of the theory of co-ordination. For it is indeed of little\nmoment to us to learn by what process of change, as a rule utterly\nsuperficial, a certain word has come to be co-ordinated with a concept\nentirely different from the one with which it had been previously\nco-ordinated. Of incomparably greater importance would be investigations\nconcerning the gradual change of the concepts themselves, although by no\nmeans as important as the real study of concepts. To be sure, such\ninvestigations are much more difficult than the study of word forms set\ndown in writing.\n\nNevertheless, on account of a historical process, which it would lead us\ntoo far afield to discuss, an idea of such word investigations has been\nformed which is wholly disproportionate to their importance. And if we\nask ourselves what part such labors have taken in the progress of human\ncivilization, we are at a loss for an answer. Students of the _science_\nof language make a sharp distinction between it and the _knowledge_ of\nlanguage, which is regarded as incomparably lower. But while a knowledge\nof language is important in at least one respect, in that it presents to\nus the cultural material set down in other languages, or makes them\naccessible in translation to those who do not know foreign languages,\nphilology is of no service in this respect at all, and the pursuit of it\nwill seem as inconceivably futile to future science as the\nscholasticism of the middle ages seems to us now.\n\nThe unwarranted importance attached to the historical study of language\nforms is paralleled by the equally unwarranted importance ascribed to\ngrammatical and orthographic correctness in the use of language. This\nperverse pedantry has been carried to such lengths that it is considered\nalmost dishonorable for any one to violate the usual forms of his mother\ntongue, or even of a foreign language, like the French. We forget that\nneither Shakespeare nor Luther nor Goethe spoke or wrote a \"correct\"\nEnglish or German, and we forget that it cannot be the object of a true\ncultivation of language to _preserve_ as accurately as possible existing\nlinguistic usage, with its imperfections, amounting at times to\nabsurdities. Its real object lies rather in the appropriate\n_development_ and _improvement_ of the language. We have already\nmentioned the fact that in one department, orthography, the true\nconception of the nature of language and of its development is gradually\nbeginning to assert itself. Among most nations efforts are being made to\nimprove orthography with a view to unambiguity, and when once sufficient\nclearness is had as to the object aimed for in spelling, there will be\nno special difficulty in finding the required means to attain it.\n\nBut in all the other departments of language we are still almost wholly\nwithout a conception of the genuine needs. Though the example of the\nEnglish language proves that we can entirely dispense with the manifold\nco-ordinations in the same sentence as appearing in the special plural\nforms of the adjective, verb, pronoun, etc., yet the idea of consciously\napplying to other languages the natural process of improvement\nunconsciously evolved in the English language seems not to have occurred\neven to the boldest language reformers. So strongly are we all under the\ndomination of the \"schoolmaster\" ideal, that is to say, the ideal of\npreserving every linguistic absurdity and impracticability simply\nbecause it is \"good usage.\"\n\nA twofold advantage will have been attained by the introduction of a\n_universal auxiliary language_ (p. 183). Recently the efforts in that\ndirection have made considerable progress. In the first place it will\nprovide a general means of communication in all matters of common human\ninterest, especially the sciences. This will mean a saving of energy\nscarcely to be estimated. In the second place, the superstitious awe of\nlanguage and our treatment of it will give way to a more appropriate\nevaluation of its technical aim. And when by the help of the artificial\nauxiliary language, we shall be able to convince ourselves daily how\nmuch simpler and completer such a language can be made than are the\n\"natural\" languages, then the need will irresistibly assert itself to\nhave these languages also participate in its advantages. The\nconsequences of such progress to human intellectual work in general\nwould be extraordinarily great. For it may be asserted that philosophy,\nthe most general of all the sciences, has hitherto made such extremely\nlimited progress only _because it was compelled to make use of the\nmedium of general language_. This is made obvious by the fact that the\nscience most closely related to it, mathematics, has made the greatest\nprogress of all, but that this progress began only after it had procured\nboth in the Indo-Arabic numerals and in the algebraic signs a language\nwhich actually realizes very approximately the ideal of unambiguous\nco-ordination between concept and sign.\n\n\n=35. Continuity.= Up to this point our discussions have been based on\nthe general concept of the _thing_, that is, of the individual\nexperience differentiated from other experiences. Here the fact of\n_being different_, which, as a general experience, led to the\ncorresponding elementary concept, appeared in the foreground in\naccordance with its generality. But in addition to it there is another\ngeneral fact of experience, which has led to just as general a concept.\nIt is the concept of _continuity_.\n\nWhen, for example, we watch the diminution of light in our room as it\ngrows dark in the evening, we can by no means say that we find it darker\nat the present moment than a moment before. We require a perceptibly\nlong time to be able to say with certainty that it is now darker than\nbefore, and throughout the whole time _we have never felt the increase_\nof darkness from moment to moment, although theoretically we are\nabsolutely convinced that this is the correct conception of the process.\n\nThis peculiar experience, our failure to perceive individual parts of a\nchange, the reality of which we realize when the difference reaches a\ncertain degree, is very general, and, like memory, is based upon a\nfundamental physiological fact. It has already been noted by _Herbart_,\nbut its significance was first recognized by _Fechner_, and has since\nthen become generally known in physiology and psychology under the name\nof _threshold_. _Next to memory the threshold determines the fundamental\nlines of our psychic life._\n\nThe threshold therefore means that whatever state we are in _a certain\nfinite amount of difference or change must be stepped over_ before we\ncan perceive the difference or change. This peculiarity appears in all\nour states or experiences. We have already given an example for the\nphenomena of light and darkness. The same is true of differences in\ncolor and of our judgments as to tone pitch and tone strength. Even the\ntransition from feeling well to feeling ill is usually imperceptible,\nand it is only when the change occurs in a very brief time that we\nbecome conscious of it.\n\nThe physical causes of these psychic phenomena need be indicated only in\nbrief. In all our experiences an existing chemico-physical state in our\nsense organs and in the central organ undergoes a change. Now\nexperiments with physical apparatus have shown that such a process\nalways requires a finite, though sometimes a very small, quantity of\nwork, or, generally speaking, energy, before it can be brought about at\nall. Even the finest scale, sensitive to a millionth of a gram, remains\nstationary when only a tenth of a millionth is placed upon it, although\nwe can _see_ a body of such minute weight under the microscope. In the\nsame way it requires a definite expenditure of energy in order to bring\nthe sense organs, or the central organ, into action, and all stimuli\nless than this limit or threshold produce no experience of their\npresence.\n\nBy this the difficult concept of continuity is evoked in our experience.\nThe transition from the light of day to the darkness of evening proceeds\n_continuously_, that is, at no point of the whole transition do we\nnotice that the state just passed is different from the present one,\nwhile the difference over a wider extent of the experience is\nunmistakable. If we wish to bring vividly to our minds the contradiction\nto other habits of thought which this involves, we need only to\nrepresent to ourselves the following instance. I will compare the thing\nA at a certain time with the thing B, which is so constructed that\nthough objectively different from A, the difference has not yet reached\nthe threshold. From experience, therefore, I must take A to be equal to\nB. Then I compare B with a thing C, which is objectively different from\nB in the same way as A is from B, though here, too, the difference is\nstill within the threshold, though very near it. I shall also have to\ntake B as equal to C. But now if I compare A directly with C, the sum of\nthe two differences oversteps the threshold value, and I find that A is\ndifferent from C. This, then, is a contradiction of the fundamental\nprinciple that if A = B and B = C, A = C. This principle is valid for\n_counted_ things, which, in consequence, are discontinuous, but not for\ncontinuous things susceptible by our senses. If in spite of this it is\napplied to continuous things or _magnitudes_ in the narrower sense, we\nmust bear in mind that it is just as much a case of an _extrapolation to\nthe non-existing ideal instance_ (p. 46) as in the case of the other\ngeneral principles, which, though they are derived from experience,\nnevertheless, for practical purposes, transcend experience in their use.\n\nThe examples cited above prove also that these relations are by no means\nconfined to the judgments we derive on the basis of immediate\nsensations. When by means of the scale we compare three weights, the\ndifferences of which lie within the limit of its sensitiveness but\napproach closely to it, we can arrive in a purely empirical and\nobjective way also at the contradiction A = B, B = C, but A [Not=] C. In\nweight and measurement, therefore, we hold fast to the principle that\nthe relations cited have no claim to validity outside the limit of their\npossible errors. Accordingly, though the non-equation of A [Not=] C can\nbe observed, the difference of both values cannot be greater than at\nutmost the sum of the two threshold values.\n\nThese considerations also give us a means of appraising the oft-repeated\nstatement that in contradistinction to the physical laws the\nmathematical laws are absolutely accurate. The mathematical laws do not\nrefer to real things, but to imaginary ideal limit cases. Consequently\nthey cannot be tested by experience at all, and the demands science\nmakes on them lie in quite a different sphere. Their nature must be such\nthat _experience should approximate them infinitely_, if certain\ndefinite well-known postulates are to be more and more fulfilled, and\nthat the various abstractions and idealizations should be so chosen as\nnot to contradict one another. Such contradictions have by no means\nalways been avoided. But we must not regard them as inherent in the\ninner organization of our mind, as Kant did. These contradictions spring\nfrom careless handling of the concept technique, by which postulates\nelsewhere rejected are treated as valid. We have already come across an\ninstance of such relations in the application of the concept of equality\nto unlimited groups (p. 84).\n\nWe must be guided by the same rules of precaution in answering the\nquestion whether the things felt as continuous--for example, space and\ntime--are \"truly\" continuous, or whether in the last analysis they must\nnot be conceived of as discontinuous. The various sense organs, and\nstill more, the various physical apparatus with which we examine given\nstates, are of very varying degrees of \"sensibility,\" that is, the\nthreshold for distinguishing the differences may be of very different\nmagnitudes. Therefore, a thing which is discontinuous for a sensitive\napparatus will behave as if it were continuous with a less sensitive\napparatus. Accordingly, we shall find so many the more things continuous\nthe less sharply developed our ability is to differentiate.\n\nWhile this circumstance makes it possible that we should regard\ndiscontinuous things as continuous, time relations in certain\ncircumstances produce the opposite effect. Even if in a process the\nchange is continuous but very rapid, and the new state remains unchanged\nfor a certain time, we easily conceive of this sequence as\ndiscontinuous. We cannot resist this view of the process when the change\noccurs in a shorter time than the threshold time of our mind for each\nstep in the process. But since this threshold changes with our general\ncondition, one and the same process can appear to us both continuous and\ndiscontinuous according to circumstances. Here, therefore, we have a\ncause through the operation of which, with advancing knowledge, more and\nmore things will become recognized as _continuous_.\n\nNow if we turn to _experience_, we find, as the sum total of our\nknowledge, that for the sake of expediency we approach everything with\nthe presumption that it is _continuous_. This aggregate experience\nfinds its expression in such sayings as \"Nature makes no jumps,\" and\nsimilar proverbial generalizations. But we must emphasize the fact once\nmore that in deciding matters in this way we deal solely with questions\nof expediency, not with questions of the nature of our mental capacity.\n\n\n=36. Measurement.= Measuring is in a certain way the opposite of\ncounting. While, in counting, the things are regarded in advance as\n_individual_, and the group, therefore, is a body compounded of\ndiscontinuous elements, measuring, on the other hand, consists in\n_co-ordinating numbers with continuous things_, that is, in applying to\ncontinuous things a concept formed upon the hypothesis of discontinuity.\n\nIt lies in the nature of such a problem that the difficulty of\nadaptation must crop out somewhere in the course of its attempted\nsolution. This is actually shown by the fact that measurement proves to\nbe an unconcluded and inconcludable operation. If, in spite of this,\nmeasurement may and must justly be denoted as one of the most important\nadvances in human thought, it follows that those fundamental\ndifficulties can practically be rendered harmless.\n\nLet us picture to ourselves some process of measurement--for example,\nthe determination of the length of a strip of paper. We place a rule\ndivided into millimeters (or some other unit) on the strip, and then we\ndetermine the unit-mark at which the strip ends. It turns out that the\nstrip does not end exactly at a unit-mark, but _between_ two\nunit-marks. And even if the rule is provided with divisions ten or a\nhundred times finer, the case remains the same. In most cases a\nmicroscopic examination will show that the end of the strip does not\ncoincide with a division. All that can be said, therefore, is that the\nlength must lie _between n and n + 1 units_, and even if a definite\nnumber is given, the scientifically trained person will supplement this\nnumber by the sign \u00b1 _f_, in which _f_ denotes the possible errors, that\nis, the limit within which the given number may be false.\n\nWe see at once how the characteristic concept of threshold, which has\nled to the conception of the continuous, immediately asserts itself when\nin connection with discontinuous numbers. The adaptation of the\nthreshold to numbers can be carried as far as it is possible to reduce\nthe threshold, but the latter can never be made to disappear entirely.\n\nThe significance of measurement therefore lies in the fact that it\napplies the operation of counting with all its advantages (see p. 85) to\n_continuous_ things, which as such do not at first lend themselves to\nenumeration. By the application of the unit measure a discontinuity is\nat first artificially established through dividing the thing into\npieces, each piece equal to the unit, or imagining it to be so divided.\nThen we count the pieces. When a quantity of liquid is _measured_ with a\nliter this general process is carried out physically. In all other less\ndirect methods of measurement the physical process is substituted by an\neasier process equally good. Thus, in the example of the strip of paper\nwe need not cut it up into pieces a millimeter in length. The divided\nrule is available for comparing the length of any number of millimeters\nthat happen to come under consideration, and we need only read off from\nthe figures on the rule the quantity of millimeters equal to the length\nof the strip, in order to infer that the strip can be cut up into an\nequal number of pieces each a millimeter in length.\n\nAfter it has been made possible to count continuous things in this way,\nthe numeration of them can then be subjected to all the mathematical\noperations first developed only for discrete, directly countable things.\nWhen we reflect that our knowledge of things has given them to us\n_preponderatingly as continuous_, we at once see what an important step\nforward has been made through the invention of measurement in the\nintellectual domination of our experience.\n\n\n=37. The Function.= The concept of continuity makes possible the\ndevelopment of another concept of greater universality, which can be\ncharacterized as an extension of the concept of causation (p. 31). The\nlatter is an expression of the experience, if A is, B is also, in which\nA is understood to be a definite thing at first conceived of as\nimmutable. Now it may happen that A is not immutable, but represents a\nconcept with continuously changing characteristics. Then, as a rule, B\nwill also be of that nature, so that _every special value or state of B\ncorresponds to every special value or state of A_.\n\nHere, in place of the reciprocal relation of two definite things, we\nhave the reciprocal relation of two more or less extended groups of\nsimilar things. If these things are continuous, as is assumed here (and\nwhich is extremely often the case), both groups or series, even though\nthey are finite, contain an endless quantity of individual cases. Such a\nrelation between two variable things is called a function. Although this\nconcept is used chiefly for the reciprocal relation of _continuous_\nthings, there is nothing to hinder its application to discrete things,\nand accordingly we distinguish between continuous and discontinuous\nfunctions.\n\nThe intellectual progress involved in the conception of the reciprocal\nrelation of entire _series_ or groups to one another, as distinguished\nfrom the conception of the relations between _individual_ things, is of\nthe utmost importance and in the most expressive manner characterizes\nthe difference between modern scientific thought and ancient thought.\nAncient geometry, for example, knew only the cases of the acute, right,\nand obtuse angled triangle, and treated them separately, while the\nmodern geometrician represents the side of the triangle as starting from\nthe angle zero and traversing the entire field of possible angles.\nAccordingly, unlike his colleague of old, he does not ask for the\nparticular principles bearing upon these particular cases, but he asks\nin what continuous relation do the sides and angles stand to one\nanother, and he lets the particular cases develop from out of one\nanother. In this way he attains a much profounder and more effectual\ninsight into the whole of the existing relations.\n\nIt is in mathematics in especial that the introduction of the concept of\ncontinuity and of the function concept arising from it has exercised an\nextraordinarily deep influence. The so-called _Higher Analysis_, or\n_Infinitesimal Analysis_, was the first result of this radical advance,\nand the _Theory of Functions_, in the most general sense, was the later\nresult. This progress rests on the fact that the magnitudes appearing in\nthe mathematical formulas were no longer regarded as certain definite\nvalues (or values to be arbitrarily determined), but as _variable_, that\nis, values which may range through all possible quantities. If we\nrepresent the relation between two things by the formula B = f(A),\nexpressed in spoken language by B _is a function_ of A, then in the old\nconception A and B are each individual things, while in the modern\nconception A and B represent an inexhaustible series of possibilities\nembracing every conceivable individual case that may be co-ordinated\nwith a corresponding case.\n\nHerein lies the essential advantage of the concept of continuity. It is\ntrue that it also introduces into calculation the above-mentioned\ncontradictions which crop up in the ever-recurring discussions\nconcerning the infinitely great and the infinitely small. The system\nintroduced by Leibnitz of calculating with _differentials_, that is,\nwith infinitely small quantities, which in most relations, however,\nstill preserve the character of finite quantities from which they are\nconsidered to have been derived, has proved to be as fruitful of\npractical results as it is difficult of intellectual mastery. We can\nbest conceive of these differentials as the expression of the law of the\nthreshold, which law gave rise to, or made possible, the relation\nbetween the continuous and the discrete.\n\n\n=38. The Application of the Functional Relation.= I have already shown\n(p. 34) how the first formulation of a causal relation which experience\nyields can be purified and elaborated by the multiplication of the\nexperience. The method described was based upon the fact that the\nnecessary and adequate factors of the result were obtained by\neliminating successively from the \"cause\" the various factors of which\nits concept was or could be compounded, and by concluding from the\nresult, that is, the presence or absence of the \"effect,\" as to the\nnecessity or superfluity of each factor.\n\nObviously the application of this process presupposes the possibility of\neliminating each factor in turn. Very often it is not possible, and then\nin place of the inadequate method of the individual case the _method of\nthe continuous functional relation_ steps in with its infinitely\ngreater effectiveness. If in most cases we cannot _eliminate_ the\nfactors one by one, there are very few instances in which it is not\npossible to _change_ them, or to observe the result in the automatically\nchanged values of the factors. But then we have the principle that for\nthe causal relation _all such factors are essential the change of which\ninvolves a change of the result_.\n\nIt is clear that this signifies a generalization of the former and more\nlimited method. For the elimination of the factor means that its value\nis reduced to zero. But now it is no longer necessary to go to this\nextreme limit; it suffices merely to influence in some way the factor to\nbe investigated.\n\nIt is true that here the difference in the result cannot be expressed\nwith a \"yes\" or a \"no,\" as before. It can only be said that it has\nchanged _partly_, more or less. From this it can be seen that the\napplication of this process requires more refined methods of\nobservation, especially for measuring, that is, for determining values\nor magnitudes. On the other hand, we must recognize how much deeper we\ncan penetrate into the knowledge of things by the application of the\nmeasuring process. Each advance in precision of measurement signifies\nthe discovery of a new stratum of scientific truth previously\ninaccessible.\n\n\n=39. The Law of Continuity.= From the fact that natural phenomena in\ngeneral proceed continuously we can deduce a number of important and\ngenerally applicable conclusions which are constantly used for the\ndevelopment of science.\n\nWhen a relation of two continuously varying values of the form A = f(B)\nis conjectured, we convince ourselves of its truth by observing for\ndifferent values of A the corresponding values of B, or reversely. If we\nfind that changes in the one correspond to changes in the other, the\nexistence of such a relation is proved, at first only for the observed\nvalues, though we never hesitate to conclude that for the values of A\nlying between the observed values, but themselves not yet observed, the\ncorresponding values of B will also lie between the observed values. For\nexample, if the temperature at a given place has been observed at\nintervals of two hours, we assume without hesitancy that in the hours\nbetween when no observations were made, the values lie between the\nobserved values. If we indicate the time in the usual manner by\nhorizontal lines and the temperature for the general periods of time by\nlongitudinal lines, the law of continuity asserts that all these\ntemperature points lie in a steady line, so that when a number of points\nlying sufficiently near one another is known, the points between can be\nderived from the steady line which may be drawn through the known\npoints. This very commonly applied process will yield the more accurate\nresults the nearer the known points are to one another, and the simpler\nthe line.\n\nThe application of the law of continuity or steadiness, therefore,\nmeans no less than that it is possible, from a finite, frequently not\neven a very large, number of individual results, to obtain the means of\npredicting the result for an infinitely large number of unexamined\ncases. The instrument derived from this law, therefore, is an eminently\n_scientific_ one.\n\nThe value of this instrument is still greater if it succeeds in\nexpressing the relation A = f(B) in strict mathematical form. First, the\nresult of the determination of a number of individual values of that\nfunction is represented as a table of co-ordinated values. By the\ngraphic process above described, or by its equivalent, the mathematical\nprocess of interpolation, this table is so extended that it also\nsupplies all the intermediate values. But this is still a case of a\nmechanical co-ordination of the corresponding values. Often we succeed,\nespecially in the relation of simple or pure concepts, in finding a\ngeneral mathematical rule by which the magnitude A can be derived from\nthe magnitude B, and reversely. This is the only instance in which we\nspeak of a natural law in the quantitative sense.\n\nThus, for example, we can observe what volume a given quantity of air\noccupies when successively subjected to different pressures. If we\narrange all these values together in a table, we can also calculate\nthe volume for all the intermediate pressures. But on close inspection\nof the corresponding numbers of pressure and volume we notice that\nthey are in inverse ratio, or that when multiplied by one another\ntheir products will be the same. If we denote the space by v and the\npressure by p, this fact assumes the mathematical form p\u00b7v = K, in\nwhich K is a definite number depending upon the quantity of air, the\nunit of pressure, etc., but remaining unchanged in an experimental\nseries in which these factors stay the same. The general functional\nequation A = f(B) becomes the definite p = K/v. And this formula\nenables us to determine by a simple calculation the volume for any\ndegree of pressure, provided the value of K has been once ascertained\nby experiment.\n\nAt first we have a right to such a calculation only within the province\nin which the experiments have been made, and the simple mathematical\nexpression of the natural law has for the time being no further\nsignificance than that of a specially convenient rule for interpolation.\nBut such a form immediately evokes a question which demands an\nexperimental answer. How far can the form be extended? That there must\nbe a limit is to be directly inferred from the consideration of the\nformula itself. For if we let p = 0, then v = infinity, both of which\nlie beyond the field of possible experience.\n\nSimilar considerations obtain in all such mathematically formulated\nnatural laws, and each time, therefore, we must ask what the _range of\nvalidity_ of such an expression is, and answer the question by\nexperiment.\n\nWhile in this discussion the mathematically formulated natural law seems\nto have the nature only of a convenient formula of interpolation, we are\nnevertheless in the habit of regarding the discovery of such a formula\nas a great intellectual accomplishment, which so impresses us that we\nfrequently call it by the name of the discoverer. Now, wherein lies the\nmore significant value of such formulations?\n\nIt lies in the fact that simple formulas are discovered only _when the\nconceptual analysis of the phenomenon has advanced far enough_. The very\nsimplicity of the formula shows that the concept formation which is at\nthe basis of it is especially serviceable. In Ptolemy's theory of the\nmotion of the planets the means for calculating their positions in\nadvance was given just as in the theory of Copernicus. But Ptolemy's\ntheory was based on the assumption that the earth stands still, and that\nthe sun and the other planets move. The assumption that the sun stands\nstill and that the earth and the other planets move greatly facilitates\nthe calculation of the position of the planets. In this lay the primary\nvalue of the advance made by Copernicus. It was not until much later\nthat it was found that a number of other actual relations could be\nrepresented much more fittingly by means of the same hypothesis, and\nthus the Copernican theory has come to be generally recognized and\napplied.\n\nThe significance of the law of continuity and its field of application\nhave by no means been exhausted by what has been said above. But later\nwe shall have a number of occasions to point out its application in\nspecial instances, and so cause its use to become a steady mental habit\nwith the beginner in scientific research.\n\n\n=40. Time and Space.= Time and space are two very general concepts,\nthough without doubt not elementary concepts. For besides the elementary\nconcept of continuity which both contain, time has the further character\nof being one-seried or one-dimensional, of not admitting of the\npossibility of return to a past point of time (absence of double points)\nand of absolute onesidedness, that is, of the fundamental difference\nbetween before and after. This last quality is the very one not found in\nthe space concept, which is in every sense symmetrical. On the other\nhand, owing to the three dimensions it has a _three_fold manifoldness.\n\nThat despite this radical distinction in the properties of space and\ntime all of our experiences can be expressed or represented within the\nconcepts of space and time, is very clear proof that experience is much\nmore limited than the formal manifoldness of the conceivable. In this\nsense space and time can be conceived as natural laws which may be\napplied to all our experiences. Here at the same time the\nsubjective-human element of the natural law becomes very clear.\n\nThe properties of time are of so simple and obvious a nature that there\nis no special science of time. What we need to know about it appears as\npart of physics, especially of mechanics. Nevertheless time plays an\nessential r\u00f4le in _phoronomy_, a subject which we shall consider\npresently. In phoronomy, however, time appears only in its simplest form\nas a one-seried continuous manifoldness.\n\nAs for space, the presence of the three dimensions conditions a great\nmanifoldness of possible relations, and hence the existence of a very\nextensive science of bodies in space, of _geometry_. Geometry is divided\ninto various parts depending upon whether or not the concept of\nmeasurement enters. When dealing with purely spacial relations apart\nfrom the concept of measurement it is called geometry of position. In\norder to introduce the element of measurement a certain hypothesis is\nnecessary which is undemonstrable, and therefore appears to be arbitrary\nand can be justified only because it is the simplest of all possible\nhypotheses. This hypothesis takes for granted that a rigid body can be\nmoved in all directions in space without changing in measure. Or, to\nstate the inverse of this hypothesis, in space those parts are called\nequal which a rigid body occupies, no matter how it is moved about.\n\nWe are not conscious of the extreme arbitrariness of this assumption\nsimply because we have become accustomed to it in school. But if we\nreflect that in daily experience the space occupied by a rigid body, say\na stick, seems to the eye to undergo radical changes as it shifts its\nposition in space and that we can maintain that hypothesis only by\ndeclaring these changes to be \"apparent,\" we recognize the arbitrariness\nwhich really resides in that assumption. We could represent all the\nrelations just as well if we were to assume that those changes are real,\nand that they are successively undone when we restore the stick to its\nformer relation to our eye. But though such a conception is\nfundamentally practicable in so far as it deals merely with the space\npicture of the stick, we nevertheless find that it would lead to such\nextreme complications with regard to other relations (for example, the\nfact that the weight of the stick is not affected by the change of the\noptic picture) that we do better if we adhere to the usual assumption\nthat the optical changes are merely apparent.\n\nIn this connection we learn what an enormous influence the various parts\nof experience exert upon one another in the development of science. In\nevery special generalization of experiences, that is, in every\nindividual scientific theory, our aim is not only to generalize this\nspecial group of experiences in themselves, but at the same time to join\nsuch other experiences to them as expedience demands. If the effect of\nthis necessity is on the one hand to render the elaboration of an\nappropriate theory more difficult, it has on the other hand the great\nadvantage of affording a choice among several theories of apparently\nlike value, and thus making possible a more precise notion of the\nreality. For example, for the understanding of the mutual movements of\nthe sun and the earth it is the same whether we assume that the sun\nmoves about the earth or the earth about the sun. It is not until we try\nto represent theoretically the position of the other planets that we see\nthe economic advantage of the second conception, and facts like\nFoucault's experiment with a pendulum can be represented only according\nto this second conception in our present state of knowledge.\n\nLikewise, the assumption on which scientific geometry goes, that space\nhas the same properties in all directions, conflicts with immediate\nexperience. In immediate experience we make a sharp distinction between\nbelow and above, although we are prepared to admit the \"homogeneity\" of\nspace in the horizontal direction. This is due, as physics teaches, to\nthe fact that we are placed in a field of gravitation which acts only\nfrom above downward and which permits free horizontal turnings, although\nit imparts a characteristic difference to the third direction. Since\nconsiderations of another kind enable us to place ourselves in a\nposition in which we ignore this field of gravitation in the\ninvestigation of space, geometry abstracts this element and disregards\nthe corresponding manifoldness. In the theory of the gravitation\npotential, on the other hand, this very manifoldness is made the subject\nof scientific investigation.\n\nThe common application of the concepts of space and time results in the\nconcept of _motion_, the science of which is called phoronomics. In\norder to make this new variable subject to measurement we must arrive at\nan agreement or convention as to the way in which to measure time. For\nsince past time can never be reproduced we actually experience only\nunextended moments, and have no means of recognizing or defining the\nequality of two periods of time by placing them side by side, as we can\nin the case of spacial magnitudes. We help ourselves by saying _that in\nuninfluenced motions equal periods of time must correspond to the equal\nchanges in space_. We regard the rotation of the earth on its axis and\nits revolution about the sun as such uninfluenced motions. The two\ndepend upon dissimilar conditions, and the empirical fact that the\nrelation of the two motions, or the relation between the day and the\nyear, remains practically the same, sustains that assumption, and at the\nsame time shows the expediency of the given definition of time.\n\n_Analytic geometry_, the application of algebra to geometric relations,\noccupies a noteworthy position, from the point of view of method, in the\nscience of space. It yields geometric results by means of calculation,\nthat is, by the application of the _algebraic_ material of symbols we\ncan obtain data concerning unknown _spacial_ relations. An explanation\nis necessary of how by a method apparently so extraneous such results as\nthese can be attained.\n\nThe answer lies again in the general principle of co-ordination, which\nin this very case receives a particularly cogent illustration. Three\nalgebraic signs, x, y, and z, are co-ordinated with the three variable\ndimensions of space. First, the same independent and constant\nvariability is ascribed to these signs, and, further, the same mutual\nrelations are assumed to subsist between them as actually exist between\nthe three-spacial dimensions. In other words, precisely the same kind of\nmanifoldness is imparted to these algebraic signs as the spacial\ndimensions possess to which they are co-ordinated, and we may therefore\nexpect that all the conclusions arising from these assumptions will find\ntheir corresponding parts in the spacial manifoldness. Accordingly, a\nco-ordinated spacial relation corresponds to every change of those\nalgebraic formulas resulting from calculation, and if such changes lead\nto an algebraically simple form, then the spacial form corresponding to\nit must show an analogous simplicity. Here, therefore, we have a case\nsuch as was described under simpler conditions on p. 86 of operations\nundertaken with one group and repeated correspondingly in the\nco-ordinated group. And it is only the great difference in the things\nof which in this case the two groups are composed--spacial relations on\nthe one side and algebraic signs on the other--that creates the\nimpression of astonishment which was felt very strongly at the invention\nof this method, and which is still felt by students with talent for\nmathematics when they first become acquainted with analytical geometry.\n\n\n=41. Recapitulation.= Before we proceed to consider the fundamentals of\nother sciences, it is well to make a general r\u00e9sum\u00e9 of the field so far\ntraversed. Since the later sciences, as we have already observed, make\nuse of the entire apparatus of the earlier sciences, the mastery of them\nmust be assured in order to render their special application possible.\n\nThis does not mean that one must have complete command of the entire\nrange of those earlier sciences in order to pursue a later one. Mere\nhuman limitations would prevent the fulfilment of such a demand. As a\nmatter of fact, successful work can be done in one of the later sciences\neven if only the most general features of the earlier ones have been\nclearly grasped. Nevertheless, the rapidity and certainty of the results\nare very considerably increased by a more thorough knowledge of the\nearlier sciences, and the investigator, accordingly, should seek a\nmiddle road between the danger of insufficient preparation for his\nspecial science and the danger of never getting to it from sheer\npreparation. In any circumstances he must be prepared always, even\nthough it be in later age, to acquire those fundamental aids so soon as\nhe feels the need of them for carrying out any special work. It is\ngenerally acceded that without logic the adequate pursuit of science is\nimpossible. Nevertheless, the opinion is widely current, even among men\nof science, that everybody has command of the needful logic without\nhaving studied it. No more than a man can learn of himself to use the\ncalculus, even if he may have discovered unaided some of its elementary\nprinciples, can he acquire certainty and readiness in the use of the\nlogical rules generally necessary, unless he has made the necessary\nstudies. It is true that the scientific works of the great pioneers and\nleaders in the special sciences furnish practical examples of such\nlogical activity. But complete freedom and security are acquired only on\nthe basis of conscious knowledge.\n\nWe have now seen how, from the physiological construction of our mental\napparatus, the process of concept formation and the experience of\nconcept connections are the basis of the whole of mental life. The laws\nof the mutual interaction of the most general or elementary concepts\noperated in the formation of the concepts, _thing_, _group_,\n_co-ordination_. Here were found the fundamentals of logic or the\nscience of concepts. A special process of abstraction yielded the\nconcept of _number_, and with it the corresponding field of\n_mathematics_, arithmetic, algebra, and the theory of numbers.\n\nBy means of the second fundamental fact of physiology, the _threshold_,\nanother elementary fact was explained, that of _continuity_. The\nco-ordination of individual things under the influence of this concept\nwas expanded into the _co-ordination of continuous phenomena-series_,\nand yielded the correspondingly more general concept of the _function_.\nFrom the application of the number concept to continuous things, the\nidea of _measurement_ resulted. In mathematics the concept of continuity\nled to higher _analysis_ and the _theory of functions_. Finally, the\nconcept of continuity proved to be an inexhaustible aid for the\nextension of scientific knowledge and for the formulation of natural\nlaws in mathematical form.\n\n\n\n\nPART III\n\nTHE PHYSICAL SCIENCES\n\n\n=42. General.= In the formal sciences we began the specialization of the\nobject from the most general concept of thing conceivable, possessing no\nother characteristic attribute than its capability of being\ndistinguished from other things; and we carried the specialization so\nfar that we could follow in its movements an object definite as to time\nand space. This object, to be sure, was defined only in that it occupied\na definite space, and accordingly had a definite form. As a matter of\nfact, the spacial thing of geometry and phoronomy reveals no further\nattributes.\n\nIt is here that the physical sciences enter into their dominion one\nafter the other, and fill the empty space of the geometric thing with\ndefinite attributes. These are the secondary qualities of Locke, of\nwhich he assumed that they do not belong so much to the bodies\nthemselves as that they merely appear to us so on account of the nature\nof our human sense organs. Now that our knowledge concerning the nature\nof those properties as well as the structure of our sense organs is\nmuch more thorough, we have more definite ideas also of the subjective\npart of the corresponding experiences, and in a large measure are able\nto separate it from the objective part.\n\nAll properties which physical bodies in contradistinction to geometric\nbodies possess can be traced back to a fundamental concept, which, in\nconjunction with the concepts explained in the former chapter, serves to\ncharacterize and distinguish the physical structure. For example, the\nfact that we can distinguish cubes of equal size but of different\nmaterial, different temperature, and different luminosity, can be traced\nback always and entirely to the different kinds of energy acting in the\ngeometric space in question. The concept of energy, therefore, plays\napproximately the same r\u00f4le in the physical sciences as the concept of\nthing in the formal sciences, and the essentials of this new field of\nscience are the comprehensive knowledge and development of this concept.\nBecause of its great importance it has long been known and applied in\nindividual forms. But the systematization of the physical sciences\nrelative to energy is a matter of only recent date.\n\n\n=43. Mechanics.= Recently many scientists have taken exception to the\ntraditional division of mechanics into _statics_, or the science of\nequilibrium, and _dynamics_, or the science of motion, because it does\nnot correspond to the essence of the thing, equilibrium being only the\nlimit-case of motion. However, the classic presentations of this\nscience are based on that division, so that it must express an essential\ndifference. This difference we can clearly recognize through the\napplication of the concept of energy to mechanics. We then learn that\nstatics is the science of work, or the energy of position, and that\ndynamics is the science of living force, or of the energy of motion.\n\nBy _work_ in the mechanical sense we mean the expenditure of force\nrequired for the locomotion of physical bodies. While a cube of lead is\ngeometrically equal to a cube of glass, we experience a great difference\nbetween them when we lift them from the floor to a table. We call the\ncube of lead heavier than the glass cube, and we find it requires more\nwork to raise the former than the latter. For psychologic reasons this\njudgment becomes especially clear when the work required to lift the\nlead cube marks the limit of our physical capacity.\n\nWork depends not only upon the difference described above, but also upon\nthe distance through which it is exerted. It increases in proportion as\nthe distance increases. In mechanics work is proportional both to the\ndistance and to that peculiar property which in the given example we\ncall _weight_. But a more general concept has been formed for that\nproperty in the mechanical sense, called _force_, of which weight\nconstitutes but a special instance. Whenever there is a resistance\ncombined with a change of place we speak of a force, _and the product\nof the force and the distance we call work_.\n\nThe cause of this kind of concept formation is the following: There are\na great number of different machines, all of them possessing the\npeculiarity that work can be put into them at a definite place and taken\nout at another place. Now, centuries of experience have shown that it is\nimpossible to obtain more work from such mechanical machines than has\nbeen put into them. As a matter of fact, the work obtained is always\nless than the work put in, and the two approach equality as the machine\napproaches perfection. It is to such ideal machines, therefore, that\n_the law of the conservation of work_ applies. This law states that,\nthough a given quantity of work may be changed in the most manifold ways\nas to direction, force, etc., it is impossible to change its _quantity_.\n\nThe reason we can judge of this fact with such certainty is because for\nmany centuries a number of the ablest mechanicians have sought for a\nsolution of the problem of perpetual motion, that is, for the\nconstruction of a machine from which more work can be gotten than is put\ninto it. All such attempts have failed. But the positive result secured\nfrom these apparently futile efforts is the law of the conservation of\nwork. The greatness and importance of this result will become apparent\nin the further course of our study.\n\nHere for the first time we meet with a law expressing the\n_quantitative_ conservation of a thing, which may none the less undergo\nthe most varied qualitative changes. With the knowledge of this fact we\ninvoluntarily combine the notion that it is the \"same\" thing that passes\nthrough all these transformations, and that it only changes its outward\nform without being changed in its essence. Such ideas, it is true, are\nwidespread, but they have a very doubtful side to them, since they\ncorrespond to no distinct concept. If we want to call the quantitative\nmagnitude of the product of the force and distance the \"essence\" of\nwork, and the determination of the force and the distance according to\nmagnitude and direction, which come under consideration for each special\nvalue, as its \"form,\" then, of course, there is no objection to be made\nto mere nomenclature. But we must bear in mind that the difference\nobtaining here lies exclusively in the fact that the amount of work\nmeasured quantitatively remains unchanged, while its factors undergo\nsimultaneous and opposite changes.\n\nThis discovery, that there is a magnitude which can be quantitatively\ndetermined, and which, as experience shows, remains unchanged, however\nmuch its factors may change, invariably results not only in a very\nsimple and clear formulation of the corresponding natural law, but also\ncorresponds to the general tendency of the human mind to work out\nconceptually \"the permanent in change.\" If, in accordance with the\nword-sense, we denote everything which persists under changing\nconditions by the name of _substance, we encounter in work the first\nsubstance_ of which we have attained knowledge in our scientific\njourneys. In the history of the evolution of human thought this\nsubstance has been preceded by others, especially by the weight and mass\nof ponderable bodies (which are also subject to a law of conservation),\nso that at present we are inclined to connect with the word substance a\ntacit secondary sense of ponderability. But this is a remnant of the\nstill very widely spread mechanistic theory of the universe, which,\nthough it has almost finished its r\u00f4le in physics, will presumably\ncontinue to persist for a long time to come in the popularly scientific\nconsciousness in accordance with the laws of collective thought.\n\n\n=44. Kinetic Energy.= The law of the conservation of work is by no means\ntrue of all cases in which work is expended or converted, but, as has\nbeen said, only of _ideal_ machines, that is, of such cases which do not\nexist in reality. But while in imperfect machines there is at least an\napproximation to this law, there are besides countless normal cases in\nwhich we cannot even speak of an approximation. When, for example, a\nstone falls to the ground from a certain height, a certain quantity of\nwork is expended, which is equal to that by means of which the stone can\nbe raised again to its original height. This quantity of work apparently\ndisappears entirely when the stone remains lying on the ground. We\nshall discuss this case later. Or the falling of the stone can be so\nguided that it can lift itself again. This happens, for instance, when,\nby fastening the stone to a thread, it is forced to move in a curved\npath, or to perform pendular oscillations. In that case, it is true, the\nstone will fall to the lowest point which the thread permits, and so\nwill there have lost its work without having done any other work in the\nmeantime. But it has entered a condition by virtue of which it raises\nitself again, so that (as before, only in the ideal limit-case) it\nreaches its former height, and so has lost no work. For this moment,\ntoo, then, the law of the conservation of work obtains. But in the\nmeantime new relations have arisen.\n\nWhat distinguishes the stone moving like a pendulum from the stone which\nsimply falls is, that at its lowest point it has not remained lying\nstill, but possesses a certain velocity. By means of this it lifts\nitself again, and after it has reached its former height, it has lost\nits velocity. _Therefore, there is a reciprocal relation between the\nwork which it loses and the velocity which it gains_, and the question\nmay therefore be put, How can this relation be represented\nmathematically? Experience teaches that in every such case a function of\nthe velocity and of another property of the body, called _mass_, can be\nestablished in such a way that this function, called the _kinetic\nenergy_ of the body, increases precisely as much as the amount of work\nthe body has expended, and _vice versa_. The sum of the kinetic energy\nof the body and of the _work_ is therefore _constant_, and the clearest\nmode of conceiving of this relation is by assuming _that work can be\ntransformed into kinetic energy and vice versa_ in such a way that given\namounts of the two magnitudes are equal or equivalent to one another.\nNaturally, this is only an abbreviated way of expressing the actual\nrelations, for it might just as well be assumed that the work really\ndisappears and the kinetic energy really originates anew, and that the\ndisappearance of the one substance only happens regularly to coincide\nwith the origin of the other. But it is this regular conjunction of\nphenomena that constitutes the sole ground of every _causal_ relation,\nand in such a sense we are justified _in regarding the disappearing work\nas the cause of the kinetic energy that arises_, and to designate this\nrelation summarily as a transformation.\n\nBy the inclusion of cases in which work is converted into kinetic energy\nthe law of the conservation of work therefore becomes _the law of the\nconservation of the sum of work and kinetic energy_. We are thereby\ncompelled to extend the concept of substance, which at first contains\nonly work, to the sum of both magnitudes, and to introduce a new name\nfor this enlarged concept.\n\nIt will soon appear that all cases of imperfect machines, in which work\ndisappears without giving rise to an equivalent amount of kinetic\nenergy, can, with a corresponding enlargement of the concept, be\nlikewise included in the law of conservation. For experience has shown\nthat in such cases something else arises, heat, light, or electric\nforce, etc. This generalized concept, which embraces all natural\nprocesses and permits the sum of all corresponding values to be\nexpressed by a law of conservation, we call _energy_. The law in\nquestion, therefore, is:\n\n_In all processes the sum of the existing energies remains unchanged._\n\nThe principle of the conservation of work in perfect machines proves to\nbe an ideal special instance of this general law. A perfect machine is\none in which work changes into nothing but _work_ of another kind, and\nnot into a different kind of energy. Then each side of the equation\nwhich expresses the general law of energy, namely,\n\nEnergy that has disappeared = energy that has arisen,\n\ncontains only the magnitude of the work, and expresses the law of the\nconservation of work. If, on the other hand, as in the case of the\npendulum, the work increasingly changes part by part into kinetic\nenergy, and _vice versa_, the equation during the first period is:\n\nWork that has disappeared = kinetic energy that has arisen,\n\nand during the second period in which the pendulum rises again,\n\nKinetic energy that has disappeared = work that has arisen.\n\nThus, while work can be called a substance only in a limited sense,\nsince its conservation is limited only to perfect machines, we may call\nenergy a substance unqualifiedly, since in every instance of which we\nknow the principle has been maintained _that a quantity of any energy\nnever disappears unless an equivalent quantity of another energy\narises_. Accordingly, this law of the conservation of energy must be\ntaken as a fundamental law of the physical sciences. But not only do all\nthe phenomena of physics, including chemistry, occur within the limits\nof the law of conservation, but until the contrary is proved the law of\nconservation must also be regarded as operative in all the later\nsciences, that is, in all the activities of organisms, so that all the\nphenomena of life must also take place within the limits of the law of\nconservation. This corresponds to the general fact, which I have\nemphasized a number of times, that all the laws of a former science find\napplication in all the following sciences, since the latter can only\ncontain concepts which by specialization, that is, by the addition of\nfurther characteristics, have sprung from the concepts of the former or\nmore general sciences.\n\n\n=45. Mass and Matter.= It has been noted above that kinetic energy\ndepends upon another magnitude beside velocity. A conception of its\nnature can be obtained when we try to put different bodies in motion.\nIn doing so the muscles of the arm perform certain quantities of work,\nand we feel whether the quantities are greater or smaller. In this way\nwe obtain a clear consciousness of the fact that different bodies\nrequire quite different quantities of work for the same velocity. The\nproperty which comes into play here is called _mass_, and mass is\nproportional to the work which the various bodies require to attain the\nsame velocity. Since the work and the velocity can be measured very\naccurately by appropriate means, mass also lends itself to a\ncorrespondingly accurate measurement.\n\nAll known ponderable bodies have mass. That means there is a regular\nconnection between the property which makes a body tend to the earth\nwith a certain definite force (called weight) and the property by virtue\nof which a body assumes certain velocities under the influences of\nmotive causes. We can readily conceive that it is possible for us to\nlearn only of such bodies as are heavy, that is, bodies which are _held_\nby the earth, since the others, if they exist at all, would naturally\nhave left the earth long ago. That all these bodies also have mass is to\nbe explained in a similar way. For a body of mass zero would at each\nimpulse assume infinitely great velocity, and could therefore never be\nthe object of our observation. Consequently, by reason of the physical\nconditions obtaining on the earth's surface, the bodies known to us must\ncombine both properties, mass and weight.\n\nThe name given to this concept of the combined presence of mass and\nweight in space is _matter_. Experience shows that there is a law of\n_conservation_ for these magnitudes also, according to which _whatever\nchanges we may produce in bodies possessing weight and mass, no change\nwill occur in the sum of their weight and mass_. According to the\nnomenclature previously introduced we must therefore call weight and\nmass substances, since they remain the same as to quantity, no matter\nwhat changes they may undergo. However, it is usual to apply the name\nsubstance to the concept of matter composed of mass and weight. In fact,\nscientists often go so far as to limit the name to this single instance\nof the various laws of conservation, and to take substance to mean\nexclusively the combination of mass and weight. This is connected with\nthe conception which we are about to discuss, that all natural phenomena\ncan ultimately be conceived as the motion of matter. Through the greater\npart of the nineteenth century this conception, called _scientific\nmaterialism_, was accepted almost without opposition. At present it is\nbeing more and more recognized that it was only an unproved assumption,\nwhich the development of science daily proves to be more untenable.\n\n\n=46. Energetic Mechanics.= In the light of our previous observations the\nbranch of science traditionally known as mechanics appears as the\nscience of work and of kinetic energy. Furthermore, statics is shown to\nbe the science of work, while dynamics, besides treating of kinetic\nenergy in itself, also treats of the phenomena of the change of work\ninto kinetic energy, and _vice versa_. We shall find the same relation\nagain later, only in more manifold forms. Every branch of physics proves\nto be the science of a special kind of energy, and to the knowledge of\neach kind of energy must be added the knowledge of the relations by\nwhich it changes to the other forms of energy and _vice versa_. It is\ntrue that in the traditional division of physics this system has not\nbeen strictly carried out, since an additional and very influential\nmotive for classification has been the regard paid to the various human\nsense organs.\n\nNevertheless this ground does not lie in the field of physics, but in\nthat of physiology, and must, therefore, be abandoned in the interest of\nstrict systematization.\n\nOf the physical sciences mechanics was the first to develop in the\ncourse of historical evolution. A number of factors contributed to this\nend--the wide distribution of mechanical phenomena, their significance\nto human life, and the comparative simplicity of the principles of\nmechanics, which made it possible to discover them at an early date.\nMost to be noted is, that of all departments of physics mechanics is the\nfirst which lent itself to comprehensive _mathematical_ treatment. It is\ntrue that the mathematical treatment of mechanics was possible only\nafter idealizing assumptions had been made--perfect machines and the\nlike--so that the results of this mathematical treatment not\ninfrequently had very little to do with reality. The mistake of losing\nsight of the physical problem and of making mechanics a chapter of\nmathematics has not always been avoided, and it is only in most recent\ntimes that the consciousness has again arisen that the classical\nmechanics, in arbitrarily limiting itself to extreme idealized cases,\nsometimes runs the risk of losing sight of the aim of science.\n\n\n=47. The Mechanistic Theories.= Because the evolution of mechanics\nantedates that of the other branches of physics, mechanics has largely\nserved as a model for the formal organization of the other physical\nsciences, just as geometry, which has been handed down to us from\nantiquity in the very elaborate form of Euclid, has largely been used as\na model for scientific work in general. Such methods of analogy prove to\nbe extremely useful at first because they serve as a guide to indicate\nwhen and where new sciences, in which all possibilities are open, can be\ngot hold of. But later on such analogies are apt to be harmful. For each\nnew science soon requires new methods, by reason of the peculiar\nmanifoldness which it has to deal with, and the finding and the\nintroduction of these new methods are easily delayed, and, as a matter\nof fact, often have been delayed, because scientists could not free\nthemselves soon enough from the old analogy.\n\nBy its being based upon memory the human mind is so constructed that it\ncannot assimilate something entirely new. The new must in some way be\nconnected with the known in order that it may be organically embodied in\nthe aggregate of concepts. Therefore, it is the first involuntary\nimpulse of our mind, in the presence of new experiences or thoughts, to\nlook about for such points at which a linking of the unknown to the\nknown seems possible. In the case of mechanics this necessity for\nfinding connecting links has acted in such a way that the attempt has\nbeen made, and is still being made, to conceive and represent all\nphysical phenomena as mechanical.\n\nThe impulse to this was first given by the extraordinary successes which\nmechanics has attained in the generalization and prediction of the\n_motions of the heavenly bodies_. The names of Copernicus, Kepler, and\nNewton mark the individual steps in the mechanization of astronomy. The\ncause of this lies in the fact that the heavenly bodies actually\napproximate very closely the ideal of the purely mechanical form with\nwhich classical mechanics deals. These successes encourage the attempt\nto apply these mental instruments that were productive of such rich\nresults to all other natural phenomena. An old theory, according to\nwhich all physical things are composed of the most minute solid\nparticles of matter called _atoms_, supported these tendencies and\ninvited the attempt to regard the little world of atoms as subject to\nthe same laws as had been found to apply so successfully to the great\nworld of the stars.\n\nThus we see how this mechanistic hypothesis, the assumption that all\nnatural phenomena can be reduced to mechanical phenomena, comes as if it\nwere a self-understood matter, and with its claim to be a profound\ninterpretation of nature it scarcely permits the question as to its\njustification to be raised at all. And the effects here have been the\nsame as I described above in cases in which inferences from analogy are\naccepted too extensively or too credulously. While it is true, no doubt,\nthat the mechanical hypothesis at first was fruitful of results in\nspecial research, because it facilitated the putting of the\nquestion--for example, we need think only of the atomic hypothesis in\nchemistry--later, the efforts to find further hypothetic help for the\ninadequacies of the hypothesis that gradually came to light, have not\ninfrequently led scientific research to pseudo-problems, that is, to\nquestions which are questions only in hypothesis, but to which no actual\nreality can be shown to correspond. Such problems, therefore, are by\ntheir very nature _insoluble_, and constitute an inexhaustible source of\ndifferences of scientific opinion.\n\nThe most flagrant of the injurious consequences of the mechanistic\nhypothesis appear in the scientific treatment of the mental phenomena.\nReady as scientists were to represent all other life phenomena, such as\ndigestion, assimilation, and even generation and propagation, as the\nconsequence of an extremely complicated play of certain atoms, their\ncourage never went so far as to apply this principle to mental life and\nto consider that by mechanics the last word had been said on the\nsubject.\n\nIt is because of this hesitancy to bring mental phenomena under the same\nmechanistic principle as all the other phenomena that the philosophical\nsystems had to search for some other means to connect the mental world\nwith the mechanical, and the efforts of the philosophers to bring about\nthis end have been most varied. Of the various doctrines that have come\ndown to us, that of the _pre-established harmony_ proposed by Leibnitz\nis in the ascendant in our day, and is now called the theory of the\n_psycho-physical parallelism_. According to this theory it is assumed\nthat the mental world exists alongside, and quite independent of, the\nmechanical, but that the things have been so prearranged that mental\nprocesses take place simultaneously with certain mechanical processes\n(according to some, with all mechanical processes) in such a way that,\nalthough the two series do not influence each other in the least, they\nalways correspond to each other precisely. How such a relation has come\nabout and how it is maintained remains unsaid, or is left to future\nexplanation.\n\nWe need only think of the content of this hypothesis with an unbiased\nmind to lose all relish for it at once. In fact, it has no other _raison\nd'\u00eatre_ than the presumption that the mental and the mechanical world\nare opposed to each other. As soon as we abandon the thesis that the\nnon-mental world is exclusively mechanical, we acquire the possibility\nagain of finding for the theory of mental phenomena a constant and\nregular connection with the theories of all other phenomena, especially\nwith the phenomena of life. Therefore it will be found most expedient in\nevery respect, instead of rendering scientific research one-sided and\nalmost blind to nonconforming facts by preconceived hypotheses, such as\nthe mechanistic hypothesis, to seek, as hitherto, from step to step, the\nnew elements of manifoldness which must be taken account of in the\nprogressive upbuilding of science and to limit ourselves faithfully to\nthem in the formation of general ideas.\n\n\n=48. Complementary Branches of Mechanics.= The field of pure or\nclassical mechanics is limited to the above two kinds of energy, work\nand kinetic energy, though these do not exhaust the manifoldness of the\nmechanical energies. Accordingly, other branches of mechanics dealing\nwith the corresponding phenomena are added to the classical mechanics\ndescribed above.\n\nIf by mechanical energies we understand all energies in which _changes\nof space are connected with changes of energy_, there are as many\ndifferent forms as there are spacial concepts that seem applicable.\n_Form_, _Volume_, and _Surface_ of bodies in space are especially\nrecognizable as the field of action for energy, which shows different\nproperties or manifoldnesses according to each of these relations.\n\nThe _energy of form_ is manifested in bodies (solid or rigid bodies)\nthat maintain a definite shape because every change of shape is\nconnected with work or with the expenditure of some other energy. If the\nchanges are small, the bodies are of such a nature that they return to\ntheir former condition of their own accord after the force exerted upon\nthem has ceased to act. This property is called _elasticity_. However,\nthe theory of elasticity, which has been extensively and rationally\ndeveloped, is regarded as belonging rather to mathematical physics in\ngeneral than to mechanics in particular. In greater changes of shape the\nenergy of form, or elastic energy, passes into other forms, and the body\ndoes not return to its former shape after the force has been removed.\n\nOther bodies have no energy of form (or only in an infinitesimally\nslight degree), so that they allow of changes of form without the\nexpenditure of work, but their volume can be changed only by work. These\nare divided into two classes. First, the _liquids_, which have a\ndefinite volume (corresponding to the definite shape of solids), the\nchanges of which in _every_ sense, both compression and expansion,\nrequire work. Secondly, the _gases_ with volume energy in only one sense\nof the word, in which only the compression of volume requires work,\nwhile in expansion a certain amount of work is thrown off. Such bodies\ncan exist only so long as the expenditure of their volume energy by\nspontaneous expansion is prevented by the presence of a counter energy,\nas, for example, the elasticity of the walls of a vessel. This tendency\nis called _pressure_.\n\nFinally, there are energy qualities at the surfaces between various\nkinds of bodies which come into play at the change of these surfaces.\nThey always lie in such a direction that the enlargement of the surfaces\nrequires work, and hence, by reason of the law of conservation of\nenergy, cannot proceed by itself. In cases where there has been an\ninverse kind of energy present, that is, one which diminishes with\nincreasing surface, it also has been active as a rule, thus bringing\nabout the disappearance of the existing boundaries.\n\nSince the seat of this kind of energy is in the surfaces (or\nsuperficies), it is called _surface-energy_. The phenomena depending\nupon it manifest themselves most clearly at the surface boundaries\nbetween _liquids_ and _gases_. They are called _capillary phenomena_.\nThis strange name, derived from the word _capilla_, hair, has its origin\nin the fact that because of surface-energy liquids rise in tubes which\nthey wet, and the narrower the tube the higher they rise. If the lumen\nof the tube is as fine as a _hair_, a considerable rise can be observed.\nThis is the entire connection between the name and the thing.\n\nThe mechanics of liquids is called _hydromechanics_, that of gases,\n_aeromechanics_, after the most familiar liquid, water, and the most\nfamiliar gas, air. The study of surface-energy under the name of the\ncapillary theory forms part of theoretical physics. While formerly this\nbranch, too, was regarded as a working part, or, rather, as a playing\npart, of mathematical problems, in more recent times extensive\nexperimental research has made its entry in this province also, and has\ndemonstrated the necessity of passing from the former abstractions or\nidealizations, which were carried altogether too far, to a better and\nprofounder regard for the actually existing complexities.\n\n\n=49. The Theory of Heat.= The various forms of energies the aggregate of\nwhich is comprehended in physics, have very different special\ncharacters. A systematic investigation has not yet been made of the\ncharacters of manifoldness by which, for example, work is distinguished\nfrom heat, electrical energy from kinetic energy, etc., nor of what are\nthe essential properties peculiar to each individual energy. We feel\ncertain that differences do exist, for otherwise the energies could not\nbe distinguished, and we feel certain that these differences are very\nimportant, for doubt seldom arises as to the kind of energy to which a\ncertain phenomenon is to be assigned. But just as we have no systematic\ntable of the elementary concepts, so we are still without a systematic\nnatural history of the forms of energy in which the peculiarities of\nevery species are characterized, and in which the entire material is so\narranged according to these characteristics that we can take a general\nsurvey of it.\n\nAs regards heat energy, its foremost and most striking characteristic is\nits physiological effect. In our skin there are organs for the\nperception of heat as well as of cold, that is, for temperatures above\nand below the temperature of the skin. However, the temperature that\nthese organs can bear without injury to themselves is of a very small\nrange, beyond which physical apparatuses of all kinds must be used, such\nas \"thermometers.\"\n\nHeat is the simplest kind of energy from the point of view of\nmanifoldness. Every heat quantity is marked by a temperature, just as a\nkinetic energy is marked by velocity. But while a velocity is determined\nin space so that velocities of equal magnitude have in addition a\nthreefold infinite manifoldness in reference to direction, a temperature\nis characterized completely and unambiguously by a simple number, the\ndegree of temperature. Two temperatures of equal degree can in no wise\nbe distinguished, since temperature possesses no other possible\nmanifoldness than degree.\n\nThe same property is found in heat energy itself. In heat energy we\nmeasure the quantity of energy itself and call it the _heat quantity_,\nwhile in some of the other kinds of energy, only the factors into which\nthey can be divided are measured, and no habitual conception of the\nenergy itself is developed. A heat quantity is likewise fully indicated\nby its measure number.\n\nThat heat is an energy, that is, that it is developed in equal\nquantities from other kinds of energy, and can change back again into\nthem, is a discovery which, despite its fundamental and general\ncharacter, was not made before the forties of the nineteenth century. As\noften happens in cases of important scientific advances, the same idea\ncame simultaneously to a number of investigators. The first to grasp and\nfully comprehend this idea was _Julius Robert Mayer_ of Heilbronn, who\npublished his results in 1842. Mayer not only showed that the imperfect\nmachines (p. 134), which limit the validity of the law of the\nconservation of work, owe this peculiarity to the fact that they\ntransform a part of the work into _heat_, and that when we take account\nof this part, the law of conservation holds perfectly good, but he also\ncalculated, with extraordinary acumen, the mechanical equivalent of heat\nfrom the then existing data of physics. That is to say, he determined\nhow many units of heat (in the measure then in use) correspond to a unit\nof work (in its specific measure) in the change from one to the other,\nand back. And this fundamental knowledge of the existence of a\nquantitatively unchangeable substance, arising from work, and capable of\nbeing transformed into it, Mayer did not limit in its application merely\nto heat. He was the first to construct a table, which he made as\ncomplete as possible, of all the forms of energy then known, and to\nassert and prove the possibility of their reciprocal change into each\nother.\n\nIn view of this relation of the quantitative equivalent of the various\nforms of energy when transformed into one another, an attempt is being\nmade at present to measure them all with the _same unit_. That is, some\neasily obtained quantity of energy is arbitrarily chosen as a unit and\nit is determined that in every other form of energy the unit shall be\nequal to the quantity obtained from that unit on its transformation into\nthe energy in question. For formal reasons the kinetic energy of a mass\nof two grams which moves with the velocity of one centimeter in a second\nhas been chosen as the unit. It is called _erg_, an abbreviation of\nenergy. The amount is very small, and for technical reasons 10^{10}\ntimes greater unit is used. To raise the temperature of a gram of water\none degree a quantity of energy equal to 41,830,000 ergs is required.\n\n\n=50. The Second Fundamental Principle.= Another fundamental discovery\nhas been made in connection with the heat form of energy, which, like\nthe law of conservation, relates to all forms of energy, but has found\nits first and most important application in heat. While the law of\nconservation answers the question, how much of the new form of energy is\ndeveloped if a given quantity of energy changes, but gives no clue as to\nwhen such a change occurs, this second law asserts the condition under\nwhich such changes arise, and is therefore called the _second\nfundamental principle_.\n\nThe discovery of this law antedates _Mayer's_ discovery of the law of\nconservation by about twenty years, and was made by a French military\nengineer, _Sadi Carnot_, who died soon afterward without having lived to\nsee the recognition his great work obtained. _Carnot_ asked himself the\nquestion, Upon what does the action of the steam engine, which had just\nthen come into use, depend? This led him first to the more general\nquestion of the action of heat engines in general. He found that no heat\nengine could work unless the heat dropped from a higher to a lower\ntemperature, just as no water wheel can work unless the water flows from\na higher to a lower level, and he determined the conditions which an\n_ideal heat engine_ must fulfil, that is, a machine in which the\ngreatest possible value in work is obtained from heat. However, an ideal\nmachine of this nature can be constructed in very different ways, and\nCarnot's discovery consists in the recognition of the fact _that the\nquantity of work obtained from the heat unit does not at all depend upon\nthe peculiar construction of the ideal machine, but is determined solely\nby the temperature between which the heat transition takes place_. This\nfollows from the following considerations:\n\nIn the first place an ideal engine must be _reversible_, that is, it\nmust be capable of working both ways, changing heat into work and work\nback into heat. Now, if we have two ideal engines between the same\ntemperatures, and if we assume that engine A produces more work from the\nsame quantity of heat than engine B, then let A move one way and let B\nmove the other way with the work obtained from A. Since B produces less\nwork from a given amount of heat, hence more heat from an equal amount\nof work, there will in the end be more heat at the higher temperature\nthan was originally there. But experience teaches _that there is no\nmeans in nature by which heat in the absence of concomitant change could\nbe caused to rise to a higher temperature_. Therefore an engine so\nconstructed as to produce this result is impossible, And B cannot be of\nsuch a nature as to produce less work from the same quantity of heat\nthan A.\n\nThe reverse is also impossible. For then we need merely couple the\nengines in the reverse way in order to obtain the same effect.\nTherefore, since B can do neither less nor more work than A, the two\nmust do the same amount of work--which was to be proved.\n\nIt is obvious that this process of proof is similar to that by which the\nlaw of conservation was established. Because the arbitrary creation of\nenergy from nothing is impossible there must be definite and immutable\nrelations of change between the forms of energy. Because energy at rest\ndoes not spontaneously pass into conditions in which it can do work,\nthe efficiencies of the machines must have definite and unchangeable\nvalues. If, for example, we could cause heat of its own accord to rise\nto a higher temperature, we could also construct a perpetual motion\nmachine which would always yield work at no expense. But this perpetual\nmotion would not be one that creates work out of nothing, but one that\nextracts it from energy at rest. A perpetual motion machine of this\nnature, too, is, according to our experience, impossible, and this\nimpossibility forms the content of the second fundamental principle.\n\nOn the face of it this apparently \"self-evident\" proposition does not\nreveal how fruitful of results it is when applied to the discovery of\nsimple but not obvious relations. It can only be said here that the\ndeductions from this principle form the chief content of the extensive\nscience of thermodynamics, which deals with the changes of heat into\nother forms of energy. We must only emphasize the fact that the\napplication of this law, as was already observed in stating it, is not\nconfined to the changes of heat alone. It is a law rather which finds\napplication in _all_ the forms of energy. For in every form of energy\nthere is a property which corresponds to temperature in heat, and upon\nthe equality or the inequality of which depends whether the energy in\nquestion is at rest or ready for transformations. This property is\ncalled the _intensity_ of the energy. In work, for instance, it is\n_force_, in volume-energy it is _pressure_. If once the intensity in a\nbody is equal, its energy is at rest, and it never again moves of its\nown accord.\n\nAnother form in which to present these relations is to make a\ndistinction between _free_ energy and energy _at rest_. If we have a\nheat quantity the temperature of which is higher than that of the\nsurrounding objects, it can be used to do work only until its\ntemperature has dropped to that of the surrounding objects. Although\nenergy in abundance is still present, there is no longer any energy\n_capable of change_, or _free_ energy. Since differences of temperature,\nlike other differences of intensity, have a constant tendency to\ndiminish, the amount of free energy on earth is constantly decreasing,\nand yet it is only this free energy that has value. For since all\nphenomena depend upon change of energy, and change of energy is possible\nonly through free energy, _free energy is the condition of all\nphenomena_.\n\n\n=51. Electricity and Magnetism.= While the knowledge of heat energy goes\nback to the most ancient periods of civilization, electrical and\nmagnetic energies are relatively young acquisitions. The highly\ndeveloped technical application of both with the rich harvests they have\nyielded belongs exclusively to most recent times.\n\nBoth these forms of energy, like those discussed above, are connected in\nthe main with ponderable \"matter,\" but in a much slighter and less\nregular measure. While it is not possible as yet to render any given\nbody free of heat (although lately the absolute zero point has been\nconsiderably approximated), freedom from electrical and magnetic energy\nis the normal condition of most bodies. This is connected with the\npeculiarity that electrical and magnetic properties are decidedly\nbi-symmetrical or _polar_. This property is not found in any other form\nof energy, and can serve as the special scientific characteristic of\nelectricity and magnetism. This peculiarity shows itself in the concepts\nof positive and negative magnetism, and positive and negative\nelectricity, and is due to the fact that two equal opposite quantities\nof electricity or magnetism, when added together, do not produce double\ntheir value, but nullify each other.[G]\n\n[G] For the sake of the layman it must be observed that those\n\"quantities\" are not energy magnitudes but factors of the electrical and\nmagnetic energies. Energy itself in its various forms is an _exclusively\npositive magnitude_, and the result of the additions of their various\namounts is always the sum, never the difference, of their numerical\nvalues. By the negative sign is understood the energy _expended_ in\ncontradistinction to the energy _received_. It is therefore nothing more\nthan the indication of a mathematical operation.\n\nThe fact that electrical and magnetic energies generally exist only in a\ntransitory state (with the notable exception of the magnetic condition\nof the earth) is probably the cause of our not having developed a sense\norgan for them, especially since their phenomena as they occur in nature\nhave only occasionally and in very rare instances (thunderstorms) an\ninfluence upon us. On the other hand, the modern development of\nelectrotechnics is based upon that property of electrical energy by\nvirtue of which large quantities of it can be conducted along a thin\nwire over great distances without any considerable loss, and at the\npoint desired can be easily changed into any other forms of energy. But\nsince the collection and conservation of large quantities of electrical\nenergy is hardly possible technically, the electrical apparatus must be\nso constructed that the quantities each time required should be produced\nat the moment they are used. The chief source of electricity is the\nchemical energy of coal, which is first transformed into heat, then into\nmechanical energy, and finally into electrical energy. This extremely\nroundabout process is necessary because a method technically practicable\nof transforming the chemical energy of coal directly into electrical\nenergy has not yet been invented. On the other hand, mechanical energy\ncan be easily and completely changed into electrical energy. Upon this\nis based the exploitation of much \"water power,\" the energy of which\ncould not be utilized but for the great capacity for change of the\nelectrical form.\n\n\n=52. Light.= The case of light in our day seems to be similar to that of\nsound, which, although it has its special sense organ in man, is yet no\nparticular form of energy, but has been found to be a combination of\nmechanical energies in an oscillatory or mutually changing state. It\nseems highly probable that light, too, is not a special form of energy,\nbut a peculiar oscillatory combination of electrical and magnetic\nenergies. It is true that the circle of proof is not yet quite closed,\nbut the gaps have become so small that the above conclusion may at any\nrate be accepted as highly probable.\n\nHowever that may be, light is an energy which, according to the known\nlaws, travels through space with tremendous rapidity. We will call it\n_radiant energy_, since the part optically visible, to which alone the\nname light in its original sense belongs, represents an extremely small\nportion of a vast field, the properties of which change quite\ncontinuously from one end to the other.\n\nRadiant energy is characterized as an oscillatory or wave-like process.\nSo long as this fact was unknown (up to the beginning of the nineteenth\ncentury) it was thought that light consisted of minute spherical\nparticles, which shot through space in a straight line with the\ntremendous velocity mentioned above. Later, in order to \"explain\" its\nwave nature, which in the meantime has come to be recognized, it was\nassumed to be due to the elastic vibrations of an all-pervading thing\ncalled _ether_, of which we know nothing else. This elastic undulatory\ntheory has been abandoned in our time in favor of an _electromagnetic_\ntheory supported by quite considerable experiential grounds. Whether it\nwill be spared the fate that has overtaken the older theories (or rather\nhypotheses) of light cannot as yet be predicted with any degree of\ncertainty.\n\nRadiant energy is of very marked importance in human relations. As light\nit serves, with the aid of the corresponding receiving organs, the eyes,\nas a more manifold means of intercommunication between our bodies and\nthe outer world than any other form of energy. The energy quantities\npenetrating to us from the extreme limits of the world space mark the\noutermost limits of which we have knowledge in any way whatsoever, and\nfinally the energy quantities radiating to us from the sun constitute\nthe supply of free energy at the expense of which all organic life on\nearth is maintained. Even the chemical energy stored up in coal\nrepresents nothing else than accumulations of former sun radiation,\nwhich had been transformed by the plants into the permanent form of\nchemical energy.\n\nVery recently other newly discovered forms of radiant energy have been\nadded to light. They are produced in manifold circumstances, and some\nbodies emit them constantly. The scientific elaboration of these\nextremely manifold and unusual phenomena has not yet been carried so far\nthat they can be reduced to a doubt-free system. But so much, it seems,\nis already apparent, that they are presumably not purely new forms of\nenergy, but rather very composite phenomena which may yield one or more\nnew energies as component parts. But despite the peculiarity of these\nnew rays, nothing certain has as yet been proved against the law of\nconservation itself.\n\n\n=53. Chemical Energy.= Since chemical energy is only one of several\nforms of energy, there seems to be no justification for allotting it to\na special science, since all the other forms of energy must be\nincorporated in physics.\n\nBut the actual existence of chemistry as a special science which has\nalready many subdivisions is justified in the first place by the\nexternal fact that in practical life and in industry chemistry occupies\na very wide field comparable, if not superior, to that of the whole of\nphysics. In the next place, from the psychological point of view, it is\nfound that the chemist's methods of reasoning and working are so\ndifferent from those of the physicist that a division seems to be in\norder for that reason also. Finally, there is in the nature of chemical\nenergy itself an important distinction which marks it off from the other\nforms.\n\nWhile, for example, there is only one form of heat or of kinetic energy,\nand in electricity there are only the two forms of polar opposites,\nchemistry, even after the greatest theoretical reduction, possesses at\nleast about eighty forms. That is, it possesses as many forms as there\nare _chemical elements_. The experiential law, that the elements cannot\nbe changed into one another,[H] also limits the corresponding changes\nof the chemical energies into one another, and thus characterizes the\nindependence of these various forms. From this results a\ndisproportionately greater manifoldness of relations, which find their\nexpression in the many thousands of the individualized chemical\nsubstances or combinations.\n\n[H] Lately changes of elements into one another have been observed in\nindividual instances, but in such peculiar circumstances that for the\npresent we need not consider these discoveries, which have only just\nbegun.\n\nThis great manifoldness and the slight regularity hitherto found in\nconnection with the properties and reciprocal relations of the numerous\nchemical elements renders modern chemistry more a descriptive than a\nrational science. It was no more than twenty years ago that an earnest\nand successful attempt was begun to apply the stricter methods of\nphysics to the investigation of chemical phenomena. These labors, so far\nas they have gone, have yielded a great many far-reaching and\ncomprehensive principles.\n\nThe significance of chemistry in human life is twofold. In the first\nplace the energy of the human body, just as that of all other living\norganisms, depends chiefly upon the action of chemical energies in the\nmost manifold forms. Of all the physical sciences, therefore, chemistry\nis the most important for biology, particularly for physiology. In the\nsecond place, as I have emphasized a number of times, it possesses the\npeculiar property which enables it to be _preserved_ for a long time\nwithout passing into other forms and being dissipated. Furthermore,\nenergy in this form permits of the most powerful _concentration_. More\nof chemical energy can be stored in a given space than of any other form\nof energy. Both these properties, then, may be considered as the reason\nwhy organic beings are constituted chiefly by means of chemical energy.\nAt any rate, it is due to these two peculiarities that chemical energy\nserves as the primary source for almost all the energy used in industry.\n\nFurther, the manifoldness of chemical energy is the cause of the\npeculiar manner in which it is transformed into other forms. In the\nother forms of energy the transformation can be effected by the body\nitself. Nothing else is required. If a stone is thrown and it hits\nagainst a wall, it loses its kinetic energy, the greater part of which\nchanges into heat. But in order to liberate the _chemical_ energy of,\nsay, coal, the coal _alone_ is not sufficient; _another_ chemical\nsubstance is required, the oxygen of the air. The interaction of the two\nsubstances produces a new substance, and it is only during this process\nthat a corresponding part of the chemical energy is liberated. There are\na few chemical processes also (allotropic and isomeric changes) in which\na single substance without the co-agency of another substance can give\noff energy. But the quantity of energy thus obtained is infinitely\nsmall as compared to that liberated by the interaction of two or more\nsubstances. Because of the necessity of two or more substances to\nco-operate in giving off chemical energy, the opportunity for the\ntransformation of chemical energy is less than for the transformation of\nthe other forms of energy, and this is the main reason why it can be\nconserved so long and so easily. All that is necessary is to prevent\ncontact with another substance. This is a problem, it is true, which\nfrom the point of view of strict theoretical rigor it is almost\nimpossible to solve. In practice, however, it can be easily solved for\nperiods of time long enough at least to require special means to enable\nus to recognize that it is only a temporary and not a fundamental\nsolution. Scientifically expressed, the cause of this is that the\n_diffusion_ of the various substances in one another can theoretically\nnever be completely eliminated, while on the other hand the velocity of\nthe diffusion over distances measured only by decimeters is extremely\nlow.\n\n\n\n\nPART IV\n\nTHE BIOLOGIC SCIENCES\n\n\n=54. Life.= Among the bodies in our environment that are ponderable and\nhave mass the animate beings are so strikingly distinguished from the\ninanimate that in most cases we have not the slightest doubt whether a\nbody belongs to the one kind or to the other, even if in some cases we\nhappen not to be familiar with its peculiar form. In the first place,\ntherefore, we must answer the question in a general way and tell what\nthe distinguishing peculiarities are that mark them off one from the\nother.\n\nThe first peculiarity is this, that living organisms are not _stable_\nbut _stationary_ forms. This distinction is based upon the fact that a\nstable form is at rest or unchangeable in all its parts, while a\nstationary body, though it seems unchangeable in its form, internally\nundergoes a constant change of its parts. Thus, a brass faucet is a\nstable body, since it not only preserves its form and function\npermanently, but consists at all times of the same material and shows\nthe same peculiarities, such as stains, defects in form, etc. It cannot\nbe said, it is true, that it will remain completely unchanged for all\ntime. Its metal suffers a gradual chemical and mechanical deterioration.\nBut this is not essential to the existence of the faucet, since the\ndeterioration varies greatly with circumstances, and if conditions are\nideal it can be reduced to zero.\n\nOn the other hand, the jet of water flowing from the faucet is a\nstationary body. In favorable circumstances it can assume a constant\nform, so that at a hasty glance it might be taken for a stable glass\nrod. On closer examination it will be found that the parts of water of\nwhich it is formed are not the same at any given instant as the instant\nbefore, each part that has flowed away being replaced by another just as\nlarge following it.\n\nFrom this difference in the nature of the two bodies results a\ndifference in their behavior. If I make a mark on the faucet with a\nfile, the mark remains permanent. But even if I sever the entire water\njet with a knife, the cut is healed the next moment, because by reason\nof the continuous flow of the water, the severed place is instantly\neliminated from the body. Owing to this nature peculiar to stationary\nbodies, they have the capacity of _being healed_ or of _regeneration_.\n\nFor a body to continue permanently in a stationary condition the\nmaterial of which it is composed must be permanently _supplied_. If we\nturn off the faucet, the water jet immediately disappears or \"dies.\"\nEvidently, therefore, a stationary body can subsist by its own means\nonly if it has the property or capacity to provide itself continually\nwith the necessary material. This material consists in the main of\nponderable or chemical substances of definite physical and chemical\nproperties, and thus the _change of substance_, _metabolism_, appears as\na necessary property of the stationary body. In order, however, that\nmetabolism should take place we must have free _energy_, or energy\nhaving the capacity to work, since it is only free energy that can cause\nsubstances to change, just as every phenomenon in the world implies the\nequalization of free energy. For a stationary body to exist\nindependently, therefore, it must have the property of being able\nspontaneously to possess itself of the necessary substances and of free\nenergy. But since, as we have already said, the energy of organisms is\nstored up and used in the main in the form of chemical energy, the two\ntasks which a stationary body has to perform, that of meeting the need\nfor substances and for energy, are as a rule externally combined. In\norganisms these two necessities combined are called _nutrition_, and\nthus we recognize in the capacity for _self-acquisition of nutrition_\nanother essential property of organisms.\n\nA third essential property of organisms is the capacity for\n_reproduction_, for the bringing forth of similar beings. It is never\nimpossible that the balance between the receipts and expenditures of a\nstationary body should, in consequence of some external causes, be\ndisturbed, even when under normal conditions it possesses the property\nof self-nutrition. If the disturbance remains below a certain point,\nthen, as we have already stated, regeneration sets in. But the\ndisturbance may rise above that point, in which case the body ceases to\nexist, or dies. Then a similar body will not arise unless the manifold\nnecessities that have led to the origin of the first will combine again\nto produce the second. That such a thing is possible, that, in fact, it\noften happens, is shown, for example, by the waves of the ocean, which\nhave a stationary character since, while they are composed of constantly\nchanging masses of water, their form remains unchanged. The waves are\ndestroyed in the breakers, but arise again and again through the action\nof the wind upon the surface of the water. But the more complex such\nbodies are the less easily they are formed, while once they have been\nformed and have found the conditions of their existence, their\npreservation is much easier.\n\nBeings, therefore, which have the capacity to form similar bodies out of\nthemselves regularly and at the right time can preserve their species\nmuch more easily than those in which this property is absent. Death has\nto a great extent lost its power over beings capable of reproduction. By\nway of illustration let us take another stationary thing, a flame. A\nflame is not an organism because it is not self-sustaining. Yet it\nmultiplies itself. And while a single little flame soon dies out, the\nsea of flame of a burning forest, which started from a single small\nflame, is well-nigh inextinguishable, and it cannot be fought in any\nother way than by letting it die its natural death and burn to the end.\n\nThus, while the fulfilment of the first two conditions, the stationary\nchange and the self-supply of food, could produce bodies, which would be\nable to exist for a longer or shorter period, but which at some time\nwould have to give way to other bodies of different form and nature, the\ncapacity for reproduction creates the condition that forms of the _same\nspecies_ continue to exist even after the existence of the individual\nhas ceased.\n\nThese three properties constitute the essential characteristics of\nanimate things or organisms.\n\nThat the organisms are all constructed upon the basis of chemical energy\nis a fact of experience which may be understood to imply that the other\nforms of energy are not capable of producing the above-mentioned\nconditions. This is due to the properties of chemical energy to which I\nhave already called attention: its great concentration and, at the same\ntime, its capacity for prolonged preservation. That chemical energy is\nthe only form of energy suitable to life is obvious from the fact that\nin airship navigation, for example, the kinetic energy required for\nsteering can be supplied only in the form of gasoline or hydrogen, that\nis, in the form of chemical energy, because any of the other forms would\nbe much too heavy. The flight of a bee or the swimming of a dolphin\ncannot be conceived of except as brought about through chemical energy.\n\nThat this chemical energy is essentially that of _carbon_ has also been\nestablished by experience, although it is not quite universal, for the\nsulphur bacteria found their household upon the energy of sulphur. The\ncause of the preference of carbon is again to be sought in its special\nfitness for the purpose, due, on the one hand, to its wide distribution,\nand, on the other hand, to the exceeding manifoldness of its\ncombinations.\n\nFinally, the construction of the organisms from a peculiar combination\nof solid and liquid substances can be proved to be equally due to\ntechnical relations.\n\nThese three last-named peculiarities are therefore to be regarded as the\nspecial characteristics of the organisms with which we are acquainted on\nthe surface of the earth in the conditions there prevailing. We need not\nregard them conceptually as unchangeable or irreplaceable. But the first\nthree characteristics, namely, the stationary nature, self-supply of\nnutrition, and reproduction, we may regard as the _essential\ncharacteristics of organisms_. They constitute the frame within which\neverything must be found which we should recognize as living in the\nwidest sense.\n\n\n=55. The Storehouse of Free Energy.= If we ask whence the organisms\nobtain the free energy which they require for the maintenance of their\nstationary existence, the answer is that _solar radiation_ alone\nfurnishes this supply. Without this permanent supply the free energies\nupon the earth, so far as our knowledge goes, would long ago have\nreached a state of equilibrium, and the earth's bodies would be stable,\nthat is, dead and not stationary and living.\n\nIt is comprehensible, therefore, that machines should have evolved in\nthe organism for _transforming the radiant energy of the sun into a\npermanent form_, and, as we have already learned, chemical energy is\npermanent, while radiant energy is an extremely transitory form of\nenergy, that is, it changes very readily. The very fact that, owing to\nthe change from day to night, the supply of radiant energy periodically\nceases, makes the storing-up of energy for the night necessary to the\nexistence of a form dependent upon it. Thus, we recognize in the\n_photochemical_ processes, that is, in the transformation of radiant\nenergy into chemical energy, the foundation of life on earth.\n\nThis work is done by the plants, which thus provide a store of free\nenergy not only for their own needs but also for all the other organisms\nwhich possess themselves directly or indirectly of the plant-chemical\nsupplies in order to utilize them for their individual purposes. In this\nmanner nourishment in the widest sense is secured for all organisms,\nbeing based upon the regular supply of free energy derived from the\nsun. This also explains the great chemical similarity of all organisms,\nwhich could not subsist if they were not so constructed as to be able to\nutilize the chemical energy in the form in which it is provided by the\nplants.\n\nOf the great stream of free energy poured out from the sun into cosmic\nspace the earth receives an extremely small portion (corresponding to\nthe bit of space it occupies in the heavenly sphere as seen from the\nsun), and the plants collect and store up only a very small fraction of\nthis portion received by the earth. Measurements have shown that in most\nfavorable circumstances a plant leaf changes only about 1/50 of the\nradiant energy it receives into chemical energy. If we consider that\nonly a small part of the surface of the earth is covered with plants and\nthat during the winter no energy from the sun is stored up at all, we\nperceive what infinite possibilities for development there still are in\narresting and storing up free energy. The part stored up by the plants\nflows from these into the countless streams, brooks, and veins of the\nother organisms, to end finally as used-up energy, or energy at rest.\nThis energy is at rest, it is true, only in relation to the earth's\nsurface. We do not know whether the radiation from the earth, which at\npresent amounts to about as much as the radiation from the sun to the\nearth, is in its turn somewhere utilized.\n\nWhile the free energy is poured out in such a stream in one direction,\nthe ponderable substances of which the organisms are made up _circulate_\nthrough plants and animals and back again. This is especially true of\n_carbon_, which is freed from its combination with oxygen, that is, from\ncarbonic acid, by the sun energy transformed in the plants. While carbon\nserves to build up the plant body and represents its supply of chemical\nenergy, the oxygen is returned to the air. These two substances are\nagain chemically combined in the various organisms and the quantities of\nenergy which were necessary for their decomposition are again available\nfor the manifold functions of life. The product of the chemical\ncombination, carbonic acid, returns to the air and is ready for renewed\ndecomposition in the plants.\n\nThus, the entire mechanism of life can be compared to a water-wheel. The\nfree energy corresponds to the water, which must flow in one direction\nthrough the wheel in order to provide it with the necessary amount of\nwork. The chemical elements of the organisms correspond to the wheel,\nwhich constantly turns in a circle as it transfers the energy of the\nfalling water to the individual parts of the machine.\n\n\n=56. The Soul.= Our observations so far have shown the organisms to be\nextremely specialized individual instances of physico-chemical machines.\nNow we have to take into consideration a property which seems markedly\nto distinguish them from the lifeless machines, and which we have\nalready encountered in the very beginning of our treatise.\n\nIt is the property which we there called _memory_, and which we will\ndefine in a very general way as the quality by virtue of which the\nrepetition in organisms of a process which has taken place a number of\ntimes is preferred to new processes, because it originates more easily\nand proceeds more smoothly. It is readily apparent that by this property\nthe organisms are enabled to travel on the sea of physical possibilities\nas if equipped with a keel, by which the voyage is made stable and the\nkeeping of the course is assured.\n\nIf we ask whether this is exclusively a quality of organisms the\nquestion cannot be answered affirmatively. Inanimate bodies also have\nsomething like the quality of adaptation. An accurate clock attains its\nvaluable qualities only after it has been going for some time, and the\nbest violin is \"raw\" until it has been \"broken in.\" An accumulator must\nbe \"formed\" before it can do its normal amount of work. All these\nprocesses are due to the fact that the repetition of the same process\nimproves the action, that is, it facilitates or increases it.\n\nAdaptation or memory, then, is not limited to organisms. In inanimate\nthings, however, this property is comparatively rare. Memory, therefore,\nis to be regarded as another property of organisms representing an\nextreme specialization of the inorganic possibilities. This is an\nimportant point of view for what follows.\n\nIn the first place, this property of adaptation facilitates and assures\nnourishment. If we take the fundamental idea developed by Darwin, that\nthat predominates in the world which by virtue of its properties endures\nthe longest time, then it is evident that a body which teleologically\npreserves and elaborates its nourishment will live longer than a similar\nbody without this property. Moreover, by the general process of\nadaptation, these \"teleological\" properties come to be more greatly\ndeveloped and more readily exercised in the body that lives longer, so\nthat its long life gives it another advantage over its rival. Thus we\ncan understand how this property of adaptation, which at first is to be\nconceived of as a purely physico-chemical quality is found developed in\nall organisms.\n\nIn its most primitive forms the quality of adaptation gives rise to the\n_phenomena of reaction_, or to _reflex_ actions, that is, to a series of\nprocesses in the organism in response to the stimulus of an outward\nenergy. This response is made in furtherance of the life of the\norganism. Reactions that serve a certain end, that is, teleological\nreactions, can naturally be developed to such stimuli alone to which the\norganism is frequently and regularly subjected. This is why adaptation\nto unusual phenomena is generally lacking, and in relation to them the\norganisms are often extremely unfit. The typical example of this is the\nmoth, which flies into the light and is burned.\n\nAs the reactions become more fixed they develop into longer and more\ncomplicated series, which then appear to us as _instinctive actions_.\nBut here, too, we find the characteristic unsuitability when unwonted\ncircumstances arise, even if the teleologic reactions to stimuli become\nmore manifold.\n\nFinally, there are the _conscious acts_ which appear to us to be the\nhighest degree of the series. It is with the teleologic regulation of\nthese conscious acts, including the very highest activities of mankind,\nthat this book deals. They are distinguished from instinctive action by\nthe fact that they no longer proceed in a single and definite series,\nbut are combined at need in the most manifold ways. But the fundamental\nfact, namely, that actions are based upon the repetition of coinciding\nexperiences, at once appears here also, since the basis of the entire\nconscious life of the soul, the formation of _concepts_, is made\npossible only through _repetition_. Thus, we are justified in regarding\nthe various degrees of mental activity from the simplest reflex\nmanifestation to the highest mental act as a connected series of\nincreasingly manifold and purposive actions proceeding from the same\nphysico-chemical and physiological foundation.\n\n\n=57. Feeling, Thinking, Acting.= For good reasons it is generally\nassumed that the organisms have not always been what they are now, but\nhave \"developed\" from previous simpler forms. It is undecided whether\noriginally there were one or several forms from which the present forms\nsprang, nor is it known how life first made its appearance on earth. So\nlong as the various assumptions with regard to this question have not\nled to decisive, actually demonstrable differences in the results, a\ndiscussion of it is fruitless, and therefore unscientific. The usual\nword evolution is non-purposive in so far as it signifies the appearance\nof something already existing. Another conception is better according to\nwhich the influence of _changed_ conditions of existence has yielded the\nmost important factor of change.\n\nThe change that the organisms undergo is always in a definite direction.\nMore and more complex and manifold forms are evolved, and the evolution\nof these forms is characterized by an ever greater specialization of the\nfunctions of life, so that every specially developed organ comes to\nperform but one function. It is true that by this means the organism is\nbetter fitted to perform those functions, but at the same time it grows\nmore susceptible to injury, since its existence depends upon the proper\nsimultaneous activity of many different organs. Such an evolution,\ntherefore, can occur only when the general conditions of life have grown\nsteadier, so that the danger of disturbance becomes less. We are\naccustomed to regard changes in this direction as higher developments,\nand the progressive simplifications of the organization (as for example\nin parasites) as backward steps.\n\nSince our opinion as to what constitutes a higher and a lower organism\nis doubtless arbitrary, let us ask whether it is not possible to find an\n_objective_ standard by which to measure the relative perfection of the\ndifferent organisms. The question must be answered in the affirmative\nwhen we take into consideration the following. Since the quantity of\navailable free energy upon the earth is limited, the organism which\ntransforms the energy at its disposal more completely and with the least\nloss into the forms of energy necessary for the function of life, must\nbe regarded as the more perfect organism. In fact, we observe that with\nincreasing complexity of the organisms there is for the most part also\nan increasing improvement in that direction, and we can therefore speak\nof some beings as more perfect than others. This view-point is\nespecially significant in the evaluation of _human_ progress, appearing,\nas it does, as the general standard of all civilization.\n\nThe perfection of the organism shows itself in relation to the outer\nworld in the development of the _sense organs_. While a single-celled\nanimal reacts almost exclusively to chemical, sometimes also to optical,\nstimuli, and receives these with the entire surface of its body, special\nparts of the body develop more and more toward perfection. These are the\nparts that respond with special ease to the appropriate stimuli, that\nis, react to them with an increasingly smaller expenditure of energy.\nThen the points at which the stimuli are received are separated from\nthose in which the reaction occurs, and the two are connected by\n_conducting paths_, the nerves, in which an energy process takes place.\nOur present knowledge of this process still leaves much to be desired.\nIt is a process which moves with fairly great but by no means\nextraordinary rapidity (about ten to thirty meters per second) along the\nconducting paths. At the one end of this path it is caused by actions of\nvarious kinds, chiefly that of the specific energy, for which the sense\norgan is developed. At the other end it discharges specific effects.\nThere is no doubt that here we have in each instance a case of energy\ntransformation connected with a _discharge_, that is, with the action of\nother energies which lie at the ends ready for change. Hence there is no\nequivalence between the different kinds of energy, the discharging and\nthe discharged, mostly not even a proportional relation, although both\nincrease and decrease simultaneously.\n\nWhat the form of the energy is that is propagated in the nerves is\nunknown. It can be either a special form which arises only under the\nconditions here present (just as, for example, a galvanic stream\ndevelops only under definite chemical and spacial conditions), or a\nspecial combination of known energies, as in sound and probably in\nlight. Some day, it is likely, we shall have a more accurate knowledge\nof the nerve process which will solve the question.\n\nWhen such a process is caused by some energy impulse from without, it\nmay produce various results. In the simplest case it discharges the\ncorresponding reaction, just as the leaves of the sensitive plant close\nwhen they are touched. Or it may give rise to a series of processes\nfollowing one another like the instinctive actions. Or, finally, it may\nbring about a series of inner processes which lead to an extreme\ndifferentiation of slight differences of this stimulus and to a\ncorresponding graded reaction with the anticipation of success. We call\nthis conscious thinking, willing, and acting.\n\nThrough the age-long effect of the blunder committed by Plato in making\na fundamental distinction between mental life and physical life, we\nexperience the utmost difficulty in habituating ourselves to the thought\nof the regular connection between the simplest physiological and the\nhighest intellectual acts. Moreover, this contrast has been accentuated\nby the mechanical hypothesis. If we abandon the mechanical hypothesis\nand adhere to the summarization of experience free from all hypotheses,\nas represented in the science of energy, this contrast disappears. For\neven if we concede the impossibility of conceiving thought as\n_mechanical_, there is no difficulty in conceiving of it as _energetic_,\nespecially since we know that mental work is connected with expenditure\nof energy and exhaustion just as physical work is. However, the\nelucidation of this subject lies almost entirely in the future since the\nidea just developed has but only begun to influence scientific work in\nthis field. But judging from the results that have already been obtained\nwe may hope for a speedy development.\n\n\n=58. Society.= The external circumstance that as an organism multiplies\nthe new being must come to life in the proximity of the older one, is in\nitself cause for the formation of closed groups confined to certain\nlocalities by animal organisms of the same species. But they become\nscattered if the advantage of their living together is not such as to\noutweigh the disadvantage of having a narrow field of competition for\nthe means of sustenance. Thus we see different plants and animals\nbehaving differently in this respect. While some species live in as\ngreat isolation as possible, others form communities, even if there is\nno mechanical tie to hold them together by a common integument.\n\nSince the second case is true of man in a highly marked degree, his\n_social_ characteristics and needs form a large and important part of\nhis life. And since, further, the socialization of man makes continuous\nheadway with increasing civilization--we need but think of the\ndevelopment of the former little groups and tribes into states and the\npresent very active internationalization of the most important affairs\nof mankind, especially of the sciences--the social problems also\noccupy an ever larger place in the organization of human life.\n\nWhat distinguishes man most essentially from the other animals, even the\nmost advanced, is his capacity for perfection, which in the lower animal\ncan be paralleled at best by its capacity for _self-preservation_. While\nthe organization of the animals within the short period of which we have\nany historical knowledge has to all appearances remained essentially\nunchanged, the world of mankind has changed in quite a remarkable way.\nThis change consists in an increasing subjection of the external world\nto human purposes, and rests upon the increasing socialization of his\ncapacities.\n\nMemory and heredity (the latter being but an extension of memory to the\noffspring, which is to be conceived of as a part of the older organism)\nsecures in the first place only the preservation of the stock and the\nrenewed development of the new individual in the average type. If a\nspecially favored individual succeeds in accomplishing greater\nachievements, he may in favorable circumstances transmit this capacity\nfor higher attainments to his offspring. But such individuals gain an\nadvantage in the struggle for existence only if the other sides of their\nactivity do not suffer curtailment as a result. With the limited amount\nof energy at the individual's disposal every extraordinary\naccomplishment involves a corresponding _one-sidedness_, and as soon as\na certain measure is slightly overstepped, it will cause a reduction of\nthe other functions which will render the individual less fit in the\nstruggle for existence. But this is true only so long as an individual\nmust live _by himself_. As soon as he forms part of a social\norganization which benefits by his particular activity, the organization\ncompensates for the personal disadvantages by its collective activity,\nand a social community not only finds room for such special\ndevelopments, but it even encourages and promotes them.\n\nWe have already seen that such manifestations occur within the organism\nitself. Higher functions, depending upon the higher susceptibility of\nthe sense organs, can only be attained at the expense of the general\nfunctions by the organ in question. We observe this fact in all socially\norganized beings, like bees and ants, which display a high degree of\nspecialization in the functions of the individual subordinate groups;\nthe specialization often being carried so far that the individual groups\ncan no longer subsist by themselves alone. It is only the organization\nas a whole that is capable of permanent existence.\n\nWhile the evolution of such superior functions involves a corresponding\ndifferentiation, and therefore a _division_ and _separation_ of the\nsuperior functions within the social structure, the necessity for\n_communication_ and for _mutual support_ results in an _approximation_\nof the individuals and the groups. In every society, therefore, the\ncentrifugal and the centripetal forces work simultaneously in\nco-operation and in opposition to one another. While the extreme\nspecialization on the one hand seems to make for the best individual\nfunctioning, on the other hand it renders the entire collective\nstructure much more dependent, and therefore much more subject to\ninjury, as is shown by the example of the queen bee, whose departure\nthreatens the existence of the entire hive. Thus a medium degree of\ndifferentiation will as a general rule produce the most permanent social\nstructure.\n\n\n=59. Language and Intercourse.= The essential value of the social\norganization resides in the fact that the work of the individual, in so\nfar as it is adapted to it, accrues to the benefit of the collective\nwhole. For this it is absolutely essential that the members of the\ncollectivity should be able to _have intercourse_ with one another in\norder that every part of the general activity may be communicated to the\nothers. This intercourse is obtained through language in the most\ngeneral sense.\n\nWe have already learned that the essence of language consists in the\nco-ordination of concept to sign. The social application of language\ndemands that the signs co-ordinated to the concepts in use should be the\nsame for all the members of the social organization. Only in this way\ncan the members make themselves mutually understood. But intelligible\nmeans of communication and division of labor impart to the social\nknowledge that is set down in writing a kind of independent existence.\nMany centuries ago the possibility ceased for one person to store in his\nmemory the entire stock of human knowledge. Nowadays we have men who are\nversed only in single parts of separate sciences, and the aggregate\nknowledge appears at first to be a unity existing only in thought. But\nbecause this knowledge is set down in signs which endure far beyond the\nlife of the individual and at the appropriate moment can unfold its\nentire power even after a long period of inactivity, it has acquired an\nexistence of a social character independent of the individual. For\nalthough it survives the individual, it cannot survive the death of\nhuman society.\n\nAs the socialization of all mankind advances to ever greater unities,\nthe linguistic limitations sprung from former stages of evolution prove\nto be a hindrance. The mother tongue, of course, forms the first and\nmost important entry for the individual to the common store of\nknowledge. But in view of the linguistic limitation of which I have just\nspoken the efforts in our day are carried on with renewed zeal to create\na _universal auxiliary language_ (p. 100) by means of which intercourse\nshould be made possible beyond the language boundaries. There have\nalready been gratifying results.[I]\n\n[I] At the present time \"Ido\" is the best. It is a highly practicable\nartificial language, and its advocates have succeeded in organizing it\nto insure its normal development. An older and still rather widespread\nform called \"Esperanto\" has failed to organize itself so as to insure\nits development and it must inevitably die out.\n\n\n=60. Civilization.= Everything which serves the social progress of\nmankind is appropriately called civilization or culture, and the\nobjective characteristic of progress consists in improved methods for\nseizing and utilizing the raw energies of nature for human purposes.\nThus it was a cultural act when a primitive man discovered that he could\nextend the radius of his muscle energy by taking a pole in his hand, and\nit was another cultural act when a primitive man discovered that by\nthrowing a stone he could send his muscle energy a distance of many\nmeters to the desired point. The effect of the knife, the spear, the\narrow, and of all the other primitive implements can be called in each\ncase a purposive transformation of energy. And at the other end of the\nscale of civilization the most abstract scientific discovery, by reason\nof its generalization and simplification, signifies a corresponding\neconomy of energy for all the coming generations that may have anything\nto do with the matter. Thus, in fact, the concept of progress as here\ndefined embraces the entire sweep of human endeavor for perfection, or\nthe entire field of culture, and at the same time it shows the great\nscientific value of the concept of energy.\n\nIf we consider further that, according to the second fundamental\nprinciple, the free energy accessible to us can only decrease, but not\nincrease, while the number of men whose existence depends directly on\nthe consumption of a due amount of free energy is constantly on the\nincrease, then we at once see the objective necessity of the development\nof civilization in that sense. His foresight puts man in a position to\nact culturally. But if we examine our present social order from this\npoint of view, we realize with horror how barbarous it still is. Not\nonly do murder and war destroy cultural values without substituting\nothers in their place, not only do the countless conflicts which take\nplace between the different nations and political organizations act\nanticulturally, but so do also the conflicts between the various social\nclasses of one nation, for they destroy quantities of free energy which\nare thus withdrawn from the total of real cultural values. At present\nmankind is in a state of development in which progress depends much less\nupon the leadership of a few distinguished individuals than upon the\ncollective labor of all workers. Proof of this is that it is coming to\nbe more and more the fact that the great scientific discoveries are made\nsimultaneously by a number of independent investigators--an indication\nthat society creates in several places the individual conditions\nrequisite for such discoveries. Thus we are living at a time when men\nare gradually approximating one another very closely in their natures,\nand when the social organization therefore demands and strives for as\nthorough an equalization as possible in the conditions of existence of\nall men.\n\n\n\n\nINDEX\n\n\n  Above and below, distinction between, 121\n\n  Abstract, concrete and, 16 ff.\n\n  Abstraction, 20\n\n  Action, conscious, 174;\n    instinctive, 174\n\n  Adaptation, 172 ff.\n\n  Aeromechanics, 147\n\n  Algebra, 80\n\n  Alikeness, definition of, 51 ff.\n\n  Allotropic changes, 161\n\n  Analysis, infinitesimal, 111\n\n  Analytic geometry, 122 ff.\n\n  Analytic judgments, 66\n\n  Anthropology, 57\n\n  Ants, specialization of, 181\n\n  Applied sciences, 57 ff.\n\n  _A priori_ judgments, 44\n\n  Aristotle, 38, 66\n\n  Aristotle's logic, 22\n\n  Arithmetic, 79 ff.\n\n  Assertions, never absolutely correct, 53\n\n  Association, 63 ff., 81\n\n  Astronomic objective, 6\n\n  Astronomy as an applied science, 58\n\n  Atomic hypothesis in chemistry, 142\n\n  Atoms, 141\n\n\n  Bees, specialization of, 181\n\n  Biological sciences, 55;\n    life most general concept in, 56\n\n  Botany, 56\n\n\n  C\u00e6sar, Julius, 76\n\n  Capillary phenomena, 146\n\n  Capillary theory, 147\n\n  Carbon, its circulation through plants and animals, 171;\n    life based on the energy of, 168\n\n  Carbonic acid, 171\n\n  Carnot, Sadi, 151\n\n  Causal relation, purification of, 34 ff.\n\n  Causation, the law of, 31 ff.\n\n  Chemical combinations, 71 ff.;\n    quantitative relations in, 74\n\n  Chemical energy, 159 ff.;\n    capable of powerful concentration, 161;\n    different forms of, 159\n\n  Chemical formulas represent concepts not sounds, 95\n\n  Chemistry, 20, 47, 55;\n    significance of, 160 ff.\n\n  Chinese script based on direct co-ordination, 93\n\n  Civilization, 184 ff.\n\n  Classification, not definite, 2;\n    purpose of, 2-4\n\n  Classification of the sciences, 53 ff.\n\n  Collective activity, 181\n\n  Combination, sequence in, 73 ff.\n\n  Combinations, theory of, 71\n\n  Combinatory schematization, 73;\n    in chemistry, 71 ff.;\n    in physics, 72\n\n  Communication, 181\n\n  Community among plants and animals, 179\n\n  Comparison, 82\n\n  Comte, Auguste, 54\n\n  Concept, the most general, 61 ff.\n\n  Concepts, arbitrary, 23;\n    complex, 23;\n    complex empirical, 23;\n    definition of, 16;\n    empirical, 18;\n    formation of, 19;\n    general, 26;\n    in ceaseless flux, 88;\n    science of, 15 ff., 122;\n    simple, 20;\n    simple and complex, 19 ff.\n\n  Conclusion, the, 24 ff.;\n    analytic, 66;\n    scientific, 27, 30, 66 ff.\n\n  Concrete and abstract, 16 ff.\n\n  Conjugacy, most general concept in formal sciences, 56\n\n  Conscious action, 174\n\n  Conscious thinking, willing, and acting, 178\n\n  Conservation of energy, the law of the, 135 ff.\n\n  Conservation of matter, 138\n\n  Conservation of the sum of work and kinetic energy, the law of the, 134\n\n  Conservation of work, the law of the, 130\n\n  Conservation, quantitative, 131\n\n  Continuity, 101 ff.;\n    the law of, 113 ff.\n\n  Co-ordinated signs, change in, 88 ff.\n\n  Co-ordination, 80 ff.;\n    a means of obtaining facts without dealing directly with the\n      corresponding realities, 87;\n    between concept and word not unambiguous, 89;\n    between concept and written sign, direct and indirect, 92 ff.;\n    identity the limit case in, 82;\n    integral numbers the best basis of, 85;\n    in use among primitive men and higher animals, 87;\n    its importance, 85;\n    methodology of the sciences based upon, 85;\n    of numbers with signs, 90 ff.;\n    possibility of unambiguous, 88\n\n  Copernican theory, 117 ff.\n\n  Copernicus, 117, 141\n\n  Corpuscular theory of light, 5, 157\n\n  Counting, 85 ff.;\n    defined, 85;\n    purpose of, 86\n\n  Culture, see Civilization\n\n\n  Darwin, his fundamental theory, 173\n\n  Deduction, 40 ff.;\n    the process of, 41 ff.\n\n  Deductive sciences, 38\n\n  Determinateness, absolute, only in ideal world, 50\n\n  Determinateness of things, the, 47 ff.\n\n  Determinism, 48, 51\n\n  Differential Calculus, see Differentials\n\n  Differentials, 112\n\n  Double numbers or double points in a group, 82\n\n  Dynamics, 128 ff.;\n    definition of, 139\n\n\n  Elasticity, 145\n\n  Elastic undulatory theory of light, see Wave theory of light\n\n  Electricity, principal source of, 156\n\n  Electricity and magnetism, 154 ff.\n\n  Electromagnetic theory of light, 157 ff.\n\n  Electrotechnics, 156\n\n  Empirical sciences, 38\n\n  Energetic mechanics, 138 ff.\n\n  Energy, a substance, 136;\n    at rest, 154;\n    free, 154;\n    importance of concept of, 128;\n    in nerves, 177;\n    the most general concept in the physical sciences, 56;\n    of form, 145;\n    of volume, 145\n\n  Energy intensity, 153\n\n  Erg, definition of, 150\n\n  Esperanto, 183, note\n\n  Euclid, 44, 140\n\n  European-American scripts based on indirect co-ordination, 93\n\n  Experience, incompleteness of, 27;\n    more limited than the conceivable, 118\n\n  Experiences, distinguished by _being familiar_, 31;\n    limited knowledge of, 31\n\n  Experiential sciences, see Empirical sciences\n\n  Extrapolation, 46, 50, 104\n\n\n  Familiarity due to recalling former similar experiences, 11\n\n  Fechner, 102\n\n  Feeling, thinking, acting, 174 ff.\n\n  Force, 129 ff., 153\n\n  Formal sciences, 54;\n    are empirical sciences, 55;\n    order most general concept in, 56\n\n  Foucault's pendulum experiment, 121\n\n  Freedom of the will, 50 ff.\n\n  Frequency of process facilitates repetition, 11 ff.\n\n  Function, 109 ff.;\n    continuous and discontinuous, 110;\n    most general concept in formal sciences, 56\n\n  Functional relation, the application of the, 112 ff.\n\n  Functions, the theory of, 111\n\n  Fundamental principle, the second, 150 ff.\n\n\n  Gases, 145\n\n  Generalization, suitable place for, in text-books, 9 ff.\n\n  Geometry, 47, 54, 119, 127;\n    ancient and modern methods of, 110 ff.\n\n  Goethe, 99\n\n  Good usage in language, 100\n\n  Grammatical correctness, importance attached to, 99\n\n  Grammatical rules, 97\n\n  Gravitation potential, the, 112\n\n  Group, the, 65 ff.;\n    double members or double points in, 82;\n    linear arrangement of members of, 75 ff.\n\n  Groups, artificial and natural, 69 ff.;\n    closed, among animals, 179;\n    infinite, equality of, 84;\n    related, 69 ff.;\n    unequivocal order of, 83\n\n\n  Heat, mechanical equivalent of, 149;\n    theory of, 147 ff.\n\n  Heat energy, 148 ff.\n\n  Heat engine, 151;\n    ideal, 151 ff.\n\n  Heat quantity, 148 ff.\n\n  Heliotrope, 90\n\n  Herbart, 102\n\n  Heredity, 180\n\n  Higher analysis, 111\n\n  Homonym, 89\n\n  Hydromechanics, 147\n\n\n  Ideal cases, 44 ff.\n\n  Ideal machines, 132\n\n  Identity, the limit case in co-ordination, 82\n\n  Ido, 183, note\n\n  Imperfection, indestructible quality of science, 4\n\n  Incompleteness, no hindrance to efficiency of science, 5\n\n  Indestructibility of matter, see Conservation of matter\n\n  Indo-Arabic notation, 91\n\n  Induction, 38;\n    the complete and the incomplete, 39\n\n  Inductive sciences, 38\n\n  Inference, by induction, 38;\n    from analogy, 140\n\n  Infinitesimal analysis, 111\n\n  Inorganic world, lack of memory and foresight in, 33\n\n  Insoluble problems, 142\n\n  Instinctive action, 174\n\n  Intercourse, language and, 182 ff.\n\n  Isolation among plants and animals, 179\n\n  Isomeric, 74\n\n  Isomeric changes, 161\n\n\n  Judgments, analytic, 66\n\n\n  Kant, 44, 66, 105\n\n  Kepler, 141\n\n  Kinetic energy, 132;\n    and work, their sum constant, 133 ff.;\n    transformed into work and _vice versa_, 134\n\n  Knowledge, aim of, 19;\n    individual, compared to telephone, 7 ff.;\n    limited, 31;\n    possibility of error in, ineradicable, 40;\n    social character of, 183\n\n\n  Language, beginnings of, 88;\n    defective in co-ordination, 96;\n    distinction between science and knowledge of, 98;\n    good usage in, 100;\n    and intercourse, 182 ff.;\n    needless inflections in, 99 ff.;\n    of words more imperfect than written language, 92;\n    purpose of its cultivation, 99;\n    the science of, 97 ff.;\n    unambiguity the ideal of, 89;\n    a universal auxiliary, 100;\n    written, 89 ff.\n\n  Leibnitz, 88;\n    his doctrine of pre-established harmony, 143;\n    inventor of differentials, 112\n\n  Life, 163 ff.;\n    the most general concept in the biological sciences, 56\n\n  Light, 5, 156 ff.\n\n  Liquids, 145\n\n  Locke, John, 21 ff., 88;\n    his elaboration of the notion of simple and complex \"ideas,\" 21;\n    his secondary qualities, 127\n\n  Logic, 54, 67 ff.;\n    content of, 19;\n    definition of, 15 ff.\n\n  Luther, 99\n\n\n  Magnetism, electricity and, 154 ff.\n\n  Man, compared to pair of sieves, 34;\n    his capacity for perfection, 180\n\n  Manifold, the science of the, 54\n\n  Mass, 132 ff., 136 ff.;\n    a substance, 138\n\n  Mathematical laws, accuracy of, 105\n\n  Mathematics, 54;\n    an empirical science, 55;\n    influence on, of concept of continuity, 111;\n    its progress after introduction of Indo-Arabic numerals and algebraic\n      signs, 101\n\n  Matter, definition of, 138\n\n  Mayer, Julius Robert, 149;\n    his discovery of the law of conservation, 151\n\n  Measurement, 107\n\n  Mechanical energies, 144\n\n  Mechanics, 55, 128 ff.;\n    complementary branches of, 144 ff.;\n    definition of, 138;\n    early development of, 139;\n    energetic, 138 ff.;\n    the first branch of physics treated mathematically, 139;\n    pure or classical, 144\n\n  Mechanistic hypothesis, the, as an interpretation of all\n      natural phenomena, 142;\n    especially injurious in study of mental phenomena, 142\n\n  Mechanistic theories, 140 ff.\n\n  Mechanistic theory of the universe, 132\n\n  Mechanization of astronomy, 141\n\n  Memory, 16, 32, 180;\n    definition of, 172;\n    general characteristic of, 61;\n    lack of, in inorganic world, 53\n\n  Metabolism, 165\n\n  Methodology of the sciences based upon co-ordination, 85\n\n  Microscope, 6\n\n  Motion, the science of, 54, 122;\n    uninfluenced, 122\n\n  Musical notation, 93\n\n\n  Names, arbitrariness of, 62;\n    signs and, 86 ff.\n\n  Natural laws, 28 ff.;\n    definition of, 28;\n    their extent dependent upon stage of knowledge in each science, 7;\n    their usual origin, 42 ff.;\n    prediction from, only approximate, 48\n\n  Natural philosophy, definition of, 1;\n    importance of, in study of science, 10;\n    place of, in text-books, 9 ff.\n\n  Negation, 68 ff.\n\n  Nerves, 177\n\n  Nervous discharge, 177\n\n  Newton, Sir Isaac, 141\n\n  Number groups, unlimited, 78\n\n  Numbers, 78 ff.;\n    theory of, 80\n\n  Numerals, co-ordination of, with signs, 86\n\n  Numerical names different in different languages, 86\n\n  Numerical signs international, 86\n\n  Nutrition, 165\n\n\n  Objective, astronomic, 6;\n    photographic, 6\n\n  Objective character of the world, 34\n\n  Optical telegraph, 90\n\n  Optics, geometric, 5\n\n  Optic signs, 90\n\n  Order, most general concept in formal sciences, 56\n\n  Organisms, standard for measuring relative perfection of, 176;\n    stationary forms, 163\n\n  Orthography, efforts to improve, 99;\n    English, defective in co-ordination, 96;\n    exaggerated importance of correctness in, 99;\n    mistakes in, 97;\n    reform of, 97\n\n\n  Parabolic curve, 48\n\n  Paradoxes of the infinite, 84\n\n  Pasigraphy, 92 ff.;\n    Chinese system of, 94\n\n  Permanent in change, the, 131\n\n  Perpetual motion, 130\n\n  Perpetual motion machine, 153\n\n  Philology, 97 ff.\n\n  Philosophy, limited progress in, 101\n\n  Phonetic writing, 33 ff.\n\n  Phoronomy, 54, 119, 122, 127\n\n  Photochemical processes, foundation of terrestial life, 169\n\n  Photographic objective, 6\n\n  Physical sciences, 55\n\n  Physics, 47, 55;\n    each branch of, treats of a special kind of energy, 139\n    the science of the different kinds of energy, 72;\n\n  Physiology, 55 ff.\n\n  Plato, his distinction between mental and physical life, 178\n\n  Polarity of electricity and magnetism, 155\n\n  Political organizations, conflicts between, 185\n\n  Prediction, 12\n\n  Pre-established harmony, 143\n\n  Pressure, 146, 154\n\n  Progress, depends on collective labor, 185;\n    economy of energy, 184;\n    evaluation of, 176\n\n  Pseudo-problems in science, 142\n\n  Psychology, 47, 55 ff.\n\n  Psycho-physical parallelism, 143\n\n  Ptolemy's system, 117\n\n  Pure science, 57\n\n\n  Quantity, the science of, see Mathematics, 54\n\n\n  Radiant energy, 157;\n    its importance to man, 158\n\n  Rational sciences, see Deductive sciences\n\n  Rays, straight lines of, 5\n\n  Reaction, teleological, 173\n\n  Reality, 16 ff.\n\n  Reflection, 5\n\n  Reflex action, 173\n\n  Refraction, 5\n\n  Repetition, basis of conscious life, 174\n\n  Reproduction, 165 ff.\n\n  Roman notation, 91\n\n\n  Science, aim of, 13 ff.;\n    comparison of, to a network, 42;\n    comparison of, to a tree or forest, 6;\n    definition of, 13;\n    eternal truth of, 6 ff.;\n    \"for its own sake,\" 13 ff.;\n    the facts of, unalterable, 8 ff.;\n    the function of, 23, 37;\n    importance of theoretical, 15;\n    its procedure, 45;\n    the study of happiness, 28\n\n  Sciences, the table of the, 54 ff.\n\n  Scientific discoveries, independent simultaneous, 185\n\n  Scientific instinct, 43\n\n  Scientific materialism, 138\n\n  Scientific written language based on direct co-ordination, 93\n\n  Self-preservation, 180\n\n  Sense organs, 176 ff.\n\n  Shakespeare, 99\n\n  Signs and names, 86 ff.\n\n  Social characteristics, importance of, 179 ff.\n\n  Social classes, conflicts between, 185\n\n  Socialization of human capacities, 180\n\n  Social order still barbarous, 185\n\n  Social organization, 180;\n    how best obtained, 182;\n    its tendency to equalize conditions, 185;\n    secures permanence among specialized individuals, 181\n\n  Social problems, 179 ff.\n\n  Society, 179 ff.;\n    centrifugal and centripetal forces in, 181 ff.;\n    division of functions in, 181\n\n  Sociology, 47, 55, 57\n\n  Solar radiation, 169\n\n  Soul, the, 171 ff.\n\n  Sound signs, advantage and disadvantage of, 89 ff.\n\n  Sound writing, 33 ff., 92 ff.\n\n  Space, four-dimensional, 77, note;\n    homogeneity of, in\n    horizontal direction, 121;\n    the science of, 54;\n    symmetrical and tri-dimensional, 118;\n    time and, 118 ff.;\n    tri-dimensional, 76\n\n  Specialization, one-sidedness of, 180 ff.\n\n  Spelling reform, 97\n\n  Stable forms, 163\n\n  Statics, 128 ff.;\n    definition of, 138 ff.\n\n  Stationary bodies, capable of regeneration, 164\n\n  Stationary forms, 163\n\n  Substance, 132\n\n  Surface-energy, 146\n\n  Syllogism, the, classic method of argumentation, 65 ff.\n\n  Synonym, 89\n\n\n  Table of the sciences, 54 ff.\n\n  Telegraph, optical, 90\n\n  \"Teleological\" properties of organisms, 173\n\n  Teleological reaction, 173\n\n  Telescope, 5\n\n  Temperature, 148\n\n  Theoretical science, importance of, 15\n\n  Theory of functions, 111\n\n  Theory of numbers, 80\n\n  Thermo-chemistry, 37\n\n  Thermo-dynamics, 153\n\n  Thing, definition of, 62 ff.\n\n  Thought conceived of as energetic, 178\n\n  Threshold, 102\n\n  Time, a form of inner life, 76;\n    measurement of, 122;\n    one-seried, or one-dimensional, 118;\n    and space, 118 ff.\n\n\n  Unambiguity, in language, 89;\n    of co-ordination of numbers to signs, 90\n\n  Universal auxiliary language, 100, 183\n\n\n  Velocity, 133\n\n  Volume energy, 145\n\n\n  War, 185\n\n  Wave surface, 6\n\n  Wave theory of light, 5, 157\n\n  Weight, 132, 137 ff.;\n    a substance, 138\n\n  Work, mechanical, 129;\n    product of the force and the distance, 130;\n    a substance in a limited sense, 136\n\n  Written language, 89 ff.\n\n  Written signs, 90\n\n\n  Zoology, 56\n\n       *       *       *       *       *\n\n\nAmerican Public Problems\n\nEDITED BY\n\nRALPH CURTIS RINGWALT\n\n\nIMMIGRATION: And Its Effects Upon the United States\n\nBy PRESCOTT F. HALL, A.B., LL.B., Secretary of the Immigration\nRestriction League. 393 pp. $1.50 net. By mail, $1.65.\n\n\"Should prove interesting to everyone. Very readable, forceful and\nconvincing. Mr. Hall considers every possible phase of this great\nquestion and does it in a masterly way that shows not only that he\nthoroughly understands it, but that he is deeply interested in it and\nhas studied everything bearing upon it.\"--_Boston Transcript._\n\n\"A readable work containing a vast amount of valuable information.\nEspecially to be commended is the discussion of the racial effects. As a\ntrustworthy general guide it should prove a godsend.\"--_N. Y. Evening\nPost._\n\n\"Earnest and unprejudiced.... Cannot fail to be of great assistance in\nclarifying and setting on a solid foundation the ideas of people who are\nnow becoming convinced that the problems of immigration in the nation\nand the municipality will soon reach a more acute stage than ever\nbefore.\"--_Philadelphia Press._\n\n\"An auspicious omen of the worth of Messrs. Henry Holt and Company's\nrecently announced series on American Public Problems.... Mr. Hall has\nbeen in close touch with the immigration movement and he writes with a\ngrasp and a fullness of information which must commend his work to every\nreader.... A handbook ... to which one may turn conveniently for\ninformation for which he would otherwise be obliged to search through\nmany a dusty document.\"--_The World To-day._\n\n\nTHE ELECTION OF SENATORS\n\nBy Professor GEORGE H. HAYNES, Author of \"Representation in State\nLegislatures.\" 300 pp. $1.50 net. By mail $1.65.\n\nShows the historical reasons for the present method, and its effect on\nthe senate and senators, and on state and local government, with a\ndetailed review of the arguments for and against direct election.\n\n\"A timely book.... Prof. Haynes is qualified for a historical and\nanalytical treatise on the subject of the Senate.\"\n\n  --_N. Y. Evening Sun._\n\n\"Well worth reading, and unique because it is devoted wholly to the\nelection of senators and to the deliberations of the Senate.\"\n\n  --_Boston Transcript._\n\n\"Able and dispassionate, and ought to be widely read.\"\n\n  --_New York Commercial._\n\n\"Of considerable popular as well as historical interest.\"--_Dial._\n\n\n  Henry Holt and Company\n  29 WEST 23D STREET  NEW YORK\n\n\nR. M. JOHNSTON'S LEADING AMERICAN SOLDIERS\n\nBiographies of Washington, Greene, Taylor, Scott, Andrew Jackson, Grant,\nSherman, Sheridan, McClellan, Meade, Lee, \"Stonewall\" Jackson, Joseph E.\nJohnson. With portraits. 1 vol. $1.75 net; by mail $1.88.\n\nThe first of a new series of biographies of leading Americans.\n\n\"Performs a real service in preserving the essentials.\"--_Review of\nReviews._\n\n\"Very interesting.... Much sound originality of treatment, and the style\nis clear.\"--_Springfield Republican._\n\n\nELIZA R. SCIDMORE'S AS THE HAGUE ORDAINS\n\nJournal of a Russian Prisoner's Wife in Japan. Illustrated from\nphotographs. $1.50 net, by mail $1.62.\n\n\"Holds a tremendous human interest.... Author writes with wit and a\ndelightfully feminine abandon.\"--_Outlook._\n\n\"This surprisingly outspoken volume ... could have been written only by\nan extraordinarily able woman who knew the inside of Russian politics\nand also had actual experience in Japanese war hospitals.\"--_Chicago\nRecord-Herald._\n\n\nW. F. JOHNSON'S FOUR CENTURIES OF THE PANAMA CANAL\n\nWith 16 illustrations and 6 colored maps. $3.00 net; by mail, $3.27.\n\n \"The most thorough and comprehensive book on the Panama\nCanal.\"--_Nation._\n\n\nJOHN L. GIVENS' MAKING A NEWSPAPER\n\nThe author was recently with the _New York Evening Sun_. $1.50 net; by\nmail $1.62.\n\nSome seventy-five leading newspapers praise this book as the best\ndetailed account of the business, editorial, reportorial and\nmanufacturing organization of a metropolitan journal. It should be\ninvaluable to those entering upon newspaper work and a revelation to the\ngeneral reader.\n\n\nTHE OPEN ROAD THE FRIENDLY TOWN\n\nCompiled by E. V. Lucas. Full gilt, illustrated cover linings, each\n(cloth) $1.50; (leather) $2.50.\n\nPretty anthologies of prose and verse from British and American authors,\nrespectively for wayfarers and the urbane.\n\n\n* * * If the reader will send his name and address the publishers will\nsend, from time to time, information regarding their new books.\n\n  HENRY HOLT AND COMPANY\n  PUBLISHERS  (x-'07)  NEW YORK\n\n\n  RECENT VOLUMES IN\n  THE AMERICAN NATURE SERIES\n\n  (Prospectus on request)\n\n=SHELL-FISH INDUSTRIES= By JAMES L. KELLOGG of Williams College.\n\nLarge 12mo. Illustrated by half-tones and original drawings. Just\npublished. $1.75 net.\n\nCovers classification, propagation and distribution. For the person who\neats oysters, clams or scallops, there is information on their\nstructure, life-histories and habits. A chapter is devoted to shell-fish\nas collectors and carriers of disease organisms. The oyster culturist\nwill find the life history of bivalves, a comparison of various culture\nmethods, and a description of oyster fields in various parts of the\nworld. Several facts concerning the habits of bivalves, here presented\nfor the first time, will be of interest to naturalists.\n\n=FISH STORIES: Alleged and Experienced, with a Little History, Natural\nand Unnatural=\n\nBy CHARLES F. HOLDER, Author of \"The Log of a Sea Angler,\" etc., and\nDAVID STARR JORDAN, Author of \"A Guide to the Study of Fishes,\" etc.\nWith colored plates and many illustrations from photographs. $1.75 net.\n\n \"A delightful miscellany, telling about fish of the strangest kind, with\nscientific description melting into accounts of personal adventure.\nNearly everything that is entertaining in the fish world is touched upon\nand science and fishing are made very readable.\"--_New York Sun._\n\n  =INSECT STORIES= By VERNON L. KELLOGG.\n  Illustrated, $1.50 net.\n\nStrange, true stories, primarily for children, but certainly for those\ngrown-ups who like to read discriminatingly to their children.\n\n\"The author is among a few scientific writers of distinction who can\ninterest the popular mind. No intelligent youth can fail to read it with\ndelight and profit.\"--_The Nation._\n\n  =THE LIFE OF A FOSSIL HUNTER= By CHARLES H. STERNBERG,\n  With introduction by Prof. H. F. Osborn, 48 Illustrations, $1.60 net.\n\nThe most interesting autobiography of the oldest and best known explorer\nin this field.\n\n\"One of the most interesting books to be found anywhere.\"--_William\nAllen White._\n\n=THE FRESH WATER AQUARIUM AND ITS INHABITANTS= By OTTO EGGELING and\nFREDERICK EHRENBERG.\n\nA Guide for the Amateur Aquarist. With 100 illustrations, large 12mo,\n$2.00 net.\n\n\"The best guide to the aquarium.\"--_The Independent._\n\n  HENRY HOLT AND COMPANY\n  PUBLISHERS  NEW YORK\n\n\n  THE THEORY OF THE THEATRE\n  And Other Principles of Dramatic Criticism\n\nBy CLAYTON HAMILTON. Author of \"Materials and Methods of Fiction.\" $1.50\nnet; by mail, $1.60.\n\nCONTENTS:\n\nTHE THEORY OF THE THEATRE.--What is a Play?--The Psychology of Theatre\nAudiences.--The Actor and the Dramatist.--Stage Conventions in Modern\nTimes.--Economy of Attention in Theatrical Performances.--Emphasis in\nthe Drama.--The Four Leading Types of Drama: Tragedy and Melodrama;\nComedy and Farce.--The Modern Social Drama.\n\nOTHER PRINCIPLES OF DRAMATIC CRITICISM.--The Public and the\nDramatist.--Dramatic Art and the Theatre Business.--The Happy Endings in\nthe Theatre.--The Boundaries of Approbation.--Imitation and Suggestion\nin the Drama.--Holding the Mirror up to Nature.--Blank Verse on the\nContemporary Stage.--Dramatic Literature and Theatric Journalism.--The\nIntention of Performance.--The Quality of New Endeavor.--The Effect of\nPlays upon the Public.--Pleasant and Unpleasant Plays.--Themes in the\nTheatre.--The Function of Imagination.\n\n\nDRAMATISTS OF TO-DAY\n\nROSTAND, HAUPTMANN, SUDERMANN, PINERO, SHAW, PHILLIPS, MAETERLINCK\n\nBy PROF. EDWARD EVERETT HALE, JR., of Union College. With gilt top,\n$1.50 net. (By mail, $1.60.)\n\nAn informal discussion of their principal plays and of the performances\nof some of them. The volume opens with a paper \"On Standards of\nCriticism,\" and concludes with \"Our Idea of Tragedy,\" and an appendix of\nall the plays of each author, with dates of their first performance or\npublication.\n\n_New York Evening Post:_ \"It is not often nowadays that a theatrical\nbook can be met with so free from gush and mere eulogy, or so weighted\nby common sense ... an excellent chronological appendix and full index\n... uncommonly useful for reference.\"\n\n_Dial:_ \"Noteworthy example of literary criticism in one of the most\ninteresting of literary fields.... Well worth reading a second time.\"\n\n\nTHE GERMAN DRAMA OF THE NINETEENTH CENTURY\n\nBy GEORG WITKOWSKI. Translated by PROF. L. E. HORNING. 12mo. $1.00.\n\nKleist, Grillparzer, Hebbel, Ludwig, Wildenbruch, Sudermann, Hauptmann,\nand minor dramatists receive attention.\n\n_New York Times Review:_ \"The translation of this brief, clear, and\nlogical account was an extremely happy idea. Nothing at the same time so\ncomprehensive and terse has appeared on the subject, and it is a subject\nof increasing interest to the English-speaking public.\"\n\n\n  HENRY HOLT AND COMPANY\n  PUBLISHERS  NEW YORK\n\n       *       *       *       *       *\n\nTranscriber's Notes:\n\n  Bold text is denoted by =equal signs=. The caret ^ indicates that the following character or [\n  {expression} is superscripted.\n\n  Mid-sentence capital letters are used by the Author to indicate the\n  beginning of a quote or question, which terminates at the end of the\n  sentence.\n\n  Typographical errors corrected:\n\n    p. 100: approprate changed to appropriate (... to a more appropriate\n            evaluation ...).\n\n    p. 108: meassure changed to measure (By the application of the unit\n            measure ...).\n\n    p. 184: correspondng changed to corresponding (... signifies a\n            corresponding economy ...).\n\n    p. 191: A single period deleted from index.\n\n    P. 188, 189: limit-case changed to limit case (2 occurrences), to\n                 mirror text (3 occurrences).\n\n  Alphabetical sequencing adjusted in index:\n\n    P. 189: Two 'Energy' entries moved after Energetic mechanics.\n\n    P. 191: Photographic objective moved below Photochemical processes.\n\n    P. 191: Physics: The order of the sub-entries swapped.\n\n    P. 192: Pure science moved down four places to end of \"P\" entries.\n\n    P. 193: Two 'Teleological' entries moved after Telegraph, optical.","title":"Natural Philosophy","tradition":"philosophy-ethics","translated_text":null,"word_count":48779}