{"collection":"philosophy-ethics","full_text_available":true,"graph_url":"https://unchartedknowledge.5gfusion.net/v2/api/page/gutenberg_39702/graph","html_url":"https://unchartedknowledge.5gfusion.net/v2/page/gutenberg_39702","id":"gutenberg_39702","license":"Public Domain (US)","rights_basis":"gutenberg_metadata_copyright_false","source":"gutenberg","source_url":"https://www.gutenberg.org/ebooks/39702","summary":null,"tags":["Category: Mathematics","Category: Philosophy & Ethics","Mathematics","Mathematics -- Philosophy"],"text":"Produced by Anna Hall, Albert L\u00c3\u00a1szl\u00c3\u00b3 and the Online\nDistributed Proofreading Team at http://www.pgdp.net (This\nfile was produced from images generously made available\nby The Internet Archive)\n\n\n\n\n\n\n\n\n\nTHE\n\nPHILOSOPHY\n\nOF\n\nMATHEMATICS.\n\n\n                          THE SCIENCE OF MATHEMATICS.\n                                       |\n                +----------------------+-------------------+\n                |                                          |\n                |                                          |\n       ABSTRACT MATHEMATICS.                    CONCRETE MATHEMATICS.\n                |                                          |\n                |                                          |\n                |                                  +-------+------+\n                |                                  |              |\n    ANALYSIS; _or_, _The Calculus_.             GEOMETRY.     MECHANICS.\n                |                                  |\n                |                                  |\n        +-------+----------+               +-------+---------+\n        |                  |               |                 |\n    =Ordinary       =Transcendental   =Synthetic=        =Analytic=\n    Analysis=;         Analysis=;     _or Special_       _or General_\n  _or_, _Calculus   _or_, _Calculus    =Geometry=.       =Geometry=.\n    of Direct         of Indirect          |                 |\n   Functions_.        Functions_.          |                 |\n        |                |                 |                 |\n     +--+--+       +-----+--+         +----+----+        +---+---+\n     |     |       |        |         |         |        |       |\n     |     |       |        |         |         |        |       |\n     |     |       |        |         |         |        |       |\n     |     |       |        |         |         |        |       |\n     | Algebra.    |    Calculus      |     Algebraic.   |   Of three\n     |             |       of         |   Trigonometry.  |  dimensions.\n     |             |   Variations.    |                  |\n     |             |                  |                  |\n  Arithmetic.   Differential       Graphical.          Of two\n                and Integral      Descriptive        dimensions.\n                 Calculus.         Geometry.\n\n\n\n\nTHE\nPHILOSOPHY\nOF\nMATHEMATICS;\n\nTRANSLATED FROM THE\nCOURS DE PHILOSOPHIE POSITIVE\nOF\nAUGUSTE COMTE,\nBY\nW. M. GILLESPIE,\nPROFESSOR OF CIVIL ENGINEERING & ADJ. PROF. OF MATHEMATICS\nIN UNION COLLEGE.\n\nNEW YORK:\nHARPER & BROTHERS, PUBLISHERS,\n82 CLIFF STREET\n1851.\n\n\nEntered, according to Act of Congress, in the year one thousand\neight hundred and fifty-one, by\n\nHARPER & BROTHERS.\n\nin the Clerk's Office of the District Court of the Southern District\nof New York.\n\n\n\n\nPREFACE.\n\n\nThe pleasure and profit which the translator has received from the great\nwork here presented, have induced him to lay it before his\nfellow-teachers and students of Mathematics in a more accessible form\nthan that in which it has hitherto appeared. The want of a comprehensive\nmap of the wide region of mathematical science--a bird's-eye view of its\nleading features, and of the true bearings and relations of all its\nparts--is felt by every thoughtful student. He is like the visitor to a\ngreat city, who gets no just idea of its extent and situation till he\nhas seen it from some commanding eminence. To have a panoramic view of\nthe whole district--presenting at one glance all the parts in due\nco-ordination, and the darkest nooks clearly shown--is invaluable to\neither traveller or student. It is this which has been most perfectly\naccomplished for mathematical science by the author whose work is here\npresented.\n\nClearness and depth, comprehensiveness and precision, have never,\nperhaps, been so remarkably united as in AUGUSTE COMTE. He views his\nsubject from an elevation which gives to each part of the complex whole\nits true position and value, while his telescopic glance loses none of\nthe needful details, and not only itself pierces to the heart of the\nmatter, but converts its opaqueness into such transparent crystal, that\nother eyes are enabled to see as deeply into it as his own.\n\nAny mathematician who peruses this volume will need no other\njustification of the high opinion here expressed; but others may\nappreciate the following endorsements of well-known authorities. _Mill_,\nin his \"Logic,\" calls the work of M. Comte \"by far the greatest yet\nproduced on the Philosophy of the sciences;\" and adds, \"of this\nadmirable work, one of the most admirable portions is that in which he\nmay truly be said to have created the Philosophy of the higher\nMathematics:\" _Morell_, in his \"Speculative Philosophy of Europe,\" says,\n\"The classification given of the sciences at large, and their regular\norder of development, is unquestionably a master-piece of scientific\nthinking, as simple as it is comprehensive;\" and _Lewes_, in his\n\"Biographical History of Philosophy,\" names Comte \"the Bacon of the\nnineteenth century,\" and says, \"I unhesitatingly record my conviction\nthat this is the greatest work of our age.\"\n\nThe complete work of M. Comte--his \"_Cours de Philosophie\nPositive_\"--fills six large octavo volumes, of six or seven hundred\npages each, two thirds of the first volume comprising the purely\nmathematical portion. The great bulk of the \"Course\" is the probable\ncause of the fewness of those to whom even this section of it is known.\nIts presentation in its present form is therefore felt by the translator\nto be a most useful contribution to mathematical progress in this\ncountry. The comprehensiveness of the style of the author--grasping all\npossible forms of an idea in one Briarean sentence, armed at all points\nagainst leaving any opening for mistake or forgetfulness--occasionally\nverges upon cumbersomeness and formality. The translator has, therefore,\nsometimes taken the liberty of breaking up or condensing a long\nsentence, and omitting a few passages not absolutely necessary, or\nreferring to the peculiar \"Positive philosophy\" of the author; but he\nhas generally aimed at a conscientious fidelity to the original. It has\noften been difficult to retain its fine shades and subtile distinctions\nof meaning, and, at the same time, replace the peculiarly appropriate\nFrench idioms by corresponding English ones. The attempt, however, has\nalways been made, though, when the best course has been at all doubtful,\nthe language of the original has been followed as closely as possible,\nand, when necessary, smoothness and grace have been unhesitatingly\nsacrificed to the higher attributes of clearness and precision.\n\nSome forms of expression may strike the reader as unusual, but they have\nbeen retained because they were characteristic, not of the mere language\nof the original, but of its spirit. When a great thinker has clothed his\nconceptions in phrases which are singular even in his own tongue, he who\nprofesses to translate him is bound faithfully to preserve such forms of\nspeech, as far as is practicable; and this has been here done with\nrespect to such peculiarities of expression as belong to the author,\nnot as a foreigner, but as an individual--not because he writes in\nFrench, but because he is Auguste Comte.\n\nThe young student of Mathematics should not attempt to read the whole of\nthis volume at once, but should peruse each portion of it in connexion\nwith the temporary subject of his special study: the first chapter of\nthe first book, for example, while he is studying Algebra; the first\nchapter of the second book, when he has made some progress in Geometry;\nand so with the rest. Passages which are obscure at the first reading\nwill brighten up at the second; and as his own studies cover a larger\nportion of the field of Mathematics, he will see more and more clearly\ntheir relations to one another, and to those which he is next to take\nup. For this end he is urgently recommended to obtain a perfect\nfamiliarity with the \"Analytical Table of Contents,\" which maps out the\nwhole subject, the grand divisions of which are also indicated in the\nTabular View facing the title-page. Corresponding heads will be found in\nthe body of the work, the principal divisions being in SMALL CAPITALS,\nand the subdivisions in _Italics_. For these details the translator\nalone is responsible.\n\n\n\n\n  ANALYTICAL TABLE OF CONTENTS.\n\n  INTRODUCTION.\n\n  Page\n\n  GENERAL CONSIDERATIONS ON MATHEMATICAL SCIENCE   17\n\n  THE OBJECT OF MATHEMATICS                        18\n    Measuring Magnitudes                           18\n      Difficulties                                 19\n      General Method                               20\n      Illustrations                                21\n        1. Falling Bodies                          21\n        2. Inaccessible Distances                  23\n        3. Astronomical Facts                      24\n\n  TRUE DEFINITION OF MATHEMATICS                   25\n    A Science, not an Art                          25\n\n  ITS TWO FUNDAMENTAL DIVISIONS                    26\n      Their different Objects                      27\n      Their different Natures                      29\n    _Concrete Mathematics_                         31\n      Geometry and Mechanics                       32\n    _Abstract Mathematics_                         33\n      The Calculus, or Analysis                    33\n\n  EXTENT OF ITS FIELD                              35\n      Its Universality                             36\n      Its Limitations                              37\n\n\n  BOOK I.\n\n  ANALYSIS.\n\n\n  CHAPTER I.\n\n  Page\n\n  GENERAL VIEW OF MATHEMATICAL ANALYSIS            45\n\n  THE TRUE IDEA OF AN EQUATION                     46\n      Division of Functions into Abstract and\n        Concrete                                   47\n      Enumeration of Abstract Functions            50\n\n  DIVISIONS OF THE CALCULUS                        53\n    _The Calculus of Values, or Arithmetic_        57\n      Its Extent                                   57\n      Its true Nature                              59\n    _The Calculus of Functions_                    61\n      Two Modes of obtaining Equations             61\n        1. By the Relations between the given\n           Quantities                              61\n        2. By the Relations between auxiliary\n           Quantities                              64\n    Corresponding Divisions of the Calculus of\n      Functions                                    67\n\n\n  CHAPTER II.\n\n  ORDINARY ANALYSIS; OR, ALGEBRA.                  69\n\n      Its Object                                   69\n      Classification of Equations                  70\n\n  ALGEBRAIC EQUATIONS                              71\n      Their Classification                         71\n\n  ALGEBRAIC RESOLUTION OF EQUATIONS                72\n      Its Limits                                   72\n      General Solution                             72\n      What we know in Algebra                      74\n\n  NUMERICAL RESOLUTION OF EQUATIONS                75\n      Its limited Usefulness                       76\n  Different Divisions of the two Systems           78\n\n  THE THEORY OF EQUATIONS                          79\n\n  THE METHOD OF INDETERMINATE COEFFICIENTS         80\n\n  IMAGINARY QUANTITIES                             81\n\n  NEGATIVE QUANTITIES                              81\n\n  THE PRINCIPLE OF HOMOGENEITY                     84\n\n\n  CHAPTER III.\n\n  TRANSCENDENTAL ANALYSIS:\n\n  Page\n\n  ITS DIFFERENT CONCEPTIONS                        88\n\n      Preliminary Remarks                          88\n      Its early History                            89\n\n  METHOD OF LEIBNITZ                               91\n      Infinitely small Elements                    91\n      _Examples_:\n        1. Tangents                                93\n        2. Rectification of an Arc                 94\n        3. Quadrature of a Curve                   95\n        4. Velocity in variable Motion             95\n        5. Distribution of Heat                    96\n      Generality of the Formulas                   97\n      Demonstration of the Method                  98\n        Illustration by Tangents                  102\n\n  METHOD OF NEWTON                                103\n      Method of Limits                            103\n        _Examples_:\n          1. Tangents                             104\n          2. Rectifications                       105\n      Fluxions and Fluents                        106\n\n  METHOD OF LAGRANGE                              108\n      Derived Functions                           108\n      An extension of ordinary Analysis           108\n      _Example_: Tangents                         109\n    _Fundamental Identity of the three Methods_   110\n    _Their comparative Value_                     113\n      That of Leibnitz                            113\n      That of Newton                              115\n      That of Lagrange                            117\n\n\n  CHAPTER IV.\n\n  Page\n\n  THE DIFFERENTIAL AND INTEGRAL CALCULUS          120\n\n  ITS TWO FUNDAMENTAL DIVISIONS                   120\n\n  THEIR RELATIONS TO EACH OTHER                   121\n        1. Use of the Differential Calculus as\n           preparatory to that of the Integral    123\n        2. Employment of the Differential\n           Calculus alone                         125\n        3. Employment of the Integral Calculus\n           alone                                  125\n            Three Classes of Questions hence\n              resulting                           126\n\n  THE DIFFERENTIAL CALCULUS                       127\n      Two Cases: Explicit and Implicit Functions  127\n        Two sub-Cases: a single Variable or\n          several                                 129\n        Two other Cases: Functions separate or\n          combined                                130\n      Reduction of all to the Differentiation of\n        the ten elementary Functions              131\n      Transformation of derived Functions for\n        new Variables                             132\n      Different Orders of Differentiation         133\n      Analytical Applications                     133\n\n  THE INTEGRAL CALCULUS                           135\n      Its fundamental Division: Explicit and\n        Implicit Functions                        135\n      Subdivisions: a single Variable or several  136\n      Calculus of partial Differences             137\n      Another Subdivision: different Orders of\n        Differentiation                           138\n      Another equivalent Distinction              140\n      _Quadratures_                               142\n        Integration of Transcendental Functions   143\n          Integration by Parts                    143\n        Integration of Algebraic Functions        143\n      Singular Solutions                          144\n      Definite Integrals                          146\n      Prospects of the Integral Calculus          148\n\n\n  CHAPTER V.\n\n  Page\n\n  THE CALCULUS OF VARIATIONS                      151\n\n  PROBLEMS GIVING RISE TO IT                      151\n      Ordinary Questions of Maxima and Minima     151\n      A new Class of Questions                    152\n        Solid of least Resistance;\n          Brachystochrone; Isoperimeters          153\n\n  ANALYTICAL NATURE OF THESE QUESTIONS            154\n\n  METHODS OF THE OLDER GEOMETERS                  155\n\n  METHOD OF LAGRANGE                              156\n      Two Classes of Questions                    157\n        1. Absolute Maxima and Minima             157\n           Equations of Limits                    159\n             A more general Consideration         159\n        2. Relative Maxima and Minima             160\n      Other Applications of the Method of\n        Variations                                162\n\n  ITS RELATIONS TO THE ORDINARY CALCULUS          163\n\n\n  CHAPTER VI.\n\n  THE CALCULUS OF FINITE DIFFERENCES              167\n\n      Its general Character                       167\n      Its true Nature                             168\n\n  GENERAL THEORY OF SERIES                        170\n      Its Identity with this Calculus             172\n\n  PERIODIC OR DISCONTINUOUS FUNCTIONS             173\n\n  APPLICATIONS OF THIS CALCULUS                   173\n      Series                                      173\n      Interpolation                               173\n      Approximate Rectification, &c.              174\n\n\n  BOOK II.\n\n  GEOMETRY.\n\n  CHAPTER I.\n\n  Page\n\n  A GENERAL VIEW OF GEOMETRY                      179\n\n      The true Nature of Geometry                 179\n      Two fundamental Ideas                       181\n        1. The Idea of Space                      181\n        2. Different kinds of Extension           182\n\n  THE FINAL OBJECT OF GEOMETRY                    184\n      Nature of Geometrical Measurement           185\n        Of Surfaces and Volumes                   185\n        Of curve Lines                            187\n        Of right Lines                            189\n\n  THE INFINITE EXTENT OF ITS FIELD                190\n      Infinity of Lines                           190\n      Infinity of Surfaces                        191\n      Infinity of Volumes                         192\n      Analytical Invention of Curves, &c.         193\n\n  EXPANSION OF ORIGINAL DEFINITION                193\n      Properties of Lines and Surfaces            195\n      Necessity of their Study                    195\n        1. To find the most suitable Property     195\n        2. To pass from the Concrete to the\n           Abstract                               197\n      Illustrations:\n        Orbits of the Planets                     198\n        Figure of the Earth                       199\n\n  THE TWO GENERAL METHODS OF GEOMETRY             202\n      Their fundamental Difference                203\n        1\u00b0. Different Questions with respect to\n          the same Figure                         204\n        2\u00b0. Similar Questions with respect to\n          different Figures                       204\n      Geometry of the Ancients                    204\n      Geometry of the Moderns                     206\n      Superiority of the Modern                   207\n      The Ancient the base of the Modern          209\n\n\n  CHAPTER II.\n\n  ANCIENT OR SYNTHETIC GEOMETRY\n\n  Page\n\n  ITS PROPER EXTENT                               212\n      Lines; Polygons; Polyhedrons                212\n      Not to be farther restricted                213\n      Improper Application of Analysis            214\n      Attempted Demonstrations of Axioms          216\n\n  GEOMETRY OF THE RIGHT LINE                      217\n\n  GRAPHICAL SOLUTIONS                             218\n      _Descriptive Geometry_                      220\n\n  ALGEBRAICAL SOLUTIONS                           224\n      _Trigonometry_                              225\n        Two Methods of introducing Angles         226\n          1. By Arcs                              226\n          2. By trigonometrical Lines             226\n        Advantages of the latter                  226\n        Its Division of trigonometrical Questions 227\n          1. Relations between Angles and\n             trigonometrical Lines                228\n          2. Relations between trigonometrical\n             Lines and Sides                      228\n        Increase of trigonometrical Lines         228\n        Study of the Relations between them       230\n\n\n  CHAPTER III.\n\n  MODERN OR ANALYTICAL GEOMETRY\n\n  Page\n\n  THE ANALYTICAL REPRESENTATION OF FIGURES        232\n      Reduction of Figure to Position             233\n      Determination of the position of a Point    234\n\n  PLANE CURVES                                    237\n      Expression of Lines by Equations            237\n      Expression of Equations by Lines            238\n      Any change in the Line changes the Equation 240\n      Every \"Definition\" of a Line is an Equation 241\n      _Choice of Co-ordinates_                    245\n        Two different points of View              245\n          1. Representation of Lines by Equations 246\n          2. Representation of Equations by Lines 246\n        Superiority of the rectilinear System     248\n          Advantages of perpendicular Axes        249\n\n  SURFACES    251\n      Determination of a Point in Space           251\n      Expression of Surfaces by Equations         253\n      Expression of Equations by Surfaces         253\n\n  CURVES IN SPACE     255\n\n  Imperfections of Analytical Geometry            258\n      Relatively to Geometry                      258\n      Relatively to Analysis                      258\n\n\n\n\nTHE\n\nPHILOSOPHY OF MATHEMATICS.\n\nINTRODUCTION.\n\nGENERAL CONSIDERATIONS.\n\n\nAlthough Mathematical Science is the most ancient and the most perfect\nof all, yet the general idea which we ought to form of it has not yet\nbeen clearly determined. Its definition and its principal divisions have\nremained till now vague and uncertain. Indeed the plural name--\"The\nMathematics\"--by which we commonly designate it, would alone suffice to\nindicate the want of unity in the common conception of it.\n\nIn truth, it was not till the commencement of the last century that the\ndifferent fundamental conceptions which constitute this great science\nwere each of them sufficiently developed to permit the true spirit of\nthe whole to manifest itself with clearness. Since that epoch the\nattention of geometers has been too exclusively absorbed by the special\nperfecting of the different branches, and by the application which they\nhave made of them to the most important laws of the universe, to allow\nthem to give due attention to the general system of the science.\n\nBut at the present time the progress of the special departments is no\nlonger so rapid as to forbid the contemplation of the whole. The science\nof mathematics is now sufficiently developed, both in itself and as to\nits most essential application, to have arrived at that state of\nconsistency in which we ought to strive to arrange its different parts\nin a single system, in order to prepare for new advances. We may even\nobserve that the last important improvements of the science have\ndirectly paved the way for this important philosophical operation, by\nimpressing on its principal parts a character of unity which did not\npreviously exist.\n\nTo form a just idea of the object of mathematical science, we may start\nfrom the indefinite and meaningless definition of it usually given, in\ncalling it \"_The science of magnitudes_,\" or, which is more definite,\n\"_The science which has for its object the measurement of magnitudes._\"\nLet us see how we can rise from this rough sketch (which is singularly\ndeficient in precision and depth, though, at bottom, just) to a\nveritable definition, worthy of the importance, the extent, and the\ndifficulty of the science.\n\n\nTHE OBJECT OF MATHEMATICS.\n\n_Measuring Magnitudes._ The question of _measuring_ a magnitude in\nitself presents to the mind no other idea than that of the simple direct\ncomparison of this magnitude with another similar magnitude, supposed to\nbe known, which it takes for the _unit_ of comparison among all others\nof the same kind. According to this definition, then, the science of\nmathematics--vast and profound as it is with reason reputed to\nbe--instead of being an immense concatenation of prolonged mental\nlabours, which offer inexhaustible occupation to our intellectual\nactivity, would seem to consist of a simple series of mechanical\nprocesses for obtaining directly the ratios of the quantities to be\nmeasured to those by which we wish to measure them, by the aid of\noperations of similar character to the superposition of lines, as\npracticed by the carpenter with his rule.\n\nThe error of this definition consists in presenting as direct an object\nwhich is almost always, on the contrary, very indirect. The _direct_\nmeasurement of a magnitude, by superposition or any similar process, is\nmost frequently an operation quite impossible for us to perform; so that\nif we had no other means for determining magnitudes than direct\ncomparisons, we should be obliged to renounce the knowledge of most of\nthose which interest us.\n\n_Difficulties._ The force of this general observation will be understood\nif we limit ourselves to consider specially the particular case which\nevidently offers the most facility--that of the measurement of one\nstraight line by another. This comparison, which is certainly the most\nsimple which we can conceive, can nevertheless scarcely ever be effected\ndirectly. In reflecting on the whole of the conditions necessary to\nrender a line susceptible of a direct measurement, we see that most\nfrequently they cannot be all fulfilled at the same time. The first and\nthe most palpable of these conditions--that of being able to pass over\nthe line from one end of it to the other, in order to apply the unit of\nmeasurement to its whole length--evidently excludes at once by far the\ngreater part of the distances which interest us the most; in the first\nplace, all the distances between the celestial bodies, or from any one\nof them to the earth; and then, too, even the greater number of\nterrestrial distances, which are so frequently inaccessible. But even if\nthis first condition be found to be fulfilled, it is still farther\nnecessary that the length be neither too great nor too small, which\nwould render a direct measurement equally impossible. The line must also\nbe suitably situated; for let it be one which we could measure with the\ngreatest facility, if it were horizontal, but conceive it to be turned\nup vertically, and it becomes impossible to measure it.\n\nThe difficulties which we have indicated in reference to measuring\nlines, exist in a very much greater degree in the measurement of\nsurfaces, volumes, velocities, times, forces, &c. It is this fact which\nmakes necessary the formation of mathematical science, as we are going\nto see; for the human mind has been compelled to renounce, in almost all\ncases, the direct measurement of magnitudes, and to seek to determine\nthem _indirectly_, and it is thus that it has been led to the creation\nof mathematics.\n\n_General Method._ The general method which is constantly employed, and\nevidently the only one conceivable, to ascertain magnitudes which do not\nadmit of a direct measurement, consists in connecting them with others\nwhich are susceptible of being determined immediately, and by means of\nwhich we succeed in discovering the first through the relations which\nsubsist between the two. Such is the precise object of mathematical\nscience viewed as a whole. In order to form a sufficiently extended idea\nof it, we must consider that this indirect determination of magnitudes\nmay be indirect in very different degrees. In a great number of cases,\nwhich are often the most important, the magnitudes, by means of which\nthe principal magnitudes sought are to be determined, cannot themselves\nbe measured directly, and must therefore, in their turn, become the\nsubject of a similar question, and so on; so that on many occasions the\nhuman mind is obliged to establish a long series of intermediates\nbetween the system of unknown magnitudes which are the final objects of\nits researches, and the system of magnitudes susceptible of direct\nmeasurement, by whose means we finally determine the first, with which\nat first they appear to have no connexion.\n\n_Illustrations._ Some examples will make clear any thing which may seem\ntoo abstract in the preceding generalities.\n\n1. _Falling Bodies._ Let us consider, in the first place, a natural\nphenomenon, very simple, indeed, but which may nevertheless give rise to\na mathematical question, really existing, and susceptible of actual\napplications--the phenomenon of the vertical fall of heavy bodies.\n\nThe mind the most unused to mathematical conceptions, in observing this\nphenomenon, perceives at once that the two _quantities_ which it\npresents--namely, the _height_ from which a body has fallen, and the\n_time_ of its fall--are necessarily connected with each other, since\nthey vary together, and simultaneously remain fixed; or, in the language\nof geometers, that they are \"_functions_\" of each other. The phenomenon,\nconsidered under this point of view, gives rise then to a mathematical\nquestion, which consists in substituting for the direct measurement of\none of these two magnitudes, when it is impossible, the measurement of\nthe other. It is thus, for example, that we may determine indirectly the\ndepth of a precipice, by merely measuring the time that a heavy body\nwould occupy in falling to its bottom, and by suitable procedures this\ninaccessible depth will be known with as much precision as if it was a\nhorizontal line placed in the most favourable circumstances for easy and\nexact measurement. On other occasions it is the height from which a body\nhas fallen which it will be easy to ascertain, while the time of the\nfall could not be observed directly; then the same phenomenon would give\nrise to the inverse question, namely, to determine the time from the\nheight; as, for example, if we wished to ascertain what would be the\nduration of the vertical fall of a body falling from the moon to the\nearth.\n\nIn this example the mathematical question is very simple, at least when\nwe do not pay attention to the variation in the intensity of gravity, or\nthe resistance of the fluid which the body passes through in its fall.\nBut, to extend the question, we have only to consider the same\nphenomenon in its greatest generality, in supposing the fall oblique,\nand in taking into the account all the principal circumstances. Then,\ninstead of offering simply two variable quantities connected with each\nother by a relation easy to follow, the phenomenon will present a much\ngreater number; namely, the space traversed, whether in a vertical or\nhorizontal direction; the time employed in traversing it; the velocity\nof the body at each point of its course; even the intensity and the\ndirection of its primitive impulse, which may also be viewed as\nvariables; and finally, in certain cases (to take every thing into the\naccount), the resistance of the medium and the intensity of gravity. All\nthese different quantities will be connected with one another, in such a\nway that each in its turn may be indirectly determined by means of the\nothers; and this will present as many distinct mathematical questions as\nthere may be co-existing magnitudes in the phenomenon under\nconsideration. Such a very slight change in the physical conditions of a\nproblem may cause (as in the above example) a mathematical research, at\nfirst very elementary, to be placed at once in the rank of the most\ndifficult questions, whose complete and rigorous solution surpasses as\nyet the utmost power of the human intellect.\n\n2. _Inaccessible Distances._ Let us take a second example from\ngeometrical phenomena. Let it be proposed to determine a distance which\nis not susceptible of direct measurement; it will be generally conceived\nas making part of a _figure_, or certain system of lines, chosen in such\na way that all its other parts may be observed directly; thus, in the\ncase which is most simple, and to which all the others may be finally\nreduced, the proposed distance will be considered as belonging to a\ntriangle, in which we can determine directly either another side and two\nangles, or two sides and one angle. Thence-forward, the knowledge of the\ndesired distance, instead of being obtained directly, will be the result\nof a mathematical calculation, which will consist in deducing it from\nthe observed elements by means of the relation which connects it with\nthem. This calculation will become successively more and more\ncomplicated, if the parts which we have supposed to be known cannot\nthemselves be determined (as is most frequently the case) except in an\nindirect manner, by the aid of new auxiliary systems, the number of\nwhich, in great operations of this kind, finally becomes very\nconsiderable. The distance being once determined, the knowledge of it\nwill frequently be sufficient for obtaining new quantities, which will\nbecome the subject of new mathematical questions. Thus, when we know at\nwhat distance any object is situated, the simple observation of its\napparent diameter will evidently permit us to determine indirectly its\nreal dimensions, however inaccessible it may be, and, by a series of\nanalogous investigations, its surface, its volume, even its weight, and\na number of other properties, a knowledge of which seemed forbidden to\nus.\n\n3. _Astronomical Facts._ It is by such calculations that man has been\nable to ascertain, not only the distances from the planets to the earth,\nand, consequently, from each other, but their actual magnitude, their\ntrue figure, even to the inequalities of their surface; and, what seemed\nstill more completely hidden from us, their respective masses, their\nmean densities, the principal circumstances of the fall of heavy bodies\non the surface of each of them, &c.\n\nBy the power of mathematical theories, all these different results, and\nmany others relative to the different classes of mathematical phenomena,\nhave required no other direct measurements than those of a very small\nnumber of straight lines, suitably chosen, and of a greater number of\nangles. We may even say, with perfect truth, so as to indicate in a word\nthe general range of the science, that if we did not fear to multiply\ncalculations unnecessarily, and if we had not, in consequence, to\nreserve them for the determination of the quantities which could not be\nmeasured directly, the determination of all the magnitudes susceptible\nof precise estimation, which the various orders of phenomena can offer\nus, could be finally reduced to the direct measurement of a single\nstraight line and of a suitable number of angles.\n\n\nTRUE DEFINITION OF MATHEMATICS.\n\nWe are now able to define mathematical science with precision, by\nassigning to it as its object the _indirect_ measurement of magnitudes,\nand by saying it constantly proposes _to determine certain magnitudes\nfrom others by means of the precise relations existing between them_.\n\nThis enunciation, instead of giving the idea of only an _art_, as do all\nthe ordinary definitions, characterizes immediately a true _science_,\nand shows it at once to be composed of an immense chain of intellectual\noperations, which may evidently become very complicated, because of the\nseries of intermediate links which it will be necessary to establish\nbetween the unknown quantities and those which admit of a direct\nmeasurement; of the number of variables coexistent in the proposed\nquestion; and of the nature of the relations between all these different\nmagnitudes furnished by the phenomena under consideration. According to\nsuch a definition, the spirit of mathematics consists in always\nregarding all the quantities which any phenomenon can present, as\nconnected and interwoven with one another, with the view of deducing\nthem from one another. Now there is evidently no phenomenon which cannot\ngive rise to considerations of this kind; whence results the naturally\nindefinite extent and even the rigorous logical universality of\nmathematical science. We shall seek farther on to circumscribe as\nexactly as possible its real extension.\n\nThe preceding explanations establish clearly the propriety of the name\nemployed to designate the science which we are considering. This\ndenomination, which has taken to-day so definite a meaning by itself\nsignifies simply _science_ in general. Such a designation, rigorously\nexact for the Greeks, who had no other real science, could be retained\nby the moderns only to indicate the mathematics as _the_ science, beyond\nall others--the science of sciences.\n\nIndeed, every true science has for its object the determination of\ncertain phenomena by means of others, in accordance with the relations\nwhich exist between them. Every _science_ consists in the co-ordination\nof facts; if the different observations were entirely isolated, there\nwould be no science. We may even say, in general terms, that _science_\nis essentially destined to dispense, so far as the different phenomena\npermit it, with all direct observation, by enabling us to deduce from\nthe smallest possible number of immediate data the greatest possible\nnumber of results. Is not this the real use, whether in speculation or\nin action, of the _laws_ which we succeed in discovering among natural\nphenomena? Mathematical science, in this point of view, merely pushes to\nthe highest possible degree the same kind of researches which are\npursued, in degrees more or less inferior, by every real science in its\nrespective sphere.\n\n\nITS TWO FUNDAMENTAL DIVISIONS.\n\nWe have thus far viewed mathematical science only as a whole, without\npaying any regard to its divisions. We must now, in order to complete\nthis general view, and to form a just idea of the philosophical\ncharacter of the science, consider its fundamental division. The\nsecondary divisions will be examined in the following chapters.\n\nThis principal division, which we are about to investigate, can be\ntruly rational, and derived from the real nature of the subject, only so\nfar as it spontaneously presents itself to us, in making the exact\nanalysis of a complete mathematical question. We will, therefore, having\ndetermined above what is the general object of mathematical labours, now\ncharacterize with precision the principal different orders of inquiries,\nof which they are constantly composed.\n\n_Their different Objects._ The complete solution of every mathematical\nquestion divides itself necessarily into two parts, of natures\nessentially distinct, and with relations invariably determinate. We have\nseen that every mathematical inquiry has for its object to determine\nunknown magnitudes, according to the relations between them and known\nmagnitudes. Now for this object, it is evidently necessary, in the first\nplace, to ascertain with precision the relations which exist between the\nquantities which we are considering. This first branch of inquiries\nconstitutes that which I call the _concrete_ part of the solution. When\nit is finished, the question changes; it is now reduced to a pure\nquestion of numbers, consisting simply in determining unknown numbers,\nwhen we know what precise relations connect them with known numbers.\nThis second branch of inquiries is what I call the _abstract_ part of\nthe solution. Hence follows the fundamental division of general\nmathematical science into _two_ great sciences--ABSTRACT MATHEMATICS,\nand CONCRETE MATHEMATICS.\n\nThis analysis may be observed in every complete mathematical question,\nhowever simple or complicated it may be. A single example will suffice\nto make it intelligible.\n\nTaking up again the phenomenon of the vertical fall of a heavy body, and\nconsidering the simplest case, we see that in order to succeed in\ndetermining, by means of one another, the height whence the body has\nfallen, and the duration of its fall, we must commence by discovering\nthe exact relation of these two quantities, or, to use the language of\ngeometers, the _equation_ which exists between them. Before this first\nresearch is completed, every attempt to determine numerically the value\nof one of these two magnitudes from the other would evidently be\npremature, for it would have no basis. It is not enough to know vaguely\nthat they depend on one another--which every one at once perceives--but\nit is necessary to determine in what this dependence consists. This\ninquiry may be very difficult, and in fact, in the present case,\nconstitutes incomparably the greater part of the problem. The true\nscientific spirit is so modern, that no one, perhaps, before Galileo,\nhad ever remarked the increase of velocity which a body experiences in\nits fall: a circumstance which excludes the hypothesis, towards which\nour mind (always involuntarily inclined to suppose in every phenomenon\nthe most simple _functions_, without any other motive than its greater\nfacility in conceiving them) would be naturally led, that the height was\nproportional to the time. In a word, this first inquiry terminated in\nthe discovery of the law of Galileo.\n\nWhen this _concrete_ part is completed, the inquiry becomes one of quite\nanother nature. Knowing that the spaces passed through by the body in\neach successive second of its fall increase as the series of odd\nnumbers, we have then a problem purely numerical and _abstract_; to\ndeduce the height from the time, or the time from the height; and this\nconsists in finding that the first of these two quantities, according to\nthe law which has been established, is a known multiple of the second\npower of the other; from which, finally, we have to calculate the value\nof the one when that of the other is given.\n\nIn this example the concrete question is more difficult than the\nabstract one. The reverse would be the case if we considered the same\nphenomenon in its greatest generality, as I have done above for another\nobject. According to the circumstances, sometimes the first, sometimes\nthe second, of these two parts will constitute the principal difficulty\nof the whole question; for the mathematical law of the phenomenon may be\nvery simple, but very difficult to obtain, or it may be easy to\ndiscover, but very complicated; so that the two great sections of\nmathematical science, when we compare them as wholes, must be regarded\nas exactly equivalent in extent and in difficulty, as well as in\nimportance, as we shall show farther on, in considering each of them\nseparately.\n\n_Their different Natures._ These two parts, essentially distinct in\ntheir _object_, as we have just seen, are no less so with regard to the\n_nature_ of the inquiries of which they are composed.\n\nThe first should be called _concrete_, since it evidently depends on the\ncharacter of the phenomena considered, and must necessarily vary when we\nexamine new phenomena; while the second is completely independent of the\nnature of the objects examined, and is concerned with only the\n_numerical_ relations which they present, for which reason it should be\ncalled _abstract_. The same relations may exist in a great number of\ndifferent phenomena, which, in spite of their extreme diversity, will\nbe viewed by the geometer as offering an analytical question\nsusceptible, when studied by itself, of being resolved once for all.\nThus, for instance, the same law which exists between the space and the\ntime of the vertical fall of a body in a vacuum, is found again in many\nother phenomena which offer no analogy with the first nor with each\nother; for it expresses the relation between the surface of a spherical\nbody and the length of its diameter; it determines, in like manner, the\ndecrease of the intensity of light or of heat in relation to the\ndistance of the objects lighted or heated, &c. The abstract part, common\nto these different mathematical questions, having been treated in\nreference to one of these, will thus have been treated for all; while\nthe concrete part will have necessarily to be again taken up for each\nquestion separately, without the solution of any one of them being able\nto give any direct aid, in that connexion, for the solution of the rest.\n\nThe abstract part of mathematics is, then, general in its nature; the\nconcrete part, special.\n\nTo present this comparison under a new point of view, we may say\nconcrete mathematics has a philosophical character, which is essentially\nexperimental, physical, phenomenal; while that of abstract mathematics\nis purely logical, rational. The concrete part of every mathematical\nquestion is necessarily founded on the consideration of the external\nworld, and could never be resolved by a simple series of intellectual\ncombinations. The abstract part, on the contrary, when it has been very\ncompletely separated, can consist only of a series of logical\ndeductions, more or less prolonged; for if we have once found the\nequations of a phenomenon, the determination of the quantities therein\nconsidered, by means of one another, is a matter for reasoning only,\nwhatever the difficulties may be. It belongs to the understanding alone\nto deduce from these equations results which are evidently contained in\nthem, although perhaps in a very involved manner, without there being\noccasion to consult anew the external world; the consideration of which,\nhaving become thenceforth foreign to the subject, ought even to be\ncarefully set aside in order to reduce the labour to its true peculiar\ndifficulty. The _abstract_ part of mathematics is then purely\ninstrumental, and is only an immense and admirable extension of natural\nlogic to a certain class of deductions. On the other hand, geometry and\nmechanics, which, as we shall see presently, constitute the _concrete_\npart, must be viewed as real natural sciences, founded on observation,\nlike all the rest, although the extreme simplicity of their phenomena\npermits an infinitely greater degree of systematization, which has\nsometimes caused a misconception of the experimental character of their\nfirst principles.\n\nWe see, by this brief general comparison, how natural and profound is\nour fundamental division of mathematical science.\n\nWe have now to circumscribe, as exactly as we can in this first sketch,\neach of these two great sections.\n\n\nCONCRETE MATHEMATICS.\n\n_Concrete Mathematics_ having for its object the discovery of the\n_equations_ of phenomena, it would seem at first that it must be\ncomposed of as many distinct sciences as we find really distinct\ncategories among natural phenomena. But we are yet very far from having\ndiscovered mathematical laws in all kinds of phenomena; we shall even\nsee, presently, that the greater part will very probably always hide\nthemselves from our investigations. In reality, in the present condition\nof the human mind, there are directly but two great general classes of\nphenomena, whose equations we constantly know; these are, firstly,\ngeometrical, and, secondly, mechanical phenomena. Thus, then, the\nconcrete part of mathematics is composed of GEOMETRY and RATIONAL\nMECHANICS.\n\nThis is sufficient, it is true, to give to it a complete character of\nlogical universality, when we consider all phenomena from the most\nelevated point of view of natural philosophy. In fact, if all the parts\nof the universe were conceived as immovable, we should evidently have\nonly geometrical phenomena to observe, since all would be reduced to\nrelations of form, magnitude, and position; then, having regard to the\nmotions which take place in it, we would have also to consider\nmechanical phenomena. Hence the universe, in the statical point of view,\npresents only geometrical phenomena; and, considered dynamically, only\nmechanical phenomena. Thus geometry and mechanics constitute the two\nfundamental natural sciences, in this sense, that all natural effects\nmay be conceived as simple necessary results, either of the laws of\nextension or of the laws of motion.\n\nBut although this conception is always logically possible, the\ndifficulty is to specialize it with the necessary precision, and to\nfollow it exactly in each of the general cases offered to us by the\nstudy of nature; that is, to effectually reduce each principal question\nof natural philosophy, for a certain determinate order of phenomena, to\nthe question of geometry or mechanics, to which we might rationally\nsuppose it should be brought. This transformation, which requires great\nprogress to have been previously made in the study of each class of\nphenomena, has thus far been really executed only for those of\nastronomy, and for a part of those considered by terrestrial physics,\nproperly so called. It is thus that astronomy, acoustics, optics, &c.,\nhave finally become applications of mathematical science to certain\norders of observations.[1] But these applications not being by their\nnature rigorously circumscribed, to confound them with the science would\nbe to assign to it a vague and indefinite domain; and this is done in\nthe usual division, so faulty in so many other respects, of the\nmathematics into \"Pure\" and \"Applied.\"\n\n  [Footnote 1: The investigation of the mathematical phenomena of the\n  laws of heat by Baron Fourier has led to the establishment, in an\n  entirely direct manner, of Thermological equations. This great\n  discovery tends to elevate our philosophical hopes as to the future\n  extensions of the legitimate applications of mathematical analysis,\n  and renders it proper, in the opinion of author, to regard\n  _Thermology_ as a third principal branch of concrete mathematics.]\n\n\nABSTRACT MATHEMATICS.\n\nThe nature of abstract mathematics (the general division of which will\nbe examined in the following chapter) is clearly and exactly determined.\nIt is composed of what is called the _Calculus_,[2] taking this word in\nits greatest extent, which reaches from the most simple numerical\noperations to the most sublime combinations of transcendental analysis.\nThe _Calculus_ has the solution of all questions relating to numbers\nfor its peculiar object. Its _starting point_ is, constantly and\nnecessarily, the knowledge of the precise relations, _i.e._, of the\n_equations_, between the different magnitudes which are simultaneously\nconsidered; that which is, on the contrary, the _stopping point_ of\nconcrete mathematics. However complicated, or however indirect these\nrelations may be, the final object of the calculus always is to obtain\nfrom them the values of the unknown quantities by means of those which\nare known. This _science_, although nearer perfection than any other, is\nreally little advanced as yet, so that this object is rarely attained in\na manner completely satisfactory.\n\n  [Footnote 2: The translator has felt justified in employing this\n  very convenient word (for which our language has no precise\n  equivalent) as an English one, in its most extended sense, in spite\n  of its being often popularly confounded with its Differential and\n  Integral department.]\n\nMathematical analysis is, then, the true rational basis of the entire\nsystem of our actual knowledge. It constitutes the first and the most\nperfect of all the fundamental sciences. The ideas with which it\noccupies itself are the most universal, the most abstract, and the most\nsimple which it is possible for us to conceive.\n\nThis peculiar nature of mathematical analysis enables us easily to\nexplain why, when it is properly employed, it is such a powerful\ninstrument, not only to give more precision to our real knowledge, which\nis self-evident, but especially to establish an infinitely more perfect\nco-ordination in the study of the phenomena which admit of that\napplication; for, our conceptions having been so generalized and\nsimplified that a single analytical question, abstractly resolved,\ncontains the _implicit_ solution of a great number of diverse physical\nquestions, the human mind must necessarily acquire by these means a\ngreater facility in perceiving relations between phenomena which at\nfirst appeared entirely distinct from one another. We thus naturally see\narise, through the medium of analysis, the most frequent and the most\nunexpected approximations between problems which at first offered no\napparent connection, and which we often end in viewing as identical.\nCould we, for example, without the aid of analysis, perceive the least\nresemblance between the determination of the direction of a curve at\neach of its points and that of the velocity acquired by a body at every\ninstant of its variable motion? and yet these questions, however\ndifferent they may be, compose but one in the eyes of the geometer.\n\nThe high relative perfection of mathematical analysis is as easily\nperceptible. This perfection is not due, as some have thought, to the\nnature of the signs which are employed as instruments of reasoning,\neminently concise and general as they are. In reality, all great\nanalytical ideas have been formed without the algebraic signs having\nbeen of any essential aid, except for working them out after the mind\nhad conceived them. The superior perfection of the science of the\ncalculus is due principally to the extreme simplicity of the ideas which\nit considers, by whatever signs they may be expressed; so that there is\nnot the least hope, by any artifice of scientific language, of\nperfecting to the same degree theories which refer to more complex\nsubjects, and which are necessarily condemned by their nature to a\ngreater or less logical inferiority.\n\n\nTHE EXTENT OF ITS FIELD.\n\nOur examination of the philosophical character of mathematical science\nwould remain incomplete, if, after having viewed its object and\ncomposition, we did not examine the real extent of its domain.\n\n_Its Universality_. For this purpose it is indispensable to perceive,\nfirst of all, that, in the purely logical point of view, this science is\nby itself necessarily and rigorously universal; for there is no question\nwhatever which may not be finally conceived as consisting in determining\ncertain quantities from others by means of certain relations, and\nconsequently as admitting of reduction, in final analysis, to a simple\nquestion of numbers. In all our researches, indeed, on whatever subject,\nour object is to arrive at numbers, at quantities, though often in a\nvery imperfect manner and by very uncertain methods. Thus, taking an\nexample in the class of subjects the least accessible to mathematics,\nthe phenomena of living bodies, even when considered (to take the most\ncomplicated case) in the state of disease, is it not manifest that all\nthe questions of therapeutics may be viewed as consisting in determining\nthe _quantities_ of the different agents which modify the organism, and\nwhich must act upon it to bring it to its normal state, admitting, for\nsome of these quantities in certain cases, values which are equal to\nzero, or negative, or even contradictory?\n\nThe fundamental idea of Descartes on the relation of the concrete to the\nabstract in mathematics, has proven, in opposition to the superficial\ndistinction of metaphysics, that all ideas of quality may be reduced to\nthose of quantity. This conception, established at first by its immortal\nauthor in relation to geometrical phenomena only, has since been\neffectually extended to mechanical phenomena, and in our days to those\nof heat. As a result of this gradual generalization, there are now no\ngeometers who do not consider it, in a purely theoretical sense, as\ncapable of being applied to all our real ideas of every sort, so that\nevery phenomenon is logically susceptible of being represented by an\n_equation_; as much so, indeed, as is a curve or a motion, excepting the\ndifficulty of discovering it, and then of _resolving_ it, which may be,\nand oftentimes are, superior to the greatest powers of the human mind.\n\n_Its Limitations_. Important as it is to comprehend the rigorous\nuniversality, in a logical point of view, of mathematical science, it is\nno less indispensable to consider now the great real _limitations_,\nwhich, through the feebleness of our intellect, narrow in a remarkable\ndegree its actual domain, in proportion as phenomena, in becoming\nspecial, become complicated.\n\nEvery question may be conceived as capable of being reduced to a pure\nquestion of numbers; but the difficulty of effecting such a\ntransformation increases so much with the complication of the phenomena\nof natural philosophy, that it soon becomes insurmountable.\n\nThis will be easily seen, if we consider that to bring a question within\nthe field of mathematical analysis, we must first have discovered the\nprecise relations which exist between the quantities which are found in\nthe phenomenon under examination, the establishment of these equations\nbeing the necessary starting point of all analytical labours. This must\nevidently be so much the more difficult as we have to do with phenomena\nwhich are more special, and therefore more complicated. We shall thus\nfind that it is only in _inorganic physics_, at the most, that we can\njustly hope ever to obtain that high degree of scientific perfection.\n\nThe _first_ condition which is necessary in order that phenomena may\nadmit of mathematical laws, susceptible of being discovered, evidently\nis, that their different quantities should admit of being expressed by\nfixed numbers. We soon find that in this respect the whole of _organic\nphysics_, and probably also the most complicated parts of inorganic\nphysics, are necessarily inaccessible, by their nature, to our\nmathematical analysis, by reason of the extreme numerical variability of\nthe corresponding phenomena. Every precise idea of fixed numbers is\ntruly out of place in the phenomena of living bodies, when we wish to\nemploy it otherwise than as a means of relieving the attention, and when\nwe attach any importance to the exact relations of the values assigned.\n\nWe ought not, however, on this account, to cease to conceive all\nphenomena as being necessarily subject to mathematical laws, which we\nare condemned to be ignorant of, only because of the too great\ncomplication of the phenomena. The most complex phenomena of living\nbodies are doubtless essentially of no other special nature than the\nsimplest phenomena of unorganized matter. If it were possible to isolate\nrigorously each of the simple causes which concur in producing a single\nphysiological phenomenon, every thing leads us to believe that it would\nshow itself endowed, in determinate circumstances, with a kind of\ninfluence and with a quantity of action as exactly fixed as we see it in\nuniversal gravitation, a veritable type of the fundamental laws of\nnature.\n\nThere is a _second_ reason why we cannot bring complicated phenomena\nunder the dominion of mathematical analysis. Even if we could ascertain\nthe mathematical law which governs each agent, taken by itself, the\ncombination of so great a number of conditions would render the\ncorresponding mathematical problem so far above our feeble means, that\nthe question would remain in most cases incapable of solution.\n\nTo appreciate this difficulty, let us consider how complicated\nmathematical questions become, even those relating to the most simple\nphenomena of unorganized bodies, when we desire to bring sufficiently\nnear together the abstract and the concrete state, having regard to all\nthe principal conditions which can exercise a real influence over the\neffect produced. We know, for example, that the very simple phenomenon\nof the flow of a fluid through a given orifice, by virtue of its gravity\nalone, has not as yet any complete mathematical solution, when we take\ninto the account all the essential circumstances. It is the same even\nwith the still more simple motion of a solid projectile in a resisting\nmedium.\n\nWhy has mathematical analysis been able to adapt itself with such\nadmirable success to the most profound study of celestial phenomena?\nBecause they are, in spite of popular appearances, much more simple than\nany others. The most complicated problem which they present, that of the\nmodification produced in the motions of two bodies tending towards each\nother by virtue of their gravitation, by the influence of a third body\nacting on both of them in the same manner, is much less complex than the\nmost simple terrestrial problem. And, nevertheless, even it presents\ndifficulties so great that we yet possess only approximate solutions of\nit. It is even easy to see that the high perfection to which solar\nastronomy has been able to elevate itself by the employment of\nmathematical science is, besides, essentially due to our having\nskilfully profited by all the particular, and, so to say, accidental\nfacilities presented by the peculiarly favourable constitution of our\nplanetary system. The planets which compose it are quite few in number,\nand their masses are in general very unequal, and much less than that of\nthe sun; they are, besides, very distant from one another; they have\nforms almost spherical; their orbits are nearly circular, and only\nslightly inclined to each other, and so on. It results from all these\ncircumstances that the perturbations are generally inconsiderable, and\nthat to calculate them it is usually sufficient to take into the\naccount, in connexion with the action of the sun on each particular\nplanet, the influence of only one other planet, capable, by its size and\nits proximity, of causing perceptible derangements.\n\nIf, however, instead of such a state of things, our solar system had\nbeen composed of a greater number of planets concentrated into a less\nspace, and nearly equal in mass; if their orbits had presented very\ndifferent inclinations, and considerable eccentricities; if these bodies\nhad been of a more complicated form, such as very eccentric ellipsoids,\nit is certain that, supposing the same law of gravitation to exist, we\nshould not yet have succeeded in subjecting the study of the celestial\nphenomena to our mathematical analysis, and probably we should not even\nhave been able to disentangle the present principal law.\n\nThese hypothetical conditions would find themselves exactly realized in\nthe highest degree in _chemical_ phenomena, if we attempted to calculate\nthem by the theory of general gravitation.\n\nOn properly weighing the preceding considerations, the reader will be\nconvinced, I think, that in reducing the future extension of the great\napplications of mathematical analysis, which are really possible, to\nthe field comprised in the different departments of inorganic physics, I\nhave rather exaggerated than contracted the extent of its actual domain.\nImportant as it was to render apparent the rigorous logical universality\nof mathematical science, it was equally so to indicate the conditions\nwhich limit for us its real extension, so as not to contribute to lead\nthe human mind astray from the true scientific direction in the study of\nthe most complicated phenomena, by the chimerical search after an\nimpossible perfection.\n\n       *       *       *       *       *\n\nHaving thus exhibited the essential object and the principal composition\nof mathematical science, as well as its general relations with the whole\nbody of natural philosophy, we have now to pass to the special\nexamination of the great sciences of which it is composed.\n\n     _Note._--ANALYSIS and GEOMETRY are the two great heads under which\n     the subject is about to be examined. To these _M. Comte_ adds\n     Rational MECHANICS; but as it is not comprised in the usual idea of\n     Mathematics, and as its discussion would be of but limited utility\n     and interest, it is not included in the present translation.\n\n\n\n\nBOOK I.\n\nANALYSIS.\n\n\n\n\nBOOK I.\n\nANALYSIS.\n\n\n\n\nCHAPTER I.\n\nGENERAL VIEW OF MATHEMATICAL ANALYSIS.\n\n\nIn the historical development of mathematical science since the time of\nDescartes, the advances of its abstract portion have always been\ndetermined by those of its concrete portion; but it is none the less\nnecessary, in order to conceive the science in a manner truly logical,\nto consider the Calculus in all its principal branches before proceeding\nto the philosophical study of Geometry and Mechanics. Its analytical\ntheories, more simple and more general than those of concrete\nmathematics, are in themselves essentially independent of the latter;\nwhile these, on the contrary, have, by their nature, a continual need of\nthe former, without the aid of which they could make scarcely any\nprogress. Although the principal conceptions of analysis retain at\npresent some very perceptible traces of their geometrical or mechanical\norigin, they are now, however, mainly freed from that primitive\ncharacter, which no longer manifests itself except in some secondary\npoints; so that it is possible (especially since the labours of\nLagrange) to present them in a dogmatic exposition, by a purely abstract\nmethod, in a single and continuous system. It is this which will be\nundertaken in the present and the five following chapters, limiting our\ninvestigations to the most general considerations upon each principal\nbranch of the science of the calculus.\n\nThe definite object of our researches in concrete mathematics being the\ndiscovery of the _equations_ which express the mathematical laws of the\nphenomenon under consideration, and these equations constituting the\ntrue starting point of the calculus, which has for its object to obtain\nfrom them the determination of certain quantities by means of others, I\nthink it indispensable, before proceeding any farther, to go more deeply\nthan has been customary into that fundamental idea of _equation_, the\ncontinual subject, either as end or as beginning, of all mathematical\nlabours. Besides the advantage of circumscribing more definitely the\ntrue field of analysis, there will result from it the important\nconsequence of tracing in a more exact manner the real line of\ndemarcation between the concrete and the abstract part of mathematics,\nwhich will complete the general exposition of the fundamental division\nestablished in the introductory chapter.\n\n\n\n\nTHE TRUE IDEA OF AN EQUATION.\n\n\nWe usually form much too vague an idea of what an _equation_ is, when we\ngive that name to every kind of relation of equality between _any_ two\nfunctions of the magnitudes which we are considering. For, though every\nequation is evidently a relation of equality, it is far from being true\nthat, reciprocally, every relation of equality is a veritable\n_equation_, of the kind of those to which, by their nature, the methods\nof analysis are applicable.\n\nThis want of precision in the logical consideration of an idea which is\nso fundamental in mathematics, brings with it the serious inconvenience\nof rendering it almost impossible to explain, in general terms, the\ngreat and fundamental difficulty which we find in establishing the\nrelation between the concrete and the abstract, and which stands out so\nprominently in each great mathematical question taken by itself. If the\nmeaning of the word _equation_ was truly as extended as we habitually\nsuppose it to be in our definition of it, it is not apparent what great\ndifficulty there could really be, in general, in establishing the\nequations of any problem whatsoever; for the whole would thus appear to\nconsist in a simple question of form, which ought never even to exact\nany great intellectual efforts, seeing that we can hardly conceive of\nany precise relation which is not immediately a certain relation of\nequality, or which cannot be readily brought thereto by some very easy\ntransformations.\n\nThus, when we admit every species of _functions_ into the definition of\n_equations_, we do not at all account for the extreme difficulty which\nwe almost always experience in putting a problem into an equation, and\nwhich so often may be compared to the efforts required by the analytical\nelaboration of the equation when once obtained. In a word, the ordinary\nabstract and general idea of an _equation_ does not at all correspond to\nthe real meaning which geometers attach to that expression in the actual\ndevelopment of the science. Here, then, is a logical fault, a defect of\ncorrelation, which it is very important to rectify.\n\n\n_Division of Functions into Abstract and Concrete._ To succeed in doing\nso, I begin by distinguishing two sorts of _functions_, _abstract_ or\nanalytical functions, and _concrete_ functions. The first alone can\nenter into veritable _equations_. We may, therefore, henceforth define\nevery _equation_, in an exact and sufficiently profound manner, as a\nrelation of equality between two _abstract_ functions of the magnitudes\nunder consideration. In order not to have to return again to this\nfundamental definition, I must add here, as an indispensable complement,\nwithout which the idea would not be sufficiently general, that these\nabstract functions may refer not only to the magnitudes which the\nproblem presents of itself, but also to all the other auxiliary\nmagnitudes which are connected with it, and which we will often be able\nto introduce, simply as a mathematical artifice, with the sole object of\nfacilitating the discovery of the equations of the phenomena. I here\nanticipate summarily the result of a general discussion of the highest\nimportance, which will be found at the end of this chapter. We will now\nreturn to the essential distinction of functions as abstract and\nconcrete.\n\nThis distinction may be established in two ways, essentially different,\nbut complementary of each other, _\u00e0 priori_ and _\u00e0 posteriori_; that is\nto say, by characterizing in a general manner the peculiar nature of\neach species of functions, and then by making the actual enumeration of\nall the abstract functions at present known, at least so far as relates\nto the elements of which they are composed.\n\n_\u00c0 priori_, the functions which I call _abstract_ are those which\nexpress a manner of dependence between magnitudes, which can be\nconceived between numbers alone, without there being need of indicating\nany phenomenon whatever in which it is realized. I name, on the other\nhand, _concrete_ functions, those for which the mode of dependence\nexpressed cannot be defined or conceived except by assigning a\ndeterminate case of physics, geometry, mechanics, &c., in which it\nactually exists.\n\nMost functions in their origin, even those which are at present the most\npurely _abstract_, have begun by being _concrete_; so that it is easy to\nmake the preceding distinction understood, by citing only the successive\ndifferent points of view under which, in proportion as the science has\nbecome formed, geometers have considered the most simple analytical\nfunctions. I will indicate powers, for example, which have in general\nbecome abstract functions only since the labours of Vieta and Descartes.\nThe functions _x\u00b2_, _x\u00b3_, which in our present analysis are so well\nconceived as simply _abstract_, were, for the geometers of antiquity,\nperfectly _concrete_ functions, expressing the relation of the\nsuperficies of a square, or the volume of a cube to the length of their\nside. These had in their eyes such a character so exclusively, that it\nwas only by means of the geometrical definitions that they discovered\nthe elementary algebraic properties of these functions, relating to the\ndecomposition of the variable into two parts, properties which were at\nthat epoch only real theorems of geometry, to which a numerical meaning\nwas not attached until long afterward.\n\nI shall have occasion to cite presently, for another reason, a new\nexample, very suitable to make apparent the fundamental distinction\nwhich I have just exhibited; it is that of circular functions, both\ndirect and inverse, which at the present time are still sometimes\nconcrete, sometimes abstract, according to the point of view under\nwhich they are regarded.\n\n_\u00c0 posteriori_, the general character which renders a function abstract\nor concrete having been established, the question as to whether a\ncertain determinate function is veritably abstract, and therefore\nsusceptible of entering into true analytical equations, becomes a simple\nquestion of fact, inasmuch as we are going to enumerate all the\nfunctions of this species.\n\n\n_Enumeration of Abstract Functions._ At first view this enumeration\nseems impossible, the distinct analytical functions being infinite in\nnumber. But when we divide them into _simple_ and _compound_, the\ndifficulty disappears; for, though the number of the different functions\nconsidered in mathematical analysis is really infinite, they are, on the\ncontrary, even at the present day, composed of a very small number of\nelementary functions, which can be easily assigned, and which are\nevidently sufficient for deciding the abstract or concrete character of\nany given function; which will be of the one or the other nature,\naccording as it shall be composed exclusively of these simple abstract\nfunctions, or as it shall include others.\n\nWe evidently have to consider, for this purpose, only the functions of a\nsingle variable, since those relative to several independent variables\nare constantly, by their nature, more or less _compound_.\n\nLet _x_ be the independent variable, _y_ the correlative variable which\ndepends upon it. The different simple modes of abstract dependence,\nwhich we can now conceive between _y_ and _x_, are expressed by the ten\nfollowing elementary formulas, in which each function is coupled with\nits _inverse_, that is, with that which would be obtained from the\ndirect function by referring _x_ to _y_, instead of referring _y_ to\n_x_.\n\n            FUNCTION.                   ITS NAME.\n\n1st couple {1\u00b0 _y_ = _a_ + _x_             _Sum._\n           {2\u00b0 _y_ = _a_ - _x_             _Difference._\n\n2d couple  {1\u00b0 _y_ = _ax_                  _Product._\n           {2\u00b0 _y_ = _a/x_                 _Quotient._\n\n3d couple  {1\u00b0 _y_ = _x^a_                 _Power._\n           {2\u00b0 _y_ = _[ath root]x_            _Root._\n\n4th couple {1\u00b0 _y_ = _a^x_                 _Exponential._\n           {2\u00b0 _y_ = _[log a]x_            _Logarithmic._\n\n5th couple {1\u00b0 _y_ = sin. _x_              _Direct Circular._\n           {2\u00b0 _y_ = arc(sin. = _x_).      _Inverse Circular._[3]\n\n  [Footnote 3: With the view of increasing as much as possible the\n  resources and the extent (now so insufficient) of mathematical\n  analysis, geometers count this last couple of functions among the\n  analytical elements. Although this inscription is strictly\n  legitimate, it is important to remark that circular functions are\n  not exactly in the same situation as the other abstract elementary\n  functions. There is this very essential difference, that the\n  functions of the four first couples are at the same time simple and\n  abstract, while the circular functions, which may manifest each\n  character in succession, according to the point of view under which\n  they are considered and the manner in which they are employed,\n  never present these two properties simultaneously.\n\n  Some other concrete functions may be usefully introduced into the\n  number of analytical elements, certain conditions being fulfilled.\n  It is thus, for example, that the labours of M. Legendre and of M.\n  Jacobi on _elliptical_ functions have truly enlarged the field of\n  analysis; and the same is true of some definite integrals obtained\n  by M. Fourier in the theory of heat.]\n\nSuch are the elements, very few in number, which directly compose all\nthe abstract functions known at the present day. Few as they are, they\nare evidently sufficient to give rise to an infinite number of\nanalytical combinations.\n\nNo rational consideration rigorously circumscribes, _\u00e0 priori_, the\npreceding table, which is only the actual expression of the present\nstate of the science. Our analytical elements are at the present day\nmore numerous than they were for Descartes, and even for Newton and\nLeibnitz: it is only a century since the last two couples have been\nintroduced into analysis by the labours of John Bernouilli and Euler.\nDoubtless new ones will be hereafter admitted; but, as I shall show\ntowards the end of this chapter, we cannot hope that they will ever be\ngreatly multiplied, their real augmentation giving rise to very great\ndifficulties.\n\nWe can now form a definite, and, at the same time, sufficiently extended\nidea of what geometers understand by a veritable _equation_. This\nexplanation is especially suited to make us understand how difficult it\nmust be really to establish the _equations_ of phenomena, since we have\neffectually succeeded in so doing only when we have been able to\nconceive the mathematical laws of these phenomena by the aid of\nfunctions entirely composed of only the mathematical elements which I\nhave just enumerated. It is clear, in fact, that it is then only that\nthe problem becomes truly abstract, and is reduced to a pure question of\nnumbers, these functions being the only simple relations which we can\nconceive between numbers, considered by themselves. Up to this period of\nthe solution, whatever the appearances may be, the question is still\nessentially concrete, and does not come within the domain of the\n_calculus_. Now the fundamental difficulty of this passage from the\n_concrete_ to the _abstract_ in general consists especially in the\ninsufficiency of this very small number of analytical elements which we\npossess, and by means of which, nevertheless, in spite of the little\nreal variety which they offer us, we must succeed in representing all\nthe precise relations which all the different natural phenomena can\nmanifest to us. Considering the infinite diversity which must\nnecessarily exist in this respect in the external world, we easily\nunderstand how far below the true difficulty our conceptions must\nfrequently be found, especially if we add that as these elements of our\nanalysis have been in the first place furnished to us by the\nmathematical consideration of the simplest phenomena, we have, _\u00e0\npriori_, no rational guarantee of their necessary suitableness to\nrepresent the mathematical law of every other class of phenomena. I will\nexplain presently the general artifice, so profoundly ingenious, by\nwhich the human mind has succeeded in diminishing, in a remarkable\ndegree, this fundamental difficulty which is presented by the relation\nof the concrete to the abstract in mathematics, without, however, its\nhaving been necessary to multiply the number of these analytical\nelements.\n\n\n\n\nTHE TWO PRINCIPAL DIVISIONS OF THE CALCULUS.\n\n\nThe preceding explanations determine with precision the true object and\nthe real field of abstract mathematics. I must now pass to the\nexamination of its principal divisions, for thus far we have considered\nthe calculus as a whole.\n\nThe first direct consideration to be presented on the composition of the\nscience of the _calculus_ consists in dividing it, in the first place,\ninto two principal branches, to which, for want of more suitable\ndenominations, I will give the names of _Algebraic calculus_, or\n_Algebra_, and of _Arithmetical calculus_, or _Arithmetic_; but with\nthe caution to take these two expressions in their most extended logical\nacceptation, in the place of the by far too restricted meaning which is\nusually attached to them.\n\nThe complete solution of every question of the _calculus_, from the most\nelementary up to the most transcendental, is necessarily composed of two\nsuccessive parts, whose nature is essentially distinct. In the first,\nthe object is to transform the proposed equations, so as to make\napparent the manner in which the unknown quantities are formed by the\nknown ones: it is this which constitutes the _algebraic_ question. In\nthe second, our object is to _find the values_ of the formulas thus\nobtained; that is, to determine directly the values of the numbers\nsought, which are already represented by certain explicit functions of\ngiven numbers: this is the _arithmetical_ question.[4] It is apparent\nthat, in every solution which is truly rational, it necessarily follows\nthe algebraical question, of which it forms the indispensable\ncomplement, since it is evidently necessary to know the mode of\ngeneration of the numbers sought for before determining their actual\nvalues for each particular case. Thus the stopping-place of the\nalgebraic part of the solution becomes the starting point of the\narithmetical part.\n\n  [Footnote 4: Suppose, for example, that a question gives the\n  following equation between an unknown magnitude x, and two known\n  magnitudes, _a_ and _b_,\n\n  _x\u00b3_ + 3_ax_ = 2_b_,\n\n  as is the case in the problem of the trisection of an angle. We see\n  at once that the dependence between _x_ on the one side, and _ab_ on\n  the other, is completely determined; but, so long as the equation\n  preserves its primitive form, we do not at all perceive in what\n  manner the unknown quantity is derived from the data. This must be\n  discovered, however, before we can think of determining its value.\n  Such is the object of the algebraic part of the solution. When, by a\n  series of transformations which have successively rendered that\n  derivation more and more apparent, we have arrived at presenting the\n  proposed equation under the form\n\n  _x_ = \u221b(_b_ + \u221a(_b\u00b2_ + _a\u00b3_)) + \u221b(_b_ - \u221a(_b\u00b2_ + _a\u00b3_)),\n\n  the work of _algebra_ is finished; and even if we could not perform\n  the arithmetical operations indicated by that formula, we would\n  nevertheless have obtained a knowledge very real, and often very\n  important. The work of _arithmetic_ will now consist in taking that\n  formula for its starting point, and finding the number _x_ when the\n  values of the numbers _a_ and _b_ are given.]\n\nWe thus see that the _algebraic_ calculus and the _arithmetical_\ncalculus differ essentially in their object. They differ no less in the\npoint of view under which they regard quantities; which are considered\nin the first as to their _relations_, and in the second as to their\n_values_. The true spirit of the calculus, in general, requires this\ndistinction to be maintained with the most severe exactitude, and the\nline of demarcation between the two periods of the solution to be\nrendered as clear and distinct as the proposed question permits. The\nattentive observation of this precept, which is too much neglected, may\nbe of much assistance, in each particular question, in directing the\nefforts of our mind, at any moment of the solution, towards the real\ncorresponding difficulty. In truth, the imperfection of the science of\nthe calculus obliges us very often (as will be explained in the next\nchapter) to intermingle algebraic and arithmetical considerations in the\nsolution of the same question. But, however impossible it may be to\nseparate clearly the two parts of the labour, yet the preceding\nindications will always enable us to avoid confounding them.\n\nIn endeavouring to sum up as succinctly as possible the distinction just\nestablished, we see that ALGEBRA may be defined, in general, as having\nfor its object the _resolution of equations_; taking this expression in\nits full logical meaning, which signifies the transformation of\n_implicit_ functions into equivalent _explicit_ ones. In the same way,\nARITHMETIC may be defined as destined to _the determination of the\nvalues of functions_. Henceforth, therefore, we will briefly say that\nALGEBRA is the _Calculus of Functions_, and ARITHMETIC the _Calculus of\nValues_.\n\nWe can now perceive how insufficient and even erroneous are the ordinary\ndefinitions. Most generally, the exaggerated importance attributed to\nSigns has led to the distinguishing the two fundamental branches of the\nscience of the Calculus by the manner of designating in each the\nsubjects of discussion, an idea which is evidently absurd in principle\nand false in fact. Even the celebrated definition given by Newton,\ncharacterizing _Algebra_ as _Universal Arithmetic_, gives certainly a\nvery false idea of the nature of algebra and of that of arithmetic.[5]\n\n  [Footnote 5: I have thought that I ought to specially notice this\n  definition, because it serves as the basis of the opinion which\n  many intelligent persons, unacquainted with mathematical science,\n  form of its abstract part, without considering that at the time of\n  this definition mathematical analysis was not sufficiently\n  developed to enable the general character of each of its principal\n  parts to be properly apprehended, which explains why Newton could\n  at that time propose a definition which at the present day he would\n  certainly reject.]\n\nHaving thus established the fundamental division of the calculus into\ntwo principal branches, I have now to compare in general terms the\nextent, the importance, and the difficulty of these two sorts of\ncalculus, so as to have hereafter to consider only the _Calculus of\nFunctions_, which is to be the principal subject of our study.\n\n\n\n\nTHE CALCULUS OF VALUES, OR ARITHMETIC.\n\n\n_Its Extent._ The _Calculus of Values, or Arithmetic_, would appear, at\nfirst view, to present a field as vast as that of _algebra_, since it\nwould seem to admit as many distinct questions as we can conceive\ndifferent algebraic formulas whose values are to be determined. But a\nvery simple reflection will show the difference. Dividing functions into\n_simple_ and _compound_, it is evident that when we know how to\ndetermine the _value_ of simple functions, the consideration of compound\nfunctions will no longer present any difficulty. In the algebraic point\nof view, a compound function plays a very different part from that of\nthe elementary functions of which it consists, and from this, indeed,\nproceed all the principal difficulties of analysis. But it is very\ndifferent with the Arithmetical Calculus. Thus the number of truly\ndistinct arithmetical operations is only that determined by the number\nof the elementary abstract functions, the very limited list of which has\nbeen given above. The determination of the values of these ten functions\nnecessarily gives that of all the functions, infinite in number, which\nare considered in the whole of mathematical analysis, such at least as\nit exists at present. There can be no new arithmetical operations\nwithout the creation of really new analytical elements, the number of\nwhich must always be extremely small. The field of _arithmetic_ is,\nthen, by its nature, exceedingly restricted, while that of algebra is\nrigorously indefinite.\n\nIt is, however, important to remark, that the domain of the _calculus of\nvalues_ is, in reality, much more extensive than it is commonly\nrepresented; for several questions truly _arithmetical_, since they\nconsist of determinations of values, are not ordinarily classed as such,\nbecause we are accustomed to treat them only as incidental in the midst\nof a body of analytical researches more or less elevated, the too high\nopinion commonly formed of the influence of signs being again the\nprincipal cause of this confusion of ideas. Thus not only the\nconstruction of a table of logarithms, but also the calculation of\ntrigonometrical tables, are true arithmetical operations of a higher\nkind. We may also cite as being in the same class, although in a very\ndistinct and more elevated order, all the methods by which we determine\ndirectly the value of any function for each particular system of values\nattributed to the quantities on which it depends, when we cannot express\nin general terms the explicit form of that function. In this point of\nview the _numerical_ solution of questions which we cannot resolve\nalgebraically, and even the calculation of \"Definite Integrals,\" whose\ngeneral integrals we do not know, really make a part, in spite of all\nappearances, of the domain of _arithmetic_, in which we must necessarily\ncomprise all that which has for its object the _determination of the\nvalues of functions_. The considerations relative to this object are, in\nfact, constantly homogeneous, whatever the _determinations_ in question,\nand are always very distinct from truly _algebraic_ considerations.\n\nTo complete a just idea of the real extent of the calculus of values, we\nmust include in it likewise that part of the general science of the\ncalculus which now bears the name of the _Theory of Numbers_, and which\nis yet so little advanced. This branch, very extensive by its nature,\nbut whose importance in the general system of science is not very\ngreat, has for its object the discovery of the properties inherent in\ndifferent numbers by virtue of their values, and independent of any\nparticular system of numeration. It forms, then, a sort of\n_transcendental arithmetic_; and to it would really apply the definition\nproposed by Newton for algebra.\n\nThe entire domain of arithmetic is, then, much more extended than is\ncommonly supposed; but this _calculus of values_ will still never be\nmore than a point, so to speak, in comparison with the _calculus of\nfunctions_, of which mathematical science essentially consists. This\ncomparative estimate will be still more apparent from some\nconsiderations which I have now to indicate respecting the true nature\nof arithmetical questions in general, when they are more profoundly\nexamined.\n\n\n_Its true Nature._ In seeking to determine with precision in what\n_determinations of values_ properly consist, we easily recognize that\nthey are nothing else but veritable _transformations_ of the functions\nto be valued; transformations which, in spite of their special end, are\nnone the less essentially of the same nature as all those taught by\nanalysis. In this point of view, the _calculus of values_ might be\nsimply conceived as an appendix, and a particular application of the\n_calculus of functions_, so that _arithmetic_ would disappear, so to\nsay, as a distinct section in the whole body of abstract mathematics.\n\nIn order thoroughly to comprehend this consideration, we must observe\nthat, when we propose to determine the _value_ of an unknown number\nwhose mode of formation is given, it is, by the mere enunciation of the\narithmetical question, already defined and expressed under a certain\nform; and that in _determining its value_ we only put its expression\nunder another determinate form, to which we are accustomed to refer the\nexact notion of each particular number by making it re-enter into the\nregular system of _numeration_. The determination of values consists so\ncompletely of a simple _transformation_, that when the primitive\nexpression of the number is found to be already conformed to the regular\nsystem of numeration, there is no longer any determination of value,\nproperly speaking, or, rather, the question is answered by the question\nitself. Let the question be to add the two numbers _one_ and _twenty_,\nwe answer it by merely repeating the enunciation of the question,[6] and\nnevertheless we think that we have _determined the value_ of the sum.\nThis signifies that in this case the first expression of the function\nhad no need of being transformed, while it would not be thus in adding\ntwenty-three and fourteen, for then the sum would not be immediately\nexpressed in a manner conformed to the rank which it occupies in the\nfixed and general scale of numeration.\n\n\n  [Footnote 6: This is less strictly true in the English system of\n  numeration than in the French, since \"twenty-one\" is our more usual\n  mode of expressing this number.]\n\nTo sum up as comprehensively as possible the preceding views, we may\nsay, that to determine the _value_ of a number is nothing else than\nputting its primitive expression under the form\n\n  _a_ + _bz_ + _cz\u00b2_ + _dz\u00b3_ + _ez\u2074_ . . . . . + _pz^m_,\n\n_z_ being generally equal to 10, and the coefficients _a_, _b_, _c_,\n_d_, &c., being subjected to the conditions of being whole numbers less\nthan _z_; capable of becoming equal to zero; but never negative. Every\narithmetical question may thus be stated as consisting in putting under\nsuch a form any abstract function whatever of different quantities,\nwhich are supposed to have themselves a similar form already. We might\nthen see in the different operations of arithmetic only simple\nparticular cases of certain algebraic transformations, excepting the\nspecial difficulties belonging to conditions relating to the nature of\nthe coefficients.\n\nIt clearly follows that abstract mathematics is essentially composed of\nthe _Calculus of Functions_, which had been already seen to be its most\nimportant, most extended, and most difficult part. It will henceforth be\nthe exclusive subject of our analytical investigations. I will therefore\nno longer delay on the _Calculus of Values_, but pass immediately to the\nexamination of the fundamental division of the _Calculus of Functions_.\n\n\n\n\nTHE CALCULUS OF FUNCTIONS, OR ALGEBRA.\n\n\n_Principle of its Fundamental Division._ We have determined, at the\nbeginning of this chapter, wherein properly consists the difficulty\nwhich we experience in putting mathematical questions into _equations_.\nIt is essentially because of the insufficiency of the very small number\nof analytical elements which we possess, that the relation of the\nconcrete to the abstract is usually so difficult to establish. Let us\nendeavour now to appreciate in a philosophical manner the general\nprocess by which the human mind has succeeded, in so great a number of\nimportant cases, in overcoming this fundamental obstacle to _The\nestablishment of Equations_.\n\n\n1. _By the Creation of new Functions._ In looking at this important\nquestion from the most general point of view, we are led at once to the\nconception of one means of facilitating the establishment of the\nequations of phenomena. Since the principal obstacle in this matter\ncomes from the too small number of our analytical elements, the whole\nquestion would seem to be reduced to creating new ones. But this means,\nthough natural, is really illusory; and though it might be useful, it is\ncertainly insufficient.\n\nIn fact, the creation of an elementary abstract function, which shall be\nveritably new, presents in itself the greatest difficulties. There is\neven something contradictory in such an idea; for a new analytical\nelement would evidently not fulfil its essential and appropriate\nconditions, if we could not immediately _determine its value_. Now, on\nthe other hand, how are we to _determine the value_ of a new function\nwhich is truly _simple_, that is, which is not formed by a combination\nof those already known? That appears almost impossible. The introduction\ninto analysis of another elementary abstract function, or rather of\nanother couple of functions (for each would be always accompanied by its\n_inverse_), supposes then, of necessity, the simultaneous creation of a\nnew arithmetical operation, which is certainly very difficult.\n\nIf we endeavour to obtain an idea of the means which the human mind\nemploys for inventing new analytical elements, by the examination of the\nprocedures by the aid of which it has actually conceived those which we\nalready possess, our observations leave us in that respect in an entire\nuncertainty, for the artifices which it has already made use of for that\npurpose are evidently exhausted. To convince ourselves of it, let us\nconsider the last couple of simple functions which has been introduced\ninto analysis, and at the formation of which we have been present, so\nto speak, namely, the fourth couple; for, as I have explained, the fifth\ncouple does not strictly give veritable new analytical elements. The\nfunction _a^x_, and, consequently, its inverse, have been formed by\nconceiving, under a new point of view, a function which had been a long\ntime known, namely, powers--when the idea of them had become\nsufficiently generalized. The consideration of a power relatively to the\nvariation of its exponent, instead of to the variation of its base, was\nsufficient to give rise to a truly novel simple function, the variation\nfollowing then an entirely different route. But this artifice, as simple\nas ingenious, can furnish nothing more; for, in turning over in the same\nmanner all our present analytical elements, we end in only making them\nreturn into one another.\n\nWe have, then, no idea as to how we could proceed to the creation of new\nelementary abstract functions which would properly satisfy all the\nnecessary conditions. This is not to say, however, that we have at\npresent attained the effectual limit established in that respect by the\nbounds of our intelligence. It is even certain that the last special\nimprovements in mathematical analysis have contributed to extend our\nresources in that respect, by introducing within the domain of the\ncalculus certain definite integrals, which in some respects supply the\nplace of new simple functions, although they are far from fulfilling all\nthe necessary conditions, which has prevented me from inserting them in\nthe table of true analytical elements. But, on the whole, I think it\nunquestionable that the number of these elements cannot increase except\nwith extreme slowness. It is therefore not from these sources that the\nhuman mind has drawn its most powerful means of facilitating, as much\nas is possible, the establishment of equations.\n\n\n2. _By the Conception of Equations between certain auxiliary\nQuantities._ This first method being set aside, there remains evidently\nbut one other: it is, seeing the impossibility of finding directly the\nequations between the quantities under consideration, to seek for\ncorresponding ones between other auxiliary quantities, connected with\nthe first according to a certain determinate law, and from the relation\nbetween which we may return to that between the primitive magnitudes.\nSuch is, in substance, the eminently fruitful conception, which the\nhuman mind has succeeded in establishing, and which constitutes its most\nadmirable instrument for the mathematical explanation of natural\nphenomena; the _analysis_, called _transcendental_.\n\nAs a general philosophical principle, the auxiliary quantities, which\nare introduced in the place of the primitive magnitudes, or concurrently\nwith them, in order to facilitate the establishment of equations, might\nbe derived according to any law whatever from the immediate elements of\nthe question. This conception has thus a much more extensive reach than\nhas been commonly attributed to it by even the most profound geometers.\nIt is extremely important for us to view it in its whole logical extent,\nfor it will perhaps be by establishing a general mode of _derivation_\ndifferent from that to which we have thus far confined ourselves\n(although it is evidently very far from being the only possible one)\nthat we shall one day succeed in essentially perfecting mathematical\nanalysis as a whole, and consequently in establishing more powerful\nmeans of investigating the laws of nature than our present processes,\nwhich are unquestionably susceptible of becoming exhausted.\n\nBut, regarding merely the present constitution of the science, the only\nauxiliary quantities habitually introduced in the place of the primitive\nquantities in the _Transcendental Analysis_ are what are called, 1\u2070,\n_infinitely small_ elements, the _differentials_ (of different orders)\nof those quantities, if we regard this analysis in the manner of\nLEIBNITZ; or, 2\u2070, the _fluxions_, the limits of the ratios of the\nsimultaneous increments of the primitive quantities compared with one\nanother, or, more briefly, the _prime and ultimate ratios_ of these\nincrements, if we adopt the conception of NEWTON; or, 3\u2070, the\n_derivatives_, properly so called, of those quantities, that is, the\ncoefficients of the different terms of their respective increments,\naccording to the conception of LAGRANGE.\n\nThese three principal methods of viewing our present transcendental\nanalysis, and all the other less distinctly characterized ones which\nhave been successively proposed, are, by their nature, necessarily\nidentical, whether in the calculation or in the application, as will be\nexplained in a general manner in the third chapter. As to their relative\nvalue, we shall there see that the conception of Leibnitz has thus far,\nin practice, an incontestable superiority, but that its logical\ncharacter is exceedingly vicious; while that the conception of Lagrange,\nadmirable by its simplicity, by its logical perfection, by the\nphilosophical unity which it has established in mathematical analysis\n(till then separated into two almost entirely independent worlds),\npresents, as yet, serious inconveniences in the applications, by\nretarding the progress of the mind. The conception of Newton occupies\nnearly middle ground in these various relations, being less rapid, but\nmore rational than that of Leibnitz; less philosophical, but more\napplicable than that of Lagrange.\n\nThis is not the place to explain the advantages of the introduction of\nthis kind of auxiliary quantities in the place of the primitive\nmagnitudes. The third chapter is devoted to this subject. At present I\nlimit myself to consider this conception in the most general manner, in\norder to deduce therefrom the fundamental division of the _calculus of\nfunctions_ into two systems essentially distinct, whose dependence, for\nthe complete solution of any one mathematical question, is invariably\ndeterminate.\n\nIn this connexion, and in the logical order of ideas, the transcendental\nanalysis presents itself as being necessarily the first, since its\ngeneral object is to facilitate the establishment of equations, an\noperation which must evidently precede the _resolution_ of those\nequations, which is the object of the ordinary analysis. But though it\nis exceedingly important to conceive in this way the true relations of\nthese two systems of analysis, it is none the less proper, in conformity\nwith the regular usage, to study the transcendental analysis after\nordinary analysis; for though the former is, at bottom, by itself\nlogically independent of the latter, or, at least, may be essentially\ndisengaged from it, yet it is clear that, since its employment in the\nsolution of questions has always more or less need of being completed by\nthe use of the ordinary analysis, we would be constrained to leave the\nquestions in suspense if this latter had not been previously studied.\n\n\n_Corresponding Divisions of the Calculus of Functions._ It follows from\nthe preceding considerations that the _Calculus of Functions_, or\n_Algebra_ (taking this word in its most extended meaning), is composed\nof two distinct fundamental branches, one of which has for its immediate\nobject the _resolution_ of equations, when they are directly established\nbetween the magnitudes themselves which are under consideration; and the\nother, starting from equations (generally much easier to form) between\nquantities indirectly connected with those of the problem, has for its\npeculiar and constant destination the deduction, by invariable\nanalytical methods, of the corresponding equations between the direct\nmagnitudes which we are considering; which brings the question within\nthe domain of the preceding calculus.\n\nThe former calculus bears most frequently the name of _Ordinary\nAnalysis_, or of _Algebra_, properly so called. The second constitutes\nwhat is called the _Transcendental Analysis_, which has been designated\nby the different denominations of _Infinitesimal Calculus_, _Calculus of\nFluxions and of Fluents_, _Calculus of Vanishing Quantities_, the\n_Differential and Integral Calculus_, &c., according to the point of\nview in which it has been conceived.\n\nIn order to remove every foreign consideration, I will propose to name\nit CALCULUS OF INDIRECT FUNCTIONS, giving to ordinary analysis the title\nof CALCULUS OF DIRECT FUNCTIONS. These expressions, which I form\nessentially by generalizing and epitomizing the ideas of Lagrange, are\nsimply intended to indicate with precision the true general character\nbelonging to each of these two forms of analysis.\n\nHaving now established the fundamental division of mathematical\nanalysis, I have next to consider separately each of its two parts,\ncommencing with the _Calculus of Direct Functions_, and reserving more\nextended developments for the different branches of the _Calculus of\nIndirect Functions_.\n\n\n\n\nCHAPTER II.\n\nORDINARY ANALYSIS, OR ALGEBRA.\n\n\nThe _Calculus of direct Functions_, or _Algebra_, is (as was shown at\nthe end of the preceding chapter) entirely sufficient for the solution\nof mathematical questions, when they are so simple that we can form\ndirectly the equations between the magnitudes themselves which we are\nconsidering, without its being necessary to introduce in their place, or\nconjointly with them, any system of auxiliary quantities _derived_ from\nthe first. It is true that in the greatest number of important cases its\nuse requires to be preceded and prepared by that of the _Calculus of\nindirect Functions_, which is intended to facilitate the establishment\nof equations. But, although algebra has then only a secondary office to\nperform, it has none the less a necessary part in the complete solution\nof the question, so that the _Calculus of direct Functions_ must\ncontinue to be, by its nature, the fundamental base of all mathematical\nanalysis. We must therefore, before going any further, consider in a\ngeneral manner the logical composition of this calculus, and the degree\nof development to which it has at the present day arrived.\n\n\n_Its Object._ The final object of this calculus being the _resolution_\n(properly so called) of _equations_, that is, the discovery of the\nmanner in which the unknown quantities are formed from the known\nquantities, in accordance with the _equations_ which exist between them,\nit naturally presents as many different departments as we can conceive\ntruly distinct classes of equations. Its appropriate extent is\nconsequently rigorously indefinite, the number of analytical functions\nsusceptible of entering into equations being in itself quite unlimited,\nalthough they are composed of only a very small number of primitive\nelements.\n\n\n_Classification of Equations._ The rational classification of equations\nmust evidently be determined by the nature of the analytical elements of\nwhich their numbers are composed; every other classification would be\nessentially arbitrary. Accordingly, analysts begin by dividing equations\nwith one or more variables into two principal classes, according as they\ncontain functions of only the first three couples (see the table in\nchapter i., page 51), or as they include also exponential or circular\nfunctions. The names of _Algebraic_ functions and _Transcendental_\nfunctions, commonly given to these two principal groups of analytical\nelements, are undoubtedly very inappropriate. But the universally\nestablished division between the corresponding equations is none the\nless very real in this sense, that the resolution of equations\ncontaining the functions called _transcendental_ necessarily presents\nmore difficulties than those of the equations called _algebraic_. Hence\nthe study of the former is as yet exceedingly imperfect, so that\nfrequently the resolution of the most simple of them is still unknown to\nus,[7] and our analytical methods have almost exclusive reference to the\nelaboration of the latter.\n\n  [Footnote 7: Simple as may seem, for example, the equation\n\n  _a^x_ + _b^x_ = _c^x_,\n\n  we do not yet know how to resolve it, which may give some idea of\n  the extreme imperfection of this part of algebra.]\n\n\n\n\nALGEBRAIC EQUATIONS.\n\n\nConsidering now only these _Algebraic_ equations, we must observe, in\nthe first place, that although they may often contain _irrational_\nfunctions of the unknown quantities as well as _rational_ functions, we\ncan always, by more or less easy transformations, make the first case\ncome under the second, so that it is with this last that analysts have\nhad to occupy themselves exclusively in order to resolve all sorts of\n_algebraic_ equations.\n\n\n_Their Classification._ In the infancy of algebra, these equations were\nclassed according to the number of their terms. But this classification\nwas evidently faulty, since it separated cases which were really\nsimilar, and brought together others which had nothing in common besides\nthis unimportant characteristic.[8] It has been retained only for\nequations with two terms, which are, in fact, capable of being resolved\nin a manner peculiar to themselves.\n\n  [Footnote 8: The same error was afterward committed, in the infancy\n  of the infinitesimal calculus, in relation to the integration of\n  differential equations.]\n\nThe classification of equations by what is called their _degrees_, is,\non the other hand, eminently natural, for this distinction rigorously\ndetermines the greater or less difficulty of their _resolution_. This\ngradation is apparent in the cases of all the equations which can be\nresolved; but it may be indicated in a general manner independently of\nthe fact of the resolution. We need only consider that the most general\nequation of each degree necessarily comprehends all those of the\ndifferent inferior degrees, as must also the formula which determines\nthe unknown quantity. Consequently, however slight we may suppose the\ndifficulty peculiar to the _degree_ which we are considering, since it\nis inevitably complicated in the execution with those presented by all\nthe preceding degrees, the resolution really offers more and more\nobstacles, in proportion as the degree of the equation is elevated.\n\n\n\n\nALGEBRAIC RESOLUTION OF EQUATIONS.\n\n\n_Its Limits._ The resolution of algebraic equations is as yet known to\nus only in the four first degrees, such is the increase of difficulty\nnoticed above. In this respect, algebra has made no considerable\nprogress since the labours of Descartes and the Italian analysts of the\nsixteenth century, although in the last two centuries there has been\nperhaps scarcely a single geometer who has not busied himself in trying\nto advance the resolution of equations. The general equation of the\nfifth degree itself has thus far resisted all attacks.\n\nThe constantly increasing complication which the formulas for resolving\nequations must necessarily present, in proportion as the degree\nincreases (the difficulty of using the formula of the fourth degree\nrendering it almost inapplicable), has determined analysts to renounce,\nby a tacit agreement, the pursuit of such researches, although they are\nfar from regarding it as impossible to obtain the resolution of\nequations of the fifth degree, and of several other higher ones.\n\n\n_General Solution._ The only question of this kind which would be really\nof great importance, at least in its logical relations, would be the\ngeneral resolution of algebraic equations of any degree whatsoever. Now,\nthe more we meditate on this subject, the more we are led to think, with\nLagrange, that it really surpasses the scope of our intelligence. We\nmust besides observe that the formula which would express the _root_ of\nan equation of the _m^{th}_ degree would necessarily include radicals of\nthe _m^{th}_ order (or functions of an equivalent multiplicity), because\nof the _m_ determinations which it must admit. Since we have seen,\nbesides, that this formula must also embrace, as a particular case, that\nformula which corresponds to every lower degree, it follows that it\nwould inevitably also contain radicals of the next lower degree, the\nnext lower to that, &c., so that, even if it were possible to discover\nit, it would almost always present too great a complication to be\ncapable of being usefully employed, unless we could succeed in\nsimplifying it, at the same time retaining all its generality, by the\nintroduction of a new class of analytical elements of which we yet have\nno idea. We have, then, reason to believe that, without having already\nhere arrived at the limits imposed by the feeble extent of our\nintelligence, we should not be long in reaching them if we actively and\nearnestly prolonged this series of investigations.\n\nIt is, besides, important to observe that, even supposing we had\nobtained the resolution of _algebraic_ equations of any degree whatever,\nwe would still have treated only a very small part of _algebra_,\nproperly so called, that is, of the calculus of direct functions,\nincluding the resolution of all the equations which can be formed by the\nknown analytical functions.\n\nFinally, we must remember that, by an undeniable law of human nature,\nour means for conceiving new questions being much more powerful than our\nresources for resolving them, or, in other words, the human mind being\nmuch more ready to inquire than to reason, we shall necessarily always\nremain _below_ the difficulty, no matter to what degree of development\nour intellectual labour may arrive. Thus, even though we should some day\ndiscover the complete resolution of all the analytical equations at\npresent known, chimerical as the supposition is, there can be no doubt\nthat, before attaining this end, and probably even as a subsidiary\nmeans, we would have already overcome the difficulty (a much smaller\none, though still very great) of conceiving new analytical elements, the\nintroduction of which would give rise to classes of equations of which,\nat present, we are completely ignorant; so that a similar imperfection\nin algebraic science would be continually reproduced, in spite of the\nreal and very important increase of the absolute mass of our knowledge.\n\n\n_What we know in Algebra._ In the present condition of algebra, the\ncomplete resolution of the equations of the first four degrees, of any\nbinomial equations, of certain particular equations of the higher\ndegrees, and of a very small number of exponential, logarithmic, or\ncircular equations, constitute the fundamental methods which are\npresented by the calculus of direct functions for the solution of\nmathematical problems. But, limited as these elements are, geometers\nhave nevertheless succeeded in treating, in a truly admirable manner, a\nvery great number of important questions, as we shall find in the course\nof the volume. The general improvements introduced within a century into\nthe total system of mathematical analysis, have had for their principal\nobject to make immeasurably useful this little knowledge which we have,\ninstead of tending to increase it. This result has been so fully\nobtained, that most frequently this calculus has no real share in the\ncomplete solution of the question, except by its most simple parts;\nthose which have reference to equations of the two first degrees, with\none or more variables.\n\n\n\n\nNUMERICAL RESOLUTION OF EQUATIONS.\n\n\nThe extreme imperfection of algebra, with respect to the resolution of\nequations, has led analysts to occupy themselves with a new class of\nquestions, whose true character should be here noted. They have busied\nthemselves in filling up the immense gap in the resolution of algebraic\nequations of the higher degrees, by what they have named the _numerical\nresolution_ of equations. Not being able to obtain, in general, the\n_formula_ which expresses what explicit function of the given quantities\nthe unknown one is, they have sought (in the absence of this kind of\nresolution, the only one really _algebraic_) to determine, independently\nof that formula, at least the _value_ of each unknown quantity, for\nvarious designated systems of particular values attributed to the given\nquantities. By the successive labours of analysts, this incomplete and\nillegitimate operation, which presents an intimate mixture of truly\nalgebraic questions with others which are purely arithmetical, has been\nrendered possible in all cases for equations of any degree and even of\nany form. The methods for this which we now possess are sufficiently\ngeneral, although the calculations to which they lead are often so\ncomplicated as to render it almost impossible to execute them. We have\nnothing else to do, then, in this part of algebra, but to simplify the\nmethods sufficiently to render them regularly applicable, which we may\nhope hereafter to effect. In this condition of the calculus of direct\nfunctions, we endeavour, in its application, so to dispose the proposed\nquestions as finally to require only this numerical resolution of the\nequations.\n\n\n_Its limited Usefulness._ Valuable as is such a resource in the absence\nof the veritable solution, it is essential not to misconceive the true\ncharacter of these methods, which analysts rightly regard as a very\nimperfect algebra. In fact, we are far from being always able to reduce\nour mathematical questions to depend finally upon only the _numerical_\nresolution of equations; that can be done only for questions quite\nisolated or truly final, that is, for the smallest number. Most\nquestions, in fact, are only preparatory, and intended to serve as an\nindispensable preparation for the solution of other questions. Now, for\nsuch an object, it is evident that it is not the actual _value_ of the\nunknown quantity which it is important to discover, but the _formula_,\nwhich shows how it is derived from the other quantities under\nconsideration. It is this which happens, for example, in a very\nextensive class of cases, whenever a certain question includes at the\nsame time several unknown quantities. We have then, first of all, to\nseparate them. By suitably employing the simple and general method so\nhappily invented by analysts, and which consists in referring all the\nother unknown quantities to one of them, the difficulty would always\ndisappear if we knew how to obtain the algebraic resolution of the\nequations under consideration, while the _numerical_ solution would then\nbe perfectly useless. It is only for want of knowing the _algebraic_\nresolution of equations with a single unknown quantity, that we are\nobliged to treat _Elimination_ as a distinct question, which forms one\nof the greatest special difficulties of common algebra. Laborious as are\nthe methods by the aid of which we overcome this difficulty, they are\nnot even applicable, in an entirely general manner, to the elimination\nof one unknown quantity between two equations of any form whatever.\n\nIn the most simple questions, and when we have really to resolve only a\nsingle equation with a single unknown quantity, this _numerical_\nresolution is none the less a very imperfect method, even when it is\nstrictly sufficient. It presents, in fact, this serious inconvenience of\nobliging us to repeat the whole series of operations for the slightest\nchange which may take place in a single one of the quantities\nconsidered, although their relations to one another remain unchanged;\nthe calculations made for one case not enabling us to dispense with any\nof those which relate to a case very slightly different. This happens\nbecause of our inability to abstract and treat separately that purely\nalgebraic part of the question which is common to all the cases which\nresult from the mere variation of the given numbers.\n\nAccording to the preceding considerations, the calculus of direct\nfunctions, viewed in its present state, divides into two very distinct\nbranches, according as its subject is the _algebraic_ resolution of\nequations or their _numerical_ resolution. The first department, the\nonly one truly satisfactory, is unhappily very limited, and will\nprobably always remain so; the second, too often insufficient, has, at\nleast, the advantage of a much greater generality. The necessity of\nclearly distinguishing these two parts is evident, because of the\nessentially different object proposed in each, and consequently the\npeculiar point of view under which quantities are therein considered.\n\n\n_Different Divisions of the two Methods of Resolution._ If, moreover, we\nconsider these parts with reference to the different methods of which\neach is composed, we find in their logical distribution an entirely\ndifferent arrangement. In fact, the first part must be divided according\nto the nature of the equations which we are able to resolve, and\nindependently of every consideration relative to the _values_ of the\nunknown quantities. In the second part, on the contrary, it is not\naccording to the _degrees_ of the equations that the methods are\nnaturally distinguished, since they are applicable to equations of any\ndegree whatever; it is according to the numerical character of the\n_values_ of the unknown quantities; for, in calculating these numbers\ndirectly, without deducing them from general formulas, different means\nwould evidently be employed when the numbers are not susceptible of\nhaving their values determined otherwise than by a series of\napproximations, always incomplete, or when they can be obtained with\nentire exactness. This distinction of _incommensurable_ and of\n_commensurable_ roots, which require quite different principles for\ntheir determination, important as it is in the numerical resolution of\nequations, is entirely insignificant in the algebraic resolution, in\nwhich the _rational_ or _irrational_ nature of the numbers which are\nobtained is a mere accident of the calculation, which cannot exercise\nany influence over the methods employed; it is, in a word, a simple\narithmetical consideration. We may say as much, though in a less degree,\nof the division of the commensurable roots themselves into _entire_ and\n_fractional_. In fine, the case is the same, in a still greater degree,\nwith the most general classification of roots, as _real_ and\n_imaginary_. All these different considerations, which are preponderant\nas to the numerical resolution of equations, and which are of no\nimportance in their algebraic resolution, render more and more sensible\nthe essentially distinct nature of these two principal parts of algebra.\n\n\n\n\nTHE THEORY OF EQUATIONS.\n\n\nThese two departments, which constitute the immediate object of the\ncalculus of direct functions, are subordinate to a third one, purely\nspeculative, from which both of them borrow their most powerful\nresources, and which has been very exactly designated by the general\nname of _Theory of Equations_, although it as yet relates only to\n_Algebraic_ equations. The numerical resolution of equations, because of\nits generality, has special need of this rational foundation.\n\nThis last and important branch of algebra is naturally divided into two\norders of questions, viz., those which refer to the _composition_ of\nequations, and those which concern their _transformation_; these latter\nhaving for their object to modify the roots of an equation without\nknowing them, in accordance with any given law, providing that this law\nis uniform in relation to all the parts.[9]\n\n  [Footnote 9: The fundamental principle on which reposes the theory\n  of equations, and which is so frequently applied in all\n  mathematical analysis--the decomposition of algebraic, rational,\n  and entire functions, of any degree whatever, into factors of the\n  first degree--is never employed except for functions of a single\n  variable, without any one having examined if it ought to be\n  extended to functions of several variables. The general\n  impossibility of such a decomposition is demonstrated by the author\n  in detail, but more properly belongs to a special treatise.]\n\n\n\n\nTHE METHOD OF INDETERMINATE COEFFICIENTS.\n\n\nTo complete this rapid general enumeration of the different essential\nparts of the calculus of direct functions, I must, lastly, mention\nexpressly one of the most fruitful and important theories of algebra\nproper, that relating to the transformation of functions into series by\nthe aid of what is called the _Method of indeterminate Coefficients_.\nThis method, so eminently analytical, and which must be regarded as one\nof the most remarkable discoveries of Descartes, has undoubtedly lost\nsome of its importance since the invention and the development of the\ninfinitesimal calculus, the place of which it might so happily take in\nsome particular respects. But the increasing extension of the\ntranscendental analysis, although it has rendered this method much less\nnecessary, has, on the other hand, multiplied its applications and\nenlarged its resources; so that by the useful combination between the\ntwo theories, which has finally been effected, the use of the method of\nindeterminate coefficients has become at present much more extensive\nthan it was even before the formation of the calculus of indirect\nfunctions.\n\n       *       *       *       *       *\n\nHaving thus sketched the general outlines of algebra proper, I have now\nto offer some considerations on several leading points in the calculus\nof direct functions, our ideas of which may be advantageously made more\nclear by a philosophical examination.\n\n\n\n\nIMAGINARY QUANTITIES.\n\n\nThe difficulties connected with several peculiar symbols to which\nalgebraic calculations sometimes lead, and especially to the expressions\ncalled _imaginary_, have been, I think, much exaggerated through purely\nmetaphysical considerations, which have been forced upon them, in the\nplace of regarding these abnormal results in their true point of view as\nsimple analytical facts. Viewing them thus, we readily see that, since\nthe spirit of mathematical analysis consists in considering magnitudes\nin reference to their relations only, and without any regard to their\ndeterminate value, analysts are obliged to admit indifferently every\nkind of expression which can be engendered by algebraic combinations.\nThe interdiction of even one expression because of its apparent\nsingularity would destroy the generality of their conceptions. The\ncommon embarrassment on this subject seems to me to proceed essentially\nfrom an unconscious confusion between the idea of _function_ and the\nidea of _value_, or, what comes to the same thing, between the\n_algebraic_ and the _arithmetical_ point of view. A thorough examination\nwould show mathematical analysis to be much more clear in its nature\nthan even mathematicians commonly suppose.\n\n\n\n\nNEGATIVE QUANTITIES.\n\n\nAs to negative quantities, which have given rise to so many misplaced\ndiscussions, as irrational as useless, we must distinguish between their\n_abstract_ signification and their _concrete_ interpretation, which have\nbeen almost always confounded up to the present day. Under the first\npoint of view, the theory of negative quantities can be established in a\ncomplete manner by a single algebraical consideration. The necessity of\nadmitting such expressions is the same as for imaginary quantities, as\nabove indicated; and their employment as an analytical artifice, to\nrender the formulas more comprehensive, is a mechanism of calculation\nwhich cannot really give rise to any serious difficulty. We may\ntherefore regard the abstract theory of negative quantities as leaving\nnothing essential to desire; it presents no obstacles but those\ninappropriately introduced by sophistical considerations.\n\nIt is far from being so, however, with their concrete theory. This\nconsists essentially in that admirable property of the signs + and-, of\nrepresenting analytically the oppositions of directions of which certain\nmagnitudes are susceptible. This _general theorem_ on the relation of\nthe concrete to the abstract in mathematics is one of the most beautiful\ndiscoveries which we owe to the genius of Descartes, who obtained it as\na simple result of properly directed philosophical observation. A great\nnumber of geometers have since striven to establish directly its general\ndemonstration, but thus far their efforts have been illusory. Their vain\nmetaphysical considerations and heterogeneous minglings of the abstract\nand the concrete have so confused the subject, that it becomes necessary\nto here distinctly enunciate the general fact. It consists in this: if,\nin any equation whatever, expressing the relation of certain quantities\nwhich are susceptible of opposition of directions, one or more of those\nquantities come to be reckoned in a direction contrary to that which\nbelonged to them when the equation was first established, it will not be\nnecessary to form directly a new equation for this second state of the\nphenomena; it will suffice to change, in the first equation, the sign of\neach of the quantities which shall have changed its direction; and the\nequation, thus modified, will always rigorously coincide with that which\nwe would have arrived at in recommencing to investigate, for this new\ncase, the analytical law of the phenomenon. The general theorem consists\nin this constant and necessary coincidence. Now, as yet, no one has\nsucceeded in directly proving this; we have assured ourselves of it only\nby a great number of geometrical and mechanical verifications, which\nare, it is true, sufficiently multiplied, and especially sufficiently\nvaried, to prevent any clear mind from having the least doubt of the\nexactitude and the generality of this essential property, but which, in\na philosophical point of view, do not at all dispense with the research\nfor so important an explanation. The extreme extent of the theorem must\nmake us comprehend both the fundamental difficulties of this research\nand the high utility for the perfecting of mathematical science which\nwould belong to the general conception of this great truth. This\nimperfection of theory, however, has not prevented geometers from making\nthe most extensive and the most important use of this property in all\nparts of concrete mathematics.\n\nIt follows from the above general enunciation of the fact, independently\nof any demonstration, that the property of which we speak must never be\napplied to magnitudes whose directions are continually varying, without\ngiving rise to a simple opposition of direction; in that case, the sign\nwith which every result of calculation is necessarily affected is not\nsusceptible of any concrete interpretation, and the attempts sometimes\nmade to establish one are erroneous. This circumstance occurs, among\nother occasions, in the case of a radius vector in geometry, and\ndiverging forces in mechanics.\n\n\n\n\nPRINCIPLE OF HOMOGENEITY.\n\n\nA second general theorem on the relation of the concrete to the abstract\nis that which is ordinarily designated under the name of _Principle of\nHomogeneity_. It is undoubtedly much less important in its applications\nthan the preceding, but it particularly merits our attention as having,\nby its nature, a still greater extent, since it is applicable to all\nphenomena without distinction, and because of the real utility which it\noften possesses for the verification of their analytical laws. I can,\nmoreover, exhibit a direct and general demonstration of it which seems\nto me very simple. It is founded on this single observation, which is\nself-evident, that the exactitude of every relation between any concrete\nmagnitudes whatsoever is independent of the value of the _units_ to\nwhich they are referred for the purpose of expressing them in numbers.\nFor example, the relation which exists between the three sides of a\nright-angled triangle is the same, whether they are measured by yards,\nor by miles, or by inches.\n\nIt follows from this general consideration, that every equation which\nexpresses the analytical law of any phenomenon must possess this\nproperty of being in no way altered, when all the quantities which are\nfound in it are made to undergo simultaneously the change corresponding\nto that which their respective units would experience. Now this change\nevidently consists in all the quantities of each sort becoming at once\n_m_ times smaller, if the unit which corresponds to them becomes _m_\ntimes greater, or reciprocally. Thus every equation which represents any\nconcrete relation whatever must possess this characteristic of remaining\nthe same, when we make _m_ times greater all the quantities which it\ncontains, and which express the magnitudes between which the relation\nexists; excepting always the numbers which designate simply the mutual\n_ratios_ of these different magnitudes, and which therefore remain\ninvariable during the change of the units. It is this property which\nconstitutes the law of Homogeneity in its most extended signification,\nthat is, of whatever analytical functions the equations may be composed.\n\nBut most frequently we consider only the cases in which the functions\nare such as are called _algebraic_, and to which the idea of _degree_ is\napplicable. In this case we can give more precision to the general\nproposition by determining the analytical character which must be\nnecessarily presented by the equation, in order that this property may\nbe verified. It is easy to see, then, that, by the modification just\nexplained, all the _terms_ of the first degree, whatever may be their\nform, rational or irrational, entire or fractional, will become _m_\ntimes greater; all those of the second degree, _m\u00b2_ times; those of the\nthird, _m\u00b3_ times, &c. Thus the terms of the same degree, however\ndifferent may be their composition, varying in the same manner, and the\nterms of different degrees varying in an unequal proportion, whatever\nsimilarity there may be in their composition, it will be necessary, to\nprevent the equation from being disturbed, that all the terms which it\ncontains should be of the same degree. It is in this that properly\nconsists the ordinary theorem of _Homogeneity_, and it is from this\ncircumstance that the general law has derived its name, which, however,\nceases to be exactly proper for all other functions.\n\nIn order to treat this subject in its whole extent, it is important to\nobserve an essential condition, to which attention must be paid in\napplying this property when the phenomenon expressed by the equation\npresents magnitudes of different natures. Thus it may happen that the\nrespective units are completely independent of each other, and then the\ntheorem of Homogeneity will hold good, either with reference to all the\ncorresponding classes of quantities, or with regard to only a single one\nor more of them. But it will happen on other occasions that the\ndifferent units will have fixed relations to one another, determined by\nthe nature of the question; then it will be necessary to pay attention\nto this subordination of the units in verifying the homogeneity, which\nwill not exist any longer in a purely algebraic sense, and the precise\nform of which will vary according to the nature of the phenomena. Thus,\nfor example, to fix our ideas, when, in the analytical expression of\ngeometrical phenomena, we are considering at once lines, areas, and\nvolumes, it will be necessary to observe that the three corresponding\nunits are necessarily so connected with each other that, according to\nthe subordination generally established in that respect, when the first\nbecomes _m_ times greater, the second becomes _m\u00b2_ times, and the third\n_m\u00b3_ times. It is with such a modification that homogeneity will exist\nin the equations, in which, if they are _algebraic_, we will have to\nestimate the degree of each term by doubling the exponents of the\nfactors which correspond to areas, and tripling those of the factors\nrelating to volumes.\n\n       *       *       *       *       *\n\nSuch are the principal general considerations relating to the _Calculus\nof Direct Functions_. We have now to pass to the philosophical\nexamination of the _Calculus of Indirect Functions_, the much superior\nimportance and extent of which claim a fuller development.\n\n\n\n\nCHAPTER III.\n\nTRANSCENDENTAL ANALYSIS:\n\nDIFFERENT MODES OF VIEWING IT.\n\n\nWe determined, in the second chapter, the philosophical character of the\ntranscendental analysis, in whatever manner it may be conceived,\nconsidering only the general nature of its actual destination as a part\nof mathematical science. This analysis has been presented by geometers\nunder several points of view, really distinct, although necessarily\nequivalent, and leading always to identical results. They may be reduced\nto three principal ones; those of LEIBNITZ, of NEWTON, and of LAGRANGE,\nof which all the others are only secondary modifications. In the present\nstate of science, each of these three general conceptions offers\nessential advantages which pertain to it exclusively, without our having\nyet succeeded in constructing a single method uniting all these\ndifferent characteristic qualities. This combination will probably be\nhereafter effected by some method founded upon the conception of\nLagrange when that important philosophical labour shall have been\naccomplished, the study of the other conceptions will have only a\nhistoric interest; but, until then, the science must be considered as in\nonly a provisional state, which requires the simultaneous consideration\nof all the various modes of viewing this calculus. Illogical as may\nappear this multiplicity of conceptions of one identical subject, still,\nwithout them all, we could form but a very insufficient idea of this\nanalysis, whether in itself, or more especially in relation to its\napplications. This want of system in the most important part of\nmathematical analysis will not appear strange if we consider, on the one\nhand, its great extent and its superior difficulty, and, on the other,\nits recent formation.\n\n\n\n\nITS EARLY HISTORY.\n\n\nIf we had to trace here the systematic history of the successive\nformation of the transcendental analysis, it would be necessary\npreviously to distinguish carefully from the _calculus of indirect\nfunctions_, properly so called, the original idea of the _infinitesimal\nmethod_, which can be conceived by itself, independently of any\n_calculus_. We should see that the first germ of this idea is found in\nthe procedure constantly employed by the Greek geometers, under the name\nof the _Method of Exhaustions_, as a means of passing from the\nproperties of straight lines to those of curves, and consisting\nessentially in substituting for the curve the auxiliary consideration of\nan inscribed or circumscribed polygon, by means of which they rose to\nthe curve itself, taking in a suitable manner the limits of the\nprimitive ratios. Incontestable as is this filiation of ideas, it would\nbe giving it a greatly exaggerated importance to see in this method of\nexhaustions the real equivalent of our modern methods, as some geometers\nhave done; for the ancients had no logical and general means for the\ndetermination of these limits, and this was commonly the greatest\ndifficulty of the question; so that their solutions were not subjected\nto abstract and invariable rules, the uniform application of which would\nlead with certainty to the knowledge sought; which is, on the contrary,\nthe principal characteristic of our transcendental analysis. In a word,\nthere still remained the task of generalizing the conceptions used by\nthe ancients, and, more especially, by considering it in a manner purely\nabstract, of reducing it to a complete system of calculation, which to\nthem was impossible.\n\nThe first idea which was produced in this new direction goes back to the\ngreat geometer Fermat, whom Lagrange has justly presented as having\nblocked out the direct formation of the transcendental analysis by his\nmethod for the determination of _maxima_ and _minima_, and for the\nfinding of _tangents_, which consisted essentially in introducing the\nauxiliary consideration of the correlative increments of the proposed\nvariables, increments afterward suppressed as equal to zero when the\nequations had undergone certain suitable transformations. But, although\nFermat was the first to conceive this analysis in a truly abstract\nmanner, it was yet far from being regularly formed into a general and\ndistinct calculus having its own notation, and especially freed from the\nsuperfluous consideration of terms which, in the analysis of Fermat,\nwere finally not taken into the account, after having nevertheless\ngreatly complicated all the operations by their presence. This is what\nLeibnitz so happily executed, half a century later, after some\nintermediate modifications of the ideas of Fermat introduced by Wallis,\nand still more by Barrow; and he has thus been the true creator of the\ntranscendental analysis, such as we now employ it. This admirable\ndiscovery was so ripe (like all the great conceptions of the human\nintellect at the moment of their manifestation), that Newton, on his\nside, had arrived, at the same time, or a little earlier, at a method\nexactly equivalent, by considering this analysis under a very different\npoint of view, which, although more logical in itself, is really less\nadapted to give to the common fundamental method all the extent and the\nfacility which have been imparted to it by the ideas of Leibnitz.\nFinally, Lagrange, putting aside the heterogeneous considerations which\nhad guided Leibnitz and Newton, has succeeded in reducing the\ntranscendental analysis, in its greatest perfection, to a purely\nalgebraic system, which only wants more aptitude for its practical\napplications.\n\nAfter this summary glance at the general history of the transcendental\nanalysis, we will proceed to the dogmatic exposition of the three\nprincipal conceptions, in order to appreciate exactly their\ncharacteristic properties, and to show the necessary identity of the\nmethods which are thence derived. Let us begin with that of Leibnitz.\n\n\n\n\nMETHOD OF LEIBNITZ.\n\n\n_Infinitely small Elements._ This consists in introducing into the\ncalculus, in order to facilitate the establishment of equations, the\ninfinitely small elements of which all the quantities, the relations\nbetween which are sought, are considered to be composed. These elements\nor _differentials_ will have certain relations to one another, which are\nconstantly and necessarily more simple and easy to discover than those\nof the primitive quantities, and by means of which we will be enabled\n(by a special calculus having for its peculiar object the elimination of\nthese auxiliary infinitesimals) to go back to the desired equations,\nwhich it would have been most frequently impossible to obtain directly.\nThis indirect analysis may have different degrees of indirectness; for,\nwhen there is too much difficulty in forming immediately the equation\nbetween the differentials of the magnitudes under consideration, a\nsecond application of the same general artifice will have to be made,\nand these differentials be treated, in their turn, as new primitive\nquantities, and a relation be sought between their infinitely small\nelements (which, with reference to the final objects of the question,\nwill be _second differentials_), and so on; the same transformation\nadmitting of being repeated any number of times, on the condition of\nfinally eliminating the constantly increasing number of infinitesimal\nquantities introduced as auxiliaries.\n\nA person not yet familiar with these considerations does not perceive at\nonce how the employment of these auxiliary quantities can facilitate the\ndiscovery of the analytical laws of phenomena; for the infinitely small\nincrements of the proposed magnitudes being of the same species with\nthem, it would seem that their relations should not be obtained with\nmore ease, inasmuch as the greater or less value of a quantity cannot,\nin fact, exercise any influence on an inquiry which is necessarily\nindependent, by its nature, of every idea of value. But it is easy,\nnevertheless, to explain very clearly, and in a quite general manner,\nhow far the question must be simplified by such an artifice. For this\npurpose, it is necessary to begin by distinguishing _different orders_\nof infinitely small quantities, a very precise idea of which may be\nobtained by considering them as being either the successive powers of\nthe same primitive infinitely small quantity, or as being quantities\nwhich may be regarded as having finite ratios with these powers; so\nthat, to take an example, the second, third, &c., differentials of any\none variable are classed as infinitely small quantities of the second\norder, the third, &c., because it is easy to discover in them finite\nmultiples of the second, third, &c., powers of a certain first\ndifferential. These preliminary ideas being established, the spirit of\nthe infinitesimal analysis consists in constantly neglecting the\ninfinitely small quantities in comparison with finite quantities, and\ngenerally the infinitely small quantities of any order whatever in\ncomparison with all those of an inferior order. It is at once apparent\nhow much such a liberty must facilitate the formation of equations\nbetween the differentials of quantities, since, in the place of these\ndifferentials, we can substitute such other elements as we may choose,\nand as will be more simple to consider, only taking care to conform to\nthis single condition, that the new elements differ from the preceding\nones only by quantities infinitely small in comparison with them. It is\nthus that it will be possible, in geometry, to treat curved lines as\ncomposed of an infinity of rectilinear elements, curved surfaces as\nformed of plane elements, and, in mechanics, variable motions as an\ninfinite series of uniform motions, succeeding one another at infinitely\nsmall intervals of time.\n\n\nEXAMPLES. Considering the importance of this admirable conception, I\nthink that I ought here to complete the illustration of its fundamental\ncharacter by the summary indication of some leading examples.\n\n\n1. _Tangents._ Let it be required to determine, for each point of a\nplane curve, the equation of which is given, the direction of its\ntangent; a question whose general solution was the primitive object of\nthe inventors of the transcendental analysis. We will consider the\ntangent as a secant joining two points infinitely near to each other;\nand then, designating by _dy_ and _dx_ the infinitely small differences\nof the co-ordinates of those two points, the elementary principles of\ngeometry will immediately give the equation _t_ = _dy_/_dx_ for the\ntrigonometrical tangent of the angle which is made with the axis of the\nabscissas by the desired tangent, this being the most simple way of\nfixing its position in a system of rectilinear co-ordinates. This\nequation, common to all curves, being established, the question is\nreduced to a simple analytical problem, which will consist in\neliminating the infinitesimals _dx_ and _dy_, which were introduced as\nauxiliaries, by determining in each particular case, by means of the\nequation of the proposed curve, the ratio of _dy_ to _dx_, which will be\nconstantly done by uniform and very simple methods.\n\n\n2. _Rectification of an Arc._ In the second place, suppose that we wish\nto know the length of the arc of any curve, considered as a function of\nthe co-ordinates of its extremities. It would be impossible to establish\ndirectly the equation between this arc s and these co-ordinates, while\nit is easy to find the corresponding relation between the differentials\nof these different magnitudes. The most simple theorems of elementary\ngeometry will in fact give at once, considering the infinitely small arc\n_ds_ as a right line, the equations\n\n  _ds\u00b2_ = _dy\u00b2_ + _dx\u00b2_, or _ds\u00b2_ = _dx\u00b2_ + _dy\u00b2_ + _dz\u00b2_,\n\naccording as the curve is of single or double curvature. In either case,\nthe question is now entirely within the domain of analysis, which, by\nthe elimination of the differentials (which is the peculiar object of\nthe calculus of indirect functions), will carry us back from this\nrelation to that which exists between the finite quantities themselves\nunder examination.\n\n\n3. _Quadrature of a Curve._ It would be the same with the quadrature of\ncurvilinear areas. If the curve is a plane one, and referred to\nrectilinear co-ordinates, we will conceive the area A comprised between\nthis curve, the axis of the abscissas, and two extreme co-ordinates, to\nincrease by an infinitely small quantity _d_A, as the result of a\ncorresponding increment of the abscissa. The relation between these two\ndifferentials can be immediately obtained with the greatest facility by\nsubstituting for the curvilinear element of the proposed area the\nrectangle formed by the extreme ordinate and the element of the\nabscissa, from which it evidently differs only by an infinitely small\nquantity of the second order. This will at once give, whatever may be\nthe curve, the very simple differential equation\n\n  _d_A = _ydx_,\n\nfrom which, when the curve is defined, the calculus of indirect\nfunctions will show how to deduce the finite equation, which is the\nimmediate object of the problem.\n\n\n4. _Velocity in Variable Motion._ In like manner, in Dynamics, when we\ndesire to know the expression for the velocity acquired at each instant\nby a body impressed with a motion varying according to any law, we will\nconsider the motion as being uniform during an infinitely small element\nof the time _t_, and we will thus immediately form the differential\nequation _de_ = _vdt_, in which _v_ designates the velocity acquired\nwhen the body has passed over the space _e_; and thence it will be easy\nto deduce, by simple and invariable analytical procedures, the formula\nwhich would give the velocity in each particular motion, in accordance\nwith the corresponding relation between the time and the space; or,\nreciprocally, what this relation would be if the mode of variation of\nthe velocity was supposed to be known, whether with respect to the space\nor to the time.\n\n\n5. _Distribution of Heat._ Lastly, to indicate another kind of\nquestions, it is by similar steps that we are able, in the study of\nthermological phenomena, according to the happy conception of M.\nFourier, to form in a very simple manner the general differential\nequation which expresses the variable distribution of heat in any body\nwhatever, subjected to any influences, by means of the single and\neasily-obtained relation, which represents the uniform distribution of\nheat in a right-angled parallelopipedon, considering (geometrically)\nevery other body as decomposed into infinitely small elements of a\nsimilar form, and (thermologically) the flow of heat as constant during\nan infinitely small element of time. Henceforth, all the questions which\ncan be presented by abstract thermology will be reduced, as in geometry\nand mechanics, to mere difficulties of analysis, which will always\nconsist in the elimination of the differentials introduced as\nauxiliaries to facilitate the establishment of the equations.\n\nExamples of such different natures are more than sufficient to give a\nclear general idea of the immense scope of the fundamental conception of\nthe transcendental analysis as formed by Leibnitz, constituting, as it\nundoubtedly does, the most lofty thought to which the human mind has as\nyet attained.\n\nIt is evident that this conception was indispensable to complete the\nfoundation of mathematical science, by enabling us to establish, in a\nbroad and fruitful manner, the relation of the concrete to the abstract.\nIn this respect it must be regarded as the necessary complement of the\ngreat fundamental idea of Descartes on the general analytical\nrepresentation of natural phenomena: an idea which did not begin to be\nworthily appreciated and suitably employed till after the formation of\nthe infinitesimal analysis, without which it could not produce, even in\ngeometry, very important results.\n\n\n_Generality of the Formulas._ Besides the admirable facility which is\ngiven by the transcendental analysis for the investigation of the\nmathematical laws of all phenomena, a second fundamental and inherent\nproperty, perhaps as important as the first, is the extreme generality\nof the differential formulas, which express in a single equation each\ndeterminate phenomenon, however varied the subjects in relation to which\nit is considered. Thus we see, in the preceding examples, that a single\ndifferential equation gives the tangents of all curves, another their\nrectifications, a third their quadratures; and in the same way, one\ninvariable formula expresses the mathematical law of every variable\nmotion; and, finally, a single equation constantly represents the\ndistribution of heat in any body and for any case. This generality,\nwhich is so exceedingly remarkable, and which is for geometers the basis\nof the most elevated considerations, is a fortunate and necessary\nconsequence of the very spirit of the transcendental analysis,\nespecially in the conception of Leibnitz. Thus the infinitesimal\nanalysis has not only furnished a general method for indirectly forming\nequations which it would have been impossible to discover in a direct\nmanner, but it has also permitted us to consider, for the mathematical\nstudy of natural phenomena, a new order of more general laws, which\nnevertheless present a clear and precise signification to every mind\nhabituated to their interpretation. By virtue of this second\ncharacteristic property, the entire system of an immense science, such\nas geometry or mechanics, has been condensed into a small number of\nanalytical formulas, from which the human mind can deduce, by certain\nand invariable rules, the solution of all particular problems.\n\n\n_Demonstration of the Method._ To complete the general exposition of the\nconception of Leibnitz, there remains to be considered the demonstration\nof the logical procedure to which it leads, and this, unfortunately, is\nthe most imperfect part of this beautiful method.\n\nIn the beginning of the infinitesimal analysis, the most celebrated\ngeometers rightly attached more importance to extending the immortal\ndiscovery of Leibnitz and multiplying its applications than to\nrigorously establishing the logical bases of its operations. They\ncontented themselves for a long time by answering the objections of\nsecond-rate geometers by the unhoped-for solution of the most difficult\nproblems; doubtless persuaded that in mathematical science, much more\nthan in any other, we may boldly welcome new methods, even when their\nrational explanation is imperfect, provided they are fruitful in\nresults, inasmuch as its much easier and more numerous verifications\nwould not permit any error to remain long undiscovered. But this state\nof things could not long exist, and it was necessary to go back to the\nvery foundations of the analysis of Leibnitz in order to prove, in a\nperfectly general manner, the rigorous exactitude of the procedures\nemployed in this method, in spite of the apparent infractions of the\nordinary rules of reasoning which it permitted.\n\nLeibnitz, urged to answer, had presented an explanation entirely\nerroneous, saying that he treated infinitely small quantities as\n_incomparables_, and that he neglected them in comparison with finite\nquantities, \"like grains of sand in comparison with the sea:\" a view\nwhich would have completely changed the nature of his analysis, by\nreducing it to a mere approximative calculus, which, under this point of\nview, would be radically vicious, since it would be impossible to\nforesee, in general, to what degree the successive operations might\nincrease these first errors, which could thus evidently attain any\namount. Leibnitz, then, did not see, except in a very confused manner,\nthe true logical foundations of the analysis which he had created. His\nearliest successors limited themselves, at first, to verifying its\nexactitude by showing the conformity of its results, in particular\napplications, to those obtained by ordinary algebra or the geometry of\nthe ancients; reproducing, according to the ancient methods, so far as\nthey were able, the solutions of some problems after they had been once\nobtained by the new method, which alone was capable of discovering them\nin the first place.\n\nWhen this great question was considered in a more general manner,\ngeometers, instead of directly attacking the difficulty, preferred to\nelude it in some way, as Euler and D'Alembert, for example, have done,\nby demonstrating the necessary and constant conformity of the conception\nof Leibnitz, viewed in all its applications, with other fundamental\nconceptions of the transcendental analysis, that of Newton especially,\nthe exactitude of which was free from any objection. Such a general\nverification is undoubtedly strictly sufficient to dissipate any\nuncertainty as to the legitimate employment of the analysis of Leibnitz.\nBut the infinitesimal method is so important--it offers still, in almost\nall its applications, such a practical superiority over the other\ngeneral conceptions which have been successively proposed--that there\nwould be a real imperfection in the philosophical character of the\nscience if it could not justify itself, and needed to be logically\nfounded on considerations of another order, which would then cease to be\nemployed.\n\nIt was, then, of real importance to establish directly and in a general\nmanner the necessary rationality of the infinitesimal method. After\nvarious attempts more or less imperfect, a distinguished geometer,\nCarnot, presented at last the true direct logical explanation of the\nmethod of Leibnitz, by showing it to be founded on the principle of the\nnecessary compensation of errors, this being, in fact, the precise and\nluminous manifestation of what Leibnitz had vaguely and confusedly\nperceived. Carnot has thus rendered the science an essential service,\nalthough, as we shall see towards the end of this chapter, all this\nlogical scaffolding of the infinitesimal method, properly so called, is\nvery probably susceptible of only a provisional existence, inasmuch as\nit is radically vicious in its nature. Still, we should not fail to\nnotice the general system of reasoning proposed by Carnot, in order to\ndirectly legitimate the analysis of Leibnitz. Here is the substance of\nit:\n\nIn establishing the differential equation of a phenomenon, we\nsubstitute, for the immediate elements of the different quantities\nconsidered, other simpler infinitesimals, which differ from them\ninfinitely little in comparison with them; and this substitution\nconstitutes the principal artifice of the method of Leibnitz, which\nwithout it would possess no real facility for the formation of\nequations. Carnot regards such an hypothesis as really producing an\nerror in the equation thus obtained, and which for this reason he calls\n_imperfect_; only, it is clear that this error must be infinitely small.\nNow, on the other hand, all the analytical operations, whether of\ndifferentiation or of integration, which are performed upon these\ndifferential equations, in order to raise them to finite equations by\neliminating all the infinitesimals which have been introduced as\nauxiliaries, produce as constantly, by their nature, as is easily seen,\nother analogous errors, so that an exact compensation takes place, and\nthe final equations, in the words of Carnot, become _perfect_. Carnot\nviews, as a certain and invariable indication of the actual\nestablishment of this necessary compensation, the complete elimination\nof the various infinitely small quantities, which is always, in fact,\nthe final object of all the operations of the transcendental analysis;\nfor if we have committed no other infractions of the general rules of\nreasoning than those thus exacted by the very nature of the\ninfinitesimal method, the infinitely small errors thus produced cannot\nhave engendered other than infinitely small errors in all the equations,\nand the relations are necessarily of a rigorous exactitude as soon as\nthey exist between finite quantities alone, since the only errors then\npossible must be finite ones, while none such can have entered. All this\ngeneral reasoning is founded on the conception of infinitesimal\nquantities, regarded as indefinitely decreasing, while those from which\nthey are derived are regarded as fixed.\n\n\n_Illustration by Tangents._ Thus, to illustrate this abstract exposition\nby a single example, let us take up again the question of _tangents_,\nwhich is the most easy to analyze completely. We will regard the\nequation _t_ = _dy/dx_, obtained above, as being affected with an\ninfinitely small error, since it would be perfectly rigorous only for\nthe secant. Now let us complete the solution by seeking, according to\nthe equation of each curve, the ratio between the differentials of the\nco-ordinates. If we suppose this equation to be _y_ = _ax\u00b2_, we shall\nevidently have\n\n  _dy_ = 2_axdx_ + _adx\u00b2_.\n\nIn this formula we shall have to neglect the term _dx\u00b2_ as an infinitely\nsmall quantity of the second order. Then the combination of the two\n_imperfect_ equations.\n\n  _t_ = _dy/dx_, _dy_ = 2_ax(dx)_,\n\nbeing sufficient to eliminate entirely the infinitesimals, the finite\nresult, _t_ = 2_ax_, will necessarily be rigorously correct, from the\neffect of the exact compensation of the two errors committed; since, by\nits finite nature, it cannot be affected by an infinitely small error,\nand this is, nevertheless, the only one which it could have, according\nto the spirit of the operations which have been executed.\n\nIt would be easy to reproduce in a uniform manner the same reasoning\nwith reference to all the other general applications of the analysis of\nLeibnitz.\n\nThis ingenious theory is undoubtedly more subtile than solid, when we\nexamine it more profoundly; but it has really no other radical logical\nfault than that of the infinitesimal method itself, of which it is, it\nseems to me, the natural development and the general explanation, so\nthat it must be adopted for as long a time as it shall be thought proper\nto employ this method directly.\n\n       *       *       *       *       *\n\nI pass now to the general exposition of the two other fundamental\nconceptions of the transcendental analysis, limiting myself in each to\nits principal idea, the philosophical character of the analysis having\nbeen sufficiently determined above in the examination of the conception\nof Leibnitz, which I have specially dwelt upon because it admits of\nbeing most easily grasped as a whole, and most rapidly described.\n\n\n\n\nMETHOD OF NEWTON.\n\n\nNewton has successively presented his own method of conceiving the\ntranscendental analysis under several different forms. That which is at\npresent the most commonly adopted was designated by Newton, sometimes\nunder the name of the _Method of prime and ultimate Ratios_, sometimes\nunder that of the _Method of Limits_.\n\n\n_Method of Limits._ The general spirit of the transcendental analysis,\nfrom this point of view, consists in introducing as auxiliaries, in the\nplace of the primitive quantities, or concurrently with them, in order\nto facilitate the establishment of equations, the _limits of the ratios_\nof the simultaneous increments of these quantities; or, in other words,\nthe _final ratios_ of these increments; limits or final ratios which can\nbe easily shown to have a determinate and finite value. A special\ncalculus, which is the equivalent of the infinitesimal calculus, is then\nemployed to pass from the equations between these limits to the\ncorresponding equations between the primitive quantities themselves.\n\nThe power which is given by such an analysis, of expressing with more\nease the mathematical laws of phenomena, depends in general on this,\nthat since the calculus applies, not to the increments themselves of the\nproposed quantities, but to the limits of the ratios of those\nincrements, we can always substitute for each increment any other\nmagnitude more easy to consider, provided that their final ratio is the\nratio of equality, or, in other words, that the limit of their ratio is\nunity. It is clear, indeed, that the calculus of limits would be in no\nway affected by this substitution. Starting from this principle, we find\nnearly the equivalent of the facilities offered by the analysis of\nLeibnitz, which are then merely conceived under another point of view.\nThus curves will be regarded as the _limits_ of a series of rectilinear\npolygons, variable motions as the _limits_ of a collection of uniform\nmotions of constantly diminishing durations, and so on.\n\n\nEXAMPLES. 1. _Tangents._ Suppose, for example, that we wish to determine\nthe direction of the tangent to a curve; we will regard it as the limit\ntowards which would tend a secant, which should turn about the given\npoint so that its second point of intersection should indefinitely\napproach the first. Representing the differences of the co-ordinates of\nthe two points by \u0394_y_ and \u0394_x_, we would have at each instant, for the\ntrigonometrical tangent of the angle which the secant makes with the\naxis of abscissas,\n\n  _t_ = \u0394_y_/\u0394_x_;\n\nfrom which, taking the limits, we will obtain, relatively to the tangent\nitself, this general formula of transcendental analysis,\n\n  _t_ = _L_(\u0394_y_/\u0394_x_),\n\nthe characteristic _L_ being employed to designate the limit. The\ncalculus of indirect functions will show how to deduce from this formula\nin each particular case, when the equation of the curve is given, the\nrelation between _t_ and _x_, by eliminating the auxiliary quantities\nwhich have been introduced. If we suppose, in order to complete the\nsolution, that the equation of the proposed curve is _y_ = _ax\u00b2_, we\nshall evidently have\n\n  \u0394_y_ = 2_ax_\u0394_x_ + _a_(\u0394_x_)\u00b2,\n\nfrom which we shall obtain\n\n  \u0394_y_/\u0394_x_ = 2_ax_ + _a_\u0394_x_.\n\nNow it is clear that the _limit_ towards which the second number tends,\nin proportion as \u0394_x_ diminishes, is 2_ax_. We shall therefore find, by\nthis method, _t_ = 2_ax_, as we obtained it for the same case by the\nmethod of Leibnitz.\n\n2. _Rectifications._ In like manner, when the rectification of a curve\nis desired, we must substitute for the increment of the arc s the chord\nof this increment, which evidently has such a connexion with it that the\nlimit of their ratio is unity; and then we find (pursuing in other\nrespects the same plan as with the method of Leibnitz) this general\nequation of rectifications:\n\n  (_L\u0394s_/\u0394_x_)\u00b2 = 1 + (_L\u0394y_/\u0394_x_)\u00b2,\n  or (_L\u0394s_/\u0394_x_)\u00b2 = 1 + (_L\u0394y_/\u0394_x_)\u00b2 + (_L\u0394z_/\u0394_x_)\u00b2,\n\naccording as the curve is plane or of double curvature. It will now be\nnecessary, for each particular curve, to pass from this equation to that\nbetween the arc and the abscissa, which depends on the transcendental\ncalculus properly so called.\n\nWe could take up, with the same facility, by the method of limits, all\nthe other general questions, the solution of which has been already\nindicated according to the infinitesimal method.\n\nSuch is, in substance, the conception which Newton formed for the\ntranscendental analysis, or, more precisely, that which Maclaurin and\nD'Alembert have presented as the most rational basis of that analysis,\nin seeking to fix and to arrange the ideas of Newton upon that subject.\n\n\n_Fluxions and Fluents._ Another distinct form under which Newton has\npresented this same method should be here noticed, and deserves\nparticularly to fix our attention, as much by its ingenious clearness in\nsome cases as by its having furnished the notation best suited to this\nmanner of viewing the transcendental analysis, and, moreover, as having\nbeen till lately the special form of the calculus of indirect functions\ncommonly adopted by the English geometers. I refer to the calculus of\n_fluxions_ and of _fluents_, founded on the general idea of\n_velocities_.\n\nTo facilitate the conception of the fundamental idea, let us consider\nevery curve as generated by a point impressed with a motion varying\naccording to any law whatever. The different quantities which the curve\ncan present, the abscissa, the ordinate, the arc, the area, &c., will be\nregarded as simultaneously produced by successive degrees during this\nmotion. The _velocity_ with which each shall have been described will be\ncalled the _fluxion_ of that quantity, which will be inversely named its\n_fluent_. Henceforth the transcendental analysis will consist, according\nto this conception, in forming directly the equations between the\nfluxions of the proposed quantities, in order to deduce therefrom, by a\nspecial calculus, the equations between the fluents themselves. What\nhas been stated respecting curves may, moreover, evidently be applied to\nany magnitudes whatever, regarded, by the aid of suitable images, as\nproduced by motion.\n\nIt is easy to understand the general and necessary identity of this\nmethod with that of limits complicated with the foreign idea of motion.\nIn fact, resuming the case of the curve, if we suppose, as we evidently\nalways may, that the motion of the describing point is uniform in a\ncertain direction, that of the abscissa, for example, then the fluxion\nof the abscissa will be constant, like the element of the time; for all\nthe other quantities generated, the motion cannot be conceived to be\nuniform, except for an infinitely small time. Now the velocity being in\ngeneral according to its mechanical conception, the ratio of each space\nto the time employed in traversing it, and this time being here\nproportional to the increment of the abscissa, it follows that the\nfluxions of the ordinate, of the arc, of the area, &c., are really\nnothing else (rejecting the intermediate consideration of time) than the\nfinal ratios of the increments of these different quantities to the\nincrement of the abscissa. This method of fluxions and fluents is, then,\nin reality, only a manner of representing, by a comparison borrowed from\nmechanics, the method of prime and ultimate ratios, which alone can be\nreduced to a calculus. It evidently, then, offers the same general\nadvantages in the various principal applications of the transcendental\nanalysis, without its being necessary to present special proofs of\nthis.\n\n\n\n\nMETHOD OF LAGRANGE.\n\n\n_Derived Functions._ The conception of Lagrange, in its admirable\nsimplicity, consists in representing the transcendental analysis as a\ngreat algebraic artifice, by which, in order to facilitate the\nestablishment of equations, we introduce, in the place of the primitive\nfunctions, or concurrently with them, their _derived_ functions; that\nis, according to the definition of Lagrange, the coefficient of the\nfirst term of the increment of each function, arranged according to the\nascending powers of the increment of its variable. The special calculus\nof indirect functions has for its constant object, here as well as in\nthe conceptions of Leibnitz and of Newton, to eliminate these\n_derivatives_ which have been thus employed as auxiliaries, in order to\ndeduce from their relations the corresponding equations between the\nprimitive magnitudes.\n\n\n_An Extension of ordinary Analysis._ The transcendental analysis is,\nthen, nothing but a simple though very considerable extension of\nordinary analysis. Geometers have long been accustomed to introduce in\nanalytical investigations, in the place of the magnitudes themselves\nwhich they wished to study, their different powers, or their logarithms,\nor their sines, &c., in order to simplify the equations, and even to\nobtain them more easily. This successive _derivation_ is an artifice of\nthe same nature, only of greater extent, and procuring, in consequence,\nmuch more important resources for this common object.\n\nBut, although we can readily conceive, _\u00e0 priori_, that the auxiliary\nconsideration of these derivatives _may_ facilitate the establishment\nof equations, it is not easy to explain why this _must_ necessarily\nfollow from this mode of derivation rather than from any other\ntransformation. Such is the weak point of the great idea of Lagrange.\nThe precise advantages of this analysis cannot as yet be grasped in an\nabstract manner, but only shown by considering separately each principal\nquestion, so that the verification is often exceedingly laborious.\n\n\nEXAMPLE. _Tangents._ This manner of conceiving the transcendental\nanalysis may be best illustrated by its application to the most simple\nof the problems above examined--that of tangents.\n\nInstead of conceiving the tangent as the prolongation of the infinitely\nsmall element of the curve, according to the notion of Leibnitz--or as\nthe limit of the secants, according to the ideas of Newton--Lagrange\nconsiders it, according to its simple geometrical character, analogous\nto the definitions of the ancients, to be a right line such that no\nother right line can pass through the point of contact between it and\nthe curve. Then, to determine its direction, we must seek the general\nexpression of its distance from the curve, measured in any direction\nwhatever--in that of the ordinate, for example--and dispose of the\narbitrary constant relating to the inclination of the right line, which\nwill necessarily enter into that expression, in such a way as to\ndiminish that separation as much as possible. Now this distance, being\nevidently equal to the difference of the two ordinates of the curve and\nof the right line, which correspond to the same new abscissa _x_ + _h_,\nwill be represented by the formula\n\n  (_f'_(_x_) - _t_)_h_ + _qh\u00b2_ + _rh\u00b3_ + etc.,\n\nin which _t_ designates, as above, the unknown trigonometrical tangent\nof the angle which the required line makes with the axis of abscissas,\nand _f'_(_x_) the derived function of the ordinate _f_(_x_). This being\nunderstood, it is easy to see that, by disposing of _t_ so as to make\nthe first term of the preceding formula equal to zero, we will render\nthe interval between the two lines the least possible, so that any other\nline for which _t_ did not have the value thus determined would\nnecessarily depart farther from the proposed curve. We have, then, for\nthe direction of the tangent sought, the general expression _t_ =\n_f'_(_x_), a result exactly equivalent to those furnished by the\nInfinitesimal Method and the Method of Limits. We have yet to find\n_f'_(_x_) in each particular curve, which is a mere question of\nanalysis, quite identical with those which are presented, at this stage\nof the operations, by the other methods.\n\nAfter these considerations upon the principal general conceptions, we\nneed not stop to examine some other theories proposed, such as Euler's\n_Calculus of Vanishing Quantities_, which are really modifications--more\nor less important, and, moreover, no longer used--of the preceding\nmethods.\n\nI have now to establish the comparison and the appreciation of these\nthree fundamental methods. Their _perfect and necessary conformity_ is\nfirst to be proven in a general manner.\n\n\n\n\nFUNDAMENTAL IDENTITY OF THE THREE METHODS.\n\n\nIt is, in the first place, evident from what precedes, considering these\nthree methods as to their actual destination, independently of their\npreliminary ideas, that they all consist in the same general logical\nartifice, which has been characterized in the first chapter; to wit,\nthe introduction of a certain system of auxiliary magnitudes, having\nuniform relations to those which are the special objects of the inquiry,\nand substituted for them expressly to facilitate the analytical\nexpression of the mathematical laws of the phenomena, although they have\nfinally to be eliminated by the aid of a special calculus. It is this\nwhich has determined me to regularly define the transcendental analysis\nas _the calculus of indirect functions_, in order to mark its true\nphilosophical character, at the same time avoiding any discussion upon\nthe best manner of conceiving and applying it. The general effect of\nthis analysis, whatever the method employed, is, then, to bring every\nmathematical question much more promptly within the power of the\n_calculus_, and thus to diminish considerably the serious difficulty\nwhich is usually presented by the passage from the concrete to the\nabstract. Whatever progress we may make, we can never hope that the\ncalculus will ever be able to grasp every question of natural\nphilosophy, geometrical, or mechanical, or thermological, &c.,\nimmediately upon its birth, which would evidently involve a\ncontradiction. Every problem will constantly require a certain\npreliminary labour to be performed, in which the calculus can be of no\nassistance, and which, by its nature, cannot be subjected to abstract\nand invariable rules; it is that which has for its special object the\nestablishment of equations, which form the indispensable starting point\nof all analytical researches. But this preliminary labour has been\nremarkably simplified by the creation of the transcendental analysis,\nwhich has thus hastened the moment at which the solution admits of the\nuniform and precise application of general and abstract methods; by\nreducing, in each case, this special labour to the investigation of\nequations between the auxiliary magnitudes; from which the calculus then\nleads to equations directly referring to the proposed magnitudes, which,\nbefore this admirable conception, it had been necessary to establish\ndirectly and separately. Whether these indirect equations are\n_differential_ equations, according to the idea of Leibnitz, or\nequations of _limits_, conformably to the conception of Newton, or,\nlastly, _derived_ equations, according to the theory of Lagrange, the\ngeneral procedure is evidently always the same.\n\nBut the coincidence of these three principal methods is not limited to\nthe common effect which they produce; it exists, besides, in the very\nmanner of obtaining it. In fact, not only do all three consider, in the\nplace of the primitive magnitudes, certain auxiliary ones, but, still\nfarther, the quantities thus introduced as subsidiary are exactly\nidentical in the three methods, which consequently differ only in the\nmanner of viewing them. This can be easily shown by taking for the\ngeneral term of comparison any one of the three conceptions, especially\nthat of Lagrange, which is the most suitable to serve as a type, as\nbeing the freest from foreign considerations. Is it not evident, by the\nvery definition of _derived functions_, that they are nothing else than\nwhat Leibnitz calls _differential coefficients_, or the ratios of the\ndifferential of each function to that of the corresponding variable,\nsince, in determining the first differential, we will be obliged, by the\nvery nature of the infinitesimal method, to limit ourselves to taking\nthe only term of the increment of the function which contains the first\npower of the infinitely small increment of the variable? In the same\nway, is not the derived function, by its nature, likewise the necessary\n_limit_ towards which tends the ratio between the increment of the\nprimitive function and that of its variable, in proportion as this last\nindefinitely diminishes, since it evidently expresses what that ratio\nbecomes when we suppose the increment of the variable to equal zero?\nThat which is designated by _dx_/_dy_ in the method of Leibnitz; that\nwhich ought to be noted as _L_(\u0394_y_/\u0394_x_) in that of Newton; and that\nwhich Lagrange has indicated by _f'_(_x_), is constantly one same\nfunction, seen from three different points of view, the considerations\nof Leibnitz and Newton properly consisting in making known two general\nnecessary properties of the derived function. The transcendental\nanalysis, examined abstractedly and in its principle, is then always the\nsame, whatever may be the conception which is adopted, and the\nprocedures of the calculus of indirect functions are necessarily\nidentical in these different methods, which in like manner must, for any\napplication whatever, lead constantly to rigorously uniform results.\n\n\n\n\nCOMPARATIVE VALUE OF THE THREE METHODS.\n\n\nIf now we endeavour to estimate the comparative value of these three\nequivalent conceptions, we shall find in each advantages and\ninconveniences which are peculiar to it, and which still prevent\ngeometers from confining themselves to any one of them, considered as\nfinal.\n\n\n_That of Leibnitz._ The conception of Leibnitz presents incontestably,\nin all its applications, a very marked superiority, by leading in a much\nmore rapid manner, and with much less mental effort, to the formation\nof equations between the auxiliary magnitudes. It is to its use that we\nowe the high perfection which has been acquired by all the general\ntheories of geometry and mechanics. Whatever may be the different\nspeculative opinions of geometers with respect to the infinitesimal\nmethod, in an abstract point of view, all tacitly agree in employing it\nby preference, as soon as they have to treat a new question, in order\nnot to complicate the necessary difficulty by this purely artificial\nobstacle proceeding from a misplaced obstinacy in adopting a less\nexpeditious course. Lagrange himself, after having reconstructed the\ntranscendental analysis on new foundations, has (with that noble\nfrankness which so well suited his genius) rendered a striking and\ndecisive homage to the characteristic properties of the conception of\nLeibnitz, by following it exclusively in the entire system of his\n_M\u00e9chanique Analytique_. Such a fact renders any comments unnecessary.\n\nBut when we consider the conception of Leibnitz in itself and in its\nlogical relations, we cannot escape admitting, with Lagrange, that it is\nradically vicious in this, that, adopting its own expressions, the\nnotion of infinitely small quantities is a _false idea_, of which it is\nin fact impossible to obtain a clear conception, however we may deceive\nourselves in that matter. Even if we adopt the ingenious idea of the\ncompensation of errors, as above explained, this involves the radical\ninconvenience of being obliged to distinguish in mathematics two classes\nof reasonings, those which are perfectly rigorous, and those in which we\ndesignedly commit errors which subsequently have to be compensated. A\nconception which leads to such strange consequences is undoubtedly very\nunsatisfactory in a logical point of view.\n\nTo say, as do some geometers, that it is possible in every case to\nreduce the infinitesimal method to that of limits, the logical character\nof which is irreproachable, would evidently be to elude the difficulty\nrather than to remove it; besides, such a transformation almost entirely\nstrips the conception of Leibnitz of its essential advantages of\nfacility and rapidity.\n\nFinally, even disregarding the preceding important considerations, the\ninfinitesimal method would no less evidently present by its nature the\nvery serious defect of breaking the unity of abstract mathematics, by\ncreating a transcendental analysis founded on principles so different\nfrom those which form the basis of the ordinary analysis. This division\nof analysis into two worlds almost entirely independent of each other,\ntends to hinder the formation of truly general analytical conceptions.\nTo fully appreciate the consequences of this, we should have to go back\nto the state of the science before Lagrange had established a general\nand complete harmony between these two great sections.\n\n\n_That of Newton._ Passing now to the conception of Newton, it is evident\nthat by its nature it is not exposed to the fundamental logical\nobjections which are called forth by the method of Leibnitz. The notion\nof _limits_ is, in fact, remarkable for its simplicity and its\nprecision. In the transcendental analysis presented in this manner, the\nequations are regarded as exact from their very origin, and the general\nrules of reasoning are as constantly observed as in ordinary analysis.\nBut, on the other hand, it is very far from offering such powerful\nresources for the solution of problems as the infinitesimal method. The\nobligation which it imposes, of never considering the increments of\nmagnitudes separately and by themselves, nor even in their ratios, but\nonly in the limits of those ratios, retards considerably the operations\nof the mind in the formation of auxiliary equations. We may even say\nthat it greatly embarrasses the purely analytical transformations. Thus\nthe transcendental analysis, considered separately from its\napplications, is far from presenting in this method the extent and the\ngenerality which have been imprinted upon it by the conception of\nLeibnitz. It is very difficult, for example, to extend the theory of\nNewton to functions of several independent variables. But it is\nespecially with reference to its applications that the relative\ninferiority of this theory is most strongly marked.\n\nSeveral Continental geometers, in adopting the method of Newton as the\nmore logical basis of the transcendental analysis, have partially\ndisguised this inferiority by a serious inconsistency, which consists in\napplying to this method the notation invented by Leibnitz for the\ninfinitesimal method, and which is really appropriate to it alone. In\ndesignating by _dy_/_dx_ that which logically ought, in the theory of\nlimits, to be denoted by _L_(\u0394_y_/\u0394_x_), and in extending to all the\nother analytical conceptions this displacement of signs, they intended,\nundoubtedly, to combine the special advantages of the two methods; but,\nin reality, they have only succeeded in causing a vicious confusion\nbetween them, a familiarity with which hinders the formation of clear\nand exact ideas of either. It would certainly be singular, considering\nthis usage in itself, that, by the mere means of signs, it could be\npossible to effect a veritable combination between two theories so\ndistinct as those under consideration.\n\nFinally, the method of limits presents also, though in a less degree,\nthe greater inconvenience, which I have above noted in reference to the\ninfinitesimal method, of establishing a total separation between the\nordinary and the transcendental analysis; for the idea of _limits_,\nthough clear and rigorous, is none the less in itself, as Lagrange has\nremarked, a foreign idea, upon which analytical theories ought not to be\ndependent.\n\n\n_That of Lagrange._ This perfect unity of analysis, and this purely\nabstract character of its fundamental notions, are found in the highest\ndegree in the conception of Lagrange, and are found there alone; it is,\nfor this reason, the most rational and the most philosophical of all.\nCarefully removing every heterogeneous consideration, Lagrange has\nreduced the transcendental analysis to its true peculiar character, that\nof presenting a very extensive class of analytical transformations,\nwhich facilitate in a remarkable degree the expression of the conditions\nof various problems. At the same time, this analysis is thus necessarily\npresented as a simple extension of ordinary analysis; it is only a\nhigher algebra. All the different parts of abstract mathematics,\npreviously so incoherent, have from that moment admitted of being\nconceived as forming a single system.\n\nUnhappily, this conception, which possesses such fundamental properties,\nindependently of its so simple and so lucid notation, and which is\nundoubtedly destined to become the final theory of transcendental\nanalysis, because of its high philosophical superiority over all the\nother methods proposed, presents in its present state too many\ndifficulties in its applications, as compared with the conception of\nNewton, and still more with that of Leibnitz, to be as yet exclusively\nadopted. Lagrange himself has succeeded only with great difficulty in\nrediscovering, by his method, the principal results already obtained by\nthe infinitesimal method for the solution of the general questions of\ngeometry and mechanics; we may judge from that what obstacles would be\nfound in treating in the same manner questions which were truly new and\nimportant. It is true that Lagrange, on several occasions, has shown\nthat difficulties call forth, from men of genius, superior efforts,\ncapable of leading to the greatest results. It was thus that, in trying\nto adapt his method to the examination of the curvature of lines, which\nseemed so far from admitting its application, he arrived at that\nbeautiful theory of contacts which has so greatly perfected that\nimportant part of geometry. But, in spite of such happy exceptions, the\nconception of Lagrange has nevertheless remained, as a whole,\nessentially unsuited to applications.\n\nThe final result of the general comparison which I have too briefly\nsketched, is, then, as already suggested, that, in order to really\nunderstand the transcendental analysis, we should not only consider it\nin its principles according to the three fundamental conceptions of\nLeibnitz, of Newton, and of Lagrange, but should besides accustom\nourselves to carry out almost indifferently, according to these three\nprincipal methods, and especially according to the first and the last,\nthe solution of all important questions, whether of the pure calculus of\nindirect functions or of its applications. This is a course which I\ncould not too strongly recommend to all those who desire to judge\nphilosophically of this admirable creation of the human mind, as well as\nto those who wish to learn to make use of this powerful instrument with\nsuccess and with facility. In all the other parts of mathematical\nscience, the consideration of different methods for a single class of\nquestions may be useful, even independently of its historical interest,\nbut it is not indispensable; here, on the contrary, it is strictly\nnecessary.\n\nHaving determined with precision, in this chapter, the philosophical\ncharacter of the calculus of indirect functions, according to the\nprincipal fundamental conceptions of which it admits, we have next to\nconsider, in the following chapter, the logical division and the general\ncomposition of this calculus.\n\n\n\n\nCHAPTER IV.\n\n\nTHE DIFFERENTIAL AND INTEGRAL CALCULUS.\n\n\nITS TWO FUNDAMENTAL DIVISIONS.\n\n\nThe _calculus of indirect functions_, in accordance with the\nconsiderations explained in the preceding chapter, is necessarily\ndivided into two parts (or, more properly, is decomposed into two\ndifferent _calculi_ entirely distinct, although intimately connected by\ntheir nature), according as it is proposed to find the relations between\nthe auxiliary magnitudes (the introduction of which constitutes the\ngeneral spirit of this calculus) by means of the relations between the\ncorresponding primitive magnitudes; or, conversely, to try to discover\nthese direct equations by means of the indirect equations originally\nestablished. Such is, in fact, constantly the double object of the\ntranscendental analysis.\n\nThese two systems have received different names, according to the point\nof view under which this analysis has been regarded. The infinitesimal\nmethod, properly so called, having been the most generally employed for\nthe reasons which have been given, almost all geometers employ\nhabitually the denominations of _Differential Calculus_ and of _Integral\nCalculus_, established by Leibnitz, and which are, in fact, very\nrational consequences of his conception. Newton, in accordance with his\nmethod, named the first the _Calculus of Fluxions_, and the second the\n_Calculus of Fluents_, expressions which were commonly employed in\nEngland. Finally, following the eminently philosophical theory founded\nby Lagrange, one would be called the _Calculus of Derived Functions_,\nand the other the _Calculus of Primitive Functions_. I will continue to\nmake use of the terms of Leibnitz, as being more convenient for the\nformation of secondary expressions, although I ought, in accordance with\nthe suggestions made in the preceding chapter, to employ concurrently\nall the different conceptions, approaching as nearly as possible to that\nof Lagrange.\n\n\n\n\nTHEIR RELATIONS TO EACH OTHER.\n\n\nThe differential calculus is evidently the logical basis of the integral\ncalculus; for we do not and cannot know how to integrate directly any\nother differential expressions than those produced by the\ndifferentiation of the ten simple functions which constitute the general\nelements of our analysis. The art of integration consists, then,\nessentially in bringing all the other cases, as far as is possible, to\nfinally depend on only this small number of fundamental integrations.\n\nIn considering the whole body of the transcendental analysis, as I have\ncharacterized it in the preceding chapter, it is not at first apparent\nwhat can be the peculiar utility of the differential calculus,\nindependently of this necessary relation with the integral calculus,\nwhich seems as if it must be, by itself, the only one directly\nindispensable. In fact, the elimination of the _infinitesimals_ or of\nthe _derivatives_, introduced as auxiliaries to facilitate the\nestablishment of equations, constituting, as we have seen, the final and\ninvariable object of the calculus of indirect functions, it is natural\nto think that the calculus which teaches how to deduce from the\nequations between these auxiliary magnitudes, those which exist between\nthe primitive magnitudes themselves, ought strictly to suffice for the\ngeneral wants of the transcendental analysis without our perceiving, at\nthe first glance, what special and constant part the solution of the\ninverse question can have in such an analysis. It would be a real error,\nthough a common one, to assign to the differential calculus, in order to\nexplain its peculiar, direct, and necessary influence, the destination\nof forming the differential equations, from which the integral calculus\nthen enables us to arrive at the finite equations; for the primitive\nformation of differential equations is not and cannot be, properly\nspeaking, the object of any calculus, since, on the contrary, it forms\nby its nature the indispensable starting point of any calculus whatever.\nHow, in particular, could the differential calculus, which in itself is\nreduced to teaching the means of _differentiating_ the different\nequations, be a general procedure for establishing them? That which in\nevery application of the transcendental analysis really facilitates the\nformation of equations, is the infinitesimal _method_, and not the\ninfinitesimal _calculus_, which is perfectly distinct from it, although\nit is its indispensable complement. Such a consideration would, then,\ngive a false idea of the special destination which characterizes the\ndifferential calculus in the general system of the transcendental\nanalysis.\n\nBut we should nevertheless very imperfectly conceive the real peculiar\nimportance of this first branch of the calculus of indirect functions,\nif we saw in it only a simple preliminary labour, having no other\ngeneral and essential object than to prepare indispensable foundations\nfor the integral calculus. As the ideas on this matter are generally\nconfused, I think that I ought here to explain in a summary manner this\nimportant relation as I view it, and to show that in every application\nof the transcendental analysis a primary, direct, and necessary part is\nconstantly assigned to the differential calculus.\n\n\n1. _Use of the Differential Calculus as preparatory to that of the\nIntegral._ In forming the differential equations of any phenomenon\nwhatever, it is very seldom that we limit ourselves to introduce\ndifferentially only those magnitudes whose relations are sought. To\nimpose that condition would be to uselessly diminish the resources\npresented by the transcendental analysis for the expression of the\nmathematical laws of phenomena. Most frequently we introduce into the\nprimitive equations, through their differentials, other magnitudes whose\nrelations are already known or supposed to be so, and without the\nconsideration of which it would be frequently impossible to establish\nequations. Thus, for example, in the general problem of the\nrectification of curves, the differential equation,\n\n  _ds_\u00b2 = _dy_\u00b2 + _dx_\u00b2, or _ds_\u00b2 = _dx_\u00b2 + _dy_\u00b2 + _dz_\u00b2,\n\nis not only established between the desired function s and the\nindependent variable _x_, to which it is referred, but, at the same\ntime, there have been introduced, as indispensable intermediaries, the\ndifferentials of one or two other functions, _y_ and _z_, which are\namong the data of the problem; it would not have been possible to form\ndirectly the equation between _ds_ and _dx_, which would, besides, be\npeculiar to each curve considered. It is the same for most questions.\nNow in these cases it is evident that the differential equation is not\nimmediately suitable for integration. It is previously necessary that\nthe differentials of the functions supposed to be known, which have\nbeen employed as intermediaries, should be entirely eliminated, in order\nthat equations may be obtained between the differentials of the\nfunctions which alone are sought and those of the really independent\nvariables, after which the question depends on only the integral\ncalculus. Now this preparatory elimination of certain differentials, in\norder to reduce the infinitesimals to the smallest number possible,\nbelongs simply to the differential calculus; for it must evidently be\ndone by determining, by means of the equations between the functions\nsupposed to be known, taken as intermediaries, the relations of their\ndifferentials, which is merely a question of differentiation. Thus, for\nexample, in the case of rectifications, it will be first necessary to\ncalculate _dy_, or _dy_ and _dz_, by differentiating the equation or the\nequations of each curve proposed; after eliminating these expressions,\nthe general differential formula above enunciated will then contain only\n_ds_ and _dx_; having arrived at this point, the elimination of the\ninfinitesimals can be completed only by the integral calculus.\n\nSuch is, then, the general office necessarily belonging to the\ndifferential calculus in the complete solution of the questions which\nexact the employment of the transcendental analysis; to produce, as far\nas is possible, the elimination of the infinitesimals, that is, to\nreduce in each case the primitive differential equations so that they\nshall contain only the differentials of the really independent\nvariables, and those of the functions sought, by causing to disappear,\nby elimination, the differentials of all the other known functions which\nmay have been taken as intermediaries at the time of the formation of\nthe differential equations of the problem which is under consideration.\n\n\n2. _Employment of the Differential Calculus alone._ For certain\nquestions, which, although few in number, have none the less, as we\nshall see hereafter, a very great importance, the magnitudes which are\nsought enter directly, and not by their differentials, into the\nprimitive differential equations, which then contain differentially only\nthe different known functions employed as intermediaries, in accordance\nwith the preceding explanation. These cases are the most favourable of\nall; for it is evident that the differential calculus is then entirely\nsufficient for the complete elimination of the infinitesimals, without\nthe question giving rise to any integration. This is what occurs, for\nexample, in the problem of _tangents_ in geometry; in that of\n_velocities_ in mechanics, &c.\n\n\n3. _Employment of the Integral Calculus alone._ Finally, some other\nquestions, the number of which is also very small, but the importance of\nwhich is no less great, present a second exceptional case, which is in\nits nature exactly the converse of the preceding. They are those in\nwhich the differential equations are found to be immediately ready for\nintegration, because they contain, at their first formation, only the\ninfinitesimals which relate to the functions sought, or to the really\nindependent variables, without its being necessary to introduce,\ndifferentially, other functions as intermediaries. If in these new cases\nwe introduce these last functions, since, by hypothesis, they will enter\ndirectly and not by their differentials, ordinary algebra will suffice\nto eliminate them, and to bring the question to depend on only the\nintegral calculus. The differential calculus will then have no special\npart in the complete solution of the problem, which will depend entirely\nupon the integral calculus. The general question of _quadratures_ offers\nan important example of this, for the differential equation being then\n_dA = ydx_, will become immediately fit for integration as soon as we\nshall have eliminated, by means of the equation of the proposed curve,\nthe intermediary function _y_, which does not enter into it\ndifferentially. The same circumstances exist in the problem of\n_cubatures_, and in some others equally important.\n\n\n_Three classes of Questions hence resulting._ As a general result of the\nprevious considerations, it is then necessary to divide into three\nclasses the mathematical questions which require the use of the\ntranscendental analysis; the _first_ class comprises the problems\nsusceptible of being entirely resolved by means of the differential\ncalculus alone, without any need of the integral calculus; the _second_,\nthose which are, on the contrary, entirely dependent upon the integral\ncalculus, without the differential calculus having any part in their\nsolution; lastly, in the _third_ and the most extensive, which\nconstitutes the normal case, the two others being only exceptional, the\ndifferential and the integral calculus have each in their turn a\ndistinct and necessary part in the complete solution of the problem, the\nformer making the primitive differential equations undergo a preparation\nwhich is indispensable for the application of the latter. Such are\nexactly their general relations, of which too indefinite and inexact\nideas are generally formed.\n\n       *       *       *       *       *\n\nLet us now take a general survey of the logical composition of each\ncalculus, beginning with the differential.\n\n\n\n\nTHE DIFFERENTIAL CALCULUS.\n\n\nIn the exposition of the transcendental analysis, it is customary to\nintermingle with the purely analytical part (which reduces itself to the\ntreatment of the abstract principles of differentiation and integration)\nthe study of its different principal applications, especially those\nwhich concern geometry. This confusion of ideas, which is a consequence\nof the actual manner in which the science has been developed, presents,\nin the dogmatic point of view, serious inconveniences in this respect,\nthat it makes it difficult properly to conceive either analysis or\ngeometry. Having to consider here the most rational co-ordination which\nis possible, I shall include, in the following sketch, only the calculus\nof indirect functions properly so called, reserving for the portion of\nthis volume which relates to the philosophical study of _concrete_\nmathematics the general examination of its great geometrical and\nmechanical applications.\n\n\n_Two Cases: explicit and implicit Functions._ The fundamental division\nof the differential calculus, or of the general subject of\ndifferentiation, consists in distinguishing two cases, according as the\nanalytical functions which are to be differentiated are _explicit_ or\n_implicit_; from which flow two parts ordinarily designated by the names\nof differentiation _of formulas_ and differentiation _of equations_. It\nis easy to understand, _\u00e0 priori_, the importance of this\nclassification. In fact, such a distinction would be illusory if the\nordinary analysis was perfect; that is, if we knew how to resolve all\nequations algebraically, for then it would be possible to render every\n_implicit_ function _explicit_; and, by differentiating it in that\nstate alone, the second part of the differential calculus would be\nimmediately comprised in the first, without giving rise to any new\ndifficulty. But the algebraical resolution of equations being, as we\nhave seen, still almost in its infancy, and as yet impossible for most\ncases, it is plain that the case is very different, since we have,\nproperly speaking, to differentiate a function without knowing it,\nalthough it is determinate. The differentiation of implicit functions\nconstitutes then, by its nature, a question truly distinct from that\npresented by explicit functions, and necessarily more complicated. It is\nthus evident that we must commence with the differentiation of formulas,\nand reduce the differentiation of equations to this primary case by\ncertain invariable analytical considerations, which need not be here\nmentioned.\n\nThese two general cases of differentiation are also distinct in another\npoint of view equally necessary, and too important to be left unnoticed.\nThe relation which is obtained between the differentials is constantly\nmore indirect, in comparison with that of the finite quantities, in the\ndifferentiation of implicit functions than in that of explicit\nfunctions. We know, in fact, from the considerations presented by\nLagrange on the general formation of differential equations, that, on\nthe one hand, the same primitive equation may give rise to a greater or\nless number of derived equations of very different forms, although at\nbottom equivalent, depending upon which of the arbitrary constants is\neliminated, which is not the case in the differentiation of explicit\nformulas; and that, on the other hand, the unlimited system of the\ndifferent primitive equations, which correspond to the same derived\nequation, presents a much more profound analytical variety than that of\nthe different functions, which admit of one same explicit differential,\nand which are distinguished from each other only by a constant term.\nImplicit functions must therefore be regarded as being in reality still\nmore modified by differentiation than explicit functions. We shall again\nmeet with this consideration relatively to the integral calculus, where\nit acquires a preponderant importance.\n\n\n_Two Sub-cases: A single Variable or several Variables._ Each of the two\nfundamental parts of the Differential Calculus is subdivided into two\nvery distinct theories, according as we are required to differentiate\nfunctions of a single variable or functions of several independent\nvariables. This second case is, by its nature, quite distinct from the\nfirst, and evidently presents more complication, even in considering\nonly explicit functions, and still more those which are implicit. As to\nthe rest, one of these cases is deduced from the other in a general\nmanner, by the aid of an invariable and very simple principle, which\nconsists in regarding the total differential of a function which is\nproduced by the simultaneous increments of the different independent\nvariables which it contains, as the sum of the partial differentials\nwhich would be produced by the separate increment of each variable in\nturn, if all the others were constant. It is necessary, besides,\ncarefully to remark, in connection with this subject, a new idea which\nis introduced by the distinction of functions into those of one variable\nand of several; it is the consideration of these different special\nderived functions, relating to each variable separately, and the number\nof which increases more and more in proportion as the order of the\nderivation becomes higher, and also when the variables become more\nnumerous. It results from this that the differential relations belonging\nto functions of several variables are, by their nature, both much more\nindirect, and especially much more indeterminate, than those relating to\nfunctions of a single variable. This is most apparent in the case of\nimplicit functions, in which, in the place of the simple arbitrary\nconstants which elimination causes to disappear when we form the proper\ndifferential equations for functions of a single variable, it is the\narbitrary functions of the proposed variables which are then eliminated;\nwhence must result special difficulties when these equations come to be\nintegrated.\n\nFinally, to complete this summary sketch of the different essential\nparts of the differential calculus proper, I should add, that in the\ndifferentiation of implicit functions, whether of a single variable or\nof several, it is necessary to make another distinction; that of the\ncase in which it is required to differentiate at once different\nfunctions of this kind, _combined_ in certain primitive equations, from\nthat in which all these functions are _separate_.\n\nThe functions are evidently, in fact, still more implicit in the first\ncase than in the second, if we consider that the same imperfection of\nordinary analysis, which forbids our converting every implicit function\ninto an equivalent explicit function, in like manner renders us unable\nto separate the functions which enter simultaneously into any system of\nequations. It is then necessary to differentiate, not only without\nknowing how to resolve the primitive equations, but even without being\nable to effect the proper eliminations among them, thus producing a new\ndifficulty.\n\n\n_Reduction of the whole to the Differentiation of the ten elementary\nFunctions._ Such, then, are the natural connection and the logical\ndistribution of the different principal theories which compose the\ngeneral system of differentiation. Since the differentiation of implicit\nfunctions is deduced from that of explicit functions by a single\nconstant principle, and the differentiation of functions of several\nvariables is reduced by another fixed principle to that of functions of\na single variable, the whole of the differential calculus is finally\nfound to rest upon the differentiation of explicit functions with a\nsingle variable, the only one which is ever executed directly. Now it is\neasy to understand that this first theory, the necessary basis of the\nentire system, consists simply in the differentiation of the ten simple\nfunctions, which are the uniform elements of all our analytical\ncombinations, and the list of which has been given in the first chapter,\non page 51; for the differentiation of compound functions is evidently\ndeduced, in an immediate and necessary manner, from that of the simple\nfunctions which compose them. It is, then, to the knowledge of these ten\nfundamental differentials, and to that of the two general principles\njust mentioned, which bring under it all the other possible cases, that\nthe whole system of differentiation is properly reduced. We see, by the\ncombination of these different considerations, how simple and how\nperfect is the entire system of the differential calculus. It certainly\nconstitutes, in its logical relations, the most interesting spectacle\nwhich mathematical analysis can present to our understanding.\n\n\n_Transformation of derived Functions for new Variables._ The general\nsketch which I have just summarily drawn would nevertheless present an\nimportant deficiency, if I did not here distinctly indicate a final\ntheory, which forms, by its nature, the indispensable complement of the\nsystem of differentiation. It is that which has for its object the\nconstant transformation of derived functions, as a result of determinate\nchanges in the independent variables, whence results the possibility of\nreferring to new variables all the general differential formulas\nprimitively established for others. This question is now resolved in the\nmost complete and the most simple manner, as are all those of which the\ndifferential calculus is composed. It is easy to conceive the general\nimportance which it must have in any of the applications of the\ntranscendental analysis, the fundamental resources of which it may be\nconsidered as augmenting, by permitting us to choose (in order to form\nthe differential equations, in the first place, with more ease) that\nsystem of independent variables which may appear to be the most\nadvantageous, although it is not to be finally retained. It is thus, for\nexample, that most of the principal questions of geometry are resolved\nmuch more easily by referring the lines and surfaces to _rectilinear_\nco-ordinates, and that we may, nevertheless, have occasion to express\nthese lines, etc., analytically by the aid of _polar_ co-ordinates, or\nin any other manner. We will then be able to commence the differential\nsolution of the problem by employing the rectilinear system, but only as\nan intermediate step, from which, by the general theory here referred\nto, we can pass to the final system, which sometimes could not have been\nconsidered directly.\n\n\n_Different Orders of Differentiation._ In the logical classification of\nthe differential calculus which has just been given, some may be\ninclined to suggest a serious omission, since I have not subdivided each\nof its four essential parts according to another general consideration,\nwhich seems at first view very important; namely, that of the higher or\nlower order of differentiation. But it is easy to understand that this\ndistinction has no real influence in the differential calculus, inasmuch\nas it does not give rise to any new difficulty. If, indeed, the\ndifferential calculus was not rigorously complete, that is, if we did\nnot know how to differentiate at will any function whatever, the\ndifferentiation to the second or higher order of each determinate\nfunction might engender special difficulties. But the perfect\nuniversality of the differential calculus plainly gives us the assurance\nof being able to differentiate, to any order whatever, all known\nfunctions whatever, the question reducing itself to a constantly\nrepeated differentiation of the first order. This distinction,\nunimportant as it is for the differential calculus, acquires, however, a\nvery great importance in the integral calculus, on account of the\nextreme imperfection of the latter.\n\n\n_Analytical Applications._ Finally, though this is not the place to\nconsider the various applications of the differential calculus, yet an\nexception may be made for those which consist in the solution of\nquestions which are purely analytical, which ought, indeed, to be\nlogically treated in continuation of a system of differentiation,\nbecause of the evident homogeneity of the considerations involved. These\nquestions may be reduced to three essential ones.\n\nFirstly, the _development into series_ of functions of one or more\nvariables, or, more generally, the transformation of functions, which\nconstitutes the most beautiful and the most important application of the\ndifferential calculus to general analysis, and which comprises, besides\nthe fundamental series discovered by Taylor, the remarkable series\ndiscovered by Maclaurin, John Bernouilli, Lagrange, &c.:\n\nSecondly, the general _theory of maxima and minima_ values for any\nfunctions whatever, of one or more variables; one of the most\ninteresting problems which analysis can present, however elementary it\nmay now have become, and to the complete solution of which the\ndifferential calculus naturally applies:\n\nThirdly, the general determination of the true value of functions which\npresent themselves under an _indeterminate_ appearance for certain\nhypotheses made on the values of the corresponding variables; which is\nthe least extensive and the least important of the three.\n\nThe first question is certainly the principal one in all points of view;\nit is also the most susceptible of receiving a new extension hereafter,\nespecially by conceiving, in a broader manner than has yet been done,\nthe employment of the differential calculus in the transformation of\nfunctions, on which subject Lagrange has left some valuable hints.\n\n       *       *       *       *       *\n\nHaving thus summarily, though perhaps too briefly, considered the chief\npoints in the differential calculus, I now proceed to an equally rapid\nexposition of a systematic outline of the Integral Calculus, properly so\ncalled, that is, the abstract subject of integration.\n\n\n\n\nTHE INTEGRAL CALCULUS.\n\n\n_Its Fundamental Division._ The fundamental division of the Integral\nCalculus is founded on the same principle as that of the Differential\nCalculus, in distinguishing the integration of _explicit_ differential\nformulas, and the integration of _implicit_ differentials or of\ndifferential equations. The separation of these two cases is even much\nmore profound in relation to integration than to differentiation. In the\ndifferential calculus, in fact, this distinction rests, as we have seen,\nonly on the extreme imperfection of ordinary analysis. But, on the other\nhand, it is easy to see that, even though all equations could be\nalgebraically resolved, differential equations would none the less\nconstitute a case of integration quite distinct from that presented by\nthe explicit differential formulas; for, limiting ourselves, for the\nsake of simplicity, to the first order, and to a single function _y_ of\na single variable _x_, if we suppose any differential equation between\n_x_, _y_, and _dy/dx_, to be resolved with reference to _dy/dx_, the\nexpression of the derived function being then generally found to contain\nthe primitive function itself, which is the object of the inquiry, the\nquestion of integration will not have at all changed its nature, and the\nsolution will not really have made any other progress than that of\nhaving brought the proposed differential equation to be of only the\nfirst degree relatively to the derived function, which is in itself of\nlittle importance. The differential would not then be determined in a\nmanner much less _implicit_ than before, as regards the integration,\nwhich would continue to present essentially the same characteristic\ndifficulty. The algebraic resolution of equations could not make the\ncase which we are considering come within the simple integration of\nexplicit differentials, except in the special cases in which the\nproposed differential equation did not contain the primitive function\nitself, which would consequently permit us, by resolving it, to find\n_dy/dx_ in terms of _x_ only, and thus to reduce the question to the\nclass of quadratures. Still greater difficulties would evidently be\nfound in differential equations of higher orders, or containing\nsimultaneously different functions of several independent variables.\n\nThe integration of differential equations is then necessarily more\ncomplicated than that of explicit differentials, by the elaboration of\nwhich last the integral calculus has been created, and upon which the\nothers have been made to depend as far as it has been possible. All the\nvarious analytical methods which have been proposed for integrating\ndifferential equations, whether it be the separation of the variables,\nthe method of multipliers, &c., have in fact for their object to reduce\nthese integrations to those of differential formulas, the only one\nwhich, by its nature, can be undertaken directly. Unfortunately,\nimperfect as is still this necessary base of the whole integral\ncalculus, the art of reducing to it the integration of differential\nequations is still less advanced.\n\n\n_Subdivisions: one variable or several._ Each of these two fundamental\nbranches of the integral calculus is next subdivided into two others (as\nin the differential calculus, and for precisely analogous reasons),\naccording as we consider functions with a _single variable_, or\nfunctions with _several independent variables_.\n\nThis distinction is, like the preceding one, still more important for\nintegration than for differentiation. This is especially remarkable in\nreference to differential equations. Indeed, those which depend on\nseveral independent variables may evidently present this characteristic\nand much more serious difficulty, that the desired function may be\ndifferentially defined by a simple relation between its different\nspecial derivatives relative to the different variables taken\nseparately. Hence results the most difficult and also the most extensive\nbranch of the integral calculus, which is commonly named the _Integral\nCalculus of partial differences_, created by D'Alembert, and in which,\naccording to the just appreciation of Lagrange, geometers ought to have\nseen a really new calculus, the philosophical character of which has not\nyet been determined with sufficient exactness. A very striking\ndifference between this case and that of equations with a single\nindependent variable consists, as has been already observed, in the\narbitrary functions which take the place of the simple arbitrary\nconstants, in order to give to the corresponding integrals all the\nproper generality.\n\nIt is scarcely necessary to say that this higher branch of\ntranscendental analysis is still entirely in its infancy, since, even in\nthe most simple case, that of an equation of the first order between the\npartial derivatives of a single function with two independent variables,\nwe are not yet completely able to reduce the integration to that of the\nordinary differential equations. The integration of functions of several\nvariables is much farther advanced in the case (infinitely more simple\nindeed) in which it has to do with only explicit differential formulas.\nWe can then, in fact, when these formulas fulfil the necessary\nconditions of integrability, always reduce their integration to\nquadratures.\n\n\n_Other Subdivisions: different Orders of Differentiation._ A new general\ndistinction, applicable as a subdivision to the integration of explicit\nor implicit differentials, with one variable or several, is drawn from\nthe _higher or lower order of the differentials_: a distinction which,\nas we have above remarked, does not give rise to any special question in\nthe differential calculus.\n\nRelatively to _explicit differentials_, whether of one variable or of\nseveral, the necessity of distinguishing their different orders belongs\nonly to the extreme imperfection of the integral calculus. In fact, if\nwe could always integrate every differential formula of the first order,\nthe integration of a formula of the second order, or of any other, would\nevidently not form a new question, since, by integrating it at first in\nthe first degree, we would arrive at the differential expression of the\nimmediately preceding order, from which, by a suitable series of\nanalogous integrations, we would be certain of finally arriving at the\nprimitive function, the final object of these operations. But the little\nknowledge which we possess on integration of even the first order causes\nquite another state of affairs, so that a higher order of differentials\nproduces new difficulties; for, having differential formulas of any\norder above the first, it may happen that we may be able to integrate\nthem, either once, or several times in succession, and that we may still\nbe unable to go back to the primitive functions, if these preliminary\nlabours have produced, for the differentials of a lower order,\nexpressions whose integrals are not known. This circumstance must occur\nso much the oftener (the number of known integrals being still very\nsmall), seeing that these successive integrals are generally very\ndifferent functions from the derivatives which have produced them.\n\nWith reference to _implicit differentials_, the distinction of orders is\nstill more important; for, besides the preceding reason, the influence\nof which is evidently analogous in this case, and is even greater, it is\neasy to perceive that the higher order of the differential equations\nnecessarily gives rise to questions of a new nature. In fact, even if we\ncould integrate every equation of the first order relating to a single\nfunction, that would not be sufficient for obtaining the final integral\nof an equation of any order whatever, inasmuch as every differential\nequation is not reducible to that of an immediately inferior order.\nThus, for example, if we have given any relation between _x_, _y_,\n_dx/dy_, and _d_\u00b2_y_/_dx_\u00b2, to determine a function _y_ of a variable\n_x_, we shall not be able to deduce from it at once, after effecting a\nfirst integration, the corresponding differential relation between _x_,\n_y_, and _dy/dx_, from which, by a second integration, we could ascend\nto the primitive equations. This would not necessarily take place, at\nleast without introducing new auxiliary functions, unless the proposed\nequation of the second order did not contain the required function _y_,\ntogether with its derivatives. As a general principle, differential\nequations will have to be regarded as presenting cases which are more\nand more _implicit_, as they are of a higher order, and which cannot be\nmade to depend on one another except by special methods, the\ninvestigation of which consequently forms a new class of questions, with\nrespect to which we as yet know scarcely any thing, even for functions\nof a single variable.[10]\n\n  [Footnote 10: The only important case of this class which has thus\n  far been completely treated is the general integration of _linear_\n  equations of any order whatever, with constant coefficients. Even\n  this case finally depends on the algebraic resolution of equations\n  of a degree equal to the order of differentiation.]\n\n_Another equivalent distinction._ Still farther, when we examine more\nprofoundly this distinction of different orders of differential\nequations, we find that it can be always made to come under a final\ngeneral distinction, relative to differential equations, which remains\nto be noticed. Differential equations with one or more independent\nvariables may contain simply a single function, or (in a case evidently\nmore complicated and more implicit, which corresponds to the\ndifferentiation of simultaneous implicit functions) we may have to\ndetermine at the same time several functions from the differential\nequations in which they are found united, together with their different\nderivatives. It is clear that such a state of the question necessarily\npresents a new special difficulty, that of separating the different\nfunctions desired, by forming for each, from the proposed differential\nequations, an isolated differential equation which does not contain the\nother functions or their derivatives. This preliminary labour, which is\nanalogous to the elimination of algebra, is evidently indispensable\nbefore attempting any direct integration, since we cannot undertake\ngenerally (except by special artifices which are very rarely applicable)\nto determine directly several distinct functions at once.\n\nNow it is easy to establish the exact and necessary coincidence of this\nnew distinction with the preceding one respecting the order of\ndifferential equations. We know, in fact, that the general method for\nisolating functions in simultaneous differential equations consists\nessentially in forming differential equations, separately in relation to\neach function, and of an order equal to the sum of all those of the\ndifferent proposed equations. This transformation can always be\neffected. On the other hand, every differential equation of any order in\nrelation to a single function might evidently always be reduced to the\nfirst order, by introducing a suitable number of auxiliary differential\nequations, containing at the same time the different anterior\nderivatives regarded as new functions to be determined. This method has,\nindeed, sometimes been actually employed with success, though it is not\nthe natural one.\n\nHere, then, are two necessarily equivalent orders of conditions in the\ngeneral theory of differential equations; the simultaneousness of a\ngreater or smaller number of functions, and the higher or lower order of\ndifferentiation of a single function. By augmenting the order of the\ndifferential equations, we can isolate all the functions; and, by\nartificially multiplying the number of the functions, we can reduce all\nthe equations to the first order. There is, consequently, in both cases,\nonly one and the same difficulty from two different points of sight.\nBut, however we may conceive it, this new difficulty is none the less\nreal, and constitutes none the less, by its nature, a marked separation\nbetween the integration of equations of the first order and that of\nequations of a higher order. I prefer to indicate the distinction under\nthis last form as being more simple, more general, and more logical.\n\n\n_Quadratures._ From the different considerations which have been\nindicated respecting the logical dependence of the various principal\nparts of the integral calculus, we see that the integration of explicit\ndifferential formulas of the first order and of a single variable is the\nnecessary basis of all other integrations, which we never succeed in\neffecting but so far as we reduce them to this elementary case,\nevidently the only one which, by its nature, is capable of being treated\ndirectly. This simple fundamental integration is often designated by the\nconvenient expression of _quadratures_, seeing that every integral of\nthis kind, S_f_(_x_)_dx_, may, in fact, be regarded as representing the\narea of a curve, the equation of which in rectilinear co-ordinates would\nbe _y_ = _f_(_x_). Such a class of questions corresponds, in the\ndifferential calculus, to the elementary case of the differentiation of\nexplicit functions of a single variable. But the integral question is,\nby its nature, very differently complicated, and especially much more\nextensive than the differential question. This latter is, in fact,\nnecessarily reduced, as we have seen, to the differentiation of the ten\nsimple functions, the elements of all which are considered in analysis.\nOn the other hand, the integration of compound functions does not\nnecessarily follow from that of the simple functions, each combination\nof which may present special difficulties with respect to the integral\ncalculus. Hence results the naturally indefinite extent, and the so\nvaried complication of the question of _quadratures_, upon which, in\nspite of all the efforts of analysts, we still possess so little\ncomplete knowledge.\n\nIn decomposing this question, as is natural, according to the different\nforms which may be assumed by the derivative function, we distinguish\nthe case of _algebraic_ functions and that of _transcendental_\nfunctions.\n\n_Integration of Transcendental Functions._ The truly analytical\nintegration of transcendental functions is as yet very little advanced,\nwhether for _exponential_, or for _logarithmic_, or for _circular_\nfunctions. But a very small number of cases of these three different\nkinds have as yet been treated, and those chosen from among the\nsimplest; and still the necessary calculations are in most cases\nextremely laborious. A circumstance which we ought particularly to\nremark in its philosophical connection is, that the different procedures\nof quadrature have no relation to any general view of integration, and\nconsist of simple artifices very incoherent with each other, and very\nnumerous, because of the very limited extent of each.\n\nOne of these artifices should, however, here be noticed, which, without\nbeing really a method of integration, is nevertheless remarkable for its\ngenerality; it is the procedure invented by John Bernouilli, and known\nunder the name of _integration by parts_, by means of which every\nintegral may be reduced to another which is sometimes found to be more\neasy to be obtained. This ingenious relation deserves to be noticed for\nanother reason, as having suggested the first idea of that\ntransformation of integrals yet unknown, which has lately received a\ngreater extension, and of which M. Fourier especially has made so new\nand important a use in the analytical questions produced by the theory\nof heat.\n\n_Integration of Algebraic Functions._ As to the integration of algebraic\nfunctions, it is farther advanced. However, we know scarcely any thing\nin relation to irrational functions, the integrals of which have been\nobtained only in extremely limited cases, and particularly by rendering\nthem rational. The integration of rational functions is thus far the\nonly theory of the integral calculus which has admitted of being treated\nin a truly complete manner; in a logical point of view, it forms, then,\nits most satisfactory part, but perhaps also the least important. It is\neven essential to remark, in order to have a just idea of the extreme\nimperfection of the integral calculus, that this case, limited as it is,\nis not entirely resolved except for what properly concerns integration\nviewed in an abstract manner; for, in the execution, the theory finds\nits progress most frequently quite stopped, independently of the\ncomplication of the calculations, by the imperfection of ordinary\nanalysis, seeing that it makes the integration finally depend upon the\nalgebraic resolution of equations, which greatly limits its use.\n\nTo grasp in a general manner the spirit of the different procedures\nwhich are employed in quadratures, we must observe that, by their\nnature, they can be primitively founded only on the differentiation of\nthe ten simple functions. The results of this, conversely considered,\nestablish as many direct theorems of the integral calculus, the only\nones which can be directly known. All the art of integration afterwards\nconsists, as has been said in the beginning of this chapter, in reducing\nall the other quadratures, so far as is possible, to this small number\nof elementary ones, which unhappily we are in most cases unable to\neffect.\n\n_Singular Solutions._ In this systematic enumeration of the various\nessential parts of the integral calculus, considered in their logical\nrelations, I have designedly neglected (in order not to break the chain\nof sequence) to consider a very important theory, which forms implicitly\na portion of the general theory of the integration of differential\nequations, but which I ought here to notice separately, as being, so to\nspeak, outside of the integral calculus, and being nevertheless of the\ngreatest interest, both by its logical perfection and by the extent of\nits applications. I refer to what are called _Singular Solutions_ of\ndifferential equations, called sometimes, but improperly, _particular_\nsolutions, which have been the subject of very remarkable investigations\nby Euler and Laplace, and of which Lagrange especially has presented\nsuch a beautiful and simple general theory. Clairaut, who first had\noccasion to remark their existence, saw in them a paradox of the\nintegral calculus, since these solutions have the peculiarity of\nsatisfying the differential equations without being comprised in the\ncorresponding general integrals. Lagrange has since explained this\nparadox in the most ingenious and most satisfactory manner, by showing\nhow such solutions are always derived from the general integral by the\nvariation of the arbitrary constants. He was also the first to suitably\nappreciate the importance of this theory, and it is with good reason\nthat he devoted to it so full a development in his \"Calculus of\nFunctions.\" In a logical point of view, this theory deserves all our\nattention by the character of perfect generality which it admits of,\nsince Lagrange has given invariable and very simple procedures for\nfinding the _singular_ solution of any differential equation which is\nsusceptible of it; and, what is no less remarkable, these procedures\nrequire no integration, consisting only of differentiations, and are\ntherefore always applicable. Differentiation has thus become, by a\nhappy artifice, a means of compensating, in certain circumstances, for\nthe imperfection of the integral calculus. Indeed, certain problems\nespecially require, by their nature, the knowledge of these _singular_\nsolutions; such, for example, in geometry, are all the questions in\nwhich a curve is to be determined from any property of its tangent or\nits osculating circle. In all cases of this kind, after having expressed\nthis property by a differential equation, it will be, in its analytical\nrelations, the _singular_ equation which will form the most important\nobject of the inquiry, since it alone will represent the required curve;\nthe general integral, which thenceforth it becomes unnecessary to know,\ndesignating only the system of the tangents, or of the osculating\ncircles of this curve. We may hence easily understand all the importance\nof this theory, which seems to me to be not as yet sufficiently\nappreciated by most geometers.\n\n_Definite Integrals._ Finally, to complete our review of the vast\ncollection of analytical researches of which is composed the integral\ncalculus, properly so called, there remains to be mentioned one theory,\nvery important in all the applications of the transcendental analysis,\nwhich I have had to leave outside of the system, as not being really\ndestined for veritable integration, and proposing, on the contrary, to\nsupply the place of the knowledge of truly analytical integrals, which\nare most generally unknown. I refer to the determination of _definite\nintegrals_.\n\nThe expression, always possible, of integrals in infinite series, may at\nfirst be viewed as a happy general means of compensating for the extreme\nimperfection of the integral calculus. But the employment of such\nseries, because of their complication, and of the difficulty of\ndiscovering the law of their terms, is commonly of only moderate utility\nin the algebraic point of view, although sometimes very essential\nrelations have been thence deduced. It is particularly in the\narithmetical point of view that this procedure acquires a great\nimportance, as a means of calculating what are called _definite\nintegrals_, that is, the values of the required functions for certain\ndeterminate values of the corresponding variables.\n\nAn inquiry of this nature exactly corresponds, in transcendental\nanalysis, to the numerical resolution of equations in ordinary analysis.\nBeing generally unable to obtain the veritable integral--named by\nopposition the _general_ or _indefinite_ integral; that is, the function\nwhich, differentiated, has produced the proposed differential\nformula--analysts have been obliged to employ themselves in determining\nat least, without knowing this function, the particular numerical values\nwhich it would take on assigning certain designated values to the\nvariables. This is evidently resolving the arithmetical question without\nhaving previously resolved the corresponding algebraic one, which most\ngenerally is the most important one. Such an analysis is, then, by its\nnature, as imperfect as we have seen the numerical resolution of\nequations to be. It presents, like this last, a vicious confusion of\narithmetical and algebraic considerations, whence result analogous\ninconveniences both in the purely logical point of view and in the\napplications. We need not here repeat the considerations suggested in\nour third chapter. But it will be understood that, unable as we almost\nalways are to obtain the true integrals, it is of the highest importance\nto have been able to obtain this solution, incomplete and necessarily\ninsufficient as it is. Now this has been fortunately attained at the\npresent day for all cases, the determination of the value of definite\nintegrals having been reduced to entirely general methods, which leave\nnothing to desire, in a great number of cases, but less complication in\nthe calculations, an object towards which are at present directed all\nthe special transformations of analysts. Regarding now this sort of\n_transcendental arithmetic_ as perfect, the difficulty in the\napplications is essentially reduced to making the proposed research\ndepend, finally, on a simple determination of definite integrals, which\nevidently cannot always be possible, whatever analytical skill may be\nemployed in effecting such a transformation.\n\n\n_Prospects of the Integral Calculus._ From the considerations indicated\nin this chapter, we see that, while the differential calculus\nconstitutes by its nature a limited and perfect system, to which nothing\nessential remains to be added, the integral calculus, or the simple\nsystem of integration, presents necessarily an inexhaustible field for\nthe activity of the human mind, independently of the indefinite\napplications of which the transcendental analysis is evidently\nsusceptible. The general argument by which I have endeavoured, in the\nsecond chapter, to make apparent the impossibility of ever discovering\nthe algebraic solution of equations of any degree and form whatsoever,\nhas undoubtedly infinitely more force with regard to the search for a\nsingle method of integration, invariably applicable to all cases. \"It\nis,\" says Lagrange, \"one of those problems whose general solution we\ncannot hope for.\" The more we meditate on this subject, the more we\nwill be convinced that such a research is utterly chimerical, as being\nfar above the feeble reach of our intelligence; although the labours of\ngeometers must certainly augment hereafter the amount of our knowledge\nrespecting integration, and thus create methods of greater generality.\nThe transcendental analysis is still too near its origin--there is\nespecially too little time since it has been conceived in a truly\nrational manner--for us now to be able to have a correct idea of what it\nwill hereafter become. But, whatever should be our legitimate hopes, let\nus not forget to consider, before all, the limits which are imposed by\nour intellectual constitution, and which, though not susceptible of a\nprecise determination, have none the less an incontestable reality.\n\nI am induced to think that, when geometers shall have exhausted the most\nimportant applications of our present transcendental analysis, instead\nof striving to impress upon it, as now conceived, a chimerical\nperfection, they will rather create new resources by changing the mode\nof derivation of the auxiliary quantities introduced in order to\nfacilitate the establishment of equations, and the formation of which\nmight follow an infinity of other laws besides the very simple relation\nwhich has been chosen, according to the conception suggested in the\nfirst chapter. The resources of this nature appear to me susceptible of\na much greater fecundity than those which would consist of merely\npushing farther our present calculus of indirect functions. It is a\nsuggestion which I submit to the geometers who have turned their\nthoughts towards the general philosophy of analysis.\n\nFinally, although, in the summary exposition which was the object of\nthis chapter, I have had to exhibit the condition of extreme\nimperfection which still belongs to the integral calculus, the student\nwould have a false idea of the general resources of the transcendental\nanalysis if he gave that consideration too great an importance. It is\nwith it, indeed, as with ordinary analysis, in which a very small amount\nof fundamental knowledge respecting the resolution of equations has been\nemployed with an immense degree of utility. Little advanced as geometers\nreally are as yet in the science of integrations, they have nevertheless\nobtained, from their scanty abstract conceptions, the solution of a\nmultitude of questions of the first importance in geometry, in\nmechanics, in thermology, &c. The philosophical explanation of this\ndouble general fact results from the necessarily preponderating\nimportance and grasp of _abstract_ branches of knowledge, the least of\nwhich is naturally found to correspond to a crowd of _concrete_\nresearches, man having no other resource for the successive extension of\nhis intellectual means than in the consideration of ideas more and more\nabstract, and still positive.\n\n       *       *       *       *       *\n\nIn order to finish the complete exposition of the philosophical\ncharacter of the transcendental analysis, there remains to be considered\na final conception, by which the immortal Lagrange has rendered this\nanalysis still better adapted to facilitate the establishment of\nequations in the most difficult problems, by considering a class of\nequations still more _indirect_ than the ordinary differential\nequations. It is the _Calculus_, or, rather, the _Method of Variations_;\nthe general appreciation of which will be our next subject.\n\n\n\n\nCHAPTER V.\n\nTHE CALCULUS OF VARIATIONS.\n\n\nIn order to grasp with more ease the philosophical character of the\n_Method of Variations_, it will be well to begin by considering in a\nsummary manner the special nature of the problems, the general\nresolution of which has rendered necessary the formation of this\nhyper-transcendental analysis. It is still too near its origin, and its\napplications have been too few, to allow us to obtain a sufficiently\nclear general idea of it from a purely abstract exposition of its\nfundamental theory.\n\n\n\n\nPROBLEMS GIVING RISE TO IT.\n\n\nThe mathematical questions which have given birth to the _Calculus of\nVariations_ consist generally in the investigation of the _maxima_ and\n_minima_ of certain indeterminate integral formulas, which express the\nanalytical law of such or such a phenomenon of geometry or mechanics,\nconsidered independently of any particular subject. Geometers for a long\ntime designated all the questions of this character by the common name\nof _Isoperimetrical Problems_, which, however, is really suitable to\nonly the smallest number of them.\n\n\n_Ordinary Questions of Maxima and Minima._ In the common theory of\n_maxima_ and _minima_, it is proposed to discover, with reference to a\ngiven function of one or more variables, what particular values must be\nassigned to these variables, in order that the corresponding value of\nthe proposed function may be a _maximum_ or a _minimum_ with respect to\nthose values which immediately precede and follow it; that is, properly\nspeaking, we seek to know at what instant the function ceases to\nincrease and commences to decrease, or reciprocally. The differential\ncalculus is perfectly sufficient, as we know, for the general resolution\nof this class of questions, by showing that the values of the different\nvariables, which suit either the maximum or minimum, must always reduce\nto zero the different first derivatives of the given function, taken\nseparately with reference to each independent variable, and by\nindicating, moreover, a suitable characteristic for distinguishing the\nmaximum from the minimum; consisting, in the case of a function of a\nsingle variable, for example, in the derived function of the second\norder taking a negative value for the maximum, and a positive value for\nthe minimum. Such are the well-known fundamental conditions belonging to\nthe greatest number of cases.\n\n\n_A new Class of Questions._ The construction of this general theory\nhaving necessarily destroyed the chief interest which questions of this\nkind had for geometers, they almost immediately rose to the\nconsideration of a new order of problems, at once much more important\nand of much greater difficulty--those of _isoperimeters_. It is, then,\nno longer _the values of the variables_ belonging to the maximum or the\nminimum of a given function that it is required to determine. It is _the\nform of the function itself_ which is required to be discovered, from\nthe condition of the maximum or of the minimum of a certain definite\nintegral, merely indicated, which depends upon that function.\n\n\n_Solid of least Resistance._ The oldest question of this nature is that\nof _the solid of least resistance_, treated by Newton in the second book\nof the Principia, in which he determines what ought to be the meridian\ncurve of a solid of revolution, in order that the resistance experienced\nby that body in the direction of its axis may be the least possible. But\nthe course pursued by Newton, from the nature of his special method of\ntranscendental analysis, had not a character sufficiently simple,\nsufficiently general, and especially sufficiently analytical, to attract\ngeometers to this new order of problems. To effect this, the application\nof the infinitesimal method was needed; and this was done, in 1695, by\nJohn Bernouilli, in proposing the celebrated problem of the\n_Brachystochrone_.\n\nThis problem, which afterwards suggested such a long series of analogous\nquestions, consists in determining the curve which a heavy body must\nfollow in order to descend from one point to another in the shortest\npossible time. Limiting the conditions to the simple fall in a vacuum,\nthe only case which was at first considered, it is easily found that the\nrequired curve must be a reversed cycloid with a horizontal base, and\nwith its origin at the highest point. But the question may become\nsingularly complicated, either by taking into account the resistance of\nthe medium, or the change in the intensity of gravity.\n\n\n_Isoperimeters._ Although this new class of problems was in the first\nplace furnished by mechanics, it is in geometry that the principal\ninvestigations of this character were subsequently made. Thus it was\nproposed to discover which, among all the curves of the same contour\ntraced between two given points, is that whose area is a maximum or\nminimum, whence has come the name of _Problem of Isoperimeters_; or it\nwas required that the maximum or minimum should belong to the surface\nproduced by the revolution of the required curve about an axis, or to\nthe corresponding volume; in other cases, it was the vertical height of\nthe center of gravity of the unknown curve, or of the surface and of the\nvolume which it might generate, which was to become a maximum or\nminimum, &c. Finally, these problems were varied and complicated almost\nto infinity by the Bernouillis, by Taylor, and especially by Euler,\nbefore Lagrange reduced their solution to an abstract and entirely\ngeneral method, the discovery of which has put a stop to the enthusiasm\nof geometers for such an order of inquiries. This is not the place for\ntracing the history of this subject. I have only enumerated some of the\nsimplest principal questions, in order to render apparent the original\ngeneral object of the method of variations.\n\n\n_Analytical Nature of these Problems._ We see that all these problems,\nconsidered in an analytical point of view, consist, by their nature, in\ndetermining what form a certain unknown function of one or more\nvariables ought to have, in order that such or such an integral,\ndependent upon that function, shall have, within assigned limits, a\nvalue which is a maximum or a minimum with respect to all those which it\nwould take if the required function had any other form whatever.\n\nThus, for example, in the problem of the _brachystochrone_, it is well\nknown that if _y_ = _f(z)_, _x_ = \u03c0(_z_), are the rectilinear equations\nof the required curve, supposing the axes of _x_ and of _y_ to be\nhorizontal, and the axis of _z_ to be vertical, the time of the fall of\na heavy body in that curve from the point whose ordinate is _z\u2081_, to\nthat whose ordinate is _z\u2082_, is expressed in general terms by the\ndefinite integral\n\n  \u222b_{_z\u2082_}^{_z\u2081_}\u221a(1 + (_f'(z))\u00b2_ + (\u03c0'(_z_))\u00b2/(2_gz_))_dz._\n\nIt is, then, necessary to find what the two unknown functions _f_ and \u03c0\nmust be, in order that this integral may be a minimum.\n\nIn the same way, to demand what is the curve among all plane\nisoperimetrical curves, which includes the greatest area, is the same\nthing as to propose to find, among all the functions _f(x)_ which can\ngive a certain constant value to the integral\n\n  \u222b_dx_\u221a(1 + (_f'(x)_ )\u00b2),\n\nthat one which renders the integral \u222b_f(x)dx_, taken between the same\nlimits, a maximum. It is evidently always so in other questions of this\nclass.\n\n\n_Methods of the older Geometers._ In the solutions which geometers\nbefore Lagrange gave of these problems, they proposed, in substance, to\nreduce them to the ordinary theory of maxima and minima. But the means\nemployed to effect this transformation consisted in special simple\nartifices peculiar to each case, and the discovery of which did not\nadmit of invariable and certain rules, so that every really new question\nconstantly reproduced analogous difficulties, without the solutions\npreviously obtained being really of any essential aid, otherwise than by\ntheir discipline and training of the mind. In a word, this branch of\nmathematics presented, then, the necessary imperfection which always\nexists when the part common to all questions of the same class has not\nyet been distinctly grasped in order to be treated in an abstract and\nthenceforth general manner.\n\n\n\n\nMETHOD OF LAGRANGE.\n\n\nLagrange, in endeavouring to bring all the different problems of\nisoperimeters to depend upon a common analysis, organized into a\ndistinct calculus, was led to conceive a new kind of differentiation, to\nwhich he has applied the characteristic \u03b4, reserving the characteristic\n_d_ for the common differentials. These differentials of a new species,\nwhich he has designated under the name of _Variations_, consist of the\ninfinitely small increments which the integrals receive, not by virtue\nof analogous increments on the part of the corresponding variables, as\nin the ordinary transcendental analysis, but by supposing that the\n_form_ of the function placed under the sign of integration undergoes an\ninfinitely small change. This distinction is easily conceived with\nreference to curves, in which we see the ordinate, or any other variable\nof the curve, admit of two sorts of differentials, evidently very\ndifferent, according as we pass from one point to another infinitely\nnear it on the same curve, or to the corresponding point of the\ninfinitely near curve produced by a certain determinate modification of\nthe first curve.[11] It is moreover clear, that the relative\n_variations_ of different magnitudes connected with each other by any\nlaws whatever are calculated, all but the characteristic, almost exactly\nin the same manner as the differentials. Finally, from the general\nnotion of _variations_ are in like manner deduced the fundamental\nprinciples of the algorithm proper to this method, consisting simply in\nthe evidently permissible liberty of transposing at will the\ncharacteristics specially appropriated to variations, before or after\nthose which correspond to the ordinary differentials.\n\n  [Footnote 11: Leibnitz had already considered the comparison of one\n  curve with an other infinitely near to it, calling it\n  \"_Differentiatio de curva in curvam_.\" But this comparison had no\n  analogy with the conception of Lagrange, the curves of Leibnitz\n  being embraced in the same general equation, from which they were\n  deduced by the simple change of an arbitrary constant.]\n\nThis abstract conception having been once formed, Lagrange was able to\nreduce with ease, and in the most general manner, all the problems of\n_Isoperimeters_ to the simple ordinary theory of _maxima_ and _minima_.\nTo obtain a clear idea of this great and happy transformation, we must\npreviously consider an essential distinction which arises in the\ndifferent questions of isoperimeters.\n\n\n_Two Classes of Questions._ These investigations must, in fact, be\ndivided into two general classes, according as the maxima and minima\ndemanded are _absolute_ or _relative_, to employ the abridged\nexpressions of geometers.\n\n\n_Questions of the first Class._ The _first case_ is that in which the\nindeterminate definite integrals, the maximum or minimum of which is\nsought, are not subjected, by the nature of the problem, to any\ncondition; as happens, for example, in the problem of the\n_brachystochrone_, in which the choice is to be made between all\nimaginable curves. The _second_ case takes place when, on the contrary,\nthe variable integrals can vary only according to certain conditions,\nwhich usually consist in other definite integrals (which depend, in like\nmanner, upon the required functions) always retaining the same given\nvalue; as, for example, in all the geometrical questions relating to\nreal _isoperimetrical_ figures, and in which, by the nature of the\nproblem, the integral relating to the length of the curve, or to the\narea of the surface, must remain constant during the variation of that\nintegral which is the object of the proposed investigation.\n\nThe _Calculus of Variations_ gives immediately the general solution of\nquestions of the former class; for it evidently follows, from the\nordinary theory of maxima and minima, that the required relation must\nreduce to zero the _variation_ of the proposed integral with reference\nto each independent variable; which gives the condition common to both\nthe maximum and the minimum: and, as a characteristic for distinguishing\nthe one from the other, that the variation of the second order of the\nsame integral must be negative for the maximum and positive for the\nminimum. Thus, for example, in the problem of the brachystochrone, we\nwill have, in order to determine the nature of the curve sought, the\nequation of condition\n\n  \u03b4\u222b_{_z\u2082_}^{_z\u2081_}\u221a([1 + (_f'(z)_)\u00b2 + (\u03c0'(_z_))\u00b2]/(2_gz_))_dz_ = 0,\n\nwhich, being decomposed into two, with respect to the two unknown\nfunctions _f_ and \u03c0, which are independent of each other, will\ncompletely express the analytical definition of the required curve. The\nonly difficulty peculiar to this new analysis consists in the\nelimination of the characteristic \u03b4, for which the calculus of\nvariations furnishes invariable and complete rules, founded, in general,\non the method of \"integration by parts,\" from which Lagrange has thus\nderived immense advantage. The constant object of this first analytical\nelaboration (which this is not the place for treating in detail) is to\narrive at real differential equations, which can always be done; and\nthereby the question comes under the ordinary transcendental analysis,\nwhich furnishes the solution, at least so far as to reduce it to pure\nalgebra if the integration can be effected. The general object of the\nmethod of variations is to effect this transformation, for which\nLagrange has established rules, which are simple, invariable, and\ncertain of success.\n\n\n_Equations of Limits._ Among the greatest special advantages of the\nmethod of variations, compared with the previous isolated solutions of\nisoperimetrical problems, is the important consideration of what\nLagrange calls _Equations of Limits_, which were entirely neglected\nbefore him, though without them the greater part of the particular\nsolutions remained necessarily incomplete. When the limits of the\nproposed integrals are to be fixed, their variations being zero, there\nis no occasion for noticing them. But it is no longer so when these\nlimits, instead of being rigorously invariable, are only subjected to\ncertain conditions; as, for example, if the two points between which the\nrequired curve is to be traced are not fixed, and have only to remain\nupon given lines or surfaces. Then it is necessary to pay attention to\nthe variation of their co-ordinates, and to establish between them the\nrelations which correspond to the equations of these lines or of these\nsurfaces.\n\n\n_A more general consideration._ This essential consideration is only the\nfinal complement of a more general and more important consideration\nrelative to the variations of different independent variables. If these\nvariables are really independent of one another, as when we compare\ntogether all the imaginable curves susceptible of being traced between\ntwo points, it will be the same with their variations, and,\nconsequently, the terms relating to each of these variations will have\nto be separately equal to zero in the general equation which expresses\nthe maximum or the minimum. But if, on the contrary, we suppose the\nvariables to be subjected to any fixed conditions, it will be necessary\nto take notice of the resulting relation between their variations, so\nthat the number of the equations into which this general equation is\nthen decomposed is always equal to only the number of the variables\nwhich remain truly independent. It is thus, for example, that instead of\nseeking for the shortest path between any two points, in choosing it\nfrom among all possible ones, it may be proposed to find only what is\nthe shortest among all those which may be taken on any given surface; a\nquestion the general solution of which forms certainly one of the most\nbeautiful applications of the method of variations.\n\n_Questions of the second Class._ Problems in which such modifying\nconditions are considered approach very nearly, in their nature, to the\nsecond general class of applications of the method of variations,\ncharacterized above as consisting in the investigation of _relative_\nmaxima and minima. There is, however, this essential difference between\nthe two cases, that in this last the modification is expressed by an\nintegral which depends upon the function sought, while in the other it\nis designated by a finite equation which is immediately given. It is\nhence apparent that the investigation of _relative_ maxima and minima is\nconstantly and necessarily more complicated than that of _absolute_\nmaxima and minima. Luckily, a very important general theory, discovered\nby the genius of the great Euler before the invention of the Calculus of\nVariations, gives a uniform and very simple means of making one of\nthese two classes of questions dependent on the other. It consists in\nthis, that if we add to the integral which is to be a maximum or a\nminimum, a constant and indeterminate multiple of that one which, by the\nnature of the problem, is to remain constant, it will be sufficient to\nseek, by the general method of Lagrange above indicated, the _absolute_\nmaximum or minimum of this whole expression. It can be easily conceived,\nindeed, that the part of the complete variation which would proceed from\nthe last integral must be equal to zero (because of the constant\ncharacter of this last) as well as the portion due to the first\nintegral, which disappears by virtue of the maximum or minimum state.\nThese two conditions evidently unite to produce, in that respect,\neffects exactly alike.\n\nSuch is a sketch of the general manner in which the method of variation\nis applied to all the different questions which compose what is called\nthe _Theory of Isoperimeters_. It will undoubtedly have been remarked in\nthis summary exposition how much use has been made in this new analysis\nof the second fundamental property of the transcendental analysis\nnoticed in the third chapter, namely, the generality of the\ninfinitesimal expressions for the representation of the same geometrical\nor mechanical phenomenon, in whatever body it may be considered. Upon\nthis generality, indeed, are founded, by their nature, all the solutions\ndue to the method of variations. If a single formula could not express\nthe length or the area of any curve whatever; if another fixed formula\ncould not designate the time of the fall of a heavy body, according to\nwhatever line it may descend, &c., how would it have been possible to\nresolve questions which unavoidably require, by their nature, the\nsimultaneous consideration of all the cases which can be determined in\neach phenomenon by the different subjects which exhibit it.\n\n\n_Other Applications of this Method._ Notwithstanding the extreme\nimportance of the theory of isoperimeters, and though the method of\nvariations had at first no other object than the logical and general\nsolution of this order of problems, we should still have but an\nincomplete idea of this beautiful analysis if we limited its destination\nto this. In fact, the abstract conception of two distinct natures of\ndifferentiation is evidently applicable not only to the cases for which\nit was created, but also to all those which present, for any reason\nwhatever, two different manners of making the same magnitudes vary. It\nis in this way that Lagrange himself has made, in his \"_M\u00e9chanique\nAnalytique_,\" an extensive and important application of his calculus of\nvariations, by employing it to distinguish the two sorts of changes\nwhich are naturally presented by the questions of rational mechanics for\nthe different points which are considered, according as we compare the\nsuccessive positions which are occupied, in virtue of its motion, by the\nsame point of each body in two consecutive instants, or as we pass from\none point of the body to another in the same instant. One of these\ncomparisons produces ordinary differentials; the other gives rise to\n_variations_, which, there as every where, are only differentials taken\nunder a new point of view. Such is the general acceptation in which we\nshould conceive the Calculus of Variations, in order suitably to\nappreciate the importance of this admirable logical instrument, the\nmost powerful that the human mind has as yet constructed.\n\nThe method of variations being only an immense extension of the general\ntranscendental analysis, I have no need of proving specially that it is\nsusceptible of being considered under the different fundamental points\nof view which the calculus of indirect functions, considered as a whole,\nadmits of. Lagrange invented the Calculus of Variations in accordance\nwith the infinitesimal conception, and, indeed, long before he undertook\nthe general reconstruction of the transcendental analysis. When he had\nexecuted this important reformation, he easily showed how it could also\nbe applied to the Calculus of Variations, which he expounded with all\nthe proper development, according to his theory of derivative functions.\nBut the more that the use of the method of variations is difficult of\ncomprehension, because of the higher degree of abstraction of the ideas\nconsidered, the more necessary is it, in its application, to economize\nthe exertions of the mind, by adopting the most direct and rapid\nanalytical conception, namely, that of Leibnitz. Accordingly, Lagrange\nhimself has constantly preferred it in the important use which he has\nmade of the Calculus of Variations in his \"Analytical Mechanics.\" In\nfact, there does not exist the least hesitation in this respect among\ngeometers.\n\n\n\n\nITS RELATIONS TO THE ORDINARY CALCULUS.\n\n\nIn order to make as clear as possible the philosophical character of the\nCalculus of Variations, I think that I should, in conclusion, briefly\nindicate a consideration which seems to me important, and by which I can\napproach it to the ordinary transcendental analysis in a higher degree\nthan Lagrange seems to me to have done.[12]\n\n  [Footnote 12: I propose hereafter to develop this new\n  consideration, in a special work upon the _Calculus of Variations_,\n  intended to present this hyper-transcendental analysis in a new\n  point of view, which I think adapted to extend its general range.]\n\nWe noticed in the preceding chapter the formation of the _calculus of\npartial differences_, created by D'Alembert, as having introduced into\nthe transcendental analysis a new elementary idea; the notion of two\nkinds of increments, distinct and independent of one another, which a\nfunction of two variables may receive by virtue of the change of each\nvariable separately. It is thus that the vertical ordinate of a surface,\nor any other magnitude which is referred to it, varies in two manners\nwhich are quite distinct, and which may follow the most different laws,\naccording as we increase either the one or the other of the two\nhorizontal co-ordinates. Now such a consideration seems to me very\nnearly allied, by its nature, to that which serves as the general basis\nof the method of variations. This last, indeed, has in reality done\nnothing but transfer to the independent variables themselves the\npeculiar conception which had been already adopted for the functions of\nthese variables; a modification which has remarkably enlarged its use. I\nthink, therefore, that so far as regards merely the fundamental\nconceptions, we may consider the calculus created by D'Alembert as\nhaving established a natural and necessary transition between the\nordinary infinitesimal calculus and the calculus of variations; such a\nderivation of which seems to be adapted to make the general notion more\nclear and simple.\n\nAccording to the different considerations indicated in this chapter, the\nmethod of variations presents itself as the highest degree of perfection\nwhich the analysis of indirect functions has yet attained. In its\nprimitive state, this last analysis presented itself as a powerful\ngeneral means of facilitating the mathematical study of natural\nphenomena, by introducing, for the expression of their laws, the\nconsideration of auxiliary magnitudes, chosen in such a manner that\ntheir relations are necessarily more simple and more easy to obtain than\nthose of the direct magnitudes. But the formation of these differential\nequations was not supposed to admit of any general and abstract rules.\nNow the Analysis of Variations, considered in the most philosophical\npoint of view, may be regarded as essentially destined, by its nature,\nto bring within the reach of the calculus the actual establishment of\nthe differential equations; for, in a great number of important and\ndifficult questions, such is the general effect of the _varied_\nequations, which, still more _indirect_ than the simple differential\nequations with respect to the special objects of the investigation, are\nalso much more easy to form, and from which we may then, by invariable\nand complete analytical methods, the object of which is to eliminate the\nnew order of auxiliary infinitesimals which have been introduced, deduce\nthose ordinary differential equations which it would often have been\nimpossible to establish directly. The method of variations forms, then,\nthe most sublime part of that vast system of mathematical analysis,\nwhich, setting out from the most simple elements of algebra, organizes,\nby an uninterrupted succession of ideas, general methods more and more\npowerful, for the study of natural philosophy, and which, in its whole,\npresents the most incomparably imposing and unequivocal monument of the\npower of the human intellect.\n\nWe must, however, also admit that the conceptions which are habitually\nconsidered in the method of variations being, by their nature, more\nindirect, more general, and especially more abstract than all others,\nthe employment of such a method exacts necessarily and continuously the\nhighest known degree of intellectual exertion, in order never to lose\nsight of the precise object of the investigation, in following\nreasonings which offer to the mind such uncertain resting-places, and in\nwhich signs are of scarcely any assistance. We must undoubtedly\nattribute in a great degree to this difficulty the little real use which\ngeometers, with the exception of Lagrange, have as yet made of such an\nadmirable conception.\n\n\n\n\nCHAPTER VI.\n\nTHE CALCULUS OF FINITE DIFFERENCES.\n\n\nThe different fundamental considerations indicated in the five preceding\nchapters constitute, in reality, all the essential bases of a complete\nexposition of mathematical analysis, regarded in the philosophical point\nof view. Nevertheless, in order not to neglect any truly important\ngeneral conception relating to this analysis, I think that I should here\nvery summarily explain the veritable character of a kind of calculus\nwhich is very extended, and which, though at bottom it really belongs to\nordinary analysis, is still regarded as being of an essentially distinct\nnature. I refer to the _Calculus of Finite Differences_, which will be\nthe special subject of this chapter.\n\n\n_Its general Character._ This calculus, created by Taylor, in his\ncelebrated work entitled _Methodus Incrementorum_, consists essentially\nin the consideration of the finite increments which functions receive as\na consequence of analogous increments on the part of the corresponding\nvariables. These increments or _differences_, which take the\ncharacteristic \u0394, to distinguish them from _differentials_, or\ninfinitely small increments, may be in their turn regarded as new\nfunctions, and become the subject of a second similar consideration, and\nso on; from which results the notion of differences of various\nsuccessive orders, analogous, at least in appearance, to the consecutive\norders of differentials. Such a calculus evidently presents, like the\ncalculus of indirect functions, two general classes of questions:\n\n1\u00b0. To determine the successive differences of all the various\nanalytical functions of one or more variables, as the result of a\ndefinite manner of increase of the independent variables, which are\ngenerally supposed to augment in arithmetical progression.\n\n2\u00b0. Reciprocally, to start from these differences, or, more generally,\nfrom any equations established between them, and go back to the\nprimitive functions themselves, or to their corresponding relations.\n\nHence follows the decomposition of this calculus into two distinct ones,\nto which are usually given the names of the _Direct_, and the _Inverse\nCalculus of Finite Differences_, the latter being also sometimes called\nthe _Integral Calculus of Finite Differences_. Each of these would,\nalso, evidently admit of a logical distribution similar to that given in\nthe fourth chapter for the differential and the integral calculus.\n\n\n_Its true Nature._ There is no doubt that Taylor thought that by such a\nconception he had founded a calculus of an entirely new nature,\nabsolutely distinct from ordinary analysis, and more general than the\ncalculus of Leibnitz, although resting on an analogous consideration. It\nis in this way, also, that almost all geometers have viewed the analysis\nof Taylor; but Lagrange, with his usual profundity, clearly perceived\nthat these properties belonged much more to the forms and to the\nnotations employed by Taylor than to the substance of his theory. In\nfact, that which constitutes the peculiar character of the analysis of\nLeibnitz, and makes of it a truly distinct and superior calculus, is the\ncircumstance that the derived functions are in general of an entirely\ndifferent nature from the primitive functions, so that they may give\nrise to more simple and more easily formed relations: whence result the\nadmirable fundamental properties of the transcendental analysis, which\nhave been already explained. But it is not so with the _differences_\nconsidered by Taylor; for these differences are, by their nature,\nfunctions essentially similar to those which have produced them, a\ncircumstance which renders them unsuitable to facilitate the\nestablishment of equations, and prevents their leading to more general\nrelations. Every equation of finite differences is truly, at bottom, an\nequation directly relating to the very magnitudes whose successive\nstates are compared. The scaffolding of new signs, which produce an\nillusion respecting the true character of these equations, disguises it,\nhowever, in a very imperfect manner, since it could always be easily\nmade apparent by replacing the _differences_ by the equivalent\ncombinations of the primitive magnitudes, of which they are really only\nthe abridged designations. Thus the calculus of Taylor never has\noffered, and never can offer, in any question of geometry or of\nmechanics, that powerful general aid which we have seen to result\nnecessarily from the analysis of Leibnitz. Lagrange has, moreover, very\nclearly proven that the pretended analogy observed between the calculus\nof differences and the infinitesimal calculus was radically vicious, in\nthis way, that the formulas belonging to the former calculus can never\nfurnish, as particular cases, those which belong to the latter, the\nnature of which is essentially distinct.\n\nFrom these considerations I am led to think that the calculus of finite\ndifferences is, in general, improperly classed with the transcendental\nanalysis proper, that is, with the calculus of indirect functions. I\nconsider it, on the contrary, in accordance with the views of Lagrange,\nto be only a very extensive and very important branch of ordinary\nanalysis, that is to say, of that which I have named the calculus of\ndirect functions, the equations which it considers being always, in\nspite of the notation, simple _direct_ equations.\n\n\n\n\nGENERAL THEORY OF SERIES.\n\n\nTo sum up as briefly as possible the preceding explanation, the calculus\nof Taylor ought to be regarded as having constantly for its true object\nthe general theory of _Series_, the most simple cases of which had alone\nbeen considered before that illustrious geometer. I ought, properly, to\nhave mentioned this important theory in treating, in the second chapter,\nof Algebra proper, of which it is such an extensive branch. But, in\norder to avoid a double reference to it, I have preferred to notice it\nonly in the consideration of the calculus of finite differences, which,\nreduced to its most simple general expression, is nothing but a complete\nlogical study of questions relating to _series_.\n\nEvery _Series_, or succession of numbers deduced from one another\naccording to any constant law, necessarily gives rise to these two\nfundamental questions:\n\n1\u00b0. The law of the series being supposed known, to find the expression\nfor its general term, so as to be able to calculate immediately any term\nwhatever without being obliged to form successively all the preceding\nterms.\n\n2\u00b0. In the same circumstances, to determine the _sum_ of any number of\nterms of the series by means of their places, so that it can be known\nwithout the necessity of continually adding these terms together.\n\nThese two fundamental questions being considered to be resolved, it may\nbe proposed, reciprocally, to find the law of a series from the form of\nits general term, or the expression of the sum. Each of these different\nproblems has so much the more extent and difficulty, as there can be\nconceived a greater number of different _laws_ for the series, according\nto the number of preceding terms on which each term directly depends,\nand according to the function which expresses that dependence. We may\neven consider series with several variable indices, as Laplace has done\nin his \"Analytical Theory of Probabilities,\" by the analysis to which he\nhas given the name of _Theory of Generating Functions_, although it is\nreally only a new and higher branch of the calculus of finite\ndifferences or of the general theory of series.\n\nThese general views which I have indicated give only an imperfect idea\nof the truly infinite extent and variety of the questions to which\ngeometers have risen by means of this single consideration of series, so\nsimple in appearance and so limited in its origin. It necessarily\npresents as many different cases as the algebraic resolution of\nequations, considered in its whole extent; and it is, by its nature,\nmuch more complicated, so much, indeed, that it always needs this last\nto conduct it to a complete solution. We may, therefore, anticipate what\nmust still be its extreme imperfection, in spite of the successive\nlabours of several geometers of the first order. We do not, indeed,\npossess as yet the complete and logical solution of any but the most\nsimple questions of this nature.\n\n\n_Its identity with this Calculus._ It is now easy to conceive the\nnecessary and perfect identity, which has been already announced,\nbetween the calculus of finite differences and the theory of series\nconsidered in all its bearings. In fact, every differentiation after the\nmanner of Taylor evidently amounts to finding the _law_ of formation of\na series with one or with several variable indices, from the expression\nof its general term; in the same way, every analogous integration may be\nregarded as having for its object the summation of a series, the general\nterm of which would be expressed by the proposed difference. In this\npoint of view, the various problems of the calculus of differences,\ndirect or inverse, resolved by Taylor and his successors, have really a\nvery great value, as treating of important questions relating to series.\nBut it is very doubtful if the form and the notation introduced by\nTaylor really give any essential facility in the solution of questions\nof this kind. It would be, perhaps, more advantageous for most cases,\nand certainly more logical, to replace the _differences_ by the terms\nthemselves, certain combinations of which they represent. As the\ncalculus of Taylor does not rest on a truly distinct fundamental idea,\nand has nothing peculiar to it but its system of signs, there could\nnever really be any important advantage in considering it as detached\nfrom ordinary analysis, of which it is, in reality, only an immense\nbranch. This consideration of _differences_, most generally useless,\neven if it does not cause complication, seems to me to retain the\ncharacter of an epoch in which, analytical ideas not being sufficiently\nfamiliar to geometers, they were naturally led to prefer the special\nforms suitable for simple numerical comparisons.\n\n\n\n\nPERIODIC OR DISCONTINUOUS FUNCTIONS.\n\n\nHowever that may be, I must not finish this general appreciation of the\ncalculus of finite differences without noticing a new conception to\nwhich it has given birth, and which has since acquired a great\nimportance. It is the consideration of those periodic or discontinuous\nfunctions which preserve the same value for an infinite series of values\nof the corresponding variables, subjected to a certain law, and which\nmust be necessarily added to the integrals of the equations of finite\ndifferences in order to render them sufficiently general, as simple\narbitrary constants are added to all quadratures in order to complete\ntheir generality. This idea, primitively introduced by Euler, has since\nbeen the subject of extended investigation by M. Fourier, who has made\nnew and important applications of it in his mathematical theory of heat.\n\n\n\n\nAPPLICATIONS OF THIS CALCULUS.\n\n\n_Series._ Among the principal general applications which have been made\nof the calculus of finite differences, it would be proper to place in\nthe first rank, as the most extended and the most important, the\nsolution of questions relating to series; if, as has been shown, the\ngeneral theory of series ought not to be considered as constituting, by\nits nature, the actual foundation of the calculus of Taylor.\n\n\n_Interpolations._ This great class of problems being then set aside, the\nmost essential of the veritable applications of the analysis of Taylor\nis, undoubtedly, thus far, the general method of _interpolations_, so\nfrequently and so usefully employed in the investigation of the\nempirical laws of natural phenomena. The question consists, as is well\nknown, in intercalating between certain given numbers other intermediate\nnumbers, subjected to the same law which we suppose to exist between the\nfirst. We can abundantly verify, in this principal application of the\ncalculus of Taylor, how truly foreign and often inconvenient is the\nconsideration of _differences_ with respect to the questions which\ndepend on that analysis. Indeed, Lagrange has replaced the formulas of\ninterpolation, deduced from the ordinary algorithm of the calculus of\nfinite differences, by much simpler general formulas, which are now\nalmost always preferred, and which have been found directly, without\nmaking any use of the notion of _differences_, which only complicates\nthe question.\n\n\n_Approximate Rectification, &c._ A last important class of applications\nof the calculus of finite differences, which deserves to be\ndistinguished from the preceding, consists in the eminently useful\nemployment made of it in geometry for determining by approximation the\nlength and the area of any curve, and in the same way the cubature of a\nbody of any form whatever. This procedure (which may besides be\nconceived abstractly as depending on the same analytical investigation\nas the question of interpolation) frequently offers a valuable\nsupplement to the entirely logical geometrical methods which often lead\nto integrations, which we do not yet know how to effect, or to\ncalculations of very complicated execution.\n\n       *       *       *       *       *\n\nSuch are the various principal considerations to be noticed with respect\nto the calculus of finite differences. This examination completes the\nproposed philosophical outline of ABSTRACT MATHEMATICS.\n\n\nCONCRETE MATHEMATICS will now be the subject of a similar labour. In it\nwe shall particularly devote ourselves to examining how it has been\npossible (supposing the general science of the calculus to be perfect),\nby invariable procedures, to reduce to pure questions of analysis all\nthe problems which can be presented by _Geometry_ and _Mechanics_, and\nthus to impress on these two fundamental bases of natural philosophy a\ndegree of precision and especially of unity; in a word, a character of\nhigh perfection, which could be communicated to them by such a course\nalone.\n\n\n\n\nBOOK II.\n\nGEOMETRY.\n\n\n\n\nBOOK II.\n\nGEOMETRY.\n\n\n\n\nCHAPTER I.\n\nGENERAL VIEW OF GEOMETRY.\n\n\n_Its true Nature._ After the general exposition of the philosophical\ncharacter of concrete mathematics, compared with that of abstract\nmathematics, given in the introductory chapter, it need not here be\nshown in a special manner that geometry must be considered as a true\nnatural science, only much more simple, and therefore much more perfect,\nthan any other. This necessary perfection of geometry, obtained\nessentially by the application of mathematical analysis, which it so\neminently admits, is apt to produce erroneous views of the real nature\nof this fundamental science, which most minds at present conceive to be\na purely logical science quite independent of observation. It is\nnevertheless evident, to any one who examines with attention the\ncharacter of geometrical reasonings, even in the present state of\nabstract geometry, that, although the facts which are considered in it\nare much more closely united than those relating to any other science,\nstill there always exists, with respect to every body studied by\ngeometers, a certain number of primitive phenomena, which, since they\nare not established by any reasoning, must be founded on observation\nalone, and which form the necessary basis of all the deductions.\n\nThe scientific superiority of geometry arises from the phenomena which\nit considers being necessarily the most universal and the most simple of\nall. Not only may all the bodies of nature give rise to geometrical\ninquiries, as well as mechanical ones, but still farther, geometrical\nphenomena would still exist, even though all the parts of the universe\nshould be considered as immovable. Geometry is then, by its nature, more\ngeneral than mechanics. At the same time, its phenomena are more simple,\nfor they are evidently independent of mechanical phenomena, while these\nlatter are always complicated with the former. The same relations hold\ngood in comparing geometry with abstract thermology.\n\nFor these reasons, in our classification we have made geometry the first\npart of concrete mathematics; that part the study of which, in addition\nto its own importance, serves as the indispensable basis of all the\nrest.\n\nBefore considering directly the philosophical study of the different\norders of inquiries which constitute our present geometry, we should\nobtain a clear and exact idea of the general destination of that\nscience, viewed in all its bearings. Such is the object of this chapter.\n\n\n_Definition._ Geometry is commonly defined in a very vague and entirely\nimproper manner, as being _the science of extension_. An improvement on\nthis would be to say that geometry has for its object the _measurement_\nof extension; but such an explanation would be very insufficient,\nalthough at bottom correct, and would be far from giving any idea of the\ntrue general character of geometrical science.\n\nTo do this, I think that I should first explain _two fundamental ideas_,\nwhich, very simple in themselves, have been singularly obscured by the\nemployment of metaphysical considerations.\n\n\n_The Idea of Space._ The first is that of _Space_. This conception\nproperly consists simply in this, that, instead of considering extension\nin the bodies themselves, we view it in an indefinite medium, which we\nregard as containing all the bodies of the universe. This notion is\nnaturally suggested to us by observation, when we think of the\n_impression_ which a body would leave in a fluid in which it had been\nplaced. It is clear, in fact, that, as regards its geometrical\nrelations, such an _impression_ may be substituted for the body itself,\nwithout altering the reasonings respecting it. As to the physical nature\nof this indefinite _space_, we are spontaneously led to represent it to\nourselves, as being entirely analogous to the actual medium in which we\nlive; so that if this medium was liquid instead of gaseous, our\ngeometrical _space_ would undoubtedly be conceived as liquid also. This\ncircumstance is, moreover, only very secondary, the essential object of\nsuch a conception being only to make us view extension separately from\nthe bodies which manifest it to us. We can easily understand in advance\nthe importance of this fundamental image, since it permits us to study\ngeometrical phenomena in themselves, abstraction being made of all the\nother phenomena which constantly accompany them in real bodies, without,\nhowever, exerting any influence over them. The regular establishment of\nthis general abstraction must be regarded as the first step which has\nbeen made in the rational study of geometry, which would have been\nimpossible if it had been necessary to consider, together with the form\nand the magnitude of bodies, all their other physical properties. The\nuse of such an hypothesis, which is perhaps the most ancient\nphilosophical conception created by the human mind, has now become so\nfamiliar to us, that we have difficulty in exactly estimating its\nimportance, by trying to appreciate the consequences which would result\nfrom its suppression.\n\n\n_Different Kinds of Extension._ The second preliminary geometrical\nconception which we have to examine is that of the different kinds of\nextension, designated by the words _volume_, _surface_, _line_, and even\n_point_, and of which the ordinary explanation is so unsatisfactory.[13]\n\n  [Footnote 13: Lacroix has justly criticised the expression of\n  _solid_, commonly used by geometers to designate a _volume_. It is\n  certain, in fact, that when we wish to consider separately a\n  certain portion of indefinite space, conceived as gaseous, we\n  mentally solidify its exterior envelope, so that a _line_ and a\n  _surface_ are habitually, to our minds, just as _solid_ as a\n  _volume_. It may also be remarked that most generally, in order\n  that bodies may penetrate one another with more facility, we are\n  obliged to imagine the interior of the _volumes_ to be hollow,\n  which renders still more sensible the impropriety of the word\n  _solid_.]\n\nAlthough it is evidently impossible to conceive any extension absolutely\ndeprived of any one of the three fundamental dimensions, it is no less\nincontestable that, in a great number of occasions, even of immediate\nutility, geometrical questions depend on only two dimensions, considered\nseparately from the third, or on a single dimension, considered\nseparately from the two others. Again, independently of this direct\nmotive, the study of extension with a single dimension, and afterwards\nwith two, clearly presents itself as an indispensable preliminary for\nfacilitating the study of complete bodies of three dimensions, the\nimmediate theory of which would be too complicated. Such are the two\ngeneral motives which oblige geometers to consider separately extension\nwith regard to one or to two dimensions, as well as relatively to all\nthree together.\n\nThe general notions of _surface_ and of _line_ have been formed by the\nhuman mind, in order that it may be able to think, in a permanent\nmanner, of extension in two directions, or in one only. The hyperbolical\nexpressions habitually employed by geometers to define these notions\ntend to convey false ideas of them; but, examined in themselves, they\nhave no other object than to permit us to reason with facility\nrespecting these two kinds of extension, making complete abstraction of\nthat which ought not to be taken into consideration. Now for this it is\nsufficient to conceive the dimension which we wish to eliminate as\nbecoming gradually smaller and smaller, the two others remaining the\nsame, until it arrives at such a degree of tenuity that it can no longer\nfix the attention. It is thus that we naturally acquire the real idea of\na _surface_, and, by a second analogous operation, the idea of a _line_,\nby repeating for breadth what we had at first done for thickness.\nFinally, if we again repeat the same operation, we arrive at the idea of\na _point_, or of an extension considered only with reference to its\nplace, abstraction being made of all magnitude, and designed\nconsequently to determine positions.\n\n_Surfaces_ evidently have, moreover, the general property of exactly\ncircumscribing volumes; and in the same way, _lines_, in their turn,\ncircumscribe _surfaces_ and are limited by _points_. But this\nconsideration, to which too much importance is often given, is only a\nsecondary one.\n\nSurfaces and lines are, then, in reality, always conceived with three\ndimensions; it would be, in fact, impossible to represent to one's self\na surface otherwise than as an extremely thin plate, and a line\notherwise than as an infinitely fine thread. It is even plain that the\ndegree of tenuity attributed by each individual to the dimensions of\nwhich he wishes to make abstraction is not constantly identical, for it\nmust depend on the degree of subtilty of his habitual geometrical\nobservations. This want of uniformity has, besides, no real\ninconvenience, since it is sufficient, in order that the ideas of\nsurface and of line should satisfy the essential condition of their\ndestination, for each one to represent to himself the dimensions which\nare to be neglected as being smaller than all those whose magnitude his\ndaily experience gives him occasion to appreciate.\n\nWe hence see how devoid of all meaning are the fantastic discussions of\nmetaphysicians upon the foundations of geometry. It should also be\nremarked that these primordial ideas are habitually presented by\ngeometers in an unphilosophical manner, since, for example, they explain\nthe notions of the different sorts of extent in an order absolutely the\ninverse of their natural dependence, which often produces the most\nserious inconveniences in elementary instruction.\n\n\n\n\nTHE FINAL OBJECT OF GEOMETRY.\n\n\nThese preliminaries being established, we can proceed directly to the\ngeneral definition of geometry, continuing to conceive this science as\nhaving for its final object the _measurement_ of extension.\n\nIt is necessary in this matter to go into a thorough explanation,\nfounded on the distinction of the three kinds of extension, since the\nnotion of _measurement_ is not exactly the same with reference to\nsurfaces and volumes as to lines.\n\n\n_Nature of Geometrical Measurement._ If we take the word _measurement_\nin its direct and general mathematical acceptation, which signifies\nsimply the determination of the value of the _ratios_ between any\nhomogeneous magnitudes, we must consider, in geometry, that the\n_measurement_ of surfaces and of volumes, unlike that of lines, is never\nconceived, even in the most simple and the most favourable cases, as\nbeing effected directly. The comparison of two lines is regarded as\ndirect; that of two surfaces or of two volumes is, on the contrary,\nalways indirect. Thus we conceive that two lines may be superposed; but\nthe superposition of two surfaces, or, still more so, of two volumes, is\nevidently impossible in most cases; and, even when it becomes rigorously\npracticable, such a comparison is never either convenient or exact. It\nis, then, very necessary to explain wherein properly consists the truly\ngeometrical measurement of a surface or of a volume.\n\n\n_Measurement of Surfaces and of Volumes._ For this we must consider\nthat, whatever may be the form of a body, there always exists a certain\nnumber of lines, more or less easy to be assigned, the length of which\nis sufficient to define exactly the magnitude of its surface or of its\nvolume. Geometry, regarding these lines as alone susceptible of being\ndirectly measured, proposes to deduce, from the simple determination of\nthem, the ratio of the surface or of the volume sought, to the unity of\nsurface, or to the unity of volume. Thus the general object of\ngeometry, with respect to surfaces and to volumes, is properly to reduce\nall comparisons of surfaces or of volumes to simple comparisons of\nlines.\n\nBesides the very great facility which such a transformation evidently\noffers for the measurement of volumes and of surfaces, there results\nfrom it, in considering it in a more extended and more scientific\nmanner, the general possibility of reducing to questions of lines all\nquestions relating to volumes and to surfaces, considered with reference\nto their magnitude. Such is often the most important use of the\ngeometrical expressions which determine surfaces and volumes in\nfunctions of the corresponding lines.\n\nIt is true that direct comparisons between surfaces or between volumes\nare sometimes employed; but such measurements are not regarded as\ngeometrical, but only as a supplement sometimes necessary, although too\nrarely applicable, to the insufficiency or to the difficulty of truly\nrational methods. It is thus that we often determine the volume of a\nbody, and in certain cases its surface, by means of its weight. In the\nsame way, on other occasions, when we can substitute for the proposed\nvolume an equivalent liquid volume, we establish directly the comparison\nof the two volumes, by profiting by the property possessed by liquid\nmasses, of assuming any desired form. But all means of this nature are\npurely mechanical, and rational geometry necessarily rejects them.\n\nTo render more sensible the difference between these modes of\ndetermination and true geometrical measurements, I will cite a single\nvery remarkable example; the manner in which Galileo determined the\nratio of the ordinary cycloid to that of the generating circle. The\ngeometry of his time was as yet insufficient for the rational solution\nof such a problem. Galileo conceived the idea of discovering that ratio\nby a direct experiment. Having weighed as exactly as possible two plates\nof the same material and of equal thickness, one of them having the form\nof a circle and the other that of the generated cycloid, he found the\nweight of the latter always triple that of the former; whence he\ninferred that the area of the cycloid is triple that of the generating\ncircle, a result agreeing with the veritable solution subsequently\nobtained by Pascal and Wallis. Such a success evidently depends on the\nextreme simplicity of the ratio sought; and we can understand the\nnecessary insufficiency of such expedients, even when they are actually\npracticable.\n\nWe see clearly, from what precedes, the nature of that part of geometry\nrelating to _volumes_ and that relating to _surfaces_. But the character\nof the geometry of _lines_ is not so apparent, since, in order to\nsimplify the exposition, we have considered the measurement of lines as\nbeing made directly. There is, therefore, needed a complementary\nexplanation with respect to them.\n\n\n_Measurement of curved Lines._ For this purpose, it is sufficient to\ndistinguish between the right line and curved lines, the measurement of\nthe first being alone regarded as direct, and that of the other as\nalways indirect. Although superposition is sometimes strictly\npracticable for curved lines, it is nevertheless evident that truly\nrational geometry must necessarily reject it, as not admitting of any\nprecision, even when it is possible. The geometry of lines has, then,\nfor its general object, to reduce in every case the measurement of\ncurved lines to that of right lines; and consequently, in the most\nextended point of view, to reduce to simple questions of right lines all\nquestions relating to the magnitude of any curves whatever. To\nunderstand the possibility of such a transformation, we must remark,\nthat in every curve there always exist certain right lines, the length\nof which must be sufficient to determine that of the curve. Thus, in a\ncircle, it is evident that from the length of the radius we must be able\nto deduce that of the circumference; in the same way, the length of an\nellipse depends on that of its two axes; the length of a cycloid upon\nthe diameter of the generating circle, &c.; and if, instead of\nconsidering the whole of each curve, we demand, more generally, the\nlength of any arc, it will be sufficient to add to the different\nrectilinear parameters, which determine the whole curve, the chord of\nthe proposed arc, or the co-ordinates of its extremities. To discover\nthe relation which exists between the length of a curved line and that\nof similar right lines, is the general problem of the part of geometry\nwhich relates to the study of lines.\n\nCombining this consideration with those previously suggested with\nrespect to volumes and to surfaces, we may form a very clear idea of the\nscience of geometry, conceived in all its parts, by assigning to it, for\nits general object, the final reduction of the comparisons of all kinds\nof extent, volumes, surfaces, or lines, to simple comparisons of right\nlines, the only comparisons regarded as capable of being made directly,\nand which indeed could not be reduced to any others more easy to effect.\nSuch a conception, at the same time, indicates clearly the veritable\ncharacter of geometry, and seems suited to show at a single glance its\nutility and its perfection.\n\n\n_Measurement of right Lines._ In order to complete this fundamental\nexplanation, I have yet to show how there can be, in geometry, a special\nsection relating to the right line, which seems at first incompatible\nwith the principle that the measurement of this class of lines must\nalways be regarded as direct.\n\nIt is so, in fact, as compared with that of curved lines, and of all the\nother objects which geometry considers. But it is evident that the\nestimation of a right line cannot be viewed as direct except so far as\nthe linear unit can be applied to it. Now this often presents\ninsurmountable difficulties, as I had occasion to show, for another\nreason, in the introductory chapter. We must, then, make the measurement\nof the proposed right line depend on other analogous measurements\ncapable of being effected directly. There is, then, necessarily a\nprimary distinct branch of geometry, exclusively devoted to the right\nline; its object is to determine certain right lines from others by\nmeans of the relations belonging to the figures resulting from their\nassemblage. This preliminary part of geometry, which is almost\nimperceptible in viewing the whole of the science, is nevertheless\nsusceptible of a great development. It is evidently of especial\nimportance, since all other geometrical measurements are referred to\nthose of right lines, and if they could not be determined, the solution\nof every question would remain unfinished.\n\nSuch, then, are the various fundamental parts of rational geometry,\narranged according to their natural dependence; the geometry of _lines_\nbeing first considered, beginning with the right line; then the geometry\nof _surfaces_, and, finally, that of _solids_.\n\n\n\n\nINFINITE EXTENT OF ITS FIELD.\n\n\nHaving determined with precision the general and final object of\ngeometrical inquiries, the science must now be considered with respect\nto the field embraced by each of its three fundamental sections.\n\nThus considered, geometry is evidently susceptible, by its nature, of an\nextension which is rigorously infinite; for the measurement of lines, of\nsurfaces, or of volumes presents necessarily as many distinct questions\nas we can conceive different figures subjected to exact definitions; and\ntheir number is evidently infinite.\n\nGeometers limited themselves at first to consider the most simple\nfigures which were directly furnished them by nature, or which were\ndeduced from these primitive elements by the least complicated\ncombinations. But they have perceived, since Descartes, that, in order\nto constitute the science in the most philosophical manner, it was\nnecessary to make it apply to all imaginable figures. This abstract\ngeometry will then inevitably comprehend as particular cases all the\ndifferent real figures which the exterior world could present. It is\nthen a fundamental principle in truly rational geometry to consider, as\nfar as possible, all figures which can be rigorously conceived.\n\nThe most superficial examination is enough to convince us that these\nfigures present a variety which is quite infinite.\n\n\n_Infinity of Lines._ With respect to curved _lines_, regarding them as\ngenerated by the motion of a point governed by a certain law, it is\nplain that we shall have, in general, as many different curves as we\nconceive different laws for this motion, which may evidently be\ndetermined by an infinity of distinct conditions; although it may\nsometimes accidentally happen that new generations produce curves which\nhave been already obtained. Thus, among plane curves, if a point moves\nso as to remain constantly at the same distance from a fixed point, it\nwill generate a _circle_; if it is the sum or the difference of its\ndistances from two fixed points which remains constant, the curve\ndescribed will be an _ellipse_ or an _hyperbola_; if it is their\nproduct, we shall have an entirely different curve; if the point departs\nequally from a fixed point and from a fixed line, it will describe a\n_parabola_; if it revolves on a circle at the same time that this circle\nrolls along a straight line, we shall have a _cycloid_; if it advances\nalong a straight line, while this line, fixed at one of its extremities,\nturns in any manner whatever, there will result what in general terms\nare called _spirals_, which of themselves evidently present as many\nperfectly distinct curves as we can suppose different relations between\nthese two motions of translation and of rotation, &c. Each of these\ndifferent curves may then furnish new ones, by the different general\nconstructions which geometers have imagined, and which give rise to\nevolutes, to epicycloids, to caustics, &c. Finally, there exists a still\ngreater variety among curves of double curvature.\n\n\n_Infinity of Surfaces._ As to _surfaces_, the figures are necessarily\nmore different still, considering them as generated by the motion of\nlines. Indeed, the figure may then vary, not only in considering, as in\ncurves, the different infinitely numerous laws to which the motion of\nthe generating line may be subjected, but also in supposing that this\nline itself may change its nature; a circumstance which has nothing\nanalogous in curves, since the points which describe them cannot have\nany distinct figure. Two classes of very different conditions may then\ncause the figures of surfaces to vary, while there exists only one for\nlines. It is useless to cite examples of this doubly infinite\nmultiplicity of surfaces. It would be sufficient to consider the extreme\nvariety of the single group of surfaces which may be generated by a\nright line, and which comprehends the whole family of cylindrical\nsurfaces, that of conical surfaces, the most general class of\ndevelopable surfaces, &c.\n\n\n_Infinity of Volumes._ With respect to _volumes_, there is no occasion\nfor any special consideration, since they are distinguished from each\nother only by the surfaces which bound them.\n\nIn order to complete this sketch, it should be added that surfaces\nthemselves furnish a new general means of conceiving new curves, since\nevery curve may be regarded as produced by the intersection of two\nsurfaces. It is in this way, indeed, that the first lines which we may\nregard as having been truly invented by geometers were obtained, since\nnature gave directly the straight line and the circle. We know that the\nellipse, the parabola, and the hyperbola, the only curves completely\nstudied by the ancients, were in their origin conceived only as\nresulting from the intersection of a cone with circular base by a plane\nin different positions. It is evident that, by the combined employment\nof these different general means for the formation of lines and of\nsurfaces, we could produce a rigorously infinitely series of distinct\nforms in starting from only a very small number of figures directly\nfurnished by observation.\n\n\n_Analytical invention of Curves, &c._ Finally, all the various direct\nmeans for the invention of figures have scarcely any farther importance,\nsince rational geometry has assumed its final character in the hands of\nDescartes. Indeed, as we shall see more fully in chapter iii., the\ninvention of figures is now reduced to the invention of equations, so\nthat nothing is more easy than to conceive new lines and new surfaces,\nby changing at will the functions introduced into the equations. This\nsimple abstract procedure is, in this respect, infinitely more fruitful\nthan all the direct resources of geometry, developed by the most\npowerful imagination, which should devote itself exclusively to that\norder of conceptions. It also explains, in the most general and the most\nstriking manner, the necessarily infinite variety of geometrical forms,\nwhich thus corresponds to the diversity of analytical functions. Lastly,\nit shows no less clearly that the different forms of surfaces must be\nstill more numerous than those of lines, since lines are represented\nanalytically by equations with two variables, while surfaces give rise\nto equations with three variables, which necessarily present a greater\ndiversity.\n\nThe preceding considerations are sufficient to show clearly the\nrigorously infinite extent of each of the three general sections of\ngeometry.\n\n\n\n\nEXPANSION OF ORIGINAL DEFINITION.\n\n\nTo complete the formation of an exact and sufficiently extended idea of\nthe nature of geometrical inquiries, it is now indispensable to return\nto the general definition above given, in order to present it under a\nnew point of view, without which the complete science would be only very\nimperfectly conceived.\n\nWhen we assign as the object of geometry the _measurement_ of all sorts\nof lines, surfaces, and volumes, that is, as has been explained, the\nreduction of all geometrical comparisons to simple comparisons of right\nlines, we have evidently the advantage of indicating a general\ndestination very precise and very easy to comprehend. But if we set\naside every definition, and examine the actual composition of the\nscience of geometry, we will at first be induced to regard the preceding\ndefinition as much too narrow; for it is certain that the greater part\nof the investigations which constitute our present geometry do not at\nall appear to have for their object the _measurement_ of extension. In\nspite of this fundamental objection, I will persist in retaining this\ndefinition; for, in fact, if, instead of confining ourselves to\nconsidering the different questions of geometry isolatedly, we endeavour\nto grasp the leading questions, in comparison with which all others,\nhowever important they may be, must be regarded as only secondary, we\nwill finally recognize that the measurement of lines, of surfaces, and\nof volumes, is the invariable object, sometimes _direct_, though most\noften _indirect_, of all geometrical labours.\n\nThis general proposition being fundamental, since it can alone give our\ndefinition all its value, it is indispensable to enter into some\ndevelopments upon this subject.\n\n\n\n\nPROPERTIES OF LINES AND SURFACES.\n\n\nWhen we examine with attention the geometrical investigations which do\nnot seem to relate to the _measurement_ of extent, we find that they\nconsist essentially in the study of the different _properties_ of each\nline or of each surface; that is, in the knowledge of the different\nmodes of generation, or at least of definition, peculiar to each figure\nconsidered. Now we can easily establish in the most general manner the\nnecessary relation of such a study to the question of _measurement_, for\nwhich the most complete knowledge of the properties of each form is an\nindispensable preliminary. This is concurrently proven by two\nconsiderations, equally fundamental, although quite distinct in their\nnature.\n\n\nNECESSITY OF THEIR STUDY: 1. _To find the most suitable Property._ The\n_first_, purely scientific, consists in remarking that, if we did not\nknow any other characteristic property of each line or surface than that\none according to which geometers had first conceived it, in most cases\nit would be impossible to succeed in the solution of questions relating\nto its _measurement_. In fact, it is easy to understand that the\ndifferent definitions which each figure admits of are not all equally\nsuitable for such an object, and that they even present the most\ncomplete oppositions in that respect. Besides, since the primitive\ndefinition of each figure was evidently not chosen with this condition\nin view, it is clear that we must not expect, in general, to find it the\nmost suitable; whence results the necessity of discovering others, that\nis, of studying as far as is possible the _properties_ of the proposed\nfigure. Let us suppose, for example, that the circle is defined to be\n\"the curve which, with the same contour, contains the greatest area.\"\nThis is certainly a very characteristic property, but we would evidently\nfind insurmountable difficulties in trying to deduce from such a\nstarting point the solution of the fundamental questions relating to the\nrectification or to the quadrature of this curve. It is clear, in\nadvance, that the property of having all its points equally distant from\na fixed point must evidently be much better adapted to inquiries of this\nnature, even though it be not precisely the most suitable. In like\nmanner, would Archimedes ever have been able to discover the quadrature\nof the parabola if he had known no other property of that curve than\nthat it was the section of a cone with a circular base, by a plane\nparallel to its generatrix? The purely speculative labours of preceding\ngeometers, in transforming this first definition, were evidently\nindispensable preliminaries to the direct solution of such a question.\nThe same is true, in a still greater degree, with respect to surfaces.\nTo form a just idea of this, we need only compare, as to the question of\ncubature or quadrature, the common definition of the sphere with that\none, no less characteristic certainly, which would consist in regarding\na spherical body, as that one which, with the same area, contains the\ngreatest volume.\n\nNo more examples are needed to show the necessity of knowing, so far as\nis possible, all the properties of each line or of each surface, in\norder to facilitate the investigation of rectifications, of quadratures,\nand of cubatures, which constitutes the final object of geometry. We may\neven say that the principal difficulty of questions of this kind\nconsists in employing in each case the property which is best adapted\nto the nature of the proposed problem. Thus, while we continue to\nindicate, for more precision, the measurement of extension as the\ngeneral destination of geometry, this first consideration, which goes to\nthe very bottom of the subject, shows clearly the necessity of including\nin it the study, as thorough as possible, of the different generations\nor definitions belonging to the same form.\n\n\n2. _To pass from the Concrete to the Abstract._ A second consideration,\nof at least equal importance, consists in such a study being\nindispensable for organizing in a rational manner the relation of the\nabstract to the concrete in geometry.\n\nThe science of geometry having to consider all imaginable figures which\nadmit of an exact definition, it necessarily results from this, as we\nhave remarked, that questions relating to any figures presented by\nnature are always implicitly comprised in this abstract geometry,\nsupposed to have attained its perfection. But when it is necessary to\nactually pass to concrete geometry, we constantly meet with a\nfundamental difficulty, that of knowing to which of the different\nabstract types we are to refer, with sufficient approximation, the real\nlines or surfaces which we have to study. Now it is for the purpose of\nestablishing such a relation that it is particularly indispensable to\nknow the greatest possible number of properties of each figure\nconsidered in geometry.\n\nIn fact, if we always confined ourselves to the single primitive\ndefinition of a line or of a surface, supposing even that we could then\n_measure_ it (which, according to the first order of considerations,\nwould generally be impossible), this knowledge would remain almost\nnecessarily barren in the application, since we should not ordinarily\nknow how to recognize that figure in nature when it presented itself\nthere; to ensure that, it would be necessary that the single\ncharacteristic, according to which geometers had conceived it, should be\nprecisely that one whose verification external circumstances would\nadmit: a coincidence which would be purely fortuitous, and on which we\ncould not count, although it might sometimes take place. It is, then,\nonly by multiplying as much as possible the characteristic properties of\neach abstract figure, that we can be assured, in advance, of recognizing\nit in the concrete state, and of thus turning to account all our\nrational labours, by verifying in each case the definition which is\nsusceptible of being directly proven. This definition is almost always\nthe only one in given circumstances, and varies, on the other hand, for\nthe same figure, with different circumstances; a double reason for its\nprevious determination.\n\n\n_Illustration: Orbits of the Planets._ The geometry of the heavens\nfurnishes us with a very memorable example in this matter, well suited\nto show the general necessity of such a study. We know that the ellipse\nwas discovered by Kepler to be the curve which the planets describe\nabout the sun, and the satellites about their planets. Now would this\nfundamental discovery, which re-created astronomy, ever have been\npossible, if geometers had been always confined to conceiving the\nellipse only as the oblique section of a circular cone by a plane? No\nsuch definition, it is evident, would admit of such a verification. The\nmost general property of the ellipse, that the sum of the distances from\nany of its points to two fixed points is a constant quantity, is\nundoubtedly much more susceptible, by its nature, of causing the curve\nto be recognized in this case, but still is not directly suitable. The\nonly characteristic which can here be immediately verified is that which\nis derived from the relation which exists in the ellipse between the\nlength of the focal distances and their direction; the only relation\nwhich admits of an astronomical interpretation, as expressing the law\nwhich connects the distance from the planet to the sun, with the time\nelapsed since the beginning of its revolution. It was, then, necessary\nthat the purely speculative labours of the Greek geometers on the\nproperties of the conic sections should have previously presented their\ngeneration under a multitude of different points of view, before Kepler\ncould thus pass from the abstract to the concrete, in choosing from\namong all these different characteristics that one which could be most\neasily proven for the planetary orbits.\n\n\n_Illustration: Figure of the Earth._ Another example of the same order,\nbut relating to surfaces, occurs in considering the important question\nof the figure of the earth. If we had never known any other property of\nthe sphere than its primitive character of having all its points equally\ndistant from an interior point, how would we ever have been able to\ndiscover that the surface of the earth was spherical? For this, it was\nnecessary previously to deduce from this definition of the sphere some\nproperties capable of being verified by observations made upon the\nsurface alone, such as the constant ratio which exists between the\nlength of the path traversed in the direction of any meridian of a\nsphere going towards a pole, and the angular height of this pole above\nthe horizon at each point. Another example, but involving a much longer\nseries of preliminary speculations, is the subsequent proof that the\nearth is not rigorously spherical, but that its form is that of an\nellipsoid of revolution.\n\nAfter such examples, it would be needless to give any others, which any\none besides may easily multiply. All of them prove that, without a very\nextended knowledge of the different properties of each figure, the\nrelation of the abstract to the concrete, in geometry, would be purely\naccidental, and that the science would consequently want one of its most\nessential foundations.\n\nSuch, then, are two general considerations which fully demonstrate the\nnecessity of introducing into geometry a great number of investigations\nwhich have not the _measurement_ of extension for their direct object;\nwhile we continue, however, to conceive such a measurement as being the\nfinal destination of all geometrical science. In this way we can retain\nthe philosophical advantages of the clearness and precision of this\ndefinition, and still include in it, in a very logical though indirect\nmanner, all known geometrical researches, in considering those which do\nnot seem to relate to the measurement of extension, as intended either\nto prepare for the solution of the final questions, or to render\npossible the application of the solutions obtained.\n\nHaving thus recognized, as a general principle, the close and necessary\nconnexion of the study of the properties of lines and surfaces with\nthose researches which constitute the final object of geometry, it is\nevident that geometers, in the progress of their labours, must by no\nmeans constrain themselves to keep such a connexion always in view.\nKnowing, once for all, how important it is to vary as much as possible\nthe manner of conceiving each figure, they should pursue that study,\nwithout considering of what immediate use such or such a special\nproperty may be for rectifications, quadratures, and cubatures. They\nwould uselessly fetter their inquiries by attaching a puerile importance\nto the continued establishment of that co-ordination.\n\nThis general exposition of the general object of geometry is so much the\nmore indispensable, since, by the very nature of the subject, this study\nof the different properties of each line and of each surface necessarily\ncomposes by far the greater part of the whole body of geometrical\nresearches. Indeed, the questions directly relating to rectifications,\nto quadratures, and to cubatures, are evidently, by themselves, very few\nin number for each figure considered. On the other hand, the study of\nthe properties of the same figure presents an unlimited field to the\nactivity of the human mind, in which it may always hope to make new\ndiscoveries. Thus, although geometers have occupied themselves for\ntwenty centuries, with more or less activity undoubtedly, but without\nany real interruption, in the study of the conic sections, they are far\nfrom regarding that so simple subject as being exhausted; and it is\ncertain, indeed, that in continuing to devote themselves to it, they\nwould not fail to find still unknown properties of those different\ncurves. If labours of this kind have slackened considerably for a\ncentury past, it is not because they are completed, but only, as will be\npresently explained, because the philosophical revolution in geometry,\nbrought about by Descartes, has singularly diminished the importance of\nsuch researches.\n\nIt results from the preceding considerations that not only is the field\nof geometry necessarily infinite because of the variety of figures to\nbe considered, but also in virtue of the diversity of the points of view\nunder the same figure may be regarded. This last conception is, indeed,\nthat which gives the broadest and most complete idea of the whole body\nof geometrical researches. We see that studies of this kind consist\nessentially, for each line or for each surface, in connecting all the\ngeometrical phenomena which it can present, with a single fundamental\nphenomenon, regarded as the primitive definition.\n\n\n\n\nTHE TWO GENERAL METHODS OF GEOMETRY.\n\n\nHaving now explained in a general and yet precise manner the final\nobject of geometry, and shown how the science, thus defined, comprehends\na very extensive class of researches which did not at first appear\nnecessarily to belong to it, there remains to be considered the _method_\nto be followed for the formation of this science. This discussion is\nindispensable to complete this first sketch of the philosophical\ncharacter of geometry. I shall here confine myself to indicating the\nmost general consideration in this matter, developing and summing up\nthis important fundamental idea in the following chapters.\n\nGeometrical questions may be treated according to _two methods_ so\ndifferent, that there result from them two sorts of geometry, so to say,\nthe philosophical character of which does not seem to me to have yet\nbeen properly apprehended. The expressions of _Synthetical Geometry_ and\n_Analytical Geometry_, habitually employed to designate them, give a\nvery false idea of them. I would much prefer the purely historical\ndenominations of _Geometry of the Ancients_ and _Geometry of the\nModerns_, which have at least the advantage of not causing their true\ncharacter to be misunderstood. But I propose to employ henceforth the\nregular expressions of _Special Geometry_ and _General Geometry_, which\nseem to me suited to characterize with precision the veritable nature of\nthe two methods.\n\n\n_Their fundamental Difference._ The fundamental difference between the\nmanner in which we conceive Geometry since Descartes, and the manner in\nwhich the geometers of antiquity treated geometrical questions, is not\nthe use of the Calculus (or Algebra), as is commonly thought to be the\ncase. On the one hand, it is certain that the use of the calculus was\nnot entirely unknown to the ancient geometers, since they used to make\ncontinual and very extensive applications of the theory of proportions,\nwhich was for them, as a means of deduction, a sort of real, though very\nimperfect and especially extremely limited equivalent for our present\nalgebra. The calculus may even be employed in a much more complete\nmanner than they have used it, in order to obtain certain geometrical\nsolutions, which will still retain all the essential character of the\nancient geometry; this occurs very frequently with respect to those\nproblems of geometry of two or of three dimensions, which are commonly\ndesignated under the name of _determinate_. On the other hand, important\nas is the influence of the calculus in our modern geometry, various\nsolutions obtained without algebra may sometimes manifest the peculiar\ncharacter which distinguishes it from the ancient geometry, although\nanalysis is generally indispensable. I will cite, as an example, the\nmethod of Roberval for tangents, the nature of which is essentially\nmodern, and which, however, leads in certain cases to complete\nsolutions, without any aid from the calculus. It is not, then, the\ninstrument of deduction employed which is the principal distinction\nbetween the two courses which the human mind can take in geometry.\n\nThe real fundamental difference, as yet imperfectly apprehended, seems\nto me to consist in the very nature of the questions considered. In\ntruth, geometry, viewed as a whole, and supposed to have attained entire\nperfection, must, as we have seen on the one hand, embrace all\nimaginable figures, and, on the other, discover all the properties of\neach figure. It admits, from this double consideration, of being treated\naccording to two essentially distinct plans; either, 1\u00b0, by grouping\ntogether all the questions, however different they may be, which relate\nto the same figure, and isolating those relating to different bodies,\nwhatever analogy there may exist between them; or, 2\u00b0, on the contrary,\nby uniting under one point of view all similar inquiries, to whatever\ndifferent figures they may relate, and separating the questions relating\nto the really different properties of the same body. In a word, the\nwhole body of geometry may be essentially arranged either with reference\nto the _bodies_ studied or to the _phenomena_ to be considered. The\nfirst plan, which is the most natural, was that of the ancients; the\nsecond, infinitely more rational, is that of the moderns since\nDescartes.\n\n\n_Geometry of the Ancients._ Indeed, the principal characteristics of the\nancient geometry is that they studied, one by one, the different lines\nand the different surfaces, not passing to the examination of a new\nfigure till they thought they had exhausted all that there was\ninteresting in the figures already known. In this way of proceeding,\nwhen they undertook the study of a new curve, the whole of the labour\nbestowed on the preceding ones could not offer directly any essential\nassistance, otherwise than by the geometrical practice to which it had\ntrained the mind. Whatever might be the real similarity of the questions\nproposed as to two different figures, the complete knowledge acquired\nfor the one could not at all dispense with taking up again the whole of\nthe investigation for the other. Thus the progress of the mind was never\nassured; so that they could not be certain, in advance, of obtaining any\nsolution whatever, however analogous the proposed problem might be to\nquestions which had been already resolved. Thus, for example, the\ndetermination of the tangents to the three conic sections did not\nfurnish any rational assistance for drawing the tangent to any other new\ncurve, such as the conchoid, the cissoid, &c. In a word, the geometry of\nthe ancients was, according to the expression proposed above,\nessentially special.\n\n\n_Geometry of the Moderns._ In the system of the moderns, geometry is, on\nthe contrary, eminently _general_, that is to say, relating to any\nfigures whatever. It is easy to understand, in the first place, that all\ngeometrical expressions of any interest may be proposed with reference\nto all imaginable figures. This is seen directly in the fundamental\nproblems--of rectifications, quadratures, and cubatures--which\nconstitute, as has been shown, the final object of geometry. But this\nremark is no less incontestable, even for investigations which relate to\nthe different _properties_ of lines and of surfaces, and of which the\nmost essential, such as the question of tangents or of tangent planes,\nthe theory of curvatures, &c., are evidently common to all figures\nwhatever. The very few investigations which are truly peculiar to\nparticular figures have only an extremely secondary importance. This\nbeing understood, modern geometry consists essentially in abstracting,\nin order to treat it by itself, in an entirely general manner, every\nquestion relating to the same geometrical phenomenon, in whatever bodies\nit may be considered. The application of the universal theories thus\nconstructed to the special determination of the phenomenon which is\ntreated of in each particular body, is now regarded as only a subaltern\nlabour, to be executed according to invariable rules, and the success of\nwhich is certain in advance. This labour is, in a word, of the same\ncharacter as the numerical calculation of an analytical formula. There\ncan be no other merit in it than that of presenting in each case the\nsolution which is necessarily furnished by the general method, with all\nthe simplicity and elegance which the line or the surface considered can\nadmit of. But no real importance is attached to any thing but the\nconception and the complete solution of a new question belonging to any\nfigure whatever. Labours of this kind are alone regarded as producing\nany real advance in science. The attention of geometers, thus relieved\nfrom the examination of the peculiarities of different figures, and\nwholly directed towards general questions, has been thereby able to\nelevate itself to the consideration of new geometrical conceptions,\nwhich, applied to the curves studied by the ancients, have led to the\ndiscovery of important properties which they had not before even\nsuspected. Such is geometry, since the radical revolution produced by\nDescartes in the general system of the science.\n\n\n_The Superiority of the modern Geometry._ The mere indication of the\nfundamental character of each of the two geometries is undoubtedly\nsufficient to make apparent the immense necessary superiority of modern\ngeometry. We may even say that, before the great conception of\nDescartes, rational geometry was not truly constituted upon definitive\nbases, whether in its abstract or concrete relations. In fact, as\nregards science, considered speculatively, it is clear that, in\ncontinuing indefinitely to follow the course of the ancients, as did the\nmoderns before Descartes, and even for a little while afterwards, by\nadding some new curves to the small number of those which they had\nstudied, the progress thus made, however rapid it might have been, would\nstill be found, after a long series of ages, to be very inconsiderable\nin comparison with the general system of geometry, seeing the infinite\nvariety of the forms which would still have remained to be studied. On\nthe contrary, at each question resolved according to the method of the\nmoderns, the number of geometrical problems to be resolved is then, once\nfor all, diminished by so much with respect to all possible bodies.\nAnother consideration is, that it resulted, from their complete want of\ngeneral methods, that the ancient geometers, in all their\ninvestigations, were entirely abandoned to their own strength, without\never having the certainty of obtaining, sooner or later, any solution\nwhatever. Though this imperfection of the science was eminently suited\nto call forth all their admirable sagacity, it necessarily rendered\ntheir progress extremely slow; we can form some idea of this by the\nconsiderable time which they employed in the study of the conic\nsections. Modern geometry, making the progress of our mind certain,\npermits us, on the contrary, to make the greatest possible use of the\nforces of our intelligence, which the ancients were often obliged to\nwaste on very unimportant questions.\n\nA no less important difference between the two systems appears when we\ncome to consider geometry in the concrete point of view. Indeed, we have\nalready remarked that the relation of the abstract to the concrete in\ngeometry can be founded upon rational bases only so far as the\ninvestigations are made to bear directly upon all imaginable figures. In\nstudying lines, only one by one, whatever may be the number, always\nnecessarily very small, of those which we shall have considered, the\napplication of such theories to figures really existing in nature will\nnever have any other than an essentially accidental character, since\nthere is nothing to assure us that these figures can really be brought\nunder the abstract types considered by geometers.\n\nThus, for example, there is certainly something fortuitous in the happy\nrelation established between the speculations of the Greek geometers\nupon the conic sections and the determination of the true planetary\norbits. In continuing geometrical researches upon the same plan, there\nwas no good reason for hoping for similar coincidences; and it would\nhave been possible, in these special studies, that the researches of\ngeometers should have been directed to abstract figures entirely\nincapable of any application, while they neglected others, susceptible\nperhaps of an important and immediate application. It is clear, at\nleast, that nothing positively guaranteed the necessary applicability of\ngeometrical speculations. It is quite another thing in the modern\ngeometry. From the single circumstance that in it we proceed by general\nquestions relating to any figures whatever, we have in advance the\nevident certainty that the figures really existing in the external world\ncould in no case escape the appropriate theory if the geometrical\nphenomenon which it considers presents itself in them.\n\nFrom these different considerations, we see that the ancient system of\ngeometry wears essentially the character of the infancy of the science,\nwhich did not begin to become completely rational till after the\nphilosophical resolution produced by Descartes. But it is evident, on\nthe other hand, that geometry could not be at first conceived except in\nthis _special_ manner. _General_ geometry would not have been possible,\nand its necessity could not even have been felt, if a long series of\nspecial labours on the most simple figures had not previously furnished\nbases for the conception of Descartes, and rendered apparent the\nimpossibility of persisting indefinitely in the primitive geometrical\nphilosophy.\n\n\n_The Ancient the Base of the Modern._ From this last consideration we\nmust infer that, although the geometry which I have called _general_\nmust be now regarded as the only true dogmatical geometry, and that to\nwhich we shall chiefly confine ourselves, the other having no longer\nmuch more than an historical interest, nevertheless it is not possible\nto entirely dispense with _special_ geometry in a rational exposition of\nthe science. We undoubtedly need not borrow directly from ancient\ngeometry all the results which it has furnished; but, from the very\nnature of the subject, it is necessarily impossible entirely to dispense\nwith the ancient method, which will always serve as the preliminary\nbasis of the science, dogmatically as well as historically. The reason\nof this is easy to understand. In fact, _general_ geometry being\nessentially founded, as we shall soon establish, upon the employment of\nthe calculus in the transformation of geometrical into analytical\nconsiderations, such a manner of proceeding could not take possession of\nthe subject immediately at its origin. We know that the application of\nmathematical analysis, from its nature, can never commence any science\nwhatever, since evidently it cannot be employed until the science has\nalready been sufficiently cultivated to establish, with respect to the\nphenomena considered, some _equations_ which can serve as starting\npoints for the analytical operations. These fundamental equations being\nonce discovered, analysis will enable us to deduce from them a multitude\nof consequences which it would have been previously impossible even to\nsuspect; it will perfect the science to an immense degree, both with\nrespect to the generality of its conceptions and to the complete\nco-ordination established between them. But mere mathematical analysis\ncould never be sufficient to form the bases of any natural science, not\neven to demonstrate them anew when they have once been established.\nNothing can dispense with the direct study of the subject, pursued up to\nthe point of the discovery of precise relations.\n\nWe thus see that the geometry of the ancients will always have, by its\nnature, a primary part, absolutely necessary and more or less extensive,\nin the complete system of geometrical knowledge. It forms a rigorously\nindispensable introduction to _general_ geometry. But it is to this that\nit must be limited in a completely dogmatic exposition. I will consider,\nthen, directly, in the following chapter, this _special_ or\n_preliminary_ geometry restricted to exactly its necessary limits, in\norder to occupy myself thenceforth only with the philosophical\nexamination of _general_ or _definitive_ geometry, the only one which is\ntruly rational, and which at present essentially composes the science.\n\n\n\n\nCHAPTER II.\n\nANCIENT OR SYNTHETIC GEOMETRY.\n\n\nThe geometrical method of the ancients necessarily constituting a\npreliminary department in the dogmatical system of geometry, designed to\nfurnish _general_ geometry with indispensable foundations, it is now\nproper to begin with determining wherein strictly consists this\npreliminary function of _special_ geometry, thus reduced to the\nnarrowest possible limits.\n\n\n\n\nITS PROPER EXTENT.\n\n\n_Lines; Polygons; Polyhedrons._ In considering it under this point of\nview, it is easy to recognize that we might restrict it to the study of\nthe right line alone for what concerns the geometry of _lines_; to the\n_quadrature_ of rectilinear plane areas; and, lastly, to the _cubature_\nof bodies terminated by plane faces. The elementary propositions\nrelating to these three fundamental questions form, in fact, the\nnecessary starting point of all geometrical inquiries; they alone cannot\nbe obtained except by a direct study of the subject; while, on the\ncontrary, the complete theory of all other figures, even that of the\ncircle, and of the surfaces and volumes which are connected with it, may\nat the present day be completely comprehended in the domain of _general_\nor _analytical_ geometry; these primitive elements at once furnishing\n_equations_ which are sufficient to allow of the application of the\ncalculus to geometrical questions, which would not have been possible\nwithout this previous condition.\n\nIt results from this consideration that, in common practice, we give to\n_elementary_ geometry more extent than would be rigorously necessary to\nit; since, besides the right line, polygons, and polyhedrons, we also\ninclude in it the circle and the \"round\" bodies; the study of which\nmight, however, be as purely analytical as that, for example, of the\nconic sections. An unreflecting veneration for antiquity contributes to\nmaintain this defect in method; but the best reason which can be given\nfor it is the serious inconvenience for ordinary instruction which there\nwould be in postponing, to so distant an epoch of mathematical\neducation, the solution of several essential questions, which are\nsusceptible of a direct and continual application to a great number of\nimportant uses. In fact, to proceed in the most rational manner, we\nshould employ the integral calculus in obtaining the interesting results\nrelating to the length or the area of the circle, or to the quadrature\nof the sphere, &c., which have been determined by the ancients from\nextremely simple considerations. This inconvenience would be of little\nimportance with regard to the persons destined to study the whole of\nmathematical science, and the advantage of proceeding in a perfectly\nlogical order would have a much greater comparative value. But the\ncontrary case being the more frequent, theories so essential have\nnecessarily been retained in elementary geometry. Perhaps the conic\nsections, the cycloid, &c., might be advantageously added in such cases.\n\n\n_Not to be farther restricted._ While this preliminary portion of\ngeometry, which cannot be founded on the application of the calculus,\nis reduced by its nature to a very limited series of fundamental\nresearches, relating to the right line, polygonal areas, and\npolyhedrons, it is certain, on the other hand, that we cannot restrict\nit any more; although, by a veritable abuse of the spirit of analysis,\nit has been recently attempted to present the establishment of the\nprincipal theorems of elementary geometry under an algebraical point of\nview. Thus some have pretended to demonstrate, by simple abstract\nconsiderations of mathematical analysis, the constant relation which\nexists between the three angles of a rectilinear triangle, the\nfundamental proposition of the theory of similar triangles, that of\nparallelopipedons, &c.; in a word, precisely the only geometrical\npropositions which cannot be obtained except by a direct study of the\nsubject, without the calculus being susceptible of having any part in\nit. Such aberrations are the unreflecting exaggerations of that natural\nand philosophical tendency which leads us to extend farther and farther\nthe influence of analysis in mathematical studies. In mechanics, the\npretended analytical demonstrations of the parallelogram of forces are\nof similar character.\n\nThe viciousness of such a manner of proceeding follows from the\nprinciples previously presented. We have already, in fact, recognized\nthat, since the calculus is not, and cannot be, any thing but a means of\ndeduction, it would indicate a radically false idea of it to wish to\nemploy it in establishing the elementary foundations of any science\nwhatever; for on what would the analytical reasonings in such an\noperation repose? A labour of this nature, very far from really\nperfecting the philosophical character of a science, would constitute a\nreturn towards the metaphysical age, in presenting real facts as mere\nlogical abstractions.\n\nWhen we examine in themselves these pretended analytical demonstrations\nof the fundamental propositions of elementary geometry, we easily verify\ntheir necessary want of meaning. They are all founded on a vicious\nmanner of conceiving the principle of _homogeneity_, the true general\nidea of which was explained in the second chapter of the preceding book.\nThese demonstrations suppose that this principle does not allow us to\nadmit the coexistence in the same equation of numbers obtained by\ndifferent concrete comparisons, which is evidently false, and contrary\nto the constant practice of geometers. Thus it is easy to recognize\nthat, by employing the law of homogeneity in this arbitrary and\nillegitimate acceptation, we could succeed in \"demonstrating,\" with\nquite as much apparent rigour, propositions whose absurdity is manifest\nat the first glance. In examining attentively, for example, the\nprocedure by the aid of which it has been attempted to prove\nanalytically that the sum of the three angles of any rectilinear\ntriangle is constantly equal to two right angles, we see that it is\nfounded on this preliminary principle that, if two triangles have two of\ntheir angles respectively equal, the third angle of the one will\nnecessarily be equal to the third angle of the other. This first point\nbeing granted, the proposed relation is immediately deduced from it in a\nvery exact and simple manner. Now the analytical consideration by which\nthis previous proposition has been attempted to be established, is of\nsuch a nature that, if it could be correct, we could rigorously deduce\nfrom it, in reproducing it conversely, this palpable absurdity, that two\nsides of a triangle are sufficient, without any angle, for the entire\ndetermination of the third side. We may make analogous remarks on all\nthe demonstrations of this sort, the sophisms of which will be thus\nverified in a perfectly apparent manner.\n\nThe more reason that we have here to consider geometry as being at the\npresent day essentially analytical, the more necessary was it to guard\nagainst this abusive exaggeration of mathematical analysis, according to\nwhich all geometrical observation would be dispensed with, in\nestablishing upon pure algebraical abstractions the very foundations of\nthis natural science.\n\n\n_Attempted Demonstrations of Axioms, &c._ Another indication that\ngeometers have too much overlooked the character of a natural science\nwhich is necessarily inherent in geometry, appears from their vain\nattempts, so long made, to _demonstrate_ rigorously, not by the aid of\nthe calculus, but by means of certain constructions, several fundamental\npropositions of elementary geometry. Whatever may be effected, it will\nevidently be impossible to avoid sometimes recurring to simple and\ndirect observation in geometry as a means of establishing various\nresults. While, in this science, the phenomena which are considered are,\nby virtue of their extreme simplicity, much more closely connected with\none another than those relating to any other physical science, some must\nstill be found which cannot be deduced, and which, on the contrary,\nserve as starting points. It may be admitted that the greatest logical\nperfection of the science is to reduce these to the smallest number\npossible, but it would be absurd to pretend to make them completely\ndisappear. I avow, moreover, that I find fewer real inconveniences in\nextending, a little beyond what would be strictly necessary, the number\nof these geometrical notions thus established by direct observation,\nprovided they are sufficiently simple, than in making them the subjects\nof complicated and indirect demonstrations, even when these\ndemonstrations may be logically irreproachable.\n\nThe true dogmatic destination of the geometry of the ancients, reduced\nto its least possible indispensable developments, having thus been\ncharacterized as exactly as possible, it is proper to consider summarily\neach of the principal parts of which it must be composed. I think that I\nmay here limit myself to considering the first and the most extensive of\nthese parts, that which has for its object the study of _the right\nline_; the two other sections, namely, the _quadrature of polygons_ and\nthe _cubature of polyhedrons_, from their limited extent, not being\ncapable of giving rise to any philosophical consideration of any\nimportance, distinct from those indicated in the preceding chapter with\nrespect to the measure of areas and of volumes in general.\n\n\n\n\nGEOMETRY OF THE RIGHT LINE.\n\n\nThe final question which we always have in view in the study of the\nright line, properly consists in determining, by means of one another,\nthe different elements of any right-lined figure whatever; which enables\nus always to know indirectly the length and position of a right line, in\nwhatever circumstances it may be placed. This fundamental problem is\nsusceptible of two general solutions, the nature of which is quite\ndistinct, the one _graphical_, the other _algebraic_. The first, though\nvery imperfect, is that which must be first considered, because it is\nspontaneously derived from the direct study of the subject; the second,\nmuch more perfect in the most important respects, cannot be studied till\nafterwards, because it is founded upon the previous knowledge of the\nother.\n\n\n\n\nGRAPHICAL SOLUTIONS.\n\n\nThe graphical solution consists in constructing at will the proposed\nfigure, either with the same dimensions, or, more usually, with\ndimensions changed in any ratio whatever. The first mode need merely be\nmentioned as being the most simple and the one which would first occur\nto the mind, for it is evidently, by its nature, almost entirely\nincapable of application. The second is, on the contrary, susceptible of\nbeing most extensively and most usefully applied. We still make an\nimportant and continual use of it at the present day, not only to\nrepresent with exactness the forms of bodies and their relative\npositions, but even for the actual determination of geometrical\nmagnitudes, when we do not need great precision. The ancients, in\nconsequence of the imperfection of their geometrical knowledge, employed\nthis procedure in a much more extensive manner, since it was for a long\ntime the only one which they could apply, even in the most important\nprecise determinations. It was thus, for example, that Aristarchus of\nSamos estimated the relative distance from the sun and from the moon to\nthe earth, by making measurements on a triangle constructed as exactly\nas possible, so as to be similar to the right-angled triangle formed by\nthe three bodies at the instant when the moon is in quadrature, and when\nan observation of the angle at the earth would consequently be\nsufficient to define the triangle. Archimedes himself, although he was\nthe first to introduce calculated determinations into geometry, several\ntimes employed similar means. The formation of trigonometry did not\ncause this method to be entirely abandoned, although it greatly\ndiminished its use; the Greeks and the Arabians continued to employ it\nfor a great number of researches, in which we now regard the use of the\ncalculus as indispensable.\n\nThis exact reproduction of any figure whatever on a different scale\ncannot present any great theoretical difficulty when all the parts of\nthe proposed figure lie in the same plane. But if we suppose, as most\nfrequently happens, that they are situated in different planes, we see,\nthen, a new order of geometrical considerations arise. The artificial\nfigure, which is constantly plane, not being capable, in that case, of\nbeing a perfectly faithful image of the real figure, it is necessary\npreviously to fix with precision the mode of representation, which gives\nrise to different systems of _Projection_.\n\nIt then remains to be determined according to what laws the geometrical\nphenomena correspond in the two figures. This consideration generates a\nnew series of geometrical investigations, the final object of which is\nproperly to discover how we can replace constructions in relief by plane\nconstructions. The ancients had to resolve several elementary questions\nof this kind for various cases in which we now employ spherical\ntrigonometry, principally for different problems relating to the\ncelestial sphere. Such was the object of their _analemmas_, and of the\nother plane figures which for a long time supplied the place of the\ncalculus. We see by this that the ancients really knew the elements of\nwhat we now name _Descriptive Geometry_, although they did not conceive\nit in a distinct and general manner.\n\nI think it proper briefly to indicate in this place the true\nphilosophical character of this \"Descriptive Geometry;\" although, being\nessentially a science of application, it ought not to be included within\nthe proper domain of this work.\n\n\n\n\nDESCRIPTIVE GEOMETRY.\n\n\nAll questions of geometry of three dimensions necessarily give rise,\nwhen we consider their graphical solution, to a common difficulty which\nis peculiar to them; that of substituting for the different\nconstructions in relief, which are necessary to resolve them directly,\nand which it is almost always impossible to execute, simple equivalent\nplane constructions, by means of which we finally obtain the same\nresults. Without this indispensable transformation, every solution of\nthis kind would be evidently incomplete and really inapplicable in\npractice, although theoretically the constructions in space are usually\npreferable as being more direct. It was in order to furnish general\nmeans for always effecting such a transformation that _Descriptive\nGeometry_ was created, and formed into a distinct and homogeneous\nsystem, by the illustrious MONGE. He invented, in the first place, a\nuniform method of representing bodies by figures traced on a single\nplane, by the aid of _projections_ on two different planes, usually\nperpendicular to each other, and one of which is supposed to turn about\ntheir common intersection so as to coincide with the other produced; in\nthis system, or in any other equivalent to it, it is sufficient to\nregard points and lines as being determined by their projections, and\nsurfaces by the projections of their generating lines. This being\nestablished, Monge--analyzing with profound sagacity the various partial\nlabours of this kind which had before been executed by a number of\nincongruous procedures, and considering also, in a general and direct\nmanner, in what any questions of that nature must consist--found that\nthey could always be reduced to a very small number of invariable\nabstract problems, capable of being resolved separately, once for all,\nby uniform operations, relating essentially some to the contacts and\nothers to the intersections of surfaces. Simple and entirely general\nmethods for the graphical solution of these two orders of problems\nhaving been formed, all the geometrical questions which may arise in any\nof the various arts of construction--stone-cutting, carpentry,\nperspective, dialling, fortification, &c.--can henceforth be treated as\nsimple particular cases of a single theory, the invariable application\nof which will always necessarily lead to an exact solution, which may be\nfacilitated in practice by profiting by the peculiar circumstances of\neach case.\n\n       *       *       *       *       *\n\nThis important creation deserves in a remarkable degree to fix the\nattention of those philosophers who consider all that the human species\nhas yet effected as a first step, and thus far the only really complete\none, towards that general renovation of human labours, which must\nimprint upon all our arts a character of precision and of rationality,\nso necessary to their future progress. Such a revolution must, in fact,\ninevitably commence with that class of industrial labours, which is\nessentially connected with that science which is the most simple, the\nmost perfect, and the most ancient. It cannot fail to extend hereafter,\nthough with less facility, to all other practical operations. Indeed\nMonge himself, who conceived the true philosophy of the arts better than\nany one else, endeavoured to sketch out a corresponding system for the\nmechanical arts.\n\nEssential as the conception of descriptive geometry really is, it is\nvery important not to deceive ourselves with respect to its true\ndestination, as did those who, in the excitement of its first discovery,\nsaw in it a means of enlarging the general and abstract domain of\nrational geometry. The result has in no way answered to these mistaken\nhopes. And, indeed, is it not evident that descriptive geometry has no\nspecial value except as a science of application, and as forming the\ntrue special theory of the geometrical arts? Considered in its abstract\nrelations, it could not introduce any truly distinct order of\ngeometrical speculations. We must not forget that, in order that a\ngeometrical question should fall within the peculiar domain of\ndescriptive geometry, it must necessarily have been previously resolved\nby speculative geometry, the solutions of which then, as we have seen,\nalways need to be prepared for practice in such a way as to supply the\nplace of constructions in relief by plane constructions; a substitution\nwhich really constitutes the only characteristic function of descriptive\ngeometry.\n\nIt is proper, however, to remark here, that, with regard to intellectual\neducation, the study of descriptive geometry possesses an important\nphilosophical peculiarity, quite independent of its high industrial\nutility. This is the advantage which it so pre-eminently offers--in\nhabituating the mind to consider very complicated geometrical\ncombinations in space, and to follow with precision their continual\ncorrespondence with the figures which are actually traced--of thus\nexercising to the utmost, in the most certain and precise manner, that\nimportant faculty of the human mind which is properly called\n\"imagination,\" and which consists, in its elementary and positive\nacceptation, in representing to ourselves, clearly and easily, a vast\nand variable collection of ideal objects, as if they were really before\nus.\n\nFinally, to complete the indication of the general nature of descriptive\ngeometry by determining its logical character, we have to observe that,\nwhile it belongs to the geometry of the ancients by the character of its\nsolutions, on the other hand it approaches the geometry of the moderns\nby the nature of the questions which compose it. These questions are in\nfact eminently remarkable for that generality which, as we saw in the\npreceding chapter, constitutes the true fundamental character of modern\ngeometry; for the methods used are always conceived as applicable to any\nfigures whatever, the peculiarity of each having only a purely secondary\ninfluence. The solutions of descriptive geometry are then graphical,\nlike most of those of the ancients, and at the same time general, like\nthose of the moderns.\n\n       *       *       *       *       *\n\nAfter this important digression, we will pursue the philosophical\nexamination of _special_ geometry, always considered as reduced to its\nleast possible development, as an indispensable introduction to\n_general_ geometry. We have now sufficiently considered the _graphical_\nsolution of the fundamental problem relating to the right line--that\nis, the determination of the different elements of any right-lined\nfigure by means of one another--and have now to examine in a special\nmanner the _algebraic_ solution.\n\n\n\n\nALGEBRAIC SOLUTIONS.\n\n\nThis kind of solution, the evident superiority of which need not here be\ndwelt upon, belongs necessarily, by the very nature of the question, to\nthe system of the ancient geometry, although the logical method which is\nemployed causes it to be generally, but very improperly, separated from\nit. We have thus an opportunity of verifying, in a very important\nrespect, what was established generally in the preceding chapter, that\nit is not by the employment of the calculus that the modern geometry is\nessentially to be distinguished from the ancient. The ancients are in\nfact the true inventors of the present trigonometry, spherical as well\nas rectilinear; it being only much less perfect in their hands, on\naccount of the extreme inferiority of their algebraical knowledge. It\nis, then, really in this chapter, and not, as it might at first be\nthought, in those which we shall afterwards devote to the philosophical\nexamination of _general_ geometry, that it is proper to consider the\ncharacter of this important preliminary theory, which is usually, though\nimproperly, included in what is called _analytical geometry_, but which\nis really only a complement of _elementary geometry_ properly so called.\n\nSince all right-lined figures can be decomposed into triangles, it is\nevidently sufficient to know how to determine the different elements of\na triangle by means of one another, which reduces _polygonometry_ to\nsimple _trigonometry_.\n\n\n\n\nTRIGONOMETRY.\n\n\nThe difficulty in resolving algebraically such a question as the above,\nconsists essentially in forming, between the angles and the sides of a\ntriangle, three distinct equations; which, when once obtained, will\nevidently reduce all trigonometrical problems to mere questions of\nanalysis.\n\n\n_How to introduce Angles._ In considering the establishment of these\nequations in the most general manner, we immediately meet with a\nfundamental distinction with respect to the manner of introducing the\nangles into the calculation, according as they are made to enter\n_directly_, by themselves or by the circular arcs which are proportional\nto them; or _indirectly_, by the chords of these arcs, which are hence\ncalled their _trigonometrical lines_. Of these two systems of\ntrigonometry the second was of necessity the only one originally\nadopted, as being the only practicable one, since the condition of\ngeometry made it easy enough to find exact relations between the sides\nof the triangles and the trigonometrical lines which represent the\nangles, while it would have been absolutely impossible at that epoch to\nestablish equations between the sides and the angles themselves.\n\n\n_Advantages of introducing Trigonometrical Lines._ At the present day,\nsince the solution can be obtained by either system indifferently, that\nmotive for preference no longer exists; but geometers have none the less\npersisted in following from choice the system primitively admitted from\nnecessity; for, the same reason which enabled these trigonometrical\nequations to be obtained with much more facility, must, in like manner,\nas it is still more easy to conceive _\u00e0 priori_, render these equations\nmuch more simple, since they then exist only between right lines,\ninstead of being established between right lines and arcs of circles.\nSuch a consideration has so much the more importance, as the question\nrelates to formulas which are eminently elementary, and destined to be\ncontinually employed in all parts of mathematical science, as well as in\nall its various applications.\n\nIt may be objected, however, that when an angle is given, it is, in\nreality, always given by itself, and not by its trigonometrical lines;\nand that when it is unknown, it is its angular value which is properly\nto be determined, and not that of any of its trigonometrical lines. It\nseems, according to this, that such lines are only useless\nintermediaries between the sides and the angles, which have to be\nfinally eliminated, and the introduction of which does not appear\ncapable of simplifying the proposed research. It is indeed important to\nexplain, with more generality and precision than is customary, the great\nreal utility of this manner of proceeding.\n\n\n_Division of Trigonometry into two Parts._ It consists in the fact that\nthe introduction of these auxiliary magnitudes divides the entire\nquestion of trigonometry into two others essentially distinct, one of\nwhich has for its object to pass from the angles to their\ntrigonometrical lines, or the converse, and the other of which proposes\nto determine the sides of the triangles by the trigonometrical lines of\ntheir angles, or the converse. Now the first of these two fundamental\nquestions is evidently susceptible, by its nature, of being entirely\ntreated and reduced to numerical tables once for all, in considering all\npossible angles, since it depends only upon those angles, and not at all\nupon the particular triangles in which they may enter in each case;\nwhile the solution of the second question must necessarily be renewed,\nat least in its arithmetical relations, for each new triangle which it\nis necessary to resolve. This is the reason why the first portion of the\ncomplete work, which would be precisely the most laborious, is no longer\ntaken into the account, being always done in advance; while, if such a\ndecomposition had not been performed, we would evidently have found\nourselves under the obligation of recommencing the entire calculation in\neach particular case. Such is the essential property of the present\ntrigonometrical system, which in fact would really present no actual\nadvantage, if it was necessary to calculate continually the\ntrigonometrical line of each angle to be considered, or the converse;\nthe intermediate agency introduced would then be more troublesome than\nconvenient.\n\nIn order to clearly comprehend the true nature of this conception, it\nwill be useful to compare it with a still more important one, designed\nto produce an analogous effect either in its algebraic, or, still more,\nin its arithmetical relations--the admirable theory of _logarithms_. In\nexamining in a philosophical manner the influence of this theory, we see\nin fact that its general result is to decompose all imaginable\narithmetical operations into two distinct parts. The first and most\ncomplicated of these is capable of being executed in advance once for\nall (since it depends only upon the numbers to be considered, and not at\nall upon the infinitely different combinations into which they can\nenter), and consists in considering all numbers as assignable powers of\na constant number. The second part of the calculation, which must of\nnecessity be recommenced for each new formula which is to have its\nvalue determined, is thenceforth reduced to executing upon these\nexponents correlative operations which are infinitely more simple. I\nconfine myself here to merely indicating this resemblance, which any one\ncan carry out for himself.\n\nWe must besides observe, as a property (secondary at the present day,\nbut all-important at its origin) of the trigonometrical system adopted,\nthe very remarkable circumstance that the determination of angles by\ntheir trigonometrical lines, or the converse, admits of an arithmetical\nsolution (the only one which is directly indispensable for the special\ndestination of trigonometry) without the previous resolution of the\ncorresponding algebraic question. It is doubtless to such a peculiarity\nthat the ancients owed the possibility of knowing trigonometry. The\ninvestigation conceived in this way was so much the more easy, inasmuch\nas tables of chords (which the ancients naturally took as the\ntrigonometrical lines) had been previously constructed for quite a\ndifferent object, in the course of the labours of Archimedes on the\nrectification of the circle, from which resulted the actual\ndetermination of a certain series of chords; so that when Hipparchus\nsubsequently invented trigonometry, he could confine himself to\ncompleting that operation by suitable intercalations; which shows\nclearly the connexion of ideas in that matter.\n\n\n_The Increase of such Trigonometrical Lines._ To complete this\nphilosophical sketch of trigonometry, it is proper now to observe that\nthe extension of the same considerations which lead us to replace angles\nor arcs of circles by straight lines, with the view of simplifying our\nequations, must also lead us to employ concurrently several\ntrigonometrical lines, instead of confining ourselves to one only (as\ndid the ancients), so as to perfect this system by choosing that one\nwhich will be algebraically the most convenient on each occasion. In\nthis point of view, it is clear that the number of these lines is in\nitself no ways limited; provided that they are determined by the arc,\nand that they determine it, whatever may be the law according to which\nthey are derived from it, they are suitable to be substituted for it in\nthe equations. The Arabians, and subsequently the moderns, in confining\nthemselves to the most simple constructions, have carried to four or\nfive the number of _direct_ trigonometrical lines, which might be\nextended much farther.\n\nBut instead of recurring to geometrical formations, which would finally\nbecome very complicated, we conceive with the utmost facility as many\nnew trigonometrical lines as the analytical transformations may require,\nby means of a remarkable artifice, which is not usually apprehended in a\nsufficiently general manner. It consists in not directly multiplying the\ntrigonometrical lines appropriate to each arc considered, but in\nintroducing new ones, by considering this arc as indirectly determined\nby all lines relating to an arc which is a very simple function of the\nfirst. It is thus, for example, that, in order to calculate an angle\nwith more facility, we will determine, instead of its sine, the sine of\nits half, or of its double, &c. Such a creation of _indirect_\ntrigonometrical lines is evidently much more fruitful than all the\ndirect geometrical methods for obtaining new ones. We may accordingly\nsay that the number of trigonometrical lines actually employed at the\npresent day by geometers is in reality unlimited, since at every\ninstant, so to say, the transformations of analysis may lead us to\naugment it by the method which I have just indicated. Special names,\nhowever, have been given to those only of these _indirect_ lines which\nrefer to the complement of the primitive arc, the others not occurring\nsufficiently often to render such denominations necessary; a\ncircumstance which has caused a common misconception of the true extent\nof the system of trigonometry.\n\n\n_Study of their Mutual Relations._ This multiplicity of trigonometrical\nlines evidently gives rise to a third fundamental question in\ntrigonometry, the study of the relations which exist between these\ndifferent lines; since, without such a knowledge, we could not make use,\nfor our analytical necessities, of this variety of auxiliary magnitudes,\nwhich, however, have no other destination. It is clear, besides, from\nthe consideration just indicated, that this essential part of\ntrigonometry, although simply preparatory, is, by its nature,\nsusceptible of an indefinite extension when we view it in its entire\ngenerality, while the two others are circumscribed within rigorously\ndefined limits.\n\nIt is needless to add that these three principal parts of trigonometry\nhave to be studied in precisely the inverse order from that in which we\nhave seen them necessarily derived from the general nature of the\nsubject; for the third is evidently independent of the two others, and\nthe second, of that which was first presented--the resolution of\ntriangles, properly so called--which must for that reason be treated in\nthe last place; which rendered so much the more important the\nconsideration of their natural succession and logical relations to one\nanother.\n\nIt is useless to consider here separately _spherical trigonometry_,\nwhich cannot give rise to any special philosophical consideration;\nsince, essential as it is by the importance and the multiplicity of its\nuses, it can be treated at the present day only as a simple application\nof rectilinear trigonometry, which furnishes directly its fundamental\nequations, by substituting for the spherical triangle the corresponding\ntrihedral angle.\n\nThis summary exposition of the philosophy of trigonometry has been here\ngiven in order to render apparent, by an important example, that\nrigorous dependence and those successive ramifications which are\npresented by what are apparently the most simple questions of elementary\ngeometry.\n\n       *       *       *       *       *\n\nHaving thus examined the peculiar character of _special_ geometry\nreduced to its only dogmatic destination, that of furnishing to general\ngeometry an indispensable preliminary basis, we have now to give all our\nattention to the true science of geometry, considered as a whole, in the\nmost rational manner. For that purpose, it is necessary to carefully\nexamine the great original idea of Descartes, upon which it is entirely\nfounded. This will be the object of the following chapter.\n\n\n\n\nCHAPTER III.\n\nMODERN OR ANALYTICAL GEOMETRY.\n\n\n_General_ (or _Analytical_) geometry being entirely founded upon the\ntransformation of geometrical considerations into equivalent analytical\nconsiderations, we must begin with examining directly and in a thorough\nmanner the beautiful conception by which Descartes has established in a\nuniform manner the constant possibility of such a co-relation. Besides\nits own extreme importance as a means of highly perfecting geometrical\nscience, or, rather, of establishing the whole of it on rational bases,\nthe philosophical study of this admirable conception must have so much\nthe greater interest in our eyes from its characterizing with perfect\nclearness the general method to be employed in organizing the relations\nof the abstract to the concrete in mathematics, by the analytical\nrepresentation of natural phenomena. There is no conception, in the\nwhole philosophy of mathematics which better deserves to fix all our\nattention.\n\n\n\n\nANALYTICAL REPRESENTATION OF FIGURES.\n\n\nIn order to succeed in expressing all imaginable geometrical phenomena\nby simple analytical relations, we must evidently, in the first place,\nestablish a general method for representing analytically the subjects\nthemselves in which these phenomena are found, that is, the lines or the\nsurfaces to be considered. The _subject_ being thus habitually\nconsidered in a purely analytical point of view, we see how it is\nthenceforth possible to conceive in the same manner the various\n_accidents_ of which it is susceptible.\n\nIn order to organize the representation of geometrical figures by\nanalytical equations, we must previously surmount a fundamental\ndifficulty; that of reducing the general elements of the various\nconceptions of geometry to simply numerical ideas; in a word, that of\nsubstituting in geometry pure considerations of _quantity_ for all\nconsiderations of _quality_.\n\n\n_Reduction of Figure to Position._ For this purpose let us observe, in\nthe first place, that all geometrical ideas relate necessarily to these\nthree universal categories: the _magnitude_, the _figure_, and the\n_position_ of the extensions to be considered. As to the first, there is\nevidently no difficulty; it enters at once into the ideas of numbers.\nWith relation to the second, it must be remarked that it will always\nadmit of being reduced to the third. For the figure of a body evidently\nresults from the mutual position of the different points of which it is\ncomposed, so that the idea of position necessarily comprehends that of\nfigure, and every circumstance of figure can be translated by a\ncircumstance of position. It is in this way, in fact, that the human\nmind has proceeded in order to arrive at the analytical representation\nof geometrical figures, their conception relating directly only to\npositions. All the elementary difficulty is then properly reduced to\nthat of referring ideas of situation to ideas of magnitude. Such is the\ndirect destination of the preliminary conception upon which Descartes\nhas established the general system of analytical geometry.\n\nHis philosophical labour, in this relation, has consisted simply in the\nentire generalization of an elementary operation, which we may regard as\nnatural to the human mind, since it is performed spontaneously, so to\nsay, in all minds, even the most uncultivated. Thus, when we have to\nindicate the situation of an object without directly pointing it out,\nthe method which we always adopt, and evidently the only one which can\nbe employed, consists in referring that object to others which are\nknown, by assigning the magnitude of the various geometrical elements,\nby which we conceive it connected with the known objects. These elements\nconstitute what Descartes, and after him all geometers, have called the\n_co-ordinates_ of each point considered. They are necessarily two in\nnumber, if it is known in advance in what plane the point is situated;\nand three, if it may be found indifferently in any region of space. As\nmany different constructions as can be imagined for determining the\nposition of a point, whether on a plane or in space, so many distinct\nsystems of co-ordinates may be conceived; they are consequently\nsusceptible of being multiplied to infinity. But, whatever may be the\nsystem adopted, we shall always have reduced the ideas of situation to\nsimple ideas of magnitude, so that we will consider the change in the\nposition of a point as produced by mere numerical variations in the\nvalues of its co-ordinates.\n\n\n_Determination of the Position of a Point._ Considering at first only\nthe least complicated case, that of _plane geometry_, it is in this way\nthat we usually determine the position of a point on a plane, by its\ndistances from two fixed right lines considered as known, which are\ncalled _axes_, and which are commonly supposed to be perpendicular to\neach other. This system is that most frequently adopted, because of its\nsimplicity; but geometers employ occasionally an infinity of others.\nThus the position of a point on a plane may be determined, 1\u00b0, by its\ndistances from two fixed points; or, 2\u00b0, by its distance from a single\nfixed point, and the direction of that distance, estimated by the\ngreater or less angle which it makes with a fixed right line, which\nconstitutes the system of what are called _polar_ co-ordinates, the most\nfrequently used after the system first mentioned; or, 3\u00b0, by the angles\nwhich the right lines drawn from the variable point to two fixed points\nmake with the right line which joins these last; or, 4\u00b0, by the\ndistances from that point to a fixed right line and a fixed point, &c.\nIn a word, there is no geometrical figure whatever from which it is not\npossible to deduce a certain system of co-ordinates more or less\nsusceptible of being employed.\n\nA general observation, which it is important to make in this connexion,\nis, that every system of co-ordinates is equivalent to determining a\npoint, in plane geometry, by the intersection of two lines, each of\nwhich is subjected to certain fixed conditions of determination; a\nsingle one of these conditions remaining variable, sometimes the one,\nsometimes the other, according to the system considered. We could not,\nindeed, conceive any other means of constructing a point than to mark it\nby the meeting of two lines. Thus, in the most common system, that of\n_rectilinear co-ordinates_, properly so called, the point is determined\nby the intersection of two right lines, each of which remains constantly\nparallel to a fixed axis, at a greater or less distance from it; in the\n_polar_ system, the position of the point is marked by the meeting of a\ncircle, of variable radius and fixed centre, with a movable right line\ncompelled to turn about this centre: in other systems, the required\npoint might be designated by the intersection of two circles, or of any\nother two lines, &c. In a word, to assign the value of one of the\nco-ordinates of a point in any system whatever, is always necessarily\nequivalent to determining a certain line on which that point must be\nsituated. The geometers of antiquity had already made this essential\nremark, which served as the base of their method of geometrical _loci_,\nof which they made so happy a use to direct their researches in the\nresolution of _determinate_ problems, in considering separately the\ninfluence of each of the two conditions by which was defined each point\nconstituting the object, direct or indirect, of the proposed question.\nIt was the general systematization of this method which was the\nimmediate motive of the labours of Descartes, which led him to create\nanalytical geometry.\n\nAfter having clearly established this preliminary conception--by means\nof which ideas of position, and thence, implicitly, all elementary\ngeometrical conceptions are capable of being reduced to simple numerical\nconsiderations--it is easy to form a direct conception, in its entire\ngenerality, of the great original idea of Descartes, relative to the\nanalytical representation of geometrical figures: it is this which forms\nthe special object of this chapter. I will continue to consider at\nfirst, for more facility, only geometry of two dimensions, which alone\nwas treated by Descartes; and will afterwards examine separately, under\nthe same point of view, the theory of surfaces and curves of double\ncurvature.\n\n\n\n\nPLANE CURVES.\n\n\n_Expression of Lines by Equations._ In accordance with the manner of\nexpressing analytically the position of a point on a plane, it can be\neasily established that, by whatever property any line may be defined,\nthat definition always admits of being replaced by a corresponding\nequation between the two variable co-ordinates of the point which\ndescribes this line; an equation which will be thenceforth the\nanalytical representation of the proposed line, every phenomenon of\nwhich will be translated by a certain algebraic modification of its\nequation. Thus, if we suppose that a point moves on a plane without its\ncourse being in any manner determined, we shall evidently have to regard\nits co-ordinates, to whatever system they may belong, as two variables\nentirely independent of one another. But if, on the contrary, this point\nis compelled to describe a certain line, we shall necessarily be\ncompelled to conceive that its co-ordinates, in all the positions which\nit can take, retain a certain permanent and precise relation to each\nother, which is consequently susceptible of being expressed by a\nsuitable equation; which will become the very clear and very rigorous\nanalytical definition of the line under consideration, since it will\nexpress an algebraical property belonging exclusively to the\nco-ordinates of all the points of this line. It is clear, indeed, that\nwhen a point is not subjected to any condition, its situation is not\ndetermined except in giving at once its two co-ordinates, independently\nof each other; while, when the point must continue upon a defined line,\na single co-ordinate is sufficient for completely fixing its position.\nThe second co-ordinate is then a determinate _function_ of the first;\nor, in other words, there must exist between them a certain _equation_,\nof a nature corresponding to that of the line on which the point is\ncompelled to remain. In a word, each of the co-ordinates of a point\nrequiring it to be situated on a certain line, we conceive reciprocally\nthat the condition, on the part of a point, of having to belong to a\nline defined in any manner whatever, is equivalent to assigning the\nvalue of one of the two co-ordinates; which is found in that case to be\nentirely dependent on the other. The analytical relation which expresses\nthis dependence may be more or less difficult to discover, but it must\nevidently be always conceived to exist, even in the cases in which our\npresent means may be insufficient to make it known. It is by this simple\nconsideration that we may demonstrate, in an entirely general\nmanner--independently of the particular verifications on which this\nfundamental conception is ordinarily established for each special\ndefinition of a line--the necessity of the analytical representation of\nlines by equations.\n\n\n_Expression of Equations by Lines._ Taking up again the same reflections\nin the inverse direction, we could show as easily the geometrical\nnecessity of the representation of every equation of two variables, in a\ndeterminate system of co-ordinates, by a certain line; of which such a\nrelation would be, in the absence of any other known property, a very\ncharacteristic definition, the scientific destination of which will be\nto fix the attention directly upon the general course of the solutions\nof the equation, which will thus be noted in the most striking and the\nmost simple manner. This picturing of equations is one of the most\nimportant fundamental advantages of analytical geometry, which has\nthereby reacted in the highest degree upon the general perfecting of\nanalysis itself; not only by assigning to purely abstract researches a\nclearly determined object and an inexhaustible career, but, in a still\nmore direct relation, by furnishing a new philosophical medium for\nanalytical meditation which could not be replaced by any other. In fact,\nthe purely algebraic discussion of an equation undoubtedly makes known\nits solutions in the most precise manner, but in considering them only\none by one, so that in this way no general view of them could be\nobtained, except as the final result of a long and laborious series of\nnumerical comparisons. On the other hand, the geometrical _locus_ of the\nequation, being only designed to represent distinctly and with perfect\nclearness the summing up of all these comparisons, permits it to be\ndirectly considered, without paying any attention to the details which\nhave furnished it. It can thereby suggest to our mind general analytical\nviews, which we should have arrived at with much difficulty in any other\nmanner, for want of a means of clearly characterizing their object. It\nis evident, for example, that the simple inspection of the logarithmic\ncurve, or of the curve _y_ = sin. _x_, makes us perceive much more\ndistinctly the general manner of the variations of logarithms with\nrespect to their numbers, or of sines with respect to their arcs, than\ncould the most attentive study of a table of logarithms or of natural\nsines. It is well known that this method has become entirely elementary\nat the present day, and that it is employed whenever it is desired to\nget a clear idea of the general character of the law which reigns in a\nseries of precise observations of any kind whatever.\n\n\n_Any Change in the Line causes a Change in the Equation._ Returning to\nthe representation of lines by equations, which is our principal object,\nwe see that this representation is, by its nature, so faithful, that the\nline could not experience any modification, however slight it might be,\nwithout causing a corresponding change in the equation. This perfect\nexactitude even gives rise oftentimes to special difficulties; for\nsince, in our system of analytical geometry, the mere displacements of\nlines affect the equations, as well as their real variations in\nmagnitude or form, we should be liable to confound them with one another\nin our analytical expressions, if geometers had not discovered an\ningenious method designed expressly to always distinguish them. This\nmethod is founded on this principle, that although it is impossible to\nchange analytically at will the position of a line with respect to the\naxes of the co-ordinates, we can change in any manner whatever the\nsituation of the axes themselves, which evidently amounts to the same;\nthen, by the aid of the very simple general formula by which this\ntransformation of the axes is produced, it becomes easy to discover\nwhether two different equations are the analytical expressions of only\nthe same line differently situated, or refer to truly distinct\ngeometrical loci; since, in the former case, one of them will pass into\nthe other by suitably changing the axes or the other constants of the\nsystem of co-ordinates employed. It must, moreover, be remarked on this\nsubject, that general inconveniences of this nature seem to be\nabsolutely inevitable in analytical geometry; for, since the ideas of\nposition are, as we have seen, the only geometrical ideas immediately\nreducible to numerical considerations, and the conceptions of figure\ncannot be thus reduced, except by seeing in them relations of situation,\nit is impossible for analysis to escape confounding, at first, the\nphenomena of figure with simple phenomena of position, which alone are\ndirectly expressed by the equations.\n\n\n_Every Definition of a Line is an Equation._ In order to complete the\nphilosophical explanation of the fundamental conception which serves as\nthe base of analytical geometry, I think that I should here indicate a\nnew general consideration, which seems to me particularly well adapted\nfor putting in the clearest point of view this necessary representation\nof lines by equations with two variables. It consists in this, that not\nonly, as we have shown, must every defined line necessarily give rise to\na certain equation between the two co-ordinates of any one of its\npoints, but, still farther, every definition of a line may be regarded\nas being already of itself an equation of that line in a suitable system\nof co-ordinates.\n\nIt is easy to establish this principle, first making a preliminary\nlogical distinction with respect to different kinds of definitions. The\nrigorously indispensable condition of every definition is that of\ndistinguishing the object defined from all others, by assigning to it a\nproperty which belongs to it exclusively. But this end may be generally\nattained in two very different ways; either by a definition which is\nsimply _characteristic_, that is, indicative of a property which,\nalthough truly exclusive, does not make known the mode of generation of\nthe object; or by a definition which is really _explanatory_, that is,\nwhich characterizes the object by a property which expresses one of its\nmodes of generation. For example, in considering the circle as the line,\nwhich, under the same contour, contains the greatest area, we have\nevidently a definition of the first kind; while in choosing the property\nof its having all its points equally distant from a fixed point, we have\na definition of the second kind. It is, besides, evident, as a general\nprinciple, that even when any object whatever is known at first only by\na _characteristic_ definition, we ought, nevertheless, to regard it as\nsusceptible of _explanatory_ definitions, which the farther study of the\nobject would necessarily lead us to discover.\n\nThis being premised, it is clear that the general observation above\nmade, which represents every definition of a line as being necessarily\nan equation of that line in a certain system of co-ordinates, cannot\napply to definitions which are simply _characteristic_; it is to be\nunderstood only of definitions which are truly _explanatory_. But, in\nconsidering only this class, the principle is easy to prove. In fact, it\nis evidently impossible to define the generation of a line without\nspecifying a certain relation between the two simple motions of\ntranslation or of rotation, into which the motion of the point which\ndescribes it will be decomposed at each instant. Now if we form the most\ngeneral conception of what constitutes _a system of co-ordinates_, and\nadmit all possible systems, it is clear that such a relation will be\nnothing else but the _equation_ of the proposed line, in a system of\nco-ordinates of a nature corresponding to that of the mode of generation\nconsidered. Thus, for example, the common definition of the _circle_ may\nevidently be regarded as being immediately the _polar equation_ of this\ncurve, taking the centre of the circle for the pole. In the same way,\nthe elementary definition of the _ellipse_ or of the _hyperbola_--as\nbeing the curve generated by a point which moves in such a manner that\nthe sum or the difference of its distances from two fixed points remains\nconstant--gives at once, for either the one or the other curve, the\nequation _y_ + _x_ = _c_, taking for the system of co-ordinates that in\nwhich the position of a point would be determined by its distances from\ntwo fixed points, and choosing for these poles the two given foci. In\nlike manner, the common definition of any _cycloid_ would furnish\ndirectly, for that curve, the equation _y_ = _mx_; adopting as the\nco-ordinates of each point the arc which it marks upon a circle of\ninvariable radius, measuring from the point of contact of that circle\nwith a fixed line, and the rectilinear distance from that point of\ncontact to a certain origin taken on that right line. We can make\nanalogous and equally easy verifications with respect to the customary\ndefinitions of spirals, of epicycloids, &c. We shall constantly find\nthat there exists a certain system of co-ordinates, in which we\nimmediately obtain a very simple equation of the proposed line, by\nmerely writing algebraically the condition imposed by the mode of\ngeneration considered.\n\nBesides its direct importance as a means of rendering perfectly apparent\nthe necessary representation of every line by an equation, the preceding\nconsideration seems to me to possess a true scientific utility, in\ncharacterizing with precision the principal general difficulty which\noccurs in the actual establishment of these equations, and in\nconsequently furnishing an interesting indication with respect to the\ncourse to be pursued in inquiries of this kind, which, by their nature,\ncould not admit of complete and invariable rules. In fact, since any\ndefinition whatever of a line, at least among those which indicate a\nmode of generation, furnishes directly the equation of that line in a\ncertain system of co-ordinates, or, rather, of itself constitutes that\nequation, it follows that the difficulty which we often experience in\ndiscovering the equation of a curve, by means of certain of its\ncharacteristic properties, a difficulty which is sometimes very great,\nmust proceed essentially only from the commonly imposed condition of\nexpressing this curve analytically by the aid of a designated system of\nco-ordinates, instead of admitting indifferently all possible systems.\nThese different systems cannot be regarded in analytical geometry as\nbeing all equally suitable; for various reasons, the most important of\nwhich will be hereafter discussed, geometers think that curves should\nalmost always be referred, as far as is possible, to _rectilinear\nco-ordinates_, properly so called. Now we see, from what precedes, that\nin many cases these particular co-ordinates will not be those with\nreference to which the equation of the curve will be found to be\ndirectly established by the proposed definition. The principal\ndifficulty presented by the formation of the equation of a line really\nconsists, then, in general, in a certain transformation of co-ordinates.\nIt is undoubtedly true that this consideration does not subject the\nestablishment of these equations to a truly complete general method, the\nsuccess of which is always certain; which, from the very nature of the\nsubject, is evidently chimerical: but such a view may throw much useful\nlight upon the course which it is proper to adopt, in order to arrive at\nthe end proposed. Thus, after having in the first place formed the\npreparatory equation, which is spontaneously derived from the definition\nwhich we are considering, it will be necessary, in order to obtain the\nequation belonging to the system of co-ordinates which must be finally\nadmitted, to endeavour to express in a function of these last\nco-ordinates those which naturally correspond to the given mode of\ngeneration. It is upon this last labour that it is evidently impossible\nto give invariable and precise precepts. We can only say that we shall\nhave so many more resources in this matter as we shall know more of true\nanalytical geometry, that is, as we shall know the algebraical\nexpression of a greater number of different algebraical phenomena.\n\n\n\n\nCHOICE OF CO-ORDINATES.\n\n\nIn order to complete the philosophical exposition of the conception\nwhich serves as the base of analytical geometry, I have yet to notice\nthe considerations relating to the choice of the system of co-ordinates\nwhich is in general the most suitable. They will give the rational\nexplanation of the preference unanimously accorded to the ordinary\nrectilinear system; a preference which has hitherto been rather the\neffect of an empirical sentiment of the superiority of this system, than\nthe exact result of a direct and thorough analysis.\n\n\n_Two different Points of View._ In order to decide clearly between all\nthe different systems of co-ordinates, it is indispensable to\ndistinguish with care the two general points of view, the converse of\none another, which belong to analytical geometry; namely, the relation\nof algebra to geometry, founded upon the representation of lines by\nequations; and, reciprocally, the relation of geometry to algebra,\nfounded on the representation of equations by lines.\n\nIt is evident that in every investigation of general geometry these two\nfundamental points of view are of necessity always found combined,\nsince we have always to pass alternately, and at insensible intervals,\nso to say, from geometrical to analytical considerations, and from\nanalytical to geometrical considerations. But the necessity of here\ntemporarily separating them is none the less real; for the answer to the\nquestion of method which we are examining is, in fact, as we shall see\npresently, very far from being the same in both these relations, so that\nwithout this distinction we could not form any clear idea of it.\n\n\n1. _Representation of Lines by Equations._ Under _the first point of\nview_--the representation of lines by equations--the only reason which\ncould lead us to prefer one system of co-ordinates to another would be\nthe greater simplicity of the equation of each line, and greater\nfacility in arriving at it. Now it is easy to see that there does not\nexist, and could not be expected to exist, any system of co-ordinates\ndeserving in that respect a constant preference over all others. In\nfact, we have above remarked that for each geometrical definition\nproposed we can conceive a system of co-ordinates in which the equation\nof the line is obtained at once, and is necessarily found to be also\nvery simple; and this system, moreover, inevitably varies with the\nnature of the characteristic property under consideration. The\nrectilinear system could not, therefore, be constantly the most\nadvantageous for this object, although it may often be very favourable;\nthere is probably no system which, in certain particular cases, should\nnot be preferred to it, as well as to every other.\n\n\n2. _Representation of Equations by Lines._ It is by no means so,\nhowever, under the _second point of view_. We can, indeed, easily\nestablish, as a general principle, that the ordinary rectilinear system\nmust necessarily be better adapted than any other to the representation\nof equations by the corresponding geometrical _loci_; that is to say,\nthat this representation is constantly more simple and more faithful in\nit than in any other.\n\nLet us consider, for this object, that, since every system of\nco-ordinates consists in determining a point by the intersection of two\nlines, the system adapted to furnish the most suitable geometrical\n_loci_ must be that in which these two lines are the simplest possible;\na consideration which confines our choice to the _rectilinear_ system.\nIn truth, there is evidently an infinite number of systems which deserve\nthat name, that is to say, which employ only right lines to determine\npoints, besides the ordinary system which assigns the distances from two\nfixed lines as co-ordinates; such, for example, would be that in which\nthe co-ordinates of each point should be the two angles which the right\nlines, which go from that point to two fixed points, make with the right\nline, which joins these last points: so that this first consideration is\nnot rigorously sufficient to explain the preference unanimously given to\nthe common system. But in examining in a more thorough manner the nature\nof every system of co-ordinates, we also perceive that each of the two\nlines, whose meeting determines the point considered, must necessarily\noffer at every instant, among its different conditions of determination,\na single variable condition, which gives rise to the corresponding\nco-ordinate, all the rest being fixed, and constituting the _axes_ of\nthe system, taking this term in its most extended mathematical\nacceptation. The variation is indispensable, in order that we may be\nable to consider all possible positions; and the fixity is no less so,\nin order that there may exist means of comparison. Thus, in all\n_rectilinear_ systems, each of the two right lines will be subjected to\na fixed condition, and the ordinate will result from the variable\ncondition.\n\n\n_Superiority of rectilinear Co-ordinates._ From these considerations it\nis evident, as a general principle, that the most favourable system for\nthe construction of geometrical _loci_ will necessarily be that in which\nthe variable condition of each right line shall be the simplest\npossible; the fixed condition being left free to be made complex, if\nnecessary to attain that object. Now, of all possible manners of\ndetermining two movable right lines, the easiest to follow geometrically\nis certainly that in which, the direction of each right line remaining\ninvariable, it only approaches or recedes, more or less, to or from a\nconstant axis. It would be, for example, evidently more difficult to\nfigure to one's self clearly the changes of place of a point which is\ndetermined by the intersection of two right lines, which each turn\naround a fixed point, making a greater or smaller angle with a certain\naxis, as in the system of co-ordinates previously noticed. Such is the\ntrue general explanation of the fundamental property possessed by the\ncommon rectilinear system, of being better adapted than any other to the\ngeometrical representation of equations, inasmuch as it is that one in\nwhich it is the easiest to conceive the change of place of a point\nresulting from the change in the value of its co-ordinates. In order to\nfeel clearly all the force of this consideration, it would be sufficient\nto carefully compare this system with the polar system, in which this\ngeometrical image, so simple and so easy to follow, of two right lines\nmoving parallel, each one of them, to its corresponding axis, is\nreplaced by the complicated picture of an infinite series of concentric\ncircles, cut by a right line compelled to turn about a fixed point. It\nis, moreover, easy to conceive in advance what must be the extreme\nimportance to analytical geometry of a property so profoundly\nelementary, which, for that reason, must be recurring at every instant,\nand take a progressively increasing value in all labours of this kind.\n\n\n_Perpendicularity of the Axes._ In pursuing farther the consideration\nwhich demonstrates the superiority of the ordinary system of\nco-ordinates over any other as to the representation of equations, we\nmay also take notice of the utility for this object of the common usage\nof taking the two axes perpendicular to each other, whenever possible,\nrather than with any other inclination. As regards the representation of\nlines by equations, this secondary circumstance is no more universally\nproper than we have seen the general nature of the system to be; since,\naccording to the particular occasion, any other inclination of the axes\nmay deserve our preference in that respect. But, in the inverse point of\nview, it is easy to see that rectangular axes constantly permit us to\nrepresent equations in a more simple and even more faithful manner; for,\nwith oblique axes, space being divided by them into regions which no\nlonger have a perfect identity, it follows that, if the geometrical\n_locus_ of the equation extends into all these regions at once, there\nwill be presented, by reason merely of this inequality of the angles,\ndifferences of figure which do not correspond to any analytical\ndiversity, and will necessarily alter the rigorous exactness of the\nrepresentation, by being confounded with the proper results of the\nalgebraic comparisons. For example, an equation like: _x^m_ + _y^m_ =\n_c_, which, by its perfect symmetry, should evidently give a curve\ncomposed of four identical quarters, will be represented, on the\ncontrary, if we take axes not rectangular, by a geometric _locus_, the\nfour parts of which will be unequal. It is plain that the only means of\navoiding all inconveniences of this kind is to suppose the angle of the\ntwo axes to be a right angle.\n\nThe preceding discussion clearly shows that, although the ordinary\nsystem of rectilinear co-ordinates has no constant superiority over all\nothers in one of the two fundamental points of view which are\ncontinually combined in analytical geometry, yet as, on the other hand,\nit is not constantly inferior, its necessary and absolute greater\naptitude for the representation of equations must cause it to generally\nreceive the preference; although it may evidently happen, in some\nparticular cases, that the necessity of simplifying equations and of\nobtaining them more easily may determine geometers to adopt a less\nperfect system. The rectilinear system is, therefore, the one by means\nof which are ordinarily constructed the most essential theories of\ngeneral geometry, intended to express analytically the most important\ngeometrical phenomena. When it is thought necessary to choose some\nother, the polar system is almost always the one which is fixed upon,\nthis system being of a nature sufficiently opposite to that of the\nrectilinear system to cause the equations, which are too complicated\nwith respect to the latter, to become, in general, sufficiently simple\nwith respect to the other. Polar co-ordinates, moreover, have often the\nadvantage of admitting of a more direct and natural concrete\nsignification; as is the case in mechanics, for the geometrical\nquestions to which the theory of circular movement gives rise, and in\nalmost all the cases of celestial geometry.\n\n       *       *       *       *       *\n\nIn order to simplify the exposition, we have thus far considered the\nfundamental conception of analytical geometry only with respect to\n_plane curves_, the general study of which was the only object of the\ngreat philosophical renovation produced by Descartes. To complete this\nimportant explanation, we have now to show summarily how this elementary\nidea was extended by Clairaut, about a century afterwards, to the\ngeneral study of _surfaces_ and _curves of double curvature_. The\nconsiderations which have been already given will permit me to limit\nmyself on this subject to the rapid examination of what is strictly\npeculiar to this new case.\n\n\n\n\nSURFACES.\n\n\n_Determination of a Point in Space._ The complete analytical\ndetermination of a point in space evidently requires the values of three\nco-ordinates to be assigned; as, for example, in the system which is\ngenerally adopted, and which corresponds to the _rectilinear_ system of\nplane geometry, distances from the point to three fixed planes, usually\nperpendicular to one another; which presents the point as the\nintersection of three planes whose direction is invariable. We might\nalso employ the distances from the movable point to three fixed points,\nwhich would determine it by the intersection of three spheres with a\ncommon centre. In like manner, the position of a point would be defined\nby giving its distance from a fixed point, and the direction of that\ndistance, by means of the two angles which this right line makes with\ntwo invariable axes; this is the _polar_ system of geometry of three\ndimensions; the point is then constructed by the intersection of a\nsphere having a fixed centre, with two right cones with circular bases,\nwhose axes and common summit do not change. In a word, there is\nevidently, in this case at least, the same infinite variety among the\nvarious possible systems of co-ordinates which we have already observed\nin geometry of two dimensions. In general, we have to conceive a point\nas being always determined by the intersection of any three surfaces\nwhatever, as it was in the former case by that of two lines: each of\nthese three surfaces has, in like manner, all its conditions of\ndetermination constant, excepting one, which gives rise to the\ncorresponding co-ordinates, whose peculiar geometrical influence is thus\nto constrain the point to be situated upon that surface.\n\nThis being premised, it is clear that if the three co-ordinates of a\npoint are entirely independent of one another, that point can take\nsuccessively all possible positions in space. But if the point is\ncompelled to remain upon a certain surface defined in any manner\nwhatever, then two co-ordinates are evidently sufficient for determining\nits situation at each instant, since the proposed surface will take the\nplace of the condition imposed by the third co-ordinate. We must then,\nin this case, under the analytical point of view, necessarily conceive\nthis last co-ordinate as a determinate function of the two others, these\nlatter remaining perfectly independent of each other. Thus there will be\na certain equation between the three variable co-ordinates, which will\nbe permanent, and which will be the only one, in order to correspond to\nthe precise degree of indetermination in the position of the point.\n\n\n_Expression of Surfaces by Equations._ This equation, more or less easy\nto be discovered, but always possible, will be the analytical definition\nof the proposed surface, since it must be verified for all the points of\nthat surface, and for them alone. If the surface undergoes any change\nwhatever, even a simple change of place, the equation must undergo a\nmore or less serious corresponding modification. In a word, all\ngeometrical phenomena relating to surfaces will admit of being\ntranslated by certain equivalent analytical conditions appropriate to\nequations of three variables; and in the establishment and\ninterpretation of this general and necessary harmony will essentially\nconsist the science of analytical geometry of three dimensions.\n\n\n_Expression of Equations by Surfaces._ Considering next this fundamental\nconception in the inverse point of view, we see in the same manner that\nevery equation of three variables may, in general, be represented\ngeometrically by a determinate surface, primitively defined by the very\ncharacteristic property, that the co-ordinates of all its points always\nretain the mutual relation enunciated in this equation. This geometrical\nlocus will evidently change, for the same equation, according to the\nsystem of co-ordinates which may serve for the construction of this\nrepresentation. In adopting, for example, the rectilinear system, it is\nclear that in the equation between the three variables, _x_, _y_, _z_,\nevery particular value attributed to _z_ will give an equation between\nat _x_ and _y_, the geometrical locus of which will be a certain line\nsituated in a plane parallel to the plane of _x_ and _y_, and at a\ndistance from this last equal to the value of _z_; so that the complete\ngeometrical locus will present itself as composed of an infinite series\nof lines superimposed in a series of parallel planes (excepting the\ninterruptions which may exist), and will consequently form a veritable\nsurface. It would be the same in considering any other system of\nco-ordinates, although the geometrical construction of the equation\nbecomes more difficult to follow.\n\nSuch is the elementary conception, the complement of the original idea\nof Descartes, on which is founded general geometry relative to surfaces.\nIt would be useless to take up here directly the other considerations\nwhich have been above indicated, with respect to lines, and which any\none can easily extend to surfaces; whether to show that every definition\nof a surface by any method of generation whatever is really a direct\nequation of that surface in a certain system of co-ordinates, or to\ndetermine among all the different systems of possible co-ordinates that\none which is generally the most convenient. I will only add, on this\nlast point, that the necessary superiority of the ordinary rectilinear\nsystem, as to the representation of equations, is evidently still more\nmarked in analytical geometry of three dimensions than in that of two,\nbecause of the incomparably greater geometrical complication which would\nresult from the choice of any other system. This can be verified in the\nmost striking manner by considering the polar system in particular,\nwhich is the most employed after the ordinary rectilinear system, for\nsurfaces as well as for plane curves, and for the same reasons.\n\nIn order to complete the general exposition of the fundamental\nconception relative to the analytical study of surfaces, a philosophical\nexamination should be made of a final improvement of the highest\nimportance, which Monge has introduced into the very elements of this\ntheory, for the classification of surfaces in natural families,\nestablished according to the mode of generation, and expressed\nalgebraically by common differential equations, or by finite equations\ncontaining arbitrary functions.\n\n\n\n\nCURVES OF DOUBLE CURVATURE.\n\n\nLet us now consider the last elementary point of view of analytical\ngeometry of three dimensions; that relating to the algebraic\nrepresentation of curves considered in space, in the most general\nmanner. In continuing to follow the principle which has been constantly\nemployed, that of the degree of indetermination of the geometrical\nlocus, corresponding to the degree of independence of the variables, it\nis evident, as a general principle, that when a point is required to be\nsituated upon some certain curve, a single co-ordinate is enough for\ncompletely determining its position, by the intersection of this curve\nwith the surface which results from this co-ordinate. Thus, in this\ncase, the two other co-ordinates of the point must be conceived as\nfunctions necessarily determinate and distinct from the first. It\nfollows that every line, considered in space, is then represented\nanalytically, no longer by a single equation, but by the system of two\nequations between the three co-ordinates of any one of its points. It is\nclear, indeed, from another point of view, that since each of these\nequations, considered separately, expresses a certain surface, their\ncombination presents the proposed line as the intersection of two\ndeterminate surfaces. Such is the most general manner of conceiving the\nalgebraic representation of a line in analytical geometry of three\ndimensions. This conception is commonly considered in too restricted a\nmanner, when we confine ourselves to considering a line as determined by\nthe system of its two _projections_ upon two of the co-ordinate planes;\na system characterized, analytically, by this peculiarity, that each of\nthe two equations of the line then contains only two of the three\nco-ordinates, instead of simultaneously including the three variables.\nThis consideration, which consists in regarding the line as the\nintersection of two cylindrical surfaces parallel to two of the three\naxes of the co-ordinates, besides the inconvenience of being confined to\nthe ordinary rectilinear system, has the fault, if we strictly confine\nourselves to it, of introducing useless difficulties into the analytical\nrepresentation of lines, since the combination of these two cylinders\nwould evidently not be always the most suitable for forming the\nequations of a line. Thus, considering this fundamental notion in its\nentire generality, it will be necessary in each case to choose, from\namong the infinite number of couples of surfaces, the intersection of\nwhich might produce the proposed curve, that one which will lend itself\nthe best to the establishment of equations, as being composed of the\nbest known surfaces. Thus, if the problem is to express analytically a\ncircle in space, it will evidently be preferable to consider it as the\nintersection of a sphere and a plane, rather than as proceeding from any\nother combination of surfaces which could equally produce it.\n\nIn truth, this manner of conceiving the representation of lines by\nequations, in analytical geometry of three dimensions, produces, by its\nnature, a necessary inconvenience, that of a certain analytical\nconfusion, consisting in this: that the same line may thus be expressed,\nwith the same system of co-ordinates, by an infinite number of different\ncouples of equations, on account of the infinite number of couples of\nsurfaces which can form it; a circumstance which may cause some\ndifficulties in recognizing this line under all the algebraical\ndisguises of which it admits. But there exists a very simple method for\ncausing this inconvenience to disappear; it consists in giving up the\nfacilities which result from this variety of geometrical constructions.\nIt suffices, in fact, whatever may be the analytical system primitively\nestablished for a certain line, to be able to deduce from it the system\ncorresponding to a single couple of surfaces uniformly generated; as,\nfor example, to that of the two cylindrical surfaces which _project_ the\nproposed line upon two of the co-ordinate planes; surfaces which will\nevidently be always identical, in whatever manner the line may have been\nobtained, and which will not vary except when that line itself shall\nchange. Now, in choosing this fixed system, which is actually the most\nsimple, we shall generally be able to deduce from the primitive\nequations those which correspond to them in this special construction,\nby transforming them, by two successive eliminations, into two\nequations, each containing only two of the variable co-ordinates, and\nthereby corresponding to the two surfaces of projection. Such is really\nthe principal destination of this sort of geometrical combination, which\nthus offers to us an invariable and certain means of recognizing the\nidentity of lines in spite of the diversity of their equations, which is\nsometimes very great.\n\n\n\n\nIMPERFECTIONS OF ANALYTICAL GEOMETRY.\n\n\nHaving now considered the fundamental conception of analytical geometry\nunder its principal elementary aspects, it is proper, in order to make\nthe sketch complete, to notice here the general imperfections yet\npresented by this conception with respect to both geometry and to\nanalysis.\n\n_Relatively to geometry_, we must remark that the equations are as yet\nadapted to represent only entire geometrical loci, and not at all\ndeterminate portions of those loci. It would, however, be necessary, in\nsome circumstances, to be able to express analytically a part of a line\nor of a surface, or even a _discontinuous_ line or surface, composed of\na series of sections belonging to distinct geometrical figures, such as\nthe contour of a polygon, or the surface of a polyhedron. Thermology,\nespecially, often gives rise to such considerations, to which our\npresent analytical geometry is necessarily inapplicable. The labours of\nM. Fourier on discontinuous functions have, however, begun to fill up\nthis great gap, and have thereby introduced a new and essential\nimprovement into the fundamental conception of Descartes. But this\nmanner of representing heterogeneous or partial figures, being founded\non the employment of trigonometrical series proceeding according to the\nsines of an infinite series of multiple arcs, or on the use of certain\ndefinite integrals equivalent to those series, and the general integral\nof which is unknown, presents as yet too much complication to admit of\nbeing immediately introduced into the system of analytical geometry.\n\n_Relatively to analysis_, we must begin by observing that our inability\nto conceive a geometrical representation of equations containing four,\nfive, or more variables, analogous to those representations which all\nequations of two or of three variables admit, must not be viewed as an\nimperfection of our system of analytical geometry, for it evidently\nbelongs to the very nature of the subject. Analysis being necessarily\nmore general than geometry, since it relates to all possible phenomena,\nit would be very unphilosophical to desire always to find among\ngeometrical phenomena alone a concrete representation of all the laws\nwhich analysis can express.\n\nThere exists, however, another imperfection of less importance, which\nmust really be viewed as proceeding from the manner in which we conceive\nanalytical geometry. It consists in the evident incompleteness of our\npresent representation of equations of two or of three variables by\nlines or surfaces, inasmuch as in the construction of the geometric\nlocus we pay regard only to the _real_ solutions of equations, without\nat all noticing any _imaginary_ solutions. The general course of these\nlast should, however, by its nature, be quite as susceptible as that of\nthe others of a geometrical representation. It follows from this\nomission that the graphic picture of the equation is constantly\nimperfect, and sometimes even so much so that there is no geometric\nrepresentation at all when the equation admits of only imaginary\nsolutions. But, even in this last case, we evidently ought to be able to\ndistinguish between equations as different in themselves as these, for\nexample,\n\n  _x\u00b2_ + _y\u00b2_ + 1 = 0, _x\u2076_ + _y\u2074_ + 1 = 0, _y\u00b2_ + _e^x_ = 0.\n\nWe know, moreover, that this principal imperfection often brings with\nit, in analytical geometry of two or of three dimensions, a number of\nsecondary inconveniences, arising from several analytical modifications\nnot corresponding to any geometrical phenomena.\n\n       *       *       *       *       *\n\nOur philosophical exposition of the fundamental conception of analytical\ngeometry shows us clearly that this science consists essentially in\ndetermining what is the general analytical expression of such or such a\ngeometrical phenomenon belonging to lines or to surfaces; and,\nreciprocally, in discovering the geometrical interpretation of such or\nsuch an analytical consideration. A detailed examination of the most\nimportant general questions would show us how geometers have succeeded\nin actually establishing this beautiful harmony, and in thus imprinting\non geometrical science, regarded as a whole, its present eminently\nperfect character of rationality and of simplicity.\n\n     _Note._--The author devotes the two following chapters of his\n     course to the more detailed examination of Analytical Geometry of\n     two and of three dimensions; but his subsequent publication of a\n     separate work upon this branch of mathematics has been thought to\n     render unnecessary the reproduction of these two chapters in the\n     present volume.\n\n\nTHE END.","title":"The philosophy of mathematics","tradition":"philosophy-ethics","translated_text":null,"word_count":65755}