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+Project Gutenberg's The philosophy of mathematics, by Auguste Comte
+
+This eBook is for the use of anyone anywhere at no cost and with
+almost no restrictions whatsoever.  You may copy it, give it away or
+re-use it under the terms of the Project Gutenberg License included
+with this eBook or online at www.gutenberg.org/license
+
+
+Title: The philosophy of mathematics
+
+Author: Auguste Comte
+
+Translator: W. M. Gillespie
+
+Release Date: May 15, 2012 [EBook #39702]
+
+Language: English
+
+Character set encoding: UTF-8
+
+*** START OF THIS PROJECT GUTENBERG EBOOK THE PHILOSOPHY OF MATHEMATICS ***
+
+
+
+
+Produced by Anna Hall, Albert László and the Online
+Distributed Proofreading Team at http://www.pgdp.net (This
+file was produced from images generously made available
+by The Internet Archive)
+
+
+
+
+
+
+
+
+
+THE
+
+PHILOSOPHY
+
+OF
+
+MATHEMATICS.
+
+
+                          THE SCIENCE OF MATHEMATICS.
+                                       |
+                +----------------------+-------------------+
+                |                                          |
+                |                                          |
+       ABSTRACT MATHEMATICS.                    CONCRETE MATHEMATICS.
+                |                                          |
+                |                                          |
+                |                                  +-------+------+
+                |                                  |              |
+    ANALYSIS; _or_, _The Calculus_.             GEOMETRY.     MECHANICS.
+                |                                  |
+                |                                  |
+        +-------+----------+               +-------+---------+
+        |                  |               |                 |
+    =Ordinary       =Transcendental   =Synthetic=        =Analytic=
+    Analysis=;         Analysis=;     _or Special_       _or General_
+  _or_, _Calculus   _or_, _Calculus    =Geometry=.       =Geometry=.
+    of Direct         of Indirect          |                 |
+   Functions_.        Functions_.          |                 |
+        |                |                 |                 |
+     +--+--+       +-----+--+         +----+----+        +---+---+
+     |     |       |        |         |         |        |       |
+     |     |       |        |         |         |        |       |
+     |     |       |        |         |         |        |       |
+     |     |       |        |         |         |        |       |
+     | Algebra.    |    Calculus      |     Algebraic.   |   Of three
+     |             |       of         |   Trigonometry.  |  dimensions.
+     |             |   Variations.    |                  |
+     |             |                  |                  |
+  Arithmetic.   Differential       Graphical.          Of two
+                and Integral      Descriptive        dimensions.
+                 Calculus.         Geometry.
+
+
+
+
+THE
+PHILOSOPHY
+OF
+MATHEMATICS;
+
+TRANSLATED FROM THE
+COURS DE PHILOSOPHIE POSITIVE
+OF
+AUGUSTE COMTE,
+BY
+W. M. GILLESPIE,
+PROFESSOR OF CIVIL ENGINEERING & ADJ. PROF. OF MATHEMATICS
+IN UNION COLLEGE.
+
+NEW YORK:
+HARPER & BROTHERS, PUBLISHERS,
+82 CLIFF STREET
+1851.
+
+
+Entered, according to Act of Congress, in the year one thousand
+eight hundred and fifty-one, by
+
+HARPER & BROTHERS.
+
+in the Clerk's Office of the District Court of the Southern District
+of New York.
+
+
+
+
+PREFACE.
+
+
+The pleasure and profit which the translator has received from the great
+work here presented, have induced him to lay it before his
+fellow-teachers and students of Mathematics in a more accessible form
+than that in which it has hitherto appeared. The want of a comprehensive
+map of the wide region of mathematical science--a bird's-eye view of its
+leading features, and of the true bearings and relations of all its
+parts--is felt by every thoughtful student. He is like the visitor to a
+great city, who gets no just idea of its extent and situation till he
+has seen it from some commanding eminence. To have a panoramic view of
+the whole district--presenting at one glance all the parts in due
+co-ordination, and the darkest nooks clearly shown--is invaluable to
+either traveller or student. It is this which has been most perfectly
+accomplished for mathematical science by the author whose work is here
+presented.
+
+Clearness and depth, comprehensiveness and precision, have never,
+perhaps, been so remarkably united as in AUGUSTE COMTE. He views his
+subject from an elevation which gives to each part of the complex whole
+its true position and value, while his telescopic glance loses none of
+the needful details, and not only itself pierces to the heart of the
+matter, but converts its opaqueness into such transparent crystal, that
+other eyes are enabled to see as deeply into it as his own.
+
+Any mathematician who peruses this volume will need no other
+justification of the high opinion here expressed; but others may
+appreciate the following endorsements of well-known authorities. _Mill_,
+in his "Logic," calls the work of M. Comte "by far the greatest yet
+produced on the Philosophy of the sciences;" and adds, "of this
+admirable work, one of the most admirable portions is that in which he
+may truly be said to have created the Philosophy of the higher
+Mathematics:" _Morell_, in his "Speculative Philosophy of Europe," says,
+"The classification given of the sciences at large, and their regular
+order of development, is unquestionably a master-piece of scientific
+thinking, as simple as it is comprehensive;" and _Lewes_, in his
+"Biographical History of Philosophy," names Comte "the Bacon of the
+nineteenth century," and says, "I unhesitatingly record my conviction
+that this is the greatest work of our age."
+
+The complete work of M. Comte--his "_Cours de Philosophie
+Positive_"--fills six large octavo volumes, of six or seven hundred
+pages each, two thirds of the first volume comprising the purely
+mathematical portion. The great bulk of the "Course" is the probable
+cause of the fewness of those to whom even this section of it is known.
+Its presentation in its present form is therefore felt by the translator
+to be a most useful contribution to mathematical progress in this
+country. The comprehensiveness of the style of the author--grasping all
+possible forms of an idea in one Briarean sentence, armed at all points
+against leaving any opening for mistake or forgetfulness--occasionally
+verges upon cumbersomeness and formality. The translator has, therefore,
+sometimes taken the liberty of breaking up or condensing a long
+sentence, and omitting a few passages not absolutely necessary, or
+referring to the peculiar "Positive philosophy" of the author; but he
+has generally aimed at a conscientious fidelity to the original. It has
+often been difficult to retain its fine shades and subtile distinctions
+of meaning, and, at the same time, replace the peculiarly appropriate
+French idioms by corresponding English ones. The attempt, however, has
+always been made, though, when the best course has been at all doubtful,
+the language of the original has been followed as closely as possible,
+and, when necessary, smoothness and grace have been unhesitatingly
+sacrificed to the higher attributes of clearness and precision.
+
+Some forms of expression may strike the reader as unusual, but they have
+been retained because they were characteristic, not of the mere language
+of the original, but of its spirit. When a great thinker has clothed his
+conceptions in phrases which are singular even in his own tongue, he who
+professes to translate him is bound faithfully to preserve such forms of
+speech, as far as is practicable; and this has been here done with
+respect to such peculiarities of expression as belong to the author,
+not as a foreigner, but as an individual--not because he writes in
+French, but because he is Auguste Comte.
+
+The young student of Mathematics should not attempt to read the whole of
+this volume at once, but should peruse each portion of it in connexion
+with the temporary subject of his special study: the first chapter of
+the first book, for example, while he is studying Algebra; the first
+chapter of the second book, when he has made some progress in Geometry;
+and so with the rest. Passages which are obscure at the first reading
+will brighten up at the second; and as his own studies cover a larger
+portion of the field of Mathematics, he will see more and more clearly
+their relations to one another, and to those which he is next to take
+up. For this end he is urgently recommended to obtain a perfect
+familiarity with the "Analytical Table of Contents," which maps out the
+whole subject, the grand divisions of which are also indicated in the
+Tabular View facing the title-page. Corresponding heads will be found in
+the body of the work, the principal divisions being in SMALL CAPITALS,
+and the subdivisions in _Italics_. For these details the translator
+alone is responsible.
+
+
+
+
+  ANALYTICAL TABLE OF CONTENTS.
+
+  INTRODUCTION.
+
+  Page
+
+  GENERAL CONSIDERATIONS ON MATHEMATICAL SCIENCE   17
+
+  THE OBJECT OF MATHEMATICS                        18
+    Measuring Magnitudes                           18
+      Difficulties                                 19
+      General Method                               20
+      Illustrations                                21
+        1. Falling Bodies                          21
+        2. Inaccessible Distances                  23
+        3. Astronomical Facts                      24
+
+  TRUE DEFINITION OF MATHEMATICS                   25
+    A Science, not an Art                          25
+
+  ITS TWO FUNDAMENTAL DIVISIONS                    26
+      Their different Objects                      27
+      Their different Natures                      29
+    _Concrete Mathematics_                         31
+      Geometry and Mechanics                       32
+    _Abstract Mathematics_                         33
+      The Calculus, or Analysis                    33
+
+  EXTENT OF ITS FIELD                              35
+      Its Universality                             36
+      Its Limitations                              37
+
+
+  BOOK I.
+
+  ANALYSIS.
+
+
+  CHAPTER I.
+
+  Page
+
+  GENERAL VIEW OF MATHEMATICAL ANALYSIS            45
+
+  THE TRUE IDEA OF AN EQUATION                     46
+      Division of Functions into Abstract and
+        Concrete                                   47
+      Enumeration of Abstract Functions            50
+
+  DIVISIONS OF THE CALCULUS                        53
+    _The Calculus of Values, or Arithmetic_        57
+      Its Extent                                   57
+      Its true Nature                              59
+    _The Calculus of Functions_                    61
+      Two Modes of obtaining Equations             61
+        1. By the Relations between the given
+           Quantities                              61
+        2. By the Relations between auxiliary
+           Quantities                              64
+    Corresponding Divisions of the Calculus of
+      Functions                                    67
+
+
+  CHAPTER II.
+
+  ORDINARY ANALYSIS; OR, ALGEBRA.                  69
+
+      Its Object                                   69
+      Classification of Equations                  70
+
+  ALGEBRAIC EQUATIONS                              71
+      Their Classification                         71
+
+  ALGEBRAIC RESOLUTION OF EQUATIONS                72
+      Its Limits                                   72
+      General Solution                             72
+      What we know in Algebra                      74
+
+  NUMERICAL RESOLUTION OF EQUATIONS                75
+      Its limited Usefulness                       76
+  Different Divisions of the two Systems           78
+
+  THE THEORY OF EQUATIONS                          79
+
+  THE METHOD OF INDETERMINATE COEFFICIENTS         80
+
+  IMAGINARY QUANTITIES                             81
+
+  NEGATIVE QUANTITIES                              81
+
+  THE PRINCIPLE OF HOMOGENEITY                     84
+
+
+  CHAPTER III.
+
+  TRANSCENDENTAL ANALYSIS:
+
+  Page
+
+  ITS DIFFERENT CONCEPTIONS                        88
+
+      Preliminary Remarks                          88
+      Its early History                            89
+
+  METHOD OF LEIBNITZ                               91
+      Infinitely small Elements                    91
+      _Examples_:
+        1. Tangents                                93
+        2. Rectification of an Arc                 94
+        3. Quadrature of a Curve                   95
+        4. Velocity in variable Motion             95
+        5. Distribution of Heat                    96
+      Generality of the Formulas                   97
+      Demonstration of the Method                  98
+        Illustration by Tangents                  102
+
+  METHOD OF NEWTON                                103
+      Method of Limits                            103
+        _Examples_:
+          1. Tangents                             104
+          2. Rectifications                       105
+      Fluxions and Fluents                        106
+
+  METHOD OF LAGRANGE                              108
+      Derived Functions                           108
+      An extension of ordinary Analysis           108
+      _Example_: Tangents                         109
+    _Fundamental Identity of the three Methods_   110
+    _Their comparative Value_                     113
+      That of Leibnitz                            113
+      That of Newton                              115
+      That of Lagrange                            117
+
+
+  CHAPTER IV.
+
+  Page
+
+  THE DIFFERENTIAL AND INTEGRAL CALCULUS          120
+
+  ITS TWO FUNDAMENTAL DIVISIONS                   120
+
+  THEIR RELATIONS TO EACH OTHER                   121
+        1. Use of the Differential Calculus as
+           preparatory to that of the Integral    123
+        2. Employment of the Differential
+           Calculus alone                         125
+        3. Employment of the Integral Calculus
+           alone                                  125
+            Three Classes of Questions hence
+              resulting                           126
+
+  THE DIFFERENTIAL CALCULUS                       127
+      Two Cases: Explicit and Implicit Functions  127
+        Two sub-Cases: a single Variable or
+          several                                 129
+        Two other Cases: Functions separate or
+          combined                                130
+      Reduction of all to the Differentiation of
+        the ten elementary Functions              131
+      Transformation of derived Functions for
+        new Variables                             132
+      Different Orders of Differentiation         133
+      Analytical Applications                     133
+
+  THE INTEGRAL CALCULUS                           135
+      Its fundamental Division: Explicit and
+        Implicit Functions                        135
+      Subdivisions: a single Variable or several  136
+      Calculus of partial Differences             137
+      Another Subdivision: different Orders of
+        Differentiation                           138
+      Another equivalent Distinction              140
+      _Quadratures_                               142
+        Integration of Transcendental Functions   143
+          Integration by Parts                    143
+        Integration of Algebraic Functions        143
+      Singular Solutions                          144
+      Definite Integrals                          146
+      Prospects of the Integral Calculus          148
+
+
+  CHAPTER V.
+
+  Page
+
+  THE CALCULUS OF VARIATIONS                      151
+
+  PROBLEMS GIVING RISE TO IT                      151
+      Ordinary Questions of Maxima and Minima     151
+      A new Class of Questions                    152
+        Solid of least Resistance;
+          Brachystochrone; Isoperimeters          153
+
+  ANALYTICAL NATURE OF THESE QUESTIONS            154
+
+  METHODS OF THE OLDER GEOMETERS                  155
+
+  METHOD OF LAGRANGE                              156
+      Two Classes of Questions                    157
+        1. Absolute Maxima and Minima             157
+           Equations of Limits                    159
+             A more general Consideration         159
+        2. Relative Maxima and Minima             160
+      Other Applications of the Method of
+        Variations                                162
+
+  ITS RELATIONS TO THE ORDINARY CALCULUS          163
+
+
+  CHAPTER VI.
+
+  THE CALCULUS OF FINITE DIFFERENCES              167
+
+      Its general Character                       167
+      Its true Nature                             168
+
+  GENERAL THEORY OF SERIES                        170
+      Its Identity with this Calculus             172
+
+  PERIODIC OR DISCONTINUOUS FUNCTIONS             173
+
+  APPLICATIONS OF THIS CALCULUS                   173
+      Series                                      173
+      Interpolation                               173
+      Approximate Rectification, &c.              174
+
+
+  BOOK II.
+
+  GEOMETRY.
+
+  CHAPTER I.
+
+  Page
+
+  A GENERAL VIEW OF GEOMETRY                      179
+
+      The true Nature of Geometry                 179
+      Two fundamental Ideas                       181
+        1. The Idea of Space                      181
+        2. Different kinds of Extension           182
+
+  THE FINAL OBJECT OF GEOMETRY                    184
+      Nature of Geometrical Measurement           185
+        Of Surfaces and Volumes                   185
+        Of curve Lines                            187
+        Of right Lines                            189
+
+  THE INFINITE EXTENT OF ITS FIELD                190
+      Infinity of Lines                           190
+      Infinity of Surfaces                        191
+      Infinity of Volumes                         192
+      Analytical Invention of Curves, &c.         193
+
+  EXPANSION OF ORIGINAL DEFINITION                193
+      Properties of Lines and Surfaces            195
+      Necessity of their Study                    195
+        1. To find the most suitable Property     195
+        2. To pass from the Concrete to the
+           Abstract                               197
+      Illustrations:
+        Orbits of the Planets                     198
+        Figure of the Earth                       199
+
+  THE TWO GENERAL METHODS OF GEOMETRY             202
+      Their fundamental Difference                203
+        1°. Different Questions with respect to
+          the same Figure                         204
+        2°. Similar Questions with respect to
+          different Figures                       204
+      Geometry of the Ancients                    204
+      Geometry of the Moderns                     206
+      Superiority of the Modern                   207
+      The Ancient the base of the Modern          209
+
+
+  CHAPTER II.
+
+  ANCIENT OR SYNTHETIC GEOMETRY
+
+  Page
+
+  ITS PROPER EXTENT                               212
+      Lines; Polygons; Polyhedrons                212
+      Not to be farther restricted                213
+      Improper Application of Analysis            214
+      Attempted Demonstrations of Axioms          216
+
+  GEOMETRY OF THE RIGHT LINE                      217
+
+  GRAPHICAL SOLUTIONS                             218
+      _Descriptive Geometry_                      220
+
+  ALGEBRAICAL SOLUTIONS                           224
+      _Trigonometry_                              225
+        Two Methods of introducing Angles         226
+          1. By Arcs                              226
+          2. By trigonometrical Lines             226
+        Advantages of the latter                  226
+        Its Division of trigonometrical Questions 227
+          1. Relations between Angles and
+             trigonometrical Lines                228
+          2. Relations between trigonometrical
+             Lines and Sides                      228
+        Increase of trigonometrical Lines         228
+        Study of the Relations between them       230
+
+
+  CHAPTER III.
+
+  MODERN OR ANALYTICAL GEOMETRY
+
+  Page
+
+  THE ANALYTICAL REPRESENTATION OF FIGURES        232
+      Reduction of Figure to Position             233
+      Determination of the position of a Point    234
+
+  PLANE CURVES                                    237
+      Expression of Lines by Equations            237
+      Expression of Equations by Lines            238
+      Any change in the Line changes the Equation 240
+      Every "Definition" of a Line is an Equation 241
+      _Choice of Co-ordinates_                    245
+        Two different points of View              245
+          1. Representation of Lines by Equations 246
+          2. Representation of Equations by Lines 246
+        Superiority of the rectilinear System     248
+          Advantages of perpendicular Axes        249
+
+  SURFACES    251
+      Determination of a Point in Space           251
+      Expression of Surfaces by Equations         253
+      Expression of Equations by Surfaces         253
+
+  CURVES IN SPACE     255
+
+  Imperfections of Analytical Geometry            258
+      Relatively to Geometry                      258
+      Relatively to Analysis                      258
+
+
+
+
+THE
+
+PHILOSOPHY OF MATHEMATICS.
+
+INTRODUCTION.
+
+GENERAL CONSIDERATIONS.
+
+
+Although Mathematical Science is the most ancient and the most perfect
+of all, yet the general idea which we ought to form of it has not yet
+been clearly determined. Its definition and its principal divisions have
+remained till now vague and uncertain. Indeed the plural name--"The
+Mathematics"--by which we commonly designate it, would alone suffice to
+indicate the want of unity in the common conception of it.
+
+In truth, it was not till the commencement of the last century that the
+different fundamental conceptions which constitute this great science
+were each of them sufficiently developed to permit the true spirit of
+the whole to manifest itself with clearness. Since that epoch the
+attention of geometers has been too exclusively absorbed by the special
+perfecting of the different branches, and by the application which they
+have made of them to the most important laws of the universe, to allow
+them to give due attention to the general system of the science.
+
+But at the present time the progress of the special departments is no
+longer so rapid as to forbid the contemplation of the whole. The science
+of mathematics is now sufficiently developed, both in itself and as to
+its most essential application, to have arrived at that state of
+consistency in which we ought to strive to arrange its different parts
+in a single system, in order to prepare for new advances. We may even
+observe that the last important improvements of the science have
+directly paved the way for this important philosophical operation, by
+impressing on its principal parts a character of unity which did not
+previously exist.
+
+To form a just idea of the object of mathematical science, we may start
+from the indefinite and meaningless definition of it usually given, in
+calling it "_The science of magnitudes_," or, which is more definite,
+"_The science which has for its object the measurement of magnitudes._"
+Let us see how we can rise from this rough sketch (which is singularly
+deficient in precision and depth, though, at bottom, just) to a
+veritable definition, worthy of the importance, the extent, and the
+difficulty of the science.
+
+
+THE OBJECT OF MATHEMATICS.
+
+_Measuring Magnitudes._ The question of _measuring_ a magnitude in
+itself presents to the mind no other idea than that of the simple direct
+comparison of this magnitude with another similar magnitude, supposed to
+be known, which it takes for the _unit_ of comparison among all others
+of the same kind. According to this definition, then, the science of
+mathematics--vast and profound as it is with reason reputed to
+be--instead of being an immense concatenation of prolonged mental
+labours, which offer inexhaustible occupation to our intellectual
+activity, would seem to consist of a simple series of mechanical
+processes for obtaining directly the ratios of the quantities to be
+measured to those by which we wish to measure them, by the aid of
+operations of similar character to the superposition of lines, as
+practiced by the carpenter with his rule.
+
+The error of this definition consists in presenting as direct an object
+which is almost always, on the contrary, very indirect. The _direct_
+measurement of a magnitude, by superposition or any similar process, is
+most frequently an operation quite impossible for us to perform; so that
+if we had no other means for determining magnitudes than direct
+comparisons, we should be obliged to renounce the knowledge of most of
+those which interest us.
+
+_Difficulties._ The force of this general observation will be understood
+if we limit ourselves to consider specially the particular case which
+evidently offers the most facility--that of the measurement of one
+straight line by another. This comparison, which is certainly the most
+simple which we can conceive, can nevertheless scarcely ever be effected
+directly. In reflecting on the whole of the conditions necessary to
+render a line susceptible of a direct measurement, we see that most
+frequently they cannot be all fulfilled at the same time. The first and
+the most palpable of these conditions--that of being able to pass over
+the line from one end of it to the other, in order to apply the unit of
+measurement to its whole length--evidently excludes at once by far the
+greater part of the distances which interest us the most; in the first
+place, all the distances between the celestial bodies, or from any one
+of them to the earth; and then, too, even the greater number of
+terrestrial distances, which are so frequently inaccessible. But even if
+this first condition be found to be fulfilled, it is still farther
+necessary that the length be neither too great nor too small, which
+would render a direct measurement equally impossible. The line must also
+be suitably situated; for let it be one which we could measure with the
+greatest facility, if it were horizontal, but conceive it to be turned
+up vertically, and it becomes impossible to measure it.
+
+The difficulties which we have indicated in reference to measuring
+lines, exist in a very much greater degree in the measurement of
+surfaces, volumes, velocities, times, forces, &c. It is this fact which
+makes necessary the formation of mathematical science, as we are going
+to see; for the human mind has been compelled to renounce, in almost all
+cases, the direct measurement of magnitudes, and to seek to determine
+them _indirectly_, and it is thus that it has been led to the creation
+of mathematics.
+
+_General Method._ The general method which is constantly employed, and
+evidently the only one conceivable, to ascertain magnitudes which do not
+admit of a direct measurement, consists in connecting them with others
+which are susceptible of being determined immediately, and by means of
+which we succeed in discovering the first through the relations which
+subsist between the two. Such is the precise object of mathematical
+science viewed as a whole. In order to form a sufficiently extended idea
+of it, we must consider that this indirect determination of magnitudes
+may be indirect in very different degrees. In a great number of cases,
+which are often the most important, the magnitudes, by means of which
+the principal magnitudes sought are to be determined, cannot themselves
+be measured directly, and must therefore, in their turn, become the
+subject of a similar question, and so on; so that on many occasions the
+human mind is obliged to establish a long series of intermediates
+between the system of unknown magnitudes which are the final objects of
+its researches, and the system of magnitudes susceptible of direct
+measurement, by whose means we finally determine the first, with which
+at first they appear to have no connexion.
+
+_Illustrations._ Some examples will make clear any thing which may seem
+too abstract in the preceding generalities.
+
+1. _Falling Bodies._ Let us consider, in the first place, a natural
+phenomenon, very simple, indeed, but which may nevertheless give rise to
+a mathematical question, really existing, and susceptible of actual
+applications--the phenomenon of the vertical fall of heavy bodies.
+
+The mind the most unused to mathematical conceptions, in observing this
+phenomenon, perceives at once that the two _quantities_ which it
+presents--namely, the _height_ from which a body has fallen, and the
+_time_ of its fall--are necessarily connected with each other, since
+they vary together, and simultaneously remain fixed; or, in the language
+of geometers, that they are "_functions_" of each other. The phenomenon,
+considered under this point of view, gives rise then to a mathematical
+question, which consists in substituting for the direct measurement of
+one of these two magnitudes, when it is impossible, the measurement of
+the other. It is thus, for example, that we may determine indirectly the
+depth of a precipice, by merely measuring the time that a heavy body
+would occupy in falling to its bottom, and by suitable procedures this
+inaccessible depth will be known with as much precision as if it was a
+horizontal line placed in the most favourable circumstances for easy and
+exact measurement. On other occasions it is the height from which a body
+has fallen which it will be easy to ascertain, while the time of the
+fall could not be observed directly; then the same phenomenon would give
+rise to the inverse question, namely, to determine the time from the
+height; as, for example, if we wished to ascertain what would be the
+duration of the vertical fall of a body falling from the moon to the
+earth.
+
+In this example the mathematical question is very simple, at least when
+we do not pay attention to the variation in the intensity of gravity, or
+the resistance of the fluid which the body passes through in its fall.
+But, to extend the question, we have only to consider the same
+phenomenon in its greatest generality, in supposing the fall oblique,
+and in taking into the account all the principal circumstances. Then,
+instead of offering simply two variable quantities connected with each
+other by a relation easy to follow, the phenomenon will present a much
+greater number; namely, the space traversed, whether in a vertical or
+horizontal direction; the time employed in traversing it; the velocity
+of the body at each point of its course; even the intensity and the
+direction of its primitive impulse, which may also be viewed as
+variables; and finally, in certain cases (to take every thing into the
+account), the resistance of the medium and the intensity of gravity. All
+these different quantities will be connected with one another, in such a
+way that each in its turn may be indirectly determined by means of the
+others; and this will present as many distinct mathematical questions as
+there may be co-existing magnitudes in the phenomenon under
+consideration. Such a very slight change in the physical conditions of a
+problem may cause (as in the above example) a mathematical research, at
+first very elementary, to be placed at once in the rank of the most
+difficult questions, whose complete and rigorous solution surpasses as
+yet the utmost power of the human intellect.
+
+2. _Inaccessible Distances._ Let us take a second example from
+geometrical phenomena. Let it be proposed to determine a distance which
+is not susceptible of direct measurement; it will be generally conceived
+as making part of a _figure_, or certain system of lines, chosen in such
+a way that all its other parts may be observed directly; thus, in the
+case which is most simple, and to which all the others may be finally
+reduced, the proposed distance will be considered as belonging to a
+triangle, in which we can determine directly either another side and two
+angles, or two sides and one angle. Thence-forward, the knowledge of the
+desired distance, instead of being obtained directly, will be the result
+of a mathematical calculation, which will consist in deducing it from
+the observed elements by means of the relation which connects it with
+them. This calculation will become successively more and more
+complicated, if the parts which we have supposed to be known cannot
+themselves be determined (as is most frequently the case) except in an
+indirect manner, by the aid of new auxiliary systems, the number of
+which, in great operations of this kind, finally becomes very
+considerable. The distance being once determined, the knowledge of it
+will frequently be sufficient for obtaining new quantities, which will
+become the subject of new mathematical questions. Thus, when we know at
+what distance any object is situated, the simple observation of its
+apparent diameter will evidently permit us to determine indirectly its
+real dimensions, however inaccessible it may be, and, by a series of
+analogous investigations, its surface, its volume, even its weight, and
+a number of other properties, a knowledge of which seemed forbidden to
+us.
+
+3. _Astronomical Facts._ It is by such calculations that man has been
+able to ascertain, not only the distances from the planets to the earth,
+and, consequently, from each other, but their actual magnitude, their
+true figure, even to the inequalities of their surface; and, what seemed
+still more completely hidden from us, their respective masses, their
+mean densities, the principal circumstances of the fall of heavy bodies
+on the surface of each of them, &c.
+
+By the power of mathematical theories, all these different results, and
+many others relative to the different classes of mathematical phenomena,
+have required no other direct measurements than those of a very small
+number of straight lines, suitably chosen, and of a greater number of
+angles. We may even say, with perfect truth, so as to indicate in a word
+the general range of the science, that if we did not fear to multiply
+calculations unnecessarily, and if we had not, in consequence, to
+reserve them for the determination of the quantities which could not be
+measured directly, the determination of all the magnitudes susceptible
+of precise estimation, which the various orders of phenomena can offer
+us, could be finally reduced to the direct measurement of a single
+straight line and of a suitable number of angles.
+
+
+TRUE DEFINITION OF MATHEMATICS.
+
+We are now able to define mathematical science with precision, by
+assigning to it as its object the _indirect_ measurement of magnitudes,
+and by saying it constantly proposes _to determine certain magnitudes
+from others by means of the precise relations existing between them_.
+
+This enunciation, instead of giving the idea of only an _art_, as do all
+the ordinary definitions, characterizes immediately a true _science_,
+and shows it at once to be composed of an immense chain of intellectual
+operations, which may evidently become very complicated, because of the
+series of intermediate links which it will be necessary to establish
+between the unknown quantities and those which admit of a direct
+measurement; of the number of variables coexistent in the proposed
+question; and of the nature of the relations between all these different
+magnitudes furnished by the phenomena under consideration. According to
+such a definition, the spirit of mathematics consists in always
+regarding all the quantities which any phenomenon can present, as
+connected and interwoven with one another, with the view of deducing
+them from one another. Now there is evidently no phenomenon which cannot
+give rise to considerations of this kind; whence results the naturally
+indefinite extent and even the rigorous logical universality of
+mathematical science. We shall seek farther on to circumscribe as
+exactly as possible its real extension.
+
+The preceding explanations establish clearly the propriety of the name
+employed to designate the science which we are considering. This
+denomination, which has taken to-day so definite a meaning by itself
+signifies simply _science_ in general. Such a designation, rigorously
+exact for the Greeks, who had no other real science, could be retained
+by the moderns only to indicate the mathematics as _the_ science, beyond
+all others--the science of sciences.
+
+Indeed, every true science has for its object the determination of
+certain phenomena by means of others, in accordance with the relations
+which exist between them. Every _science_ consists in the co-ordination
+of facts; if the different observations were entirely isolated, there
+would be no science. We may even say, in general terms, that _science_
+is essentially destined to dispense, so far as the different phenomena
+permit it, with all direct observation, by enabling us to deduce from
+the smallest possible number of immediate data the greatest possible
+number of results. Is not this the real use, whether in speculation or
+in action, of the _laws_ which we succeed in discovering among natural
+phenomena? Mathematical science, in this point of view, merely pushes to
+the highest possible degree the same kind of researches which are
+pursued, in degrees more or less inferior, by every real science in its
+respective sphere.
+
+
+ITS TWO FUNDAMENTAL DIVISIONS.
+
+We have thus far viewed mathematical science only as a whole, without
+paying any regard to its divisions. We must now, in order to complete
+this general view, and to form a just idea of the philosophical
+character of the science, consider its fundamental division. The
+secondary divisions will be examined in the following chapters.
+
+This principal division, which we are about to investigate, can be
+truly rational, and derived from the real nature of the subject, only so
+far as it spontaneously presents itself to us, in making the exact
+analysis of a complete mathematical question. We will, therefore, having
+determined above what is the general object of mathematical labours, now
+characterize with precision the principal different orders of inquiries,
+of which they are constantly composed.
+
+_Their different Objects._ The complete solution of every mathematical
+question divides itself necessarily into two parts, of natures
+essentially distinct, and with relations invariably determinate. We have
+seen that every mathematical inquiry has for its object to determine
+unknown magnitudes, according to the relations between them and known
+magnitudes. Now for this object, it is evidently necessary, in the first
+place, to ascertain with precision the relations which exist between the
+quantities which we are considering. This first branch of inquiries
+constitutes that which I call the _concrete_ part of the solution. When
+it is finished, the question changes; it is now reduced to a pure
+question of numbers, consisting simply in determining unknown numbers,
+when we know what precise relations connect them with known numbers.
+This second branch of inquiries is what I call the _abstract_ part of
+the solution. Hence follows the fundamental division of general
+mathematical science into _two_ great sciences--ABSTRACT MATHEMATICS,
+and CONCRETE MATHEMATICS.
+
+This analysis may be observed in every complete mathematical question,
+however simple or complicated it may be. A single example will suffice
+to make it intelligible.
+
+Taking up again the phenomenon of the vertical fall of a heavy body, and
+considering the simplest case, we see that in order to succeed in
+determining, by means of one another, the height whence the body has
+fallen, and the duration of its fall, we must commence by discovering
+the exact relation of these two quantities, or, to use the language of
+geometers, the _equation_ which exists between them. Before this first
+research is completed, every attempt to determine numerically the value
+of one of these two magnitudes from the other would evidently be
+premature, for it would have no basis. It is not enough to know vaguely
+that they depend on one another--which every one at once perceives--but
+it is necessary to determine in what this dependence consists. This
+inquiry may be very difficult, and in fact, in the present case,
+constitutes incomparably the greater part of the problem. The true
+scientific spirit is so modern, that no one, perhaps, before Galileo,
+had ever remarked the increase of velocity which a body experiences in
+its fall: a circumstance which excludes the hypothesis, towards which
+our mind (always involuntarily inclined to suppose in every phenomenon
+the most simple _functions_, without any other motive than its greater
+facility in conceiving them) would be naturally led, that the height was
+proportional to the time. In a word, this first inquiry terminated in
+the discovery of the law of Galileo.
+
+When this _concrete_ part is completed, the inquiry becomes one of quite
+another nature. Knowing that the spaces passed through by the body in
+each successive second of its fall increase as the series of odd
+numbers, we have then a problem purely numerical and _abstract_; to
+deduce the height from the time, or the time from the height; and this
+consists in finding that the first of these two quantities, according to
+the law which has been established, is a known multiple of the second
+power of the other; from which, finally, we have to calculate the value
+of the one when that of the other is given.
+
+In this example the concrete question is more difficult than the
+abstract one. The reverse would be the case if we considered the same
+phenomenon in its greatest generality, as I have done above for another
+object. According to the circumstances, sometimes the first, sometimes
+the second, of these two parts will constitute the principal difficulty
+of the whole question; for the mathematical law of the phenomenon may be
+very simple, but very difficult to obtain, or it may be easy to
+discover, but very complicated; so that the two great sections of
+mathematical science, when we compare them as wholes, must be regarded
+as exactly equivalent in extent and in difficulty, as well as in
+importance, as we shall show farther on, in considering each of them
+separately.
+
+_Their different Natures._ These two parts, essentially distinct in
+their _object_, as we have just seen, are no less so with regard to the
+_nature_ of the inquiries of which they are composed.
+
+The first should be called _concrete_, since it evidently depends on the
+character of the phenomena considered, and must necessarily vary when we
+examine new phenomena; while the second is completely independent of the
+nature of the objects examined, and is concerned with only the
+_numerical_ relations which they present, for which reason it should be
+called _abstract_. The same relations may exist in a great number of
+different phenomena, which, in spite of their extreme diversity, will
+be viewed by the geometer as offering an analytical question
+susceptible, when studied by itself, of being resolved once for all.
+Thus, for instance, the same law which exists between the space and the
+time of the vertical fall of a body in a vacuum, is found again in many
+other phenomena which offer no analogy with the first nor with each
+other; for it expresses the relation between the surface of a spherical
+body and the length of its diameter; it determines, in like manner, the
+decrease of the intensity of light or of heat in relation to the
+distance of the objects lighted or heated, &c. The abstract part, common
+to these different mathematical questions, having been treated in
+reference to one of these, will thus have been treated for all; while
+the concrete part will have necessarily to be again taken up for each
+question separately, without the solution of any one of them being able
+to give any direct aid, in that connexion, for the solution of the rest.
+
+The abstract part of mathematics is, then, general in its nature; the
+concrete part, special.
+
+To present this comparison under a new point of view, we may say
+concrete mathematics has a philosophical character, which is essentially
+experimental, physical, phenomenal; while that of abstract mathematics
+is purely logical, rational. The concrete part of every mathematical
+question is necessarily founded on the consideration of the external
+world, and could never be resolved by a simple series of intellectual
+combinations. The abstract part, on the contrary, when it has been very
+completely separated, can consist only of a series of logical
+deductions, more or less prolonged; for if we have once found the
+equations of a phenomenon, the determination of the quantities therein
+considered, by means of one another, is a matter for reasoning only,
+whatever the difficulties may be. It belongs to the understanding alone
+to deduce from these equations results which are evidently contained in
+them, although perhaps in a very involved manner, without there being
+occasion to consult anew the external world; the consideration of which,
+having become thenceforth foreign to the subject, ought even to be
+carefully set aside in order to reduce the labour to its true peculiar
+difficulty. The _abstract_ part of mathematics is then purely
+instrumental, and is only an immense and admirable extension of natural
+logic to a certain class of deductions. On the other hand, geometry and
+mechanics, which, as we shall see presently, constitute the _concrete_
+part, must be viewed as real natural sciences, founded on observation,
+like all the rest, although the extreme simplicity of their phenomena
+permits an infinitely greater degree of systematization, which has
+sometimes caused a misconception of the experimental character of their
+first principles.
+
+We see, by this brief general comparison, how natural and profound is
+our fundamental division of mathematical science.
+
+We have now to circumscribe, as exactly as we can in this first sketch,
+each of these two great sections.
+
+
+CONCRETE MATHEMATICS.
+
+_Concrete Mathematics_ having for its object the discovery of the
+_equations_ of phenomena, it would seem at first that it must be
+composed of as many distinct sciences as we find really distinct
+categories among natural phenomena. But we are yet very far from having
+discovered mathematical laws in all kinds of phenomena; we shall even
+see, presently, that the greater part will very probably always hide
+themselves from our investigations. In reality, in the present condition
+of the human mind, there are directly but two great general classes of
+phenomena, whose equations we constantly know; these are, firstly,
+geometrical, and, secondly, mechanical phenomena. Thus, then, the
+concrete part of mathematics is composed of GEOMETRY and RATIONAL
+MECHANICS.
+
+This is sufficient, it is true, to give to it a complete character of
+logical universality, when we consider all phenomena from the most
+elevated point of view of natural philosophy. In fact, if all the parts
+of the universe were conceived as immovable, we should evidently have
+only geometrical phenomena to observe, since all would be reduced to
+relations of form, magnitude, and position; then, having regard to the
+motions which take place in it, we would have also to consider
+mechanical phenomena. Hence the universe, in the statical point of view,
+presents only geometrical phenomena; and, considered dynamically, only
+mechanical phenomena. Thus geometry and mechanics constitute the two
+fundamental natural sciences, in this sense, that all natural effects
+may be conceived as simple necessary results, either of the laws of
+extension or of the laws of motion.
+
+But although this conception is always logically possible, the
+difficulty is to specialize it with the necessary precision, and to
+follow it exactly in each of the general cases offered to us by the
+study of nature; that is, to effectually reduce each principal question
+of natural philosophy, for a certain determinate order of phenomena, to
+the question of geometry or mechanics, to which we might rationally
+suppose it should be brought. This transformation, which requires great
+progress to have been previously made in the study of each class of
+phenomena, has thus far been really executed only for those of
+astronomy, and for a part of those considered by terrestrial physics,
+properly so called. It is thus that astronomy, acoustics, optics, &c.,
+have finally become applications of mathematical science to certain
+orders of observations.[1] But these applications not being by their
+nature rigorously circumscribed, to confound them with the science would
+be to assign to it a vague and indefinite domain; and this is done in
+the usual division, so faulty in so many other respects, of the
+mathematics into "Pure" and "Applied."
+
+  [Footnote 1: The investigation of the mathematical phenomena of the
+  laws of heat by Baron Fourier has led to the establishment, in an
+  entirely direct manner, of Thermological equations. This great
+  discovery tends to elevate our philosophical hopes as to the future
+  extensions of the legitimate applications of mathematical analysis,
+  and renders it proper, in the opinion of author, to regard
+  _Thermology_ as a third principal branch of concrete mathematics.]
+
+
+ABSTRACT MATHEMATICS.
+
+The nature of abstract mathematics (the general division of which will
+be examined in the following chapter) is clearly and exactly determined.
+It is composed of what is called the _Calculus_,[2] taking this word in
+its greatest extent, which reaches from the most simple numerical
+operations to the most sublime combinations of transcendental analysis.
+The _Calculus_ has the solution of all questions relating to numbers
+for its peculiar object. Its _starting point_ is, constantly and
+necessarily, the knowledge of the precise relations, _i.e._, of the
+_equations_, between the different magnitudes which are simultaneously
+considered; that which is, on the contrary, the _stopping point_ of
+concrete mathematics. However complicated, or however indirect these
+relations may be, the final object of the calculus always is to obtain
+from them the values of the unknown quantities by means of those which
+are known. This _science_, although nearer perfection than any other, is
+really little advanced as yet, so that this object is rarely attained in
+a manner completely satisfactory.
+
+  [Footnote 2: The translator has felt justified in employing this
+  very convenient word (for which our language has no precise
+  equivalent) as an English one, in its most extended sense, in spite
+  of its being often popularly confounded with its Differential and
+  Integral department.]
+
+Mathematical analysis is, then, the true rational basis of the entire
+system of our actual knowledge. It constitutes the first and the most
+perfect of all the fundamental sciences. The ideas with which it
+occupies itself are the most universal, the most abstract, and the most
+simple which it is possible for us to conceive.
+
+This peculiar nature of mathematical analysis enables us easily to
+explain why, when it is properly employed, it is such a powerful
+instrument, not only to give more precision to our real knowledge, which
+is self-evident, but especially to establish an infinitely more perfect
+co-ordination in the study of the phenomena which admit of that
+application; for, our conceptions having been so generalized and
+simplified that a single analytical question, abstractly resolved,
+contains the _implicit_ solution of a great number of diverse physical
+questions, the human mind must necessarily acquire by these means a
+greater facility in perceiving relations between phenomena which at
+first appeared entirely distinct from one another. We thus naturally see
+arise, through the medium of analysis, the most frequent and the most
+unexpected approximations between problems which at first offered no
+apparent connection, and which we often end in viewing as identical.
+Could we, for example, without the aid of analysis, perceive the least
+resemblance between the determination of the direction of a curve at
+each of its points and that of the velocity acquired by a body at every
+instant of its variable motion? and yet these questions, however
+different they may be, compose but one in the eyes of the geometer.
+
+The high relative perfection of mathematical analysis is as easily
+perceptible. This perfection is not due, as some have thought, to the
+nature of the signs which are employed as instruments of reasoning,
+eminently concise and general as they are. In reality, all great
+analytical ideas have been formed without the algebraic signs having
+been of any essential aid, except for working them out after the mind
+had conceived them. The superior perfection of the science of the
+calculus is due principally to the extreme simplicity of the ideas which
+it considers, by whatever signs they may be expressed; so that there is
+not the least hope, by any artifice of scientific language, of
+perfecting to the same degree theories which refer to more complex
+subjects, and which are necessarily condemned by their nature to a
+greater or less logical inferiority.
+
+
+THE EXTENT OF ITS FIELD.
+
+Our examination of the philosophical character of mathematical science
+would remain incomplete, if, after having viewed its object and
+composition, we did not examine the real extent of its domain.
+
+_Its Universality_. For this purpose it is indispensable to perceive,
+first of all, that, in the purely logical point of view, this science is
+by itself necessarily and rigorously universal; for there is no question
+whatever which may not be finally conceived as consisting in determining
+certain quantities from others by means of certain relations, and
+consequently as admitting of reduction, in final analysis, to a simple
+question of numbers. In all our researches, indeed, on whatever subject,
+our object is to arrive at numbers, at quantities, though often in a
+very imperfect manner and by very uncertain methods. Thus, taking an
+example in the class of subjects the least accessible to mathematics,
+the phenomena of living bodies, even when considered (to take the most
+complicated case) in the state of disease, is it not manifest that all
+the questions of therapeutics may be viewed as consisting in determining
+the _quantities_ of the different agents which modify the organism, and
+which must act upon it to bring it to its normal state, admitting, for
+some of these quantities in certain cases, values which are equal to
+zero, or negative, or even contradictory?
+
+The fundamental idea of Descartes on the relation of the concrete to the
+abstract in mathematics, has proven, in opposition to the superficial
+distinction of metaphysics, that all ideas of quality may be reduced to
+those of quantity. This conception, established at first by its immortal
+author in relation to geometrical phenomena only, has since been
+effectually extended to mechanical phenomena, and in our days to those
+of heat. As a result of this gradual generalization, there are now no
+geometers who do not consider it, in a purely theoretical sense, as
+capable of being applied to all our real ideas of every sort, so that
+every phenomenon is logically susceptible of being represented by an
+_equation_; as much so, indeed, as is a curve or a motion, excepting the
+difficulty of discovering it, and then of _resolving_ it, which may be,
+and oftentimes are, superior to the greatest powers of the human mind.
+
+_Its Limitations_. Important as it is to comprehend the rigorous
+universality, in a logical point of view, of mathematical science, it is
+no less indispensable to consider now the great real _limitations_,
+which, through the feebleness of our intellect, narrow in a remarkable
+degree its actual domain, in proportion as phenomena, in becoming
+special, become complicated.
+
+Every question may be conceived as capable of being reduced to a pure
+question of numbers; but the difficulty of effecting such a
+transformation increases so much with the complication of the phenomena
+of natural philosophy, that it soon becomes insurmountable.
+
+This will be easily seen, if we consider that to bring a question within
+the field of mathematical analysis, we must first have discovered the
+precise relations which exist between the quantities which are found in
+the phenomenon under examination, the establishment of these equations
+being the necessary starting point of all analytical labours. This must
+evidently be so much the more difficult as we have to do with phenomena
+which are more special, and therefore more complicated. We shall thus
+find that it is only in _inorganic physics_, at the most, that we can
+justly hope ever to obtain that high degree of scientific perfection.
+
+The _first_ condition which is necessary in order that phenomena may
+admit of mathematical laws, susceptible of being discovered, evidently
+is, that their different quantities should admit of being expressed by
+fixed numbers. We soon find that in this respect the whole of _organic
+physics_, and probably also the most complicated parts of inorganic
+physics, are necessarily inaccessible, by their nature, to our
+mathematical analysis, by reason of the extreme numerical variability of
+the corresponding phenomena. Every precise idea of fixed numbers is
+truly out of place in the phenomena of living bodies, when we wish to
+employ it otherwise than as a means of relieving the attention, and when
+we attach any importance to the exact relations of the values assigned.
+
+We ought not, however, on this account, to cease to conceive all
+phenomena as being necessarily subject to mathematical laws, which we
+are condemned to be ignorant of, only because of the too great
+complication of the phenomena. The most complex phenomena of living
+bodies are doubtless essentially of no other special nature than the
+simplest phenomena of unorganized matter. If it were possible to isolate
+rigorously each of the simple causes which concur in producing a single
+physiological phenomenon, every thing leads us to believe that it would
+show itself endowed, in determinate circumstances, with a kind of
+influence and with a quantity of action as exactly fixed as we see it in
+universal gravitation, a veritable type of the fundamental laws of
+nature.
+
+There is a _second_ reason why we cannot bring complicated phenomena
+under the dominion of mathematical analysis. Even if we could ascertain
+the mathematical law which governs each agent, taken by itself, the
+combination of so great a number of conditions would render the
+corresponding mathematical problem so far above our feeble means, that
+the question would remain in most cases incapable of solution.
+
+To appreciate this difficulty, let us consider how complicated
+mathematical questions become, even those relating to the most simple
+phenomena of unorganized bodies, when we desire to bring sufficiently
+near together the abstract and the concrete state, having regard to all
+the principal conditions which can exercise a real influence over the
+effect produced. We know, for example, that the very simple phenomenon
+of the flow of a fluid through a given orifice, by virtue of its gravity
+alone, has not as yet any complete mathematical solution, when we take
+into the account all the essential circumstances. It is the same even
+with the still more simple motion of a solid projectile in a resisting
+medium.
+
+Why has mathematical analysis been able to adapt itself with such
+admirable success to the most profound study of celestial phenomena?
+Because they are, in spite of popular appearances, much more simple than
+any others. The most complicated problem which they present, that of the
+modification produced in the motions of two bodies tending towards each
+other by virtue of their gravitation, by the influence of a third body
+acting on both of them in the same manner, is much less complex than the
+most simple terrestrial problem. And, nevertheless, even it presents
+difficulties so great that we yet possess only approximate solutions of
+it. It is even easy to see that the high perfection to which solar
+astronomy has been able to elevate itself by the employment of
+mathematical science is, besides, essentially due to our having
+skilfully profited by all the particular, and, so to say, accidental
+facilities presented by the peculiarly favourable constitution of our
+planetary system. The planets which compose it are quite few in number,
+and their masses are in general very unequal, and much less than that of
+the sun; they are, besides, very distant from one another; they have
+forms almost spherical; their orbits are nearly circular, and only
+slightly inclined to each other, and so on. It results from all these
+circumstances that the perturbations are generally inconsiderable, and
+that to calculate them it is usually sufficient to take into the
+account, in connexion with the action of the sun on each particular
+planet, the influence of only one other planet, capable, by its size and
+its proximity, of causing perceptible derangements.
+
+If, however, instead of such a state of things, our solar system had
+been composed of a greater number of planets concentrated into a less
+space, and nearly equal in mass; if their orbits had presented very
+different inclinations, and considerable eccentricities; if these bodies
+had been of a more complicated form, such as very eccentric ellipsoids,
+it is certain that, supposing the same law of gravitation to exist, we
+should not yet have succeeded in subjecting the study of the celestial
+phenomena to our mathematical analysis, and probably we should not even
+have been able to disentangle the present principal law.
+
+These hypothetical conditions would find themselves exactly realized in
+the highest degree in _chemical_ phenomena, if we attempted to calculate
+them by the theory of general gravitation.
+
+On properly weighing the preceding considerations, the reader will be
+convinced, I think, that in reducing the future extension of the great
+applications of mathematical analysis, which are really possible, to
+the field comprised in the different departments of inorganic physics, I
+have rather exaggerated than contracted the extent of its actual domain.
+Important as it was to render apparent the rigorous logical universality
+of mathematical science, it was equally so to indicate the conditions
+which limit for us its real extension, so as not to contribute to lead
+the human mind astray from the true scientific direction in the study of
+the most complicated phenomena, by the chimerical search after an
+impossible perfection.
+
+       *       *       *       *       *
+
+Having thus exhibited the essential object and the principal composition
+of mathematical science, as well as its general relations with the whole
+body of natural philosophy, we have now to pass to the special
+examination of the great sciences of which it is composed.
+
+     _Note._--ANALYSIS and GEOMETRY are the two great heads under which
+     the subject is about to be examined. To these _M. Comte_ adds
+     Rational MECHANICS; but as it is not comprised in the usual idea of
+     Mathematics, and as its discussion would be of but limited utility
+     and interest, it is not included in the present translation.
+
+
+
+
+BOOK I.
+
+ANALYSIS.
+
+
+
+
+BOOK I.
+
+ANALYSIS.
+
+
+
+
+CHAPTER I.
+
+GENERAL VIEW OF MATHEMATICAL ANALYSIS.
+
+
+In the historical development of mathematical science since the time of
+Descartes, the advances of its abstract portion have always been
+determined by those of its concrete portion; but it is none the less
+necessary, in order to conceive the science in a manner truly logical,
+to consider the Calculus in all its principal branches before proceeding
+to the philosophical study of Geometry and Mechanics. Its analytical
+theories, more simple and more general than those of concrete
+mathematics, are in themselves essentially independent of the latter;
+while these, on the contrary, have, by their nature, a continual need of
+the former, without the aid of which they could make scarcely any
+progress. Although the principal conceptions of analysis retain at
+present some very perceptible traces of their geometrical or mechanical
+origin, they are now, however, mainly freed from that primitive
+character, which no longer manifests itself except in some secondary
+points; so that it is possible (especially since the labours of
+Lagrange) to present them in a dogmatic exposition, by a purely abstract
+method, in a single and continuous system. It is this which will be
+undertaken in the present and the five following chapters, limiting our
+investigations to the most general considerations upon each principal
+branch of the science of the calculus.
+
+The definite object of our researches in concrete mathematics being the
+discovery of the _equations_ which express the mathematical laws of the
+phenomenon under consideration, and these equations constituting the
+true starting point of the calculus, which has for its object to obtain
+from them the determination of certain quantities by means of others, I
+think it indispensable, before proceeding any farther, to go more deeply
+than has been customary into that fundamental idea of _equation_, the
+continual subject, either as end or as beginning, of all mathematical
+labours. Besides the advantage of circumscribing more definitely the
+true field of analysis, there will result from it the important
+consequence of tracing in a more exact manner the real line of
+demarcation between the concrete and the abstract part of mathematics,
+which will complete the general exposition of the fundamental division
+established in the introductory chapter.
+
+
+
+
+THE TRUE IDEA OF AN EQUATION.
+
+
+We usually form much too vague an idea of what an _equation_ is, when we
+give that name to every kind of relation of equality between _any_ two
+functions of the magnitudes which we are considering. For, though every
+equation is evidently a relation of equality, it is far from being true
+that, reciprocally, every relation of equality is a veritable
+_equation_, of the kind of those to which, by their nature, the methods
+of analysis are applicable.
+
+This want of precision in the logical consideration of an idea which is
+so fundamental in mathematics, brings with it the serious inconvenience
+of rendering it almost impossible to explain, in general terms, the
+great and fundamental difficulty which we find in establishing the
+relation between the concrete and the abstract, and which stands out so
+prominently in each great mathematical question taken by itself. If the
+meaning of the word _equation_ was truly as extended as we habitually
+suppose it to be in our definition of it, it is not apparent what great
+difficulty there could really be, in general, in establishing the
+equations of any problem whatsoever; for the whole would thus appear to
+consist in a simple question of form, which ought never even to exact
+any great intellectual efforts, seeing that we can hardly conceive of
+any precise relation which is not immediately a certain relation of
+equality, or which cannot be readily brought thereto by some very easy
+transformations.
+
+Thus, when we admit every species of _functions_ into the definition of
+_equations_, we do not at all account for the extreme difficulty which
+we almost always experience in putting a problem into an equation, and
+which so often may be compared to the efforts required by the analytical
+elaboration of the equation when once obtained. In a word, the ordinary
+abstract and general idea of an _equation_ does not at all correspond to
+the real meaning which geometers attach to that expression in the actual
+development of the science. Here, then, is a logical fault, a defect of
+correlation, which it is very important to rectify.
+
+
+_Division of Functions into Abstract and Concrete._ To succeed in doing
+so, I begin by distinguishing two sorts of _functions_, _abstract_ or
+analytical functions, and _concrete_ functions. The first alone can
+enter into veritable _equations_. We may, therefore, henceforth define
+every _equation_, in an exact and sufficiently profound manner, as a
+relation of equality between two _abstract_ functions of the magnitudes
+under consideration. In order not to have to return again to this
+fundamental definition, I must add here, as an indispensable complement,
+without which the idea would not be sufficiently general, that these
+abstract functions may refer not only to the magnitudes which the
+problem presents of itself, but also to all the other auxiliary
+magnitudes which are connected with it, and which we will often be able
+to introduce, simply as a mathematical artifice, with the sole object of
+facilitating the discovery of the equations of the phenomena. I here
+anticipate summarily the result of a general discussion of the highest
+importance, which will be found at the end of this chapter. We will now
+return to the essential distinction of functions as abstract and
+concrete.
+
+This distinction may be established in two ways, essentially different,
+but complementary of each other, _à priori_ and _à posteriori_; that is
+to say, by characterizing in a general manner the peculiar nature of
+each species of functions, and then by making the actual enumeration of
+all the abstract functions at present known, at least so far as relates
+to the elements of which they are composed.
+
+_À priori_, the functions which I call _abstract_ are those which
+express a manner of dependence between magnitudes, which can be
+conceived between numbers alone, without there being need of indicating
+any phenomenon whatever in which it is realized. I name, on the other
+hand, _concrete_ functions, those for which the mode of dependence
+expressed cannot be defined or conceived except by assigning a
+determinate case of physics, geometry, mechanics, &c., in which it
+actually exists.
+
+Most functions in their origin, even those which are at present the most
+purely _abstract_, have begun by being _concrete_; so that it is easy to
+make the preceding distinction understood, by citing only the successive
+different points of view under which, in proportion as the science has
+become formed, geometers have considered the most simple analytical
+functions. I will indicate powers, for example, which have in general
+become abstract functions only since the labours of Vieta and Descartes.
+The functions _x²_, _x³_, which in our present analysis are so well
+conceived as simply _abstract_, were, for the geometers of antiquity,
+perfectly _concrete_ functions, expressing the relation of the
+superficies of a square, or the volume of a cube to the length of their
+side. These had in their eyes such a character so exclusively, that it
+was only by means of the geometrical definitions that they discovered
+the elementary algebraic properties of these functions, relating to the
+decomposition of the variable into two parts, properties which were at
+that epoch only real theorems of geometry, to which a numerical meaning
+was not attached until long afterward.
+
+I shall have occasion to cite presently, for another reason, a new
+example, very suitable to make apparent the fundamental distinction
+which I have just exhibited; it is that of circular functions, both
+direct and inverse, which at the present time are still sometimes
+concrete, sometimes abstract, according to the point of view under
+which they are regarded.
+
+_À posteriori_, the general character which renders a function abstract
+or concrete having been established, the question as to whether a
+certain determinate function is veritably abstract, and therefore
+susceptible of entering into true analytical equations, becomes a simple
+question of fact, inasmuch as we are going to enumerate all the
+functions of this species.
+
+
+_Enumeration of Abstract Functions._ At first view this enumeration
+seems impossible, the distinct analytical functions being infinite in
+number. But when we divide them into _simple_ and _compound_, the
+difficulty disappears; for, though the number of the different functions
+considered in mathematical analysis is really infinite, they are, on the
+contrary, even at the present day, composed of a very small number of
+elementary functions, which can be easily assigned, and which are
+evidently sufficient for deciding the abstract or concrete character of
+any given function; which will be of the one or the other nature,
+according as it shall be composed exclusively of these simple abstract
+functions, or as it shall include others.
+
+We evidently have to consider, for this purpose, only the functions of a
+single variable, since those relative to several independent variables
+are constantly, by their nature, more or less _compound_.
+
+Let _x_ be the independent variable, _y_ the correlative variable which
+depends upon it. The different simple modes of abstract dependence,
+which we can now conceive between _y_ and _x_, are expressed by the ten
+following elementary formulas, in which each function is coupled with
+its _inverse_, that is, with that which would be obtained from the
+direct function by referring _x_ to _y_, instead of referring _y_ to
+_x_.
+
+            FUNCTION.                   ITS NAME.
+
+1st couple {1° _y_ = _a_ + _x_             _Sum._
+           {2° _y_ = _a_ - _x_             _Difference._
+
+2d couple  {1° _y_ = _ax_                  _Product._
+           {2° _y_ = _a/x_                 _Quotient._
+
+3d couple  {1° _y_ = _x^a_                 _Power._
+           {2° _y_ = _[ath root]x_            _Root._
+
+4th couple {1° _y_ = _a^x_                 _Exponential._
+           {2° _y_ = _[log a]x_            _Logarithmic._
+
+5th couple {1° _y_ = sin. _x_              _Direct Circular._
+           {2° _y_ = arc(sin. = _x_).      _Inverse Circular._[3]
+
+  [Footnote 3: With the view of increasing as much as possible the
+  resources and the extent (now so insufficient) of mathematical
+  analysis, geometers count this last couple of functions among the
+  analytical elements. Although this inscription is strictly
+  legitimate, it is important to remark that circular functions are
+  not exactly in the same situation as the other abstract elementary
+  functions. There is this very essential difference, that the
+  functions of the four first couples are at the same time simple and
+  abstract, while the circular functions, which may manifest each
+  character in succession, according to the point of view under which
+  they are considered and the manner in which they are employed,
+  never present these two properties simultaneously.
+
+  Some other concrete functions may be usefully introduced into the
+  number of analytical elements, certain conditions being fulfilled.
+  It is thus, for example, that the labours of M. Legendre and of M.
+  Jacobi on _elliptical_ functions have truly enlarged the field of
+  analysis; and the same is true of some definite integrals obtained
+  by M. Fourier in the theory of heat.]
+
+Such are the elements, very few in number, which directly compose all
+the abstract functions known at the present day. Few as they are, they
+are evidently sufficient to give rise to an infinite number of
+analytical combinations.
+
+No rational consideration rigorously circumscribes, _à priori_, the
+preceding table, which is only the actual expression of the present
+state of the science. Our analytical elements are at the present day
+more numerous than they were for Descartes, and even for Newton and
+Leibnitz: it is only a century since the last two couples have been
+introduced into analysis by the labours of John Bernouilli and Euler.
+Doubtless new ones will be hereafter admitted; but, as I shall show
+towards the end of this chapter, we cannot hope that they will ever be
+greatly multiplied, their real augmentation giving rise to very great
+difficulties.
+
+We can now form a definite, and, at the same time, sufficiently extended
+idea of what geometers understand by a veritable _equation_. This
+explanation is especially suited to make us understand how difficult it
+must be really to establish the _equations_ of phenomena, since we have
+effectually succeeded in so doing only when we have been able to
+conceive the mathematical laws of these phenomena by the aid of
+functions entirely composed of only the mathematical elements which I
+have just enumerated. It is clear, in fact, that it is then only that
+the problem becomes truly abstract, and is reduced to a pure question of
+numbers, these functions being the only simple relations which we can
+conceive between numbers, considered by themselves. Up to this period of
+the solution, whatever the appearances may be, the question is still
+essentially concrete, and does not come within the domain of the
+_calculus_. Now the fundamental difficulty of this passage from the
+_concrete_ to the _abstract_ in general consists especially in the
+insufficiency of this very small number of analytical elements which we
+possess, and by means of which, nevertheless, in spite of the little
+real variety which they offer us, we must succeed in representing all
+the precise relations which all the different natural phenomena can
+manifest to us. Considering the infinite diversity which must
+necessarily exist in this respect in the external world, we easily
+understand how far below the true difficulty our conceptions must
+frequently be found, especially if we add that as these elements of our
+analysis have been in the first place furnished to us by the
+mathematical consideration of the simplest phenomena, we have, _à
+priori_, no rational guarantee of their necessary suitableness to
+represent the mathematical law of every other class of phenomena. I will
+explain presently the general artifice, so profoundly ingenious, by
+which the human mind has succeeded in diminishing, in a remarkable
+degree, this fundamental difficulty which is presented by the relation
+of the concrete to the abstract in mathematics, without, however, its
+having been necessary to multiply the number of these analytical
+elements.
+
+
+
+
+THE TWO PRINCIPAL DIVISIONS OF THE CALCULUS.
+
+
+The preceding explanations determine with precision the true object and
+the real field of abstract mathematics. I must now pass to the
+examination of its principal divisions, for thus far we have considered
+the calculus as a whole.
+
+The first direct consideration to be presented on the composition of the
+science of the _calculus_ consists in dividing it, in the first place,
+into two principal branches, to which, for want of more suitable
+denominations, I will give the names of _Algebraic calculus_, or
+_Algebra_, and of _Arithmetical calculus_, or _Arithmetic_; but with
+the caution to take these two expressions in their most extended logical
+acceptation, in the place of the by far too restricted meaning which is
+usually attached to them.
+
+The complete solution of every question of the _calculus_, from the most
+elementary up to the most transcendental, is necessarily composed of two
+successive parts, whose nature is essentially distinct. In the first,
+the object is to transform the proposed equations, so as to make
+apparent the manner in which the unknown quantities are formed by the
+known ones: it is this which constitutes the _algebraic_ question. In
+the second, our object is to _find the values_ of the formulas thus
+obtained; that is, to determine directly the values of the numbers
+sought, which are already represented by certain explicit functions of
+given numbers: this is the _arithmetical_ question.[4] It is apparent
+that, in every solution which is truly rational, it necessarily follows
+the algebraical question, of which it forms the indispensable
+complement, since it is evidently necessary to know the mode of
+generation of the numbers sought for before determining their actual
+values for each particular case. Thus the stopping-place of the
+algebraic part of the solution becomes the starting point of the
+arithmetical part.
+
+  [Footnote 4: Suppose, for example, that a question gives the
+  following equation between an unknown magnitude x, and two known
+  magnitudes, _a_ and _b_,
+
+  _x³_ + 3_ax_ = 2_b_,
+
+  as is the case in the problem of the trisection of an angle. We see
+  at once that the dependence between _x_ on the one side, and _ab_ on
+  the other, is completely determined; but, so long as the equation
+  preserves its primitive form, we do not at all perceive in what
+  manner the unknown quantity is derived from the data. This must be
+  discovered, however, before we can think of determining its value.
+  Such is the object of the algebraic part of the solution. When, by a
+  series of transformations which have successively rendered that
+  derivation more and more apparent, we have arrived at presenting the
+  proposed equation under the form
+
+  _x_ = ∛(_b_ + √(_b²_ + _a³_)) + ∛(_b_ - √(_b²_ + _a³_)),
+
+  the work of _algebra_ is finished; and even if we could not perform
+  the arithmetical operations indicated by that formula, we would
+  nevertheless have obtained a knowledge very real, and often very
+  important. The work of _arithmetic_ will now consist in taking that
+  formula for its starting point, and finding the number _x_ when the
+  values of the numbers _a_ and _b_ are given.]
+
+We thus see that the _algebraic_ calculus and the _arithmetical_
+calculus differ essentially in their object. They differ no less in the
+point of view under which they regard quantities; which are considered
+in the first as to their _relations_, and in the second as to their
+_values_. The true spirit of the calculus, in general, requires this
+distinction to be maintained with the most severe exactitude, and the
+line of demarcation between the two periods of the solution to be
+rendered as clear and distinct as the proposed question permits. The
+attentive observation of this precept, which is too much neglected, may
+be of much assistance, in each particular question, in directing the
+efforts of our mind, at any moment of the solution, towards the real
+corresponding difficulty. In truth, the imperfection of the science of
+the calculus obliges us very often (as will be explained in the next
+chapter) to intermingle algebraic and arithmetical considerations in the
+solution of the same question. But, however impossible it may be to
+separate clearly the two parts of the labour, yet the preceding
+indications will always enable us to avoid confounding them.
+
+In endeavouring to sum up as succinctly as possible the distinction just
+established, we see that ALGEBRA may be defined, in general, as having
+for its object the _resolution of equations_; taking this expression in
+its full logical meaning, which signifies the transformation of
+_implicit_ functions into equivalent _explicit_ ones. In the same way,
+ARITHMETIC may be defined as destined to _the determination of the
+values of functions_. Henceforth, therefore, we will briefly say that
+ALGEBRA is the _Calculus of Functions_, and ARITHMETIC the _Calculus of
+Values_.
+
+We can now perceive how insufficient and even erroneous are the ordinary
+definitions. Most generally, the exaggerated importance attributed to
+Signs has led to the distinguishing the two fundamental branches of the
+science of the Calculus by the manner of designating in each the
+subjects of discussion, an idea which is evidently absurd in principle
+and false in fact. Even the celebrated definition given by Newton,
+characterizing _Algebra_ as _Universal Arithmetic_, gives certainly a
+very false idea of the nature of algebra and of that of arithmetic.[5]
+
+  [Footnote 5: I have thought that I ought to specially notice this
+  definition, because it serves as the basis of the opinion which
+  many intelligent persons, unacquainted with mathematical science,
+  form of its abstract part, without considering that at the time of
+  this definition mathematical analysis was not sufficiently
+  developed to enable the general character of each of its principal
+  parts to be properly apprehended, which explains why Newton could
+  at that time propose a definition which at the present day he would
+  certainly reject.]
+
+Having thus established the fundamental division of the calculus into
+two principal branches, I have now to compare in general terms the
+extent, the importance, and the difficulty of these two sorts of
+calculus, so as to have hereafter to consider only the _Calculus of
+Functions_, which is to be the principal subject of our study.
+
+
+
+
+THE CALCULUS OF VALUES, OR ARITHMETIC.
+
+
+_Its Extent._ The _Calculus of Values, or Arithmetic_, would appear, at
+first view, to present a field as vast as that of _algebra_, since it
+would seem to admit as many distinct questions as we can conceive
+different algebraic formulas whose values are to be determined. But a
+very simple reflection will show the difference. Dividing functions into
+_simple_ and _compound_, it is evident that when we know how to
+determine the _value_ of simple functions, the consideration of compound
+functions will no longer present any difficulty. In the algebraic point
+of view, a compound function plays a very different part from that of
+the elementary functions of which it consists, and from this, indeed,
+proceed all the principal difficulties of analysis. But it is very
+different with the Arithmetical Calculus. Thus the number of truly
+distinct arithmetical operations is only that determined by the number
+of the elementary abstract functions, the very limited list of which has
+been given above. The determination of the values of these ten functions
+necessarily gives that of all the functions, infinite in number, which
+are considered in the whole of mathematical analysis, such at least as
+it exists at present. There can be no new arithmetical operations
+without the creation of really new analytical elements, the number of
+which must always be extremely small. The field of _arithmetic_ is,
+then, by its nature, exceedingly restricted, while that of algebra is
+rigorously indefinite.
+
+It is, however, important to remark, that the domain of the _calculus of
+values_ is, in reality, much more extensive than it is commonly
+represented; for several questions truly _arithmetical_, since they
+consist of determinations of values, are not ordinarily classed as such,
+because we are accustomed to treat them only as incidental in the midst
+of a body of analytical researches more or less elevated, the too high
+opinion commonly formed of the influence of signs being again the
+principal cause of this confusion of ideas. Thus not only the
+construction of a table of logarithms, but also the calculation of
+trigonometrical tables, are true arithmetical operations of a higher
+kind. We may also cite as being in the same class, although in a very
+distinct and more elevated order, all the methods by which we determine
+directly the value of any function for each particular system of values
+attributed to the quantities on which it depends, when we cannot express
+in general terms the explicit form of that function. In this point of
+view the _numerical_ solution of questions which we cannot resolve
+algebraically, and even the calculation of "Definite Integrals," whose
+general integrals we do not know, really make a part, in spite of all
+appearances, of the domain of _arithmetic_, in which we must necessarily
+comprise all that which has for its object the _determination of the
+values of functions_. The considerations relative to this object are, in
+fact, constantly homogeneous, whatever the _determinations_ in question,
+and are always very distinct from truly _algebraic_ considerations.
+
+To complete a just idea of the real extent of the calculus of values, we
+must include in it likewise that part of the general science of the
+calculus which now bears the name of the _Theory of Numbers_, and which
+is yet so little advanced. This branch, very extensive by its nature,
+but whose importance in the general system of science is not very
+great, has for its object the discovery of the properties inherent in
+different numbers by virtue of their values, and independent of any
+particular system of numeration. It forms, then, a sort of
+_transcendental arithmetic_; and to it would really apply the definition
+proposed by Newton for algebra.
+
+The entire domain of arithmetic is, then, much more extended than is
+commonly supposed; but this _calculus of values_ will still never be
+more than a point, so to speak, in comparison with the _calculus of
+functions_, of which mathematical science essentially consists. This
+comparative estimate will be still more apparent from some
+considerations which I have now to indicate respecting the true nature
+of arithmetical questions in general, when they are more profoundly
+examined.
+
+
+_Its true Nature._ In seeking to determine with precision in what
+_determinations of values_ properly consist, we easily recognize that
+they are nothing else but veritable _transformations_ of the functions
+to be valued; transformations which, in spite of their special end, are
+none the less essentially of the same nature as all those taught by
+analysis. In this point of view, the _calculus of values_ might be
+simply conceived as an appendix, and a particular application of the
+_calculus of functions_, so that _arithmetic_ would disappear, so to
+say, as a distinct section in the whole body of abstract mathematics.
+
+In order thoroughly to comprehend this consideration, we must observe
+that, when we propose to determine the _value_ of an unknown number
+whose mode of formation is given, it is, by the mere enunciation of the
+arithmetical question, already defined and expressed under a certain
+form; and that in _determining its value_ we only put its expression
+under another determinate form, to which we are accustomed to refer the
+exact notion of each particular number by making it re-enter into the
+regular system of _numeration_. The determination of values consists so
+completely of a simple _transformation_, that when the primitive
+expression of the number is found to be already conformed to the regular
+system of numeration, there is no longer any determination of value,
+properly speaking, or, rather, the question is answered by the question
+itself. Let the question be to add the two numbers _one_ and _twenty_,
+we answer it by merely repeating the enunciation of the question,[6] and
+nevertheless we think that we have _determined the value_ of the sum.
+This signifies that in this case the first expression of the function
+had no need of being transformed, while it would not be thus in adding
+twenty-three and fourteen, for then the sum would not be immediately
+expressed in a manner conformed to the rank which it occupies in the
+fixed and general scale of numeration.
+
+
+  [Footnote 6: This is less strictly true in the English system of
+  numeration than in the French, since "twenty-one" is our more usual
+  mode of expressing this number.]
+
+To sum up as comprehensively as possible the preceding views, we may
+say, that to determine the _value_ of a number is nothing else than
+putting its primitive expression under the form
+
+  _a_ + _bz_ + _cz²_ + _dz³_ + _ez⁴_ . . . . . + _pz^m_,
+
+_z_ being generally equal to 10, and the coefficients _a_, _b_, _c_,
+_d_, &c., being subjected to the conditions of being whole numbers less
+than _z_; capable of becoming equal to zero; but never negative. Every
+arithmetical question may thus be stated as consisting in putting under
+such a form any abstract function whatever of different quantities,
+which are supposed to have themselves a similar form already. We might
+then see in the different operations of arithmetic only simple
+particular cases of certain algebraic transformations, excepting the
+special difficulties belonging to conditions relating to the nature of
+the coefficients.
+
+It clearly follows that abstract mathematics is essentially composed of
+the _Calculus of Functions_, which had been already seen to be its most
+important, most extended, and most difficult part. It will henceforth be
+the exclusive subject of our analytical investigations. I will therefore
+no longer delay on the _Calculus of Values_, but pass immediately to the
+examination of the fundamental division of the _Calculus of Functions_.
+
+
+
+
+THE CALCULUS OF FUNCTIONS, OR ALGEBRA.
+
+
+_Principle of its Fundamental Division._ We have determined, at the
+beginning of this chapter, wherein properly consists the difficulty
+which we experience in putting mathematical questions into _equations_.
+It is essentially because of the insufficiency of the very small number
+of analytical elements which we possess, that the relation of the
+concrete to the abstract is usually so difficult to establish. Let us
+endeavour now to appreciate in a philosophical manner the general
+process by which the human mind has succeeded, in so great a number of
+important cases, in overcoming this fundamental obstacle to _The
+establishment of Equations_.
+
+
+1. _By the Creation of new Functions._ In looking at this important
+question from the most general point of view, we are led at once to the
+conception of one means of facilitating the establishment of the
+equations of phenomena. Since the principal obstacle in this matter
+comes from the too small number of our analytical elements, the whole
+question would seem to be reduced to creating new ones. But this means,
+though natural, is really illusory; and though it might be useful, it is
+certainly insufficient.
+
+In fact, the creation of an elementary abstract function, which shall be
+veritably new, presents in itself the greatest difficulties. There is
+even something contradictory in such an idea; for a new analytical
+element would evidently not fulfil its essential and appropriate
+conditions, if we could not immediately _determine its value_. Now, on
+the other hand, how are we to _determine the value_ of a new function
+which is truly _simple_, that is, which is not formed by a combination
+of those already known? That appears almost impossible. The introduction
+into analysis of another elementary abstract function, or rather of
+another couple of functions (for each would be always accompanied by its
+_inverse_), supposes then, of necessity, the simultaneous creation of a
+new arithmetical operation, which is certainly very difficult.
+
+If we endeavour to obtain an idea of the means which the human mind
+employs for inventing new analytical elements, by the examination of the
+procedures by the aid of which it has actually conceived those which we
+already possess, our observations leave us in that respect in an entire
+uncertainty, for the artifices which it has already made use of for that
+purpose are evidently exhausted. To convince ourselves of it, let us
+consider the last couple of simple functions which has been introduced
+into analysis, and at the formation of which we have been present, so
+to speak, namely, the fourth couple; for, as I have explained, the fifth
+couple does not strictly give veritable new analytical elements. The
+function _a^x_, and, consequently, its inverse, have been formed by
+conceiving, under a new point of view, a function which had been a long
+time known, namely, powers--when the idea of them had become
+sufficiently generalized. The consideration of a power relatively to the
+variation of its exponent, instead of to the variation of its base, was
+sufficient to give rise to a truly novel simple function, the variation
+following then an entirely different route. But this artifice, as simple
+as ingenious, can furnish nothing more; for, in turning over in the same
+manner all our present analytical elements, we end in only making them
+return into one another.
+
+We have, then, no idea as to how we could proceed to the creation of new
+elementary abstract functions which would properly satisfy all the
+necessary conditions. This is not to say, however, that we have at
+present attained the effectual limit established in that respect by the
+bounds of our intelligence. It is even certain that the last special
+improvements in mathematical analysis have contributed to extend our
+resources in that respect, by introducing within the domain of the
+calculus certain definite integrals, which in some respects supply the
+place of new simple functions, although they are far from fulfilling all
+the necessary conditions, which has prevented me from inserting them in
+the table of true analytical elements. But, on the whole, I think it
+unquestionable that the number of these elements cannot increase except
+with extreme slowness. It is therefore not from these sources that the
+human mind has drawn its most powerful means of facilitating, as much
+as is possible, the establishment of equations.
+
+
+2. _By the Conception of Equations between certain auxiliary
+Quantities._ This first method being set aside, there remains evidently
+but one other: it is, seeing the impossibility of finding directly the
+equations between the quantities under consideration, to seek for
+corresponding ones between other auxiliary quantities, connected with
+the first according to a certain determinate law, and from the relation
+between which we may return to that between the primitive magnitudes.
+Such is, in substance, the eminently fruitful conception, which the
+human mind has succeeded in establishing, and which constitutes its most
+admirable instrument for the mathematical explanation of natural
+phenomena; the _analysis_, called _transcendental_.
+
+As a general philosophical principle, the auxiliary quantities, which
+are introduced in the place of the primitive magnitudes, or concurrently
+with them, in order to facilitate the establishment of equations, might
+be derived according to any law whatever from the immediate elements of
+the question. This conception has thus a much more extensive reach than
+has been commonly attributed to it by even the most profound geometers.
+It is extremely important for us to view it in its whole logical extent,
+for it will perhaps be by establishing a general mode of _derivation_
+different from that to which we have thus far confined ourselves
+(although it is evidently very far from being the only possible one)
+that we shall one day succeed in essentially perfecting mathematical
+analysis as a whole, and consequently in establishing more powerful
+means of investigating the laws of nature than our present processes,
+which are unquestionably susceptible of becoming exhausted.
+
+But, regarding merely the present constitution of the science, the only
+auxiliary quantities habitually introduced in the place of the primitive
+quantities in the _Transcendental Analysis_ are what are called, 1⁰,
+_infinitely small_ elements, the _differentials_ (of different orders)
+of those quantities, if we regard this analysis in the manner of
+LEIBNITZ; or, 2⁰, the _fluxions_, the limits of the ratios of the
+simultaneous increments of the primitive quantities compared with one
+another, or, more briefly, the _prime and ultimate ratios_ of these
+increments, if we adopt the conception of NEWTON; or, 3⁰, the
+_derivatives_, properly so called, of those quantities, that is, the
+coefficients of the different terms of their respective increments,
+according to the conception of LAGRANGE.
+
+These three principal methods of viewing our present transcendental
+analysis, and all the other less distinctly characterized ones which
+have been successively proposed, are, by their nature, necessarily
+identical, whether in the calculation or in the application, as will be
+explained in a general manner in the third chapter. As to their relative
+value, we shall there see that the conception of Leibnitz has thus far,
+in practice, an incontestable superiority, but that its logical
+character is exceedingly vicious; while that the conception of Lagrange,
+admirable by its simplicity, by its logical perfection, by the
+philosophical unity which it has established in mathematical analysis
+(till then separated into two almost entirely independent worlds),
+presents, as yet, serious inconveniences in the applications, by
+retarding the progress of the mind. The conception of Newton occupies
+nearly middle ground in these various relations, being less rapid, but
+more rational than that of Leibnitz; less philosophical, but more
+applicable than that of Lagrange.
+
+This is not the place to explain the advantages of the introduction of
+this kind of auxiliary quantities in the place of the primitive
+magnitudes. The third chapter is devoted to this subject. At present I
+limit myself to consider this conception in the most general manner, in
+order to deduce therefrom the fundamental division of the _calculus of
+functions_ into two systems essentially distinct, whose dependence, for
+the complete solution of any one mathematical question, is invariably
+determinate.
+
+In this connexion, and in the logical order of ideas, the transcendental
+analysis presents itself as being necessarily the first, since its
+general object is to facilitate the establishment of equations, an
+operation which must evidently precede the _resolution_ of those
+equations, which is the object of the ordinary analysis. But though it
+is exceedingly important to conceive in this way the true relations of
+these two systems of analysis, it is none the less proper, in conformity
+with the regular usage, to study the transcendental analysis after
+ordinary analysis; for though the former is, at bottom, by itself
+logically independent of the latter, or, at least, may be essentially
+disengaged from it, yet it is clear that, since its employment in the
+solution of questions has always more or less need of being completed by
+the use of the ordinary analysis, we would be constrained to leave the
+questions in suspense if this latter had not been previously studied.
+
+
+_Corresponding Divisions of the Calculus of Functions._ It follows from
+the preceding considerations that the _Calculus of Functions_, or
+_Algebra_ (taking this word in its most extended meaning), is composed
+of two distinct fundamental branches, one of which has for its immediate
+object the _resolution_ of equations, when they are directly established
+between the magnitudes themselves which are under consideration; and the
+other, starting from equations (generally much easier to form) between
+quantities indirectly connected with those of the problem, has for its
+peculiar and constant destination the deduction, by invariable
+analytical methods, of the corresponding equations between the direct
+magnitudes which we are considering; which brings the question within
+the domain of the preceding calculus.
+
+The former calculus bears most frequently the name of _Ordinary
+Analysis_, or of _Algebra_, properly so called. The second constitutes
+what is called the _Transcendental Analysis_, which has been designated
+by the different denominations of _Infinitesimal Calculus_, _Calculus of
+Fluxions and of Fluents_, _Calculus of Vanishing Quantities_, the
+_Differential and Integral Calculus_, &c., according to the point of
+view in which it has been conceived.
+
+In order to remove every foreign consideration, I will propose to name
+it CALCULUS OF INDIRECT FUNCTIONS, giving to ordinary analysis the title
+of CALCULUS OF DIRECT FUNCTIONS. These expressions, which I form
+essentially by generalizing and epitomizing the ideas of Lagrange, are
+simply intended to indicate with precision the true general character
+belonging to each of these two forms of analysis.
+
+Having now established the fundamental division of mathematical
+analysis, I have next to consider separately each of its two parts,
+commencing with the _Calculus of Direct Functions_, and reserving more
+extended developments for the different branches of the _Calculus of
+Indirect Functions_.
+
+
+
+
+CHAPTER II.
+
+ORDINARY ANALYSIS, OR ALGEBRA.
+
+
+The _Calculus of direct Functions_, or _Algebra_, is (as was shown at
+the end of the preceding chapter) entirely sufficient for the solution
+of mathematical questions, when they are so simple that we can form
+directly the equations between the magnitudes themselves which we are
+considering, without its being necessary to introduce in their place, or
+conjointly with them, any system of auxiliary quantities _derived_ from
+the first. It is true that in the greatest number of important cases its
+use requires to be preceded and prepared by that of the _Calculus of
+indirect Functions_, which is intended to facilitate the establishment
+of equations. But, although algebra has then only a secondary office to
+perform, it has none the less a necessary part in the complete solution
+of the question, so that the _Calculus of direct Functions_ must
+continue to be, by its nature, the fundamental base of all mathematical
+analysis. We must therefore, before going any further, consider in a
+general manner the logical composition of this calculus, and the degree
+of development to which it has at the present day arrived.
+
+
+_Its Object._ The final object of this calculus being the _resolution_
+(properly so called) of _equations_, that is, the discovery of the
+manner in which the unknown quantities are formed from the known
+quantities, in accordance with the _equations_ which exist between them,
+it naturally presents as many different departments as we can conceive
+truly distinct classes of equations. Its appropriate extent is
+consequently rigorously indefinite, the number of analytical functions
+susceptible of entering into equations being in itself quite unlimited,
+although they are composed of only a very small number of primitive
+elements.
+
+
+_Classification of Equations._ The rational classification of equations
+must evidently be determined by the nature of the analytical elements of
+which their numbers are composed; every other classification would be
+essentially arbitrary. Accordingly, analysts begin by dividing equations
+with one or more variables into two principal classes, according as they
+contain functions of only the first three couples (see the table in
+chapter i., page 51), or as they include also exponential or circular
+functions. The names of _Algebraic_ functions and _Transcendental_
+functions, commonly given to these two principal groups of analytical
+elements, are undoubtedly very inappropriate. But the universally
+established division between the corresponding equations is none the
+less very real in this sense, that the resolution of equations
+containing the functions called _transcendental_ necessarily presents
+more difficulties than those of the equations called _algebraic_. Hence
+the study of the former is as yet exceedingly imperfect, so that
+frequently the resolution of the most simple of them is still unknown to
+us,[7] and our analytical methods have almost exclusive reference to the
+elaboration of the latter.
+
+  [Footnote 7: Simple as may seem, for example, the equation
+
+  _a^x_ + _b^x_ = _c^x_,
+
+  we do not yet know how to resolve it, which may give some idea of
+  the extreme imperfection of this part of algebra.]
+
+
+
+
+ALGEBRAIC EQUATIONS.
+
+
+Considering now only these _Algebraic_ equations, we must observe, in
+the first place, that although they may often contain _irrational_
+functions of the unknown quantities as well as _rational_ functions, we
+can always, by more or less easy transformations, make the first case
+come under the second, so that it is with this last that analysts have
+had to occupy themselves exclusively in order to resolve all sorts of
+_algebraic_ equations.
+
+
+_Their Classification._ In the infancy of algebra, these equations were
+classed according to the number of their terms. But this classification
+was evidently faulty, since it separated cases which were really
+similar, and brought together others which had nothing in common besides
+this unimportant characteristic.[8] It has been retained only for
+equations with two terms, which are, in fact, capable of being resolved
+in a manner peculiar to themselves.
+
+  [Footnote 8: The same error was afterward committed, in the infancy
+  of the infinitesimal calculus, in relation to the integration of
+  differential equations.]
+
+The classification of equations by what is called their _degrees_, is,
+on the other hand, eminently natural, for this distinction rigorously
+determines the greater or less difficulty of their _resolution_. This
+gradation is apparent in the cases of all the equations which can be
+resolved; but it may be indicated in a general manner independently of
+the fact of the resolution. We need only consider that the most general
+equation of each degree necessarily comprehends all those of the
+different inferior degrees, as must also the formula which determines
+the unknown quantity. Consequently, however slight we may suppose the
+difficulty peculiar to the _degree_ which we are considering, since it
+is inevitably complicated in the execution with those presented by all
+the preceding degrees, the resolution really offers more and more
+obstacles, in proportion as the degree of the equation is elevated.
+
+
+
+
+ALGEBRAIC RESOLUTION OF EQUATIONS.
+
+
+_Its Limits._ The resolution of algebraic equations is as yet known to
+us only in the four first degrees, such is the increase of difficulty
+noticed above. In this respect, algebra has made no considerable
+progress since the labours of Descartes and the Italian analysts of the
+sixteenth century, although in the last two centuries there has been
+perhaps scarcely a single geometer who has not busied himself in trying
+to advance the resolution of equations. The general equation of the
+fifth degree itself has thus far resisted all attacks.
+
+The constantly increasing complication which the formulas for resolving
+equations must necessarily present, in proportion as the degree
+increases (the difficulty of using the formula of the fourth degree
+rendering it almost inapplicable), has determined analysts to renounce,
+by a tacit agreement, the pursuit of such researches, although they are
+far from regarding it as impossible to obtain the resolution of
+equations of the fifth degree, and of several other higher ones.
+
+
+_General Solution._ The only question of this kind which would be really
+of great importance, at least in its logical relations, would be the
+general resolution of algebraic equations of any degree whatsoever. Now,
+the more we meditate on this subject, the more we are led to think, with
+Lagrange, that it really surpasses the scope of our intelligence. We
+must besides observe that the formula which would express the _root_ of
+an equation of the _m^{th}_ degree would necessarily include radicals of
+the _m^{th}_ order (or functions of an equivalent multiplicity), because
+of the _m_ determinations which it must admit. Since we have seen,
+besides, that this formula must also embrace, as a particular case, that
+formula which corresponds to every lower degree, it follows that it
+would inevitably also contain radicals of the next lower degree, the
+next lower to that, &c., so that, even if it were possible to discover
+it, it would almost always present too great a complication to be
+capable of being usefully employed, unless we could succeed in
+simplifying it, at the same time retaining all its generality, by the
+introduction of a new class of analytical elements of which we yet have
+no idea. We have, then, reason to believe that, without having already
+here arrived at the limits imposed by the feeble extent of our
+intelligence, we should not be long in reaching them if we actively and
+earnestly prolonged this series of investigations.
+
+It is, besides, important to observe that, even supposing we had
+obtained the resolution of _algebraic_ equations of any degree whatever,
+we would still have treated only a very small part of _algebra_,
+properly so called, that is, of the calculus of direct functions,
+including the resolution of all the equations which can be formed by the
+known analytical functions.
+
+Finally, we must remember that, by an undeniable law of human nature,
+our means for conceiving new questions being much more powerful than our
+resources for resolving them, or, in other words, the human mind being
+much more ready to inquire than to reason, we shall necessarily always
+remain _below_ the difficulty, no matter to what degree of development
+our intellectual labour may arrive. Thus, even though we should some day
+discover the complete resolution of all the analytical equations at
+present known, chimerical as the supposition is, there can be no doubt
+that, before attaining this end, and probably even as a subsidiary
+means, we would have already overcome the difficulty (a much smaller
+one, though still very great) of conceiving new analytical elements, the
+introduction of which would give rise to classes of equations of which,
+at present, we are completely ignorant; so that a similar imperfection
+in algebraic science would be continually reproduced, in spite of the
+real and very important increase of the absolute mass of our knowledge.
+
+
+_What we know in Algebra._ In the present condition of algebra, the
+complete resolution of the equations of the first four degrees, of any
+binomial equations, of certain particular equations of the higher
+degrees, and of a very small number of exponential, logarithmic, or
+circular equations, constitute the fundamental methods which are
+presented by the calculus of direct functions for the solution of
+mathematical problems. But, limited as these elements are, geometers
+have nevertheless succeeded in treating, in a truly admirable manner, a
+very great number of important questions, as we shall find in the course
+of the volume. The general improvements introduced within a century into
+the total system of mathematical analysis, have had for their principal
+object to make immeasurably useful this little knowledge which we have,
+instead of tending to increase it. This result has been so fully
+obtained, that most frequently this calculus has no real share in the
+complete solution of the question, except by its most simple parts;
+those which have reference to equations of the two first degrees, with
+one or more variables.
+
+
+
+
+NUMERICAL RESOLUTION OF EQUATIONS.
+
+
+The extreme imperfection of algebra, with respect to the resolution of
+equations, has led analysts to occupy themselves with a new class of
+questions, whose true character should be here noted. They have busied
+themselves in filling up the immense gap in the resolution of algebraic
+equations of the higher degrees, by what they have named the _numerical
+resolution_ of equations. Not being able to obtain, in general, the
+_formula_ which expresses what explicit function of the given quantities
+the unknown one is, they have sought (in the absence of this kind of
+resolution, the only one really _algebraic_) to determine, independently
+of that formula, at least the _value_ of each unknown quantity, for
+various designated systems of particular values attributed to the given
+quantities. By the successive labours of analysts, this incomplete and
+illegitimate operation, which presents an intimate mixture of truly
+algebraic questions with others which are purely arithmetical, has been
+rendered possible in all cases for equations of any degree and even of
+any form. The methods for this which we now possess are sufficiently
+general, although the calculations to which they lead are often so
+complicated as to render it almost impossible to execute them. We have
+nothing else to do, then, in this part of algebra, but to simplify the
+methods sufficiently to render them regularly applicable, which we may
+hope hereafter to effect. In this condition of the calculus of direct
+functions, we endeavour, in its application, so to dispose the proposed
+questions as finally to require only this numerical resolution of the
+equations.
+
+
+_Its limited Usefulness._ Valuable as is such a resource in the absence
+of the veritable solution, it is essential not to misconceive the true
+character of these methods, which analysts rightly regard as a very
+imperfect algebra. In fact, we are far from being always able to reduce
+our mathematical questions to depend finally upon only the _numerical_
+resolution of equations; that can be done only for questions quite
+isolated or truly final, that is, for the smallest number. Most
+questions, in fact, are only preparatory, and intended to serve as an
+indispensable preparation for the solution of other questions. Now, for
+such an object, it is evident that it is not the actual _value_ of the
+unknown quantity which it is important to discover, but the _formula_,
+which shows how it is derived from the other quantities under
+consideration. It is this which happens, for example, in a very
+extensive class of cases, whenever a certain question includes at the
+same time several unknown quantities. We have then, first of all, to
+separate them. By suitably employing the simple and general method so
+happily invented by analysts, and which consists in referring all the
+other unknown quantities to one of them, the difficulty would always
+disappear if we knew how to obtain the algebraic resolution of the
+equations under consideration, while the _numerical_ solution would then
+be perfectly useless. It is only for want of knowing the _algebraic_
+resolution of equations with a single unknown quantity, that we are
+obliged to treat _Elimination_ as a distinct question, which forms one
+of the greatest special difficulties of common algebra. Laborious as are
+the methods by the aid of which we overcome this difficulty, they are
+not even applicable, in an entirely general manner, to the elimination
+of one unknown quantity between two equations of any form whatever.
+
+In the most simple questions, and when we have really to resolve only a
+single equation with a single unknown quantity, this _numerical_
+resolution is none the less a very imperfect method, even when it is
+strictly sufficient. It presents, in fact, this serious inconvenience of
+obliging us to repeat the whole series of operations for the slightest
+change which may take place in a single one of the quantities
+considered, although their relations to one another remain unchanged;
+the calculations made for one case not enabling us to dispense with any
+of those which relate to a case very slightly different. This happens
+because of our inability to abstract and treat separately that purely
+algebraic part of the question which is common to all the cases which
+result from the mere variation of the given numbers.
+
+According to the preceding considerations, the calculus of direct
+functions, viewed in its present state, divides into two very distinct
+branches, according as its subject is the _algebraic_ resolution of
+equations or their _numerical_ resolution. The first department, the
+only one truly satisfactory, is unhappily very limited, and will
+probably always remain so; the second, too often insufficient, has, at
+least, the advantage of a much greater generality. The necessity of
+clearly distinguishing these two parts is evident, because of the
+essentially different object proposed in each, and consequently the
+peculiar point of view under which quantities are therein considered.
+
+
+_Different Divisions of the two Methods of Resolution._ If, moreover, we
+consider these parts with reference to the different methods of which
+each is composed, we find in their logical distribution an entirely
+different arrangement. In fact, the first part must be divided according
+to the nature of the equations which we are able to resolve, and
+independently of every consideration relative to the _values_ of the
+unknown quantities. In the second part, on the contrary, it is not
+according to the _degrees_ of the equations that the methods are
+naturally distinguished, since they are applicable to equations of any
+degree whatever; it is according to the numerical character of the
+_values_ of the unknown quantities; for, in calculating these numbers
+directly, without deducing them from general formulas, different means
+would evidently be employed when the numbers are not susceptible of
+having their values determined otherwise than by a series of
+approximations, always incomplete, or when they can be obtained with
+entire exactness. This distinction of _incommensurable_ and of
+_commensurable_ roots, which require quite different principles for
+their determination, important as it is in the numerical resolution of
+equations, is entirely insignificant in the algebraic resolution, in
+which the _rational_ or _irrational_ nature of the numbers which are
+obtained is a mere accident of the calculation, which cannot exercise
+any influence over the methods employed; it is, in a word, a simple
+arithmetical consideration. We may say as much, though in a less degree,
+of the division of the commensurable roots themselves into _entire_ and
+_fractional_. In fine, the case is the same, in a still greater degree,
+with the most general classification of roots, as _real_ and
+_imaginary_. All these different considerations, which are preponderant
+as to the numerical resolution of equations, and which are of no
+importance in their algebraic resolution, render more and more sensible
+the essentially distinct nature of these two principal parts of algebra.
+
+
+
+
+THE THEORY OF EQUATIONS.
+
+
+These two departments, which constitute the immediate object of the
+calculus of direct functions, are subordinate to a third one, purely
+speculative, from which both of them borrow their most powerful
+resources, and which has been very exactly designated by the general
+name of _Theory of Equations_, although it as yet relates only to
+_Algebraic_ equations. The numerical resolution of equations, because of
+its generality, has special need of this rational foundation.
+
+This last and important branch of algebra is naturally divided into two
+orders of questions, viz., those which refer to the _composition_ of
+equations, and those which concern their _transformation_; these latter
+having for their object to modify the roots of an equation without
+knowing them, in accordance with any given law, providing that this law
+is uniform in relation to all the parts.[9]
+
+  [Footnote 9: The fundamental principle on which reposes the theory
+  of equations, and which is so frequently applied in all
+  mathematical analysis--the decomposition of algebraic, rational,
+  and entire functions, of any degree whatever, into factors of the
+  first degree--is never employed except for functions of a single
+  variable, without any one having examined if it ought to be
+  extended to functions of several variables. The general
+  impossibility of such a decomposition is demonstrated by the author
+  in detail, but more properly belongs to a special treatise.]
+
+
+
+
+THE METHOD OF INDETERMINATE COEFFICIENTS.
+
+
+To complete this rapid general enumeration of the different essential
+parts of the calculus of direct functions, I must, lastly, mention
+expressly one of the most fruitful and important theories of algebra
+proper, that relating to the transformation of functions into series by
+the aid of what is called the _Method of indeterminate Coefficients_.
+This method, so eminently analytical, and which must be regarded as one
+of the most remarkable discoveries of Descartes, has undoubtedly lost
+some of its importance since the invention and the development of the
+infinitesimal calculus, the place of which it might so happily take in
+some particular respects. But the increasing extension of the
+transcendental analysis, although it has rendered this method much less
+necessary, has, on the other hand, multiplied its applications and
+enlarged its resources; so that by the useful combination between the
+two theories, which has finally been effected, the use of the method of
+indeterminate coefficients has become at present much more extensive
+than it was even before the formation of the calculus of indirect
+functions.
+
+       *       *       *       *       *
+
+Having thus sketched the general outlines of algebra proper, I have now
+to offer some considerations on several leading points in the calculus
+of direct functions, our ideas of which may be advantageously made more
+clear by a philosophical examination.
+
+
+
+
+IMAGINARY QUANTITIES.
+
+
+The difficulties connected with several peculiar symbols to which
+algebraic calculations sometimes lead, and especially to the expressions
+called _imaginary_, have been, I think, much exaggerated through purely
+metaphysical considerations, which have been forced upon them, in the
+place of regarding these abnormal results in their true point of view as
+simple analytical facts. Viewing them thus, we readily see that, since
+the spirit of mathematical analysis consists in considering magnitudes
+in reference to their relations only, and without any regard to their
+determinate value, analysts are obliged to admit indifferently every
+kind of expression which can be engendered by algebraic combinations.
+The interdiction of even one expression because of its apparent
+singularity would destroy the generality of their conceptions. The
+common embarrassment on this subject seems to me to proceed essentially
+from an unconscious confusion between the idea of _function_ and the
+idea of _value_, or, what comes to the same thing, between the
+_algebraic_ and the _arithmetical_ point of view. A thorough examination
+would show mathematical analysis to be much more clear in its nature
+than even mathematicians commonly suppose.
+
+
+
+
+NEGATIVE QUANTITIES.
+
+
+As to negative quantities, which have given rise to so many misplaced
+discussions, as irrational as useless, we must distinguish between their
+_abstract_ signification and their _concrete_ interpretation, which have
+been almost always confounded up to the present day. Under the first
+point of view, the theory of negative quantities can be established in a
+complete manner by a single algebraical consideration. The necessity of
+admitting such expressions is the same as for imaginary quantities, as
+above indicated; and their employment as an analytical artifice, to
+render the formulas more comprehensive, is a mechanism of calculation
+which cannot really give rise to any serious difficulty. We may
+therefore regard the abstract theory of negative quantities as leaving
+nothing essential to desire; it presents no obstacles but those
+inappropriately introduced by sophistical considerations.
+
+It is far from being so, however, with their concrete theory. This
+consists essentially in that admirable property of the signs + and-, of
+representing analytically the oppositions of directions of which certain
+magnitudes are susceptible. This _general theorem_ on the relation of
+the concrete to the abstract in mathematics is one of the most beautiful
+discoveries which we owe to the genius of Descartes, who obtained it as
+a simple result of properly directed philosophical observation. A great
+number of geometers have since striven to establish directly its general
+demonstration, but thus far their efforts have been illusory. Their vain
+metaphysical considerations and heterogeneous minglings of the abstract
+and the concrete have so confused the subject, that it becomes necessary
+to here distinctly enunciate the general fact. It consists in this: if,
+in any equation whatever, expressing the relation of certain quantities
+which are susceptible of opposition of directions, one or more of those
+quantities come to be reckoned in a direction contrary to that which
+belonged to them when the equation was first established, it will not be
+necessary to form directly a new equation for this second state of the
+phenomena; it will suffice to change, in the first equation, the sign of
+each of the quantities which shall have changed its direction; and the
+equation, thus modified, will always rigorously coincide with that which
+we would have arrived at in recommencing to investigate, for this new
+case, the analytical law of the phenomenon. The general theorem consists
+in this constant and necessary coincidence. Now, as yet, no one has
+succeeded in directly proving this; we have assured ourselves of it only
+by a great number of geometrical and mechanical verifications, which
+are, it is true, sufficiently multiplied, and especially sufficiently
+varied, to prevent any clear mind from having the least doubt of the
+exactitude and the generality of this essential property, but which, in
+a philosophical point of view, do not at all dispense with the research
+for so important an explanation. The extreme extent of the theorem must
+make us comprehend both the fundamental difficulties of this research
+and the high utility for the perfecting of mathematical science which
+would belong to the general conception of this great truth. This
+imperfection of theory, however, has not prevented geometers from making
+the most extensive and the most important use of this property in all
+parts of concrete mathematics.
+
+It follows from the above general enunciation of the fact, independently
+of any demonstration, that the property of which we speak must never be
+applied to magnitudes whose directions are continually varying, without
+giving rise to a simple opposition of direction; in that case, the sign
+with which every result of calculation is necessarily affected is not
+susceptible of any concrete interpretation, and the attempts sometimes
+made to establish one are erroneous. This circumstance occurs, among
+other occasions, in the case of a radius vector in geometry, and
+diverging forces in mechanics.
+
+
+
+
+PRINCIPLE OF HOMOGENEITY.
+
+
+A second general theorem on the relation of the concrete to the abstract
+is that which is ordinarily designated under the name of _Principle of
+Homogeneity_. It is undoubtedly much less important in its applications
+than the preceding, but it particularly merits our attention as having,
+by its nature, a still greater extent, since it is applicable to all
+phenomena without distinction, and because of the real utility which it
+often possesses for the verification of their analytical laws. I can,
+moreover, exhibit a direct and general demonstration of it which seems
+to me very simple. It is founded on this single observation, which is
+self-evident, that the exactitude of every relation between any concrete
+magnitudes whatsoever is independent of the value of the _units_ to
+which they are referred for the purpose of expressing them in numbers.
+For example, the relation which exists between the three sides of a
+right-angled triangle is the same, whether they are measured by yards,
+or by miles, or by inches.
+
+It follows from this general consideration, that every equation which
+expresses the analytical law of any phenomenon must possess this
+property of being in no way altered, when all the quantities which are
+found in it are made to undergo simultaneously the change corresponding
+to that which their respective units would experience. Now this change
+evidently consists in all the quantities of each sort becoming at once
+_m_ times smaller, if the unit which corresponds to them becomes _m_
+times greater, or reciprocally. Thus every equation which represents any
+concrete relation whatever must possess this characteristic of remaining
+the same, when we make _m_ times greater all the quantities which it
+contains, and which express the magnitudes between which the relation
+exists; excepting always the numbers which designate simply the mutual
+_ratios_ of these different magnitudes, and which therefore remain
+invariable during the change of the units. It is this property which
+constitutes the law of Homogeneity in its most extended signification,
+that is, of whatever analytical functions the equations may be composed.
+
+But most frequently we consider only the cases in which the functions
+are such as are called _algebraic_, and to which the idea of _degree_ is
+applicable. In this case we can give more precision to the general
+proposition by determining the analytical character which must be
+necessarily presented by the equation, in order that this property may
+be verified. It is easy to see, then, that, by the modification just
+explained, all the _terms_ of the first degree, whatever may be their
+form, rational or irrational, entire or fractional, will become _m_
+times greater; all those of the second degree, _m²_ times; those of the
+third, _m³_ times, &c. Thus the terms of the same degree, however
+different may be their composition, varying in the same manner, and the
+terms of different degrees varying in an unequal proportion, whatever
+similarity there may be in their composition, it will be necessary, to
+prevent the equation from being disturbed, that all the terms which it
+contains should be of the same degree. It is in this that properly
+consists the ordinary theorem of _Homogeneity_, and it is from this
+circumstance that the general law has derived its name, which, however,
+ceases to be exactly proper for all other functions.
+
+In order to treat this subject in its whole extent, it is important to
+observe an essential condition, to which attention must be paid in
+applying this property when the phenomenon expressed by the equation
+presents magnitudes of different natures. Thus it may happen that the
+respective units are completely independent of each other, and then the
+theorem of Homogeneity will hold good, either with reference to all the
+corresponding classes of quantities, or with regard to only a single one
+or more of them. But it will happen on other occasions that the
+different units will have fixed relations to one another, determined by
+the nature of the question; then it will be necessary to pay attention
+to this subordination of the units in verifying the homogeneity, which
+will not exist any longer in a purely algebraic sense, and the precise
+form of which will vary according to the nature of the phenomena. Thus,
+for example, to fix our ideas, when, in the analytical expression of
+geometrical phenomena, we are considering at once lines, areas, and
+volumes, it will be necessary to observe that the three corresponding
+units are necessarily so connected with each other that, according to
+the subordination generally established in that respect, when the first
+becomes _m_ times greater, the second becomes _m²_ times, and the third
+_m³_ times. It is with such a modification that homogeneity will exist
+in the equations, in which, if they are _algebraic_, we will have to
+estimate the degree of each term by doubling the exponents of the
+factors which correspond to areas, and tripling those of the factors
+relating to volumes.
+
+       *       *       *       *       *
+
+Such are the principal general considerations relating to the _Calculus
+of Direct Functions_. We have now to pass to the philosophical
+examination of the _Calculus of Indirect Functions_, the much superior
+importance and extent of which claim a fuller development.
+
+
+
+
+CHAPTER III.
+
+TRANSCENDENTAL ANALYSIS:
+
+DIFFERENT MODES OF VIEWING IT.
+
+
+We determined, in the second chapter, the philosophical character of the
+transcendental analysis, in whatever manner it may be conceived,
+considering only the general nature of its actual destination as a part
+of mathematical science. This analysis has been presented by geometers
+under several points of view, really distinct, although necessarily
+equivalent, and leading always to identical results. They may be reduced
+to three principal ones; those of LEIBNITZ, of NEWTON, and of LAGRANGE,
+of which all the others are only secondary modifications. In the present
+state of science, each of these three general conceptions offers
+essential advantages which pertain to it exclusively, without our having
+yet succeeded in constructing a single method uniting all these
+different characteristic qualities. This combination will probably be
+hereafter effected by some method founded upon the conception of
+Lagrange when that important philosophical labour shall have been
+accomplished, the study of the other conceptions will have only a
+historic interest; but, until then, the science must be considered as in
+only a provisional state, which requires the simultaneous consideration
+of all the various modes of viewing this calculus. Illogical as may
+appear this multiplicity of conceptions of one identical subject, still,
+without them all, we could form but a very insufficient idea of this
+analysis, whether in itself, or more especially in relation to its
+applications. This want of system in the most important part of
+mathematical analysis will not appear strange if we consider, on the one
+hand, its great extent and its superior difficulty, and, on the other,
+its recent formation.
+
+
+
+
+ITS EARLY HISTORY.
+
+
+If we had to trace here the systematic history of the successive
+formation of the transcendental analysis, it would be necessary
+previously to distinguish carefully from the _calculus of indirect
+functions_, properly so called, the original idea of the _infinitesimal
+method_, which can be conceived by itself, independently of any
+_calculus_. We should see that the first germ of this idea is found in
+the procedure constantly employed by the Greek geometers, under the name
+of the _Method of Exhaustions_, as a means of passing from the
+properties of straight lines to those of curves, and consisting
+essentially in substituting for the curve the auxiliary consideration of
+an inscribed or circumscribed polygon, by means of which they rose to
+the curve itself, taking in a suitable manner the limits of the
+primitive ratios. Incontestable as is this filiation of ideas, it would
+be giving it a greatly exaggerated importance to see in this method of
+exhaustions the real equivalent of our modern methods, as some geometers
+have done; for the ancients had no logical and general means for the
+determination of these limits, and this was commonly the greatest
+difficulty of the question; so that their solutions were not subjected
+to abstract and invariable rules, the uniform application of which would
+lead with certainty to the knowledge sought; which is, on the contrary,
+the principal characteristic of our transcendental analysis. In a word,
+there still remained the task of generalizing the conceptions used by
+the ancients, and, more especially, by considering it in a manner purely
+abstract, of reducing it to a complete system of calculation, which to
+them was impossible.
+
+The first idea which was produced in this new direction goes back to the
+great geometer Fermat, whom Lagrange has justly presented as having
+blocked out the direct formation of the transcendental analysis by his
+method for the determination of _maxima_ and _minima_, and for the
+finding of _tangents_, which consisted essentially in introducing the
+auxiliary consideration of the correlative increments of the proposed
+variables, increments afterward suppressed as equal to zero when the
+equations had undergone certain suitable transformations. But, although
+Fermat was the first to conceive this analysis in a truly abstract
+manner, it was yet far from being regularly formed into a general and
+distinct calculus having its own notation, and especially freed from the
+superfluous consideration of terms which, in the analysis of Fermat,
+were finally not taken into the account, after having nevertheless
+greatly complicated all the operations by their presence. This is what
+Leibnitz so happily executed, half a century later, after some
+intermediate modifications of the ideas of Fermat introduced by Wallis,
+and still more by Barrow; and he has thus been the true creator of the
+transcendental analysis, such as we now employ it. This admirable
+discovery was so ripe (like all the great conceptions of the human
+intellect at the moment of their manifestation), that Newton, on his
+side, had arrived, at the same time, or a little earlier, at a method
+exactly equivalent, by considering this analysis under a very different
+point of view, which, although more logical in itself, is really less
+adapted to give to the common fundamental method all the extent and the
+facility which have been imparted to it by the ideas of Leibnitz.
+Finally, Lagrange, putting aside the heterogeneous considerations which
+had guided Leibnitz and Newton, has succeeded in reducing the
+transcendental analysis, in its greatest perfection, to a purely
+algebraic system, which only wants more aptitude for its practical
+applications.
+
+After this summary glance at the general history of the transcendental
+analysis, we will proceed to the dogmatic exposition of the three
+principal conceptions, in order to appreciate exactly their
+characteristic properties, and to show the necessary identity of the
+methods which are thence derived. Let us begin with that of Leibnitz.
+
+
+
+
+METHOD OF LEIBNITZ.
+
+
+_Infinitely small Elements._ This consists in introducing into the
+calculus, in order to facilitate the establishment of equations, the
+infinitely small elements of which all the quantities, the relations
+between which are sought, are considered to be composed. These elements
+or _differentials_ will have certain relations to one another, which are
+constantly and necessarily more simple and easy to discover than those
+of the primitive quantities, and by means of which we will be enabled
+(by a special calculus having for its peculiar object the elimination of
+these auxiliary infinitesimals) to go back to the desired equations,
+which it would have been most frequently impossible to obtain directly.
+This indirect analysis may have different degrees of indirectness; for,
+when there is too much difficulty in forming immediately the equation
+between the differentials of the magnitudes under consideration, a
+second application of the same general artifice will have to be made,
+and these differentials be treated, in their turn, as new primitive
+quantities, and a relation be sought between their infinitely small
+elements (which, with reference to the final objects of the question,
+will be _second differentials_), and so on; the same transformation
+admitting of being repeated any number of times, on the condition of
+finally eliminating the constantly increasing number of infinitesimal
+quantities introduced as auxiliaries.
+
+A person not yet familiar with these considerations does not perceive at
+once how the employment of these auxiliary quantities can facilitate the
+discovery of the analytical laws of phenomena; for the infinitely small
+increments of the proposed magnitudes being of the same species with
+them, it would seem that their relations should not be obtained with
+more ease, inasmuch as the greater or less value of a quantity cannot,
+in fact, exercise any influence on an inquiry which is necessarily
+independent, by its nature, of every idea of value. But it is easy,
+nevertheless, to explain very clearly, and in a quite general manner,
+how far the question must be simplified by such an artifice. For this
+purpose, it is necessary to begin by distinguishing _different orders_
+of infinitely small quantities, a very precise idea of which may be
+obtained by considering them as being either the successive powers of
+the same primitive infinitely small quantity, or as being quantities
+which may be regarded as having finite ratios with these powers; so
+that, to take an example, the second, third, &c., differentials of any
+one variable are classed as infinitely small quantities of the second
+order, the third, &c., because it is easy to discover in them finite
+multiples of the second, third, &c., powers of a certain first
+differential. These preliminary ideas being established, the spirit of
+the infinitesimal analysis consists in constantly neglecting the
+infinitely small quantities in comparison with finite quantities, and
+generally the infinitely small quantities of any order whatever in
+comparison with all those of an inferior order. It is at once apparent
+how much such a liberty must facilitate the formation of equations
+between the differentials of quantities, since, in the place of these
+differentials, we can substitute such other elements as we may choose,
+and as will be more simple to consider, only taking care to conform to
+this single condition, that the new elements differ from the preceding
+ones only by quantities infinitely small in comparison with them. It is
+thus that it will be possible, in geometry, to treat curved lines as
+composed of an infinity of rectilinear elements, curved surfaces as
+formed of plane elements, and, in mechanics, variable motions as an
+infinite series of uniform motions, succeeding one another at infinitely
+small intervals of time.
+
+
+EXAMPLES. Considering the importance of this admirable conception, I
+think that I ought here to complete the illustration of its fundamental
+character by the summary indication of some leading examples.
+
+
+1. _Tangents._ Let it be required to determine, for each point of a
+plane curve, the equation of which is given, the direction of its
+tangent; a question whose general solution was the primitive object of
+the inventors of the transcendental analysis. We will consider the
+tangent as a secant joining two points infinitely near to each other;
+and then, designating by _dy_ and _dx_ the infinitely small differences
+of the co-ordinates of those two points, the elementary principles of
+geometry will immediately give the equation _t_ = _dy_/_dx_ for the
+trigonometrical tangent of the angle which is made with the axis of the
+abscissas by the desired tangent, this being the most simple way of
+fixing its position in a system of rectilinear co-ordinates. This
+equation, common to all curves, being established, the question is
+reduced to a simple analytical problem, which will consist in
+eliminating the infinitesimals _dx_ and _dy_, which were introduced as
+auxiliaries, by determining in each particular case, by means of the
+equation of the proposed curve, the ratio of _dy_ to _dx_, which will be
+constantly done by uniform and very simple methods.
+
+
+2. _Rectification of an Arc._ In the second place, suppose that we wish
+to know the length of the arc of any curve, considered as a function of
+the co-ordinates of its extremities. It would be impossible to establish
+directly the equation between this arc s and these co-ordinates, while
+it is easy to find the corresponding relation between the differentials
+of these different magnitudes. The most simple theorems of elementary
+geometry will in fact give at once, considering the infinitely small arc
+_ds_ as a right line, the equations
+
+  _ds²_ = _dy²_ + _dx²_, or _ds²_ = _dx²_ + _dy²_ + _dz²_,
+
+according as the curve is of single or double curvature. In either case,
+the question is now entirely within the domain of analysis, which, by
+the elimination of the differentials (which is the peculiar object of
+the calculus of indirect functions), will carry us back from this
+relation to that which exists between the finite quantities themselves
+under examination.
+
+
+3. _Quadrature of a Curve._ It would be the same with the quadrature of
+curvilinear areas. If the curve is a plane one, and referred to
+rectilinear co-ordinates, we will conceive the area A comprised between
+this curve, the axis of the abscissas, and two extreme co-ordinates, to
+increase by an infinitely small quantity _d_A, as the result of a
+corresponding increment of the abscissa. The relation between these two
+differentials can be immediately obtained with the greatest facility by
+substituting for the curvilinear element of the proposed area the
+rectangle formed by the extreme ordinate and the element of the
+abscissa, from which it evidently differs only by an infinitely small
+quantity of the second order. This will at once give, whatever may be
+the curve, the very simple differential equation
+
+  _d_A = _ydx_,
+
+from which, when the curve is defined, the calculus of indirect
+functions will show how to deduce the finite equation, which is the
+immediate object of the problem.
+
+
+4. _Velocity in Variable Motion._ In like manner, in Dynamics, when we
+desire to know the expression for the velocity acquired at each instant
+by a body impressed with a motion varying according to any law, we will
+consider the motion as being uniform during an infinitely small element
+of the time _t_, and we will thus immediately form the differential
+equation _de_ = _vdt_, in which _v_ designates the velocity acquired
+when the body has passed over the space _e_; and thence it will be easy
+to deduce, by simple and invariable analytical procedures, the formula
+which would give the velocity in each particular motion, in accordance
+with the corresponding relation between the time and the space; or,
+reciprocally, what this relation would be if the mode of variation of
+the velocity was supposed to be known, whether with respect to the space
+or to the time.
+
+
+5. _Distribution of Heat._ Lastly, to indicate another kind of
+questions, it is by similar steps that we are able, in the study of
+thermological phenomena, according to the happy conception of M.
+Fourier, to form in a very simple manner the general differential
+equation which expresses the variable distribution of heat in any body
+whatever, subjected to any influences, by means of the single and
+easily-obtained relation, which represents the uniform distribution of
+heat in a right-angled parallelopipedon, considering (geometrically)
+every other body as decomposed into infinitely small elements of a
+similar form, and (thermologically) the flow of heat as constant during
+an infinitely small element of time. Henceforth, all the questions which
+can be presented by abstract thermology will be reduced, as in geometry
+and mechanics, to mere difficulties of analysis, which will always
+consist in the elimination of the differentials introduced as
+auxiliaries to facilitate the establishment of the equations.
+
+Examples of such different natures are more than sufficient to give a
+clear general idea of the immense scope of the fundamental conception of
+the transcendental analysis as formed by Leibnitz, constituting, as it
+undoubtedly does, the most lofty thought to which the human mind has as
+yet attained.
+
+It is evident that this conception was indispensable to complete the
+foundation of mathematical science, by enabling us to establish, in a
+broad and fruitful manner, the relation of the concrete to the abstract.
+In this respect it must be regarded as the necessary complement of the
+great fundamental idea of Descartes on the general analytical
+representation of natural phenomena: an idea which did not begin to be
+worthily appreciated and suitably employed till after the formation of
+the infinitesimal analysis, without which it could not produce, even in
+geometry, very important results.
+
+
+_Generality of the Formulas._ Besides the admirable facility which is
+given by the transcendental analysis for the investigation of the
+mathematical laws of all phenomena, a second fundamental and inherent
+property, perhaps as important as the first, is the extreme generality
+of the differential formulas, which express in a single equation each
+determinate phenomenon, however varied the subjects in relation to which
+it is considered. Thus we see, in the preceding examples, that a single
+differential equation gives the tangents of all curves, another their
+rectifications, a third their quadratures; and in the same way, one
+invariable formula expresses the mathematical law of every variable
+motion; and, finally, a single equation constantly represents the
+distribution of heat in any body and for any case. This generality,
+which is so exceedingly remarkable, and which is for geometers the basis
+of the most elevated considerations, is a fortunate and necessary
+consequence of the very spirit of the transcendental analysis,
+especially in the conception of Leibnitz. Thus the infinitesimal
+analysis has not only furnished a general method for indirectly forming
+equations which it would have been impossible to discover in a direct
+manner, but it has also permitted us to consider, for the mathematical
+study of natural phenomena, a new order of more general laws, which
+nevertheless present a clear and precise signification to every mind
+habituated to their interpretation. By virtue of this second
+characteristic property, the entire system of an immense science, such
+as geometry or mechanics, has been condensed into a small number of
+analytical formulas, from which the human mind can deduce, by certain
+and invariable rules, the solution of all particular problems.
+
+
+_Demonstration of the Method._ To complete the general exposition of the
+conception of Leibnitz, there remains to be considered the demonstration
+of the logical procedure to which it leads, and this, unfortunately, is
+the most imperfect part of this beautiful method.
+
+In the beginning of the infinitesimal analysis, the most celebrated
+geometers rightly attached more importance to extending the immortal
+discovery of Leibnitz and multiplying its applications than to
+rigorously establishing the logical bases of its operations. They
+contented themselves for a long time by answering the objections of
+second-rate geometers by the unhoped-for solution of the most difficult
+problems; doubtless persuaded that in mathematical science, much more
+than in any other, we may boldly welcome new methods, even when their
+rational explanation is imperfect, provided they are fruitful in
+results, inasmuch as its much easier and more numerous verifications
+would not permit any error to remain long undiscovered. But this state
+of things could not long exist, and it was necessary to go back to the
+very foundations of the analysis of Leibnitz in order to prove, in a
+perfectly general manner, the rigorous exactitude of the procedures
+employed in this method, in spite of the apparent infractions of the
+ordinary rules of reasoning which it permitted.
+
+Leibnitz, urged to answer, had presented an explanation entirely
+erroneous, saying that he treated infinitely small quantities as
+_incomparables_, and that he neglected them in comparison with finite
+quantities, "like grains of sand in comparison with the sea:" a view
+which would have completely changed the nature of his analysis, by
+reducing it to a mere approximative calculus, which, under this point of
+view, would be radically vicious, since it would be impossible to
+foresee, in general, to what degree the successive operations might
+increase these first errors, which could thus evidently attain any
+amount. Leibnitz, then, did not see, except in a very confused manner,
+the true logical foundations of the analysis which he had created. His
+earliest successors limited themselves, at first, to verifying its
+exactitude by showing the conformity of its results, in particular
+applications, to those obtained by ordinary algebra or the geometry of
+the ancients; reproducing, according to the ancient methods, so far as
+they were able, the solutions of some problems after they had been once
+obtained by the new method, which alone was capable of discovering them
+in the first place.
+
+When this great question was considered in a more general manner,
+geometers, instead of directly attacking the difficulty, preferred to
+elude it in some way, as Euler and D'Alembert, for example, have done,
+by demonstrating the necessary and constant conformity of the conception
+of Leibnitz, viewed in all its applications, with other fundamental
+conceptions of the transcendental analysis, that of Newton especially,
+the exactitude of which was free from any objection. Such a general
+verification is undoubtedly strictly sufficient to dissipate any
+uncertainty as to the legitimate employment of the analysis of Leibnitz.
+But the infinitesimal method is so important--it offers still, in almost
+all its applications, such a practical superiority over the other
+general conceptions which have been successively proposed--that there
+would be a real imperfection in the philosophical character of the
+science if it could not justify itself, and needed to be logically
+founded on considerations of another order, which would then cease to be
+employed.
+
+It was, then, of real importance to establish directly and in a general
+manner the necessary rationality of the infinitesimal method. After
+various attempts more or less imperfect, a distinguished geometer,
+Carnot, presented at last the true direct logical explanation of the
+method of Leibnitz, by showing it to be founded on the principle of the
+necessary compensation of errors, this being, in fact, the precise and
+luminous manifestation of what Leibnitz had vaguely and confusedly
+perceived. Carnot has thus rendered the science an essential service,
+although, as we shall see towards the end of this chapter, all this
+logical scaffolding of the infinitesimal method, properly so called, is
+very probably susceptible of only a provisional existence, inasmuch as
+it is radically vicious in its nature. Still, we should not fail to
+notice the general system of reasoning proposed by Carnot, in order to
+directly legitimate the analysis of Leibnitz. Here is the substance of
+it:
+
+In establishing the differential equation of a phenomenon, we
+substitute, for the immediate elements of the different quantities
+considered, other simpler infinitesimals, which differ from them
+infinitely little in comparison with them; and this substitution
+constitutes the principal artifice of the method of Leibnitz, which
+without it would possess no real facility for the formation of
+equations. Carnot regards such an hypothesis as really producing an
+error in the equation thus obtained, and which for this reason he calls
+_imperfect_; only, it is clear that this error must be infinitely small.
+Now, on the other hand, all the analytical operations, whether of
+differentiation or of integration, which are performed upon these
+differential equations, in order to raise them to finite equations by
+eliminating all the infinitesimals which have been introduced as
+auxiliaries, produce as constantly, by their nature, as is easily seen,
+other analogous errors, so that an exact compensation takes place, and
+the final equations, in the words of Carnot, become _perfect_. Carnot
+views, as a certain and invariable indication of the actual
+establishment of this necessary compensation, the complete elimination
+of the various infinitely small quantities, which is always, in fact,
+the final object of all the operations of the transcendental analysis;
+for if we have committed no other infractions of the general rules of
+reasoning than those thus exacted by the very nature of the
+infinitesimal method, the infinitely small errors thus produced cannot
+have engendered other than infinitely small errors in all the equations,
+and the relations are necessarily of a rigorous exactitude as soon as
+they exist between finite quantities alone, since the only errors then
+possible must be finite ones, while none such can have entered. All this
+general reasoning is founded on the conception of infinitesimal
+quantities, regarded as indefinitely decreasing, while those from which
+they are derived are regarded as fixed.
+
+
+_Illustration by Tangents._ Thus, to illustrate this abstract exposition
+by a single example, let us take up again the question of _tangents_,
+which is the most easy to analyze completely. We will regard the
+equation _t_ = _dy/dx_, obtained above, as being affected with an
+infinitely small error, since it would be perfectly rigorous only for
+the secant. Now let us complete the solution by seeking, according to
+the equation of each curve, the ratio between the differentials of the
+co-ordinates. If we suppose this equation to be _y_ = _ax²_, we shall
+evidently have
+
+  _dy_ = 2_axdx_ + _adx²_.
+
+In this formula we shall have to neglect the term _dx²_ as an infinitely
+small quantity of the second order. Then the combination of the two
+_imperfect_ equations.
+
+  _t_ = _dy/dx_, _dy_ = 2_ax(dx)_,
+
+being sufficient to eliminate entirely the infinitesimals, the finite
+result, _t_ = 2_ax_, will necessarily be rigorously correct, from the
+effect of the exact compensation of the two errors committed; since, by
+its finite nature, it cannot be affected by an infinitely small error,
+and this is, nevertheless, the only one which it could have, according
+to the spirit of the operations which have been executed.
+
+It would be easy to reproduce in a uniform manner the same reasoning
+with reference to all the other general applications of the analysis of
+Leibnitz.
+
+This ingenious theory is undoubtedly more subtile than solid, when we
+examine it more profoundly; but it has really no other radical logical
+fault than that of the infinitesimal method itself, of which it is, it
+seems to me, the natural development and the general explanation, so
+that it must be adopted for as long a time as it shall be thought proper
+to employ this method directly.
+
+       *       *       *       *       *
+
+I pass now to the general exposition of the two other fundamental
+conceptions of the transcendental analysis, limiting myself in each to
+its principal idea, the philosophical character of the analysis having
+been sufficiently determined above in the examination of the conception
+of Leibnitz, which I have specially dwelt upon because it admits of
+being most easily grasped as a whole, and most rapidly described.
+
+
+
+
+METHOD OF NEWTON.
+
+
+Newton has successively presented his own method of conceiving the
+transcendental analysis under several different forms. That which is at
+present the most commonly adopted was designated by Newton, sometimes
+under the name of the _Method of prime and ultimate Ratios_, sometimes
+under that of the _Method of Limits_.
+
+
+_Method of Limits._ The general spirit of the transcendental analysis,
+from this point of view, consists in introducing as auxiliaries, in the
+place of the primitive quantities, or concurrently with them, in order
+to facilitate the establishment of equations, the _limits of the ratios_
+of the simultaneous increments of these quantities; or, in other words,
+the _final ratios_ of these increments; limits or final ratios which can
+be easily shown to have a determinate and finite value. A special
+calculus, which is the equivalent of the infinitesimal calculus, is then
+employed to pass from the equations between these limits to the
+corresponding equations between the primitive quantities themselves.
+
+The power which is given by such an analysis, of expressing with more
+ease the mathematical laws of phenomena, depends in general on this,
+that since the calculus applies, not to the increments themselves of the
+proposed quantities, but to the limits of the ratios of those
+increments, we can always substitute for each increment any other
+magnitude more easy to consider, provided that their final ratio is the
+ratio of equality, or, in other words, that the limit of their ratio is
+unity. It is clear, indeed, that the calculus of limits would be in no
+way affected by this substitution. Starting from this principle, we find
+nearly the equivalent of the facilities offered by the analysis of
+Leibnitz, which are then merely conceived under another point of view.
+Thus curves will be regarded as the _limits_ of a series of rectilinear
+polygons, variable motions as the _limits_ of a collection of uniform
+motions of constantly diminishing durations, and so on.
+
+
+EXAMPLES. 1. _Tangents._ Suppose, for example, that we wish to determine
+the direction of the tangent to a curve; we will regard it as the limit
+towards which would tend a secant, which should turn about the given
+point so that its second point of intersection should indefinitely
+approach the first. Representing the differences of the co-ordinates of
+the two points by Δ_y_ and Δ_x_, we would have at each instant, for the
+trigonometrical tangent of the angle which the secant makes with the
+axis of abscissas,
+
+  _t_ = Δ_y_/Δ_x_;
+
+from which, taking the limits, we will obtain, relatively to the tangent
+itself, this general formula of transcendental analysis,
+
+  _t_ = _L_(Δ_y_/Δ_x_),
+
+the characteristic _L_ being employed to designate the limit. The
+calculus of indirect functions will show how to deduce from this formula
+in each particular case, when the equation of the curve is given, the
+relation between _t_ and _x_, by eliminating the auxiliary quantities
+which have been introduced. If we suppose, in order to complete the
+solution, that the equation of the proposed curve is _y_ = _ax²_, we
+shall evidently have
+
+  Δ_y_ = 2_ax_Δ_x_ + _a_(Δ_x_)²,
+
+from which we shall obtain
+
+  Δ_y_/Δ_x_ = 2_ax_ + _a_Δ_x_.
+
+Now it is clear that the _limit_ towards which the second number tends,
+in proportion as Δ_x_ diminishes, is 2_ax_. We shall therefore find, by
+this method, _t_ = 2_ax_, as we obtained it for the same case by the
+method of Leibnitz.
+
+2. _Rectifications._ In like manner, when the rectification of a curve
+is desired, we must substitute for the increment of the arc s the chord
+of this increment, which evidently has such a connexion with it that the
+limit of their ratio is unity; and then we find (pursuing in other
+respects the same plan as with the method of Leibnitz) this general
+equation of rectifications:
+
+  (_LΔs_/Δ_x_)² = 1 + (_LΔy_/Δ_x_)²,
+  or (_LΔs_/Δ_x_)² = 1 + (_LΔy_/Δ_x_)² + (_LΔz_/Δ_x_)²,
+
+according as the curve is plane or of double curvature. It will now be
+necessary, for each particular curve, to pass from this equation to that
+between the arc and the abscissa, which depends on the transcendental
+calculus properly so called.
+
+We could take up, with the same facility, by the method of limits, all
+the other general questions, the solution of which has been already
+indicated according to the infinitesimal method.
+
+Such is, in substance, the conception which Newton formed for the
+transcendental analysis, or, more precisely, that which Maclaurin and
+D'Alembert have presented as the most rational basis of that analysis,
+in seeking to fix and to arrange the ideas of Newton upon that subject.
+
+
+_Fluxions and Fluents._ Another distinct form under which Newton has
+presented this same method should be here noticed, and deserves
+particularly to fix our attention, as much by its ingenious clearness in
+some cases as by its having furnished the notation best suited to this
+manner of viewing the transcendental analysis, and, moreover, as having
+been till lately the special form of the calculus of indirect functions
+commonly adopted by the English geometers. I refer to the calculus of
+_fluxions_ and of _fluents_, founded on the general idea of
+_velocities_.
+
+To facilitate the conception of the fundamental idea, let us consider
+every curve as generated by a point impressed with a motion varying
+according to any law whatever. The different quantities which the curve
+can present, the abscissa, the ordinate, the arc, the area, &c., will be
+regarded as simultaneously produced by successive degrees during this
+motion. The _velocity_ with which each shall have been described will be
+called the _fluxion_ of that quantity, which will be inversely named its
+_fluent_. Henceforth the transcendental analysis will consist, according
+to this conception, in forming directly the equations between the
+fluxions of the proposed quantities, in order to deduce therefrom, by a
+special calculus, the equations between the fluents themselves. What
+has been stated respecting curves may, moreover, evidently be applied to
+any magnitudes whatever, regarded, by the aid of suitable images, as
+produced by motion.
+
+It is easy to understand the general and necessary identity of this
+method with that of limits complicated with the foreign idea of motion.
+In fact, resuming the case of the curve, if we suppose, as we evidently
+always may, that the motion of the describing point is uniform in a
+certain direction, that of the abscissa, for example, then the fluxion
+of the abscissa will be constant, like the element of the time; for all
+the other quantities generated, the motion cannot be conceived to be
+uniform, except for an infinitely small time. Now the velocity being in
+general according to its mechanical conception, the ratio of each space
+to the time employed in traversing it, and this time being here
+proportional to the increment of the abscissa, it follows that the
+fluxions of the ordinate, of the arc, of the area, &c., are really
+nothing else (rejecting the intermediate consideration of time) than the
+final ratios of the increments of these different quantities to the
+increment of the abscissa. This method of fluxions and fluents is, then,
+in reality, only a manner of representing, by a comparison borrowed from
+mechanics, the method of prime and ultimate ratios, which alone can be
+reduced to a calculus. It evidently, then, offers the same general
+advantages in the various principal applications of the transcendental
+analysis, without its being necessary to present special proofs of
+this.
+
+
+
+
+METHOD OF LAGRANGE.
+
+
+_Derived Functions._ The conception of Lagrange, in its admirable
+simplicity, consists in representing the transcendental analysis as a
+great algebraic artifice, by which, in order to facilitate the
+establishment of equations, we introduce, in the place of the primitive
+functions, or concurrently with them, their _derived_ functions; that
+is, according to the definition of Lagrange, the coefficient of the
+first term of the increment of each function, arranged according to the
+ascending powers of the increment of its variable. The special calculus
+of indirect functions has for its constant object, here as well as in
+the conceptions of Leibnitz and of Newton, to eliminate these
+_derivatives_ which have been thus employed as auxiliaries, in order to
+deduce from their relations the corresponding equations between the
+primitive magnitudes.
+
+
+_An Extension of ordinary Analysis._ The transcendental analysis is,
+then, nothing but a simple though very considerable extension of
+ordinary analysis. Geometers have long been accustomed to introduce in
+analytical investigations, in the place of the magnitudes themselves
+which they wished to study, their different powers, or their logarithms,
+or their sines, &c., in order to simplify the equations, and even to
+obtain them more easily. This successive _derivation_ is an artifice of
+the same nature, only of greater extent, and procuring, in consequence,
+much more important resources for this common object.
+
+But, although we can readily conceive, _à priori_, that the auxiliary
+consideration of these derivatives _may_ facilitate the establishment
+of equations, it is not easy to explain why this _must_ necessarily
+follow from this mode of derivation rather than from any other
+transformation. Such is the weak point of the great idea of Lagrange.
+The precise advantages of this analysis cannot as yet be grasped in an
+abstract manner, but only shown by considering separately each principal
+question, so that the verification is often exceedingly laborious.
+
+
+EXAMPLE. _Tangents._ This manner of conceiving the transcendental
+analysis may be best illustrated by its application to the most simple
+of the problems above examined--that of tangents.
+
+Instead of conceiving the tangent as the prolongation of the infinitely
+small element of the curve, according to the notion of Leibnitz--or as
+the limit of the secants, according to the ideas of Newton--Lagrange
+considers it, according to its simple geometrical character, analogous
+to the definitions of the ancients, to be a right line such that no
+other right line can pass through the point of contact between it and
+the curve. Then, to determine its direction, we must seek the general
+expression of its distance from the curve, measured in any direction
+whatever--in that of the ordinate, for example--and dispose of the
+arbitrary constant relating to the inclination of the right line, which
+will necessarily enter into that expression, in such a way as to
+diminish that separation as much as possible. Now this distance, being
+evidently equal to the difference of the two ordinates of the curve and
+of the right line, which correspond to the same new abscissa _x_ + _h_,
+will be represented by the formula
+
+  (_f'_(_x_) - _t_)_h_ + _qh²_ + _rh³_ + etc.,
+
+in which _t_ designates, as above, the unknown trigonometrical tangent
+of the angle which the required line makes with the axis of abscissas,
+and _f'_(_x_) the derived function of the ordinate _f_(_x_). This being
+understood, it is easy to see that, by disposing of _t_ so as to make
+the first term of the preceding formula equal to zero, we will render
+the interval between the two lines the least possible, so that any other
+line for which _t_ did not have the value thus determined would
+necessarily depart farther from the proposed curve. We have, then, for
+the direction of the tangent sought, the general expression _t_ =
+_f'_(_x_), a result exactly equivalent to those furnished by the
+Infinitesimal Method and the Method of Limits. We have yet to find
+_f'_(_x_) in each particular curve, which is a mere question of
+analysis, quite identical with those which are presented, at this stage
+of the operations, by the other methods.
+
+After these considerations upon the principal general conceptions, we
+need not stop to examine some other theories proposed, such as Euler's
+_Calculus of Vanishing Quantities_, which are really modifications--more
+or less important, and, moreover, no longer used--of the preceding
+methods.
+
+I have now to establish the comparison and the appreciation of these
+three fundamental methods. Their _perfect and necessary conformity_ is
+first to be proven in a general manner.
+
+
+
+
+FUNDAMENTAL IDENTITY OF THE THREE METHODS.
+
+
+It is, in the first place, evident from what precedes, considering these
+three methods as to their actual destination, independently of their
+preliminary ideas, that they all consist in the same general logical
+artifice, which has been characterized in the first chapter; to wit,
+the introduction of a certain system of auxiliary magnitudes, having
+uniform relations to those which are the special objects of the inquiry,
+and substituted for them expressly to facilitate the analytical
+expression of the mathematical laws of the phenomena, although they have
+finally to be eliminated by the aid of a special calculus. It is this
+which has determined me to regularly define the transcendental analysis
+as _the calculus of indirect functions_, in order to mark its true
+philosophical character, at the same time avoiding any discussion upon
+the best manner of conceiving and applying it. The general effect of
+this analysis, whatever the method employed, is, then, to bring every
+mathematical question much more promptly within the power of the
+_calculus_, and thus to diminish considerably the serious difficulty
+which is usually presented by the passage from the concrete to the
+abstract. Whatever progress we may make, we can never hope that the
+calculus will ever be able to grasp every question of natural
+philosophy, geometrical, or mechanical, or thermological, &c.,
+immediately upon its birth, which would evidently involve a
+contradiction. Every problem will constantly require a certain
+preliminary labour to be performed, in which the calculus can be of no
+assistance, and which, by its nature, cannot be subjected to abstract
+and invariable rules; it is that which has for its special object the
+establishment of equations, which form the indispensable starting point
+of all analytical researches. But this preliminary labour has been
+remarkably simplified by the creation of the transcendental analysis,
+which has thus hastened the moment at which the solution admits of the
+uniform and precise application of general and abstract methods; by
+reducing, in each case, this special labour to the investigation of
+equations between the auxiliary magnitudes; from which the calculus then
+leads to equations directly referring to the proposed magnitudes, which,
+before this admirable conception, it had been necessary to establish
+directly and separately. Whether these indirect equations are
+_differential_ equations, according to the idea of Leibnitz, or
+equations of _limits_, conformably to the conception of Newton, or,
+lastly, _derived_ equations, according to the theory of Lagrange, the
+general procedure is evidently always the same.
+
+But the coincidence of these three principal methods is not limited to
+the common effect which they produce; it exists, besides, in the very
+manner of obtaining it. In fact, not only do all three consider, in the
+place of the primitive magnitudes, certain auxiliary ones, but, still
+farther, the quantities thus introduced as subsidiary are exactly
+identical in the three methods, which consequently differ only in the
+manner of viewing them. This can be easily shown by taking for the
+general term of comparison any one of the three conceptions, especially
+that of Lagrange, which is the most suitable to serve as a type, as
+being the freest from foreign considerations. Is it not evident, by the
+very definition of _derived functions_, that they are nothing else than
+what Leibnitz calls _differential coefficients_, or the ratios of the
+differential of each function to that of the corresponding variable,
+since, in determining the first differential, we will be obliged, by the
+very nature of the infinitesimal method, to limit ourselves to taking
+the only term of the increment of the function which contains the first
+power of the infinitely small increment of the variable? In the same
+way, is not the derived function, by its nature, likewise the necessary
+_limit_ towards which tends the ratio between the increment of the
+primitive function and that of its variable, in proportion as this last
+indefinitely diminishes, since it evidently expresses what that ratio
+becomes when we suppose the increment of the variable to equal zero?
+That which is designated by _dx_/_dy_ in the method of Leibnitz; that
+which ought to be noted as _L_(Δ_y_/Δ_x_) in that of Newton; and that
+which Lagrange has indicated by _f'_(_x_), is constantly one same
+function, seen from three different points of view, the considerations
+of Leibnitz and Newton properly consisting in making known two general
+necessary properties of the derived function. The transcendental
+analysis, examined abstractedly and in its principle, is then always the
+same, whatever may be the conception which is adopted, and the
+procedures of the calculus of indirect functions are necessarily
+identical in these different methods, which in like manner must, for any
+application whatever, lead constantly to rigorously uniform results.
+
+
+
+
+COMPARATIVE VALUE OF THE THREE METHODS.
+
+
+If now we endeavour to estimate the comparative value of these three
+equivalent conceptions, we shall find in each advantages and
+inconveniences which are peculiar to it, and which still prevent
+geometers from confining themselves to any one of them, considered as
+final.
+
+
+_That of Leibnitz._ The conception of Leibnitz presents incontestably,
+in all its applications, a very marked superiority, by leading in a much
+more rapid manner, and with much less mental effort, to the formation
+of equations between the auxiliary magnitudes. It is to its use that we
+owe the high perfection which has been acquired by all the general
+theories of geometry and mechanics. Whatever may be the different
+speculative opinions of geometers with respect to the infinitesimal
+method, in an abstract point of view, all tacitly agree in employing it
+by preference, as soon as they have to treat a new question, in order
+not to complicate the necessary difficulty by this purely artificial
+obstacle proceeding from a misplaced obstinacy in adopting a less
+expeditious course. Lagrange himself, after having reconstructed the
+transcendental analysis on new foundations, has (with that noble
+frankness which so well suited his genius) rendered a striking and
+decisive homage to the characteristic properties of the conception of
+Leibnitz, by following it exclusively in the entire system of his
+_Méchanique Analytique_. Such a fact renders any comments unnecessary.
+
+But when we consider the conception of Leibnitz in itself and in its
+logical relations, we cannot escape admitting, with Lagrange, that it is
+radically vicious in this, that, adopting its own expressions, the
+notion of infinitely small quantities is a _false idea_, of which it is
+in fact impossible to obtain a clear conception, however we may deceive
+ourselves in that matter. Even if we adopt the ingenious idea of the
+compensation of errors, as above explained, this involves the radical
+inconvenience of being obliged to distinguish in mathematics two classes
+of reasonings, those which are perfectly rigorous, and those in which we
+designedly commit errors which subsequently have to be compensated. A
+conception which leads to such strange consequences is undoubtedly very
+unsatisfactory in a logical point of view.
+
+To say, as do some geometers, that it is possible in every case to
+reduce the infinitesimal method to that of limits, the logical character
+of which is irreproachable, would evidently be to elude the difficulty
+rather than to remove it; besides, such a transformation almost entirely
+strips the conception of Leibnitz of its essential advantages of
+facility and rapidity.
+
+Finally, even disregarding the preceding important considerations, the
+infinitesimal method would no less evidently present by its nature the
+very serious defect of breaking the unity of abstract mathematics, by
+creating a transcendental analysis founded on principles so different
+from those which form the basis of the ordinary analysis. This division
+of analysis into two worlds almost entirely independent of each other,
+tends to hinder the formation of truly general analytical conceptions.
+To fully appreciate the consequences of this, we should have to go back
+to the state of the science before Lagrange had established a general
+and complete harmony between these two great sections.
+
+
+_That of Newton._ Passing now to the conception of Newton, it is evident
+that by its nature it is not exposed to the fundamental logical
+objections which are called forth by the method of Leibnitz. The notion
+of _limits_ is, in fact, remarkable for its simplicity and its
+precision. In the transcendental analysis presented in this manner, the
+equations are regarded as exact from their very origin, and the general
+rules of reasoning are as constantly observed as in ordinary analysis.
+But, on the other hand, it is very far from offering such powerful
+resources for the solution of problems as the infinitesimal method. The
+obligation which it imposes, of never considering the increments of
+magnitudes separately and by themselves, nor even in their ratios, but
+only in the limits of those ratios, retards considerably the operations
+of the mind in the formation of auxiliary equations. We may even say
+that it greatly embarrasses the purely analytical transformations. Thus
+the transcendental analysis, considered separately from its
+applications, is far from presenting in this method the extent and the
+generality which have been imprinted upon it by the conception of
+Leibnitz. It is very difficult, for example, to extend the theory of
+Newton to functions of several independent variables. But it is
+especially with reference to its applications that the relative
+inferiority of this theory is most strongly marked.
+
+Several Continental geometers, in adopting the method of Newton as the
+more logical basis of the transcendental analysis, have partially
+disguised this inferiority by a serious inconsistency, which consists in
+applying to this method the notation invented by Leibnitz for the
+infinitesimal method, and which is really appropriate to it alone. In
+designating by _dy_/_dx_ that which logically ought, in the theory of
+limits, to be denoted by _L_(Δ_y_/Δ_x_), and in extending to all the
+other analytical conceptions this displacement of signs, they intended,
+undoubtedly, to combine the special advantages of the two methods; but,
+in reality, they have only succeeded in causing a vicious confusion
+between them, a familiarity with which hinders the formation of clear
+and exact ideas of either. It would certainly be singular, considering
+this usage in itself, that, by the mere means of signs, it could be
+possible to effect a veritable combination between two theories so
+distinct as those under consideration.
+
+Finally, the method of limits presents also, though in a less degree,
+the greater inconvenience, which I have above noted in reference to the
+infinitesimal method, of establishing a total separation between the
+ordinary and the transcendental analysis; for the idea of _limits_,
+though clear and rigorous, is none the less in itself, as Lagrange has
+remarked, a foreign idea, upon which analytical theories ought not to be
+dependent.
+
+
+_That of Lagrange._ This perfect unity of analysis, and this purely
+abstract character of its fundamental notions, are found in the highest
+degree in the conception of Lagrange, and are found there alone; it is,
+for this reason, the most rational and the most philosophical of all.
+Carefully removing every heterogeneous consideration, Lagrange has
+reduced the transcendental analysis to its true peculiar character, that
+of presenting a very extensive class of analytical transformations,
+which facilitate in a remarkable degree the expression of the conditions
+of various problems. At the same time, this analysis is thus necessarily
+presented as a simple extension of ordinary analysis; it is only a
+higher algebra. All the different parts of abstract mathematics,
+previously so incoherent, have from that moment admitted of being
+conceived as forming a single system.
+
+Unhappily, this conception, which possesses such fundamental properties,
+independently of its so simple and so lucid notation, and which is
+undoubtedly destined to become the final theory of transcendental
+analysis, because of its high philosophical superiority over all the
+other methods proposed, presents in its present state too many
+difficulties in its applications, as compared with the conception of
+Newton, and still more with that of Leibnitz, to be as yet exclusively
+adopted. Lagrange himself has succeeded only with great difficulty in
+rediscovering, by his method, the principal results already obtained by
+the infinitesimal method for the solution of the general questions of
+geometry and mechanics; we may judge from that what obstacles would be
+found in treating in the same manner questions which were truly new and
+important. It is true that Lagrange, on several occasions, has shown
+that difficulties call forth, from men of genius, superior efforts,
+capable of leading to the greatest results. It was thus that, in trying
+to adapt his method to the examination of the curvature of lines, which
+seemed so far from admitting its application, he arrived at that
+beautiful theory of contacts which has so greatly perfected that
+important part of geometry. But, in spite of such happy exceptions, the
+conception of Lagrange has nevertheless remained, as a whole,
+essentially unsuited to applications.
+
+The final result of the general comparison which I have too briefly
+sketched, is, then, as already suggested, that, in order to really
+understand the transcendental analysis, we should not only consider it
+in its principles according to the three fundamental conceptions of
+Leibnitz, of Newton, and of Lagrange, but should besides accustom
+ourselves to carry out almost indifferently, according to these three
+principal methods, and especially according to the first and the last,
+the solution of all important questions, whether of the pure calculus of
+indirect functions or of its applications. This is a course which I
+could not too strongly recommend to all those who desire to judge
+philosophically of this admirable creation of the human mind, as well as
+to those who wish to learn to make use of this powerful instrument with
+success and with facility. In all the other parts of mathematical
+science, the consideration of different methods for a single class of
+questions may be useful, even independently of its historical interest,
+but it is not indispensable; here, on the contrary, it is strictly
+necessary.
+
+Having determined with precision, in this chapter, the philosophical
+character of the calculus of indirect functions, according to the
+principal fundamental conceptions of which it admits, we have next to
+consider, in the following chapter, the logical division and the general
+composition of this calculus.
+
+
+
+
+CHAPTER IV.
+
+
+THE DIFFERENTIAL AND INTEGRAL CALCULUS.
+
+
+ITS TWO FUNDAMENTAL DIVISIONS.
+
+
+The _calculus of indirect functions_, in accordance with the
+considerations explained in the preceding chapter, is necessarily
+divided into two parts (or, more properly, is decomposed into two
+different _calculi_ entirely distinct, although intimately connected by
+their nature), according as it is proposed to find the relations between
+the auxiliary magnitudes (the introduction of which constitutes the
+general spirit of this calculus) by means of the relations between the
+corresponding primitive magnitudes; or, conversely, to try to discover
+these direct equations by means of the indirect equations originally
+established. Such is, in fact, constantly the double object of the
+transcendental analysis.
+
+These two systems have received different names, according to the point
+of view under which this analysis has been regarded. The infinitesimal
+method, properly so called, having been the most generally employed for
+the reasons which have been given, almost all geometers employ
+habitually the denominations of _Differential Calculus_ and of _Integral
+Calculus_, established by Leibnitz, and which are, in fact, very
+rational consequences of his conception. Newton, in accordance with his
+method, named the first the _Calculus of Fluxions_, and the second the
+_Calculus of Fluents_, expressions which were commonly employed in
+England. Finally, following the eminently philosophical theory founded
+by Lagrange, one would be called the _Calculus of Derived Functions_,
+and the other the _Calculus of Primitive Functions_. I will continue to
+make use of the terms of Leibnitz, as being more convenient for the
+formation of secondary expressions, although I ought, in accordance with
+the suggestions made in the preceding chapter, to employ concurrently
+all the different conceptions, approaching as nearly as possible to that
+of Lagrange.
+
+
+
+
+THEIR RELATIONS TO EACH OTHER.
+
+
+The differential calculus is evidently the logical basis of the integral
+calculus; for we do not and cannot know how to integrate directly any
+other differential expressions than those produced by the
+differentiation of the ten simple functions which constitute the general
+elements of our analysis. The art of integration consists, then,
+essentially in bringing all the other cases, as far as is possible, to
+finally depend on only this small number of fundamental integrations.
+
+In considering the whole body of the transcendental analysis, as I have
+characterized it in the preceding chapter, it is not at first apparent
+what can be the peculiar utility of the differential calculus,
+independently of this necessary relation with the integral calculus,
+which seems as if it must be, by itself, the only one directly
+indispensable. In fact, the elimination of the _infinitesimals_ or of
+the _derivatives_, introduced as auxiliaries to facilitate the
+establishment of equations, constituting, as we have seen, the final and
+invariable object of the calculus of indirect functions, it is natural
+to think that the calculus which teaches how to deduce from the
+equations between these auxiliary magnitudes, those which exist between
+the primitive magnitudes themselves, ought strictly to suffice for the
+general wants of the transcendental analysis without our perceiving, at
+the first glance, what special and constant part the solution of the
+inverse question can have in such an analysis. It would be a real error,
+though a common one, to assign to the differential calculus, in order to
+explain its peculiar, direct, and necessary influence, the destination
+of forming the differential equations, from which the integral calculus
+then enables us to arrive at the finite equations; for the primitive
+formation of differential equations is not and cannot be, properly
+speaking, the object of any calculus, since, on the contrary, it forms
+by its nature the indispensable starting point of any calculus whatever.
+How, in particular, could the differential calculus, which in itself is
+reduced to teaching the means of _differentiating_ the different
+equations, be a general procedure for establishing them? That which in
+every application of the transcendental analysis really facilitates the
+formation of equations, is the infinitesimal _method_, and not the
+infinitesimal _calculus_, which is perfectly distinct from it, although
+it is its indispensable complement. Such a consideration would, then,
+give a false idea of the special destination which characterizes the
+differential calculus in the general system of the transcendental
+analysis.
+
+But we should nevertheless very imperfectly conceive the real peculiar
+importance of this first branch of the calculus of indirect functions,
+if we saw in it only a simple preliminary labour, having no other
+general and essential object than to prepare indispensable foundations
+for the integral calculus. As the ideas on this matter are generally
+confused, I think that I ought here to explain in a summary manner this
+important relation as I view it, and to show that in every application
+of the transcendental analysis a primary, direct, and necessary part is
+constantly assigned to the differential calculus.
+
+
+1. _Use of the Differential Calculus as preparatory to that of the
+Integral._ In forming the differential equations of any phenomenon
+whatever, it is very seldom that we limit ourselves to introduce
+differentially only those magnitudes whose relations are sought. To
+impose that condition would be to uselessly diminish the resources
+presented by the transcendental analysis for the expression of the
+mathematical laws of phenomena. Most frequently we introduce into the
+primitive equations, through their differentials, other magnitudes whose
+relations are already known or supposed to be so, and without the
+consideration of which it would be frequently impossible to establish
+equations. Thus, for example, in the general problem of the
+rectification of curves, the differential equation,
+
+  _ds_² = _dy_² + _dx_², or _ds_² = _dx_² + _dy_² + _dz_²,
+
+is not only established between the desired function s and the
+independent variable _x_, to which it is referred, but, at the same
+time, there have been introduced, as indispensable intermediaries, the
+differentials of one or two other functions, _y_ and _z_, which are
+among the data of the problem; it would not have been possible to form
+directly the equation between _ds_ and _dx_, which would, besides, be
+peculiar to each curve considered. It is the same for most questions.
+Now in these cases it is evident that the differential equation is not
+immediately suitable for integration. It is previously necessary that
+the differentials of the functions supposed to be known, which have
+been employed as intermediaries, should be entirely eliminated, in order
+that equations may be obtained between the differentials of the
+functions which alone are sought and those of the really independent
+variables, after which the question depends on only the integral
+calculus. Now this preparatory elimination of certain differentials, in
+order to reduce the infinitesimals to the smallest number possible,
+belongs simply to the differential calculus; for it must evidently be
+done by determining, by means of the equations between the functions
+supposed to be known, taken as intermediaries, the relations of their
+differentials, which is merely a question of differentiation. Thus, for
+example, in the case of rectifications, it will be first necessary to
+calculate _dy_, or _dy_ and _dz_, by differentiating the equation or the
+equations of each curve proposed; after eliminating these expressions,
+the general differential formula above enunciated will then contain only
+_ds_ and _dx_; having arrived at this point, the elimination of the
+infinitesimals can be completed only by the integral calculus.
+
+Such is, then, the general office necessarily belonging to the
+differential calculus in the complete solution of the questions which
+exact the employment of the transcendental analysis; to produce, as far
+as is possible, the elimination of the infinitesimals, that is, to
+reduce in each case the primitive differential equations so that they
+shall contain only the differentials of the really independent
+variables, and those of the functions sought, by causing to disappear,
+by elimination, the differentials of all the other known functions which
+may have been taken as intermediaries at the time of the formation of
+the differential equations of the problem which is under consideration.
+
+
+2. _Employment of the Differential Calculus alone._ For certain
+questions, which, although few in number, have none the less, as we
+shall see hereafter, a very great importance, the magnitudes which are
+sought enter directly, and not by their differentials, into the
+primitive differential equations, which then contain differentially only
+the different known functions employed as intermediaries, in accordance
+with the preceding explanation. These cases are the most favourable of
+all; for it is evident that the differential calculus is then entirely
+sufficient for the complete elimination of the infinitesimals, without
+the question giving rise to any integration. This is what occurs, for
+example, in the problem of _tangents_ in geometry; in that of
+_velocities_ in mechanics, &c.
+
+
+3. _Employment of the Integral Calculus alone._ Finally, some other
+questions, the number of which is also very small, but the importance of
+which is no less great, present a second exceptional case, which is in
+its nature exactly the converse of the preceding. They are those in
+which the differential equations are found to be immediately ready for
+integration, because they contain, at their first formation, only the
+infinitesimals which relate to the functions sought, or to the really
+independent variables, without its being necessary to introduce,
+differentially, other functions as intermediaries. If in these new cases
+we introduce these last functions, since, by hypothesis, they will enter
+directly and not by their differentials, ordinary algebra will suffice
+to eliminate them, and to bring the question to depend on only the
+integral calculus. The differential calculus will then have no special
+part in the complete solution of the problem, which will depend entirely
+upon the integral calculus. The general question of _quadratures_ offers
+an important example of this, for the differential equation being then
+_dA = ydx_, will become immediately fit for integration as soon as we
+shall have eliminated, by means of the equation of the proposed curve,
+the intermediary function _y_, which does not enter into it
+differentially. The same circumstances exist in the problem of
+_cubatures_, and in some others equally important.
+
+
+_Three classes of Questions hence resulting._ As a general result of the
+previous considerations, it is then necessary to divide into three
+classes the mathematical questions which require the use of the
+transcendental analysis; the _first_ class comprises the problems
+susceptible of being entirely resolved by means of the differential
+calculus alone, without any need of the integral calculus; the _second_,
+those which are, on the contrary, entirely dependent upon the integral
+calculus, without the differential calculus having any part in their
+solution; lastly, in the _third_ and the most extensive, which
+constitutes the normal case, the two others being only exceptional, the
+differential and the integral calculus have each in their turn a
+distinct and necessary part in the complete solution of the problem, the
+former making the primitive differential equations undergo a preparation
+which is indispensable for the application of the latter. Such are
+exactly their general relations, of which too indefinite and inexact
+ideas are generally formed.
+
+       *       *       *       *       *
+
+Let us now take a general survey of the logical composition of each
+calculus, beginning with the differential.
+
+
+
+
+THE DIFFERENTIAL CALCULUS.
+
+
+In the exposition of the transcendental analysis, it is customary to
+intermingle with the purely analytical part (which reduces itself to the
+treatment of the abstract principles of differentiation and integration)
+the study of its different principal applications, especially those
+which concern geometry. This confusion of ideas, which is a consequence
+of the actual manner in which the science has been developed, presents,
+in the dogmatic point of view, serious inconveniences in this respect,
+that it makes it difficult properly to conceive either analysis or
+geometry. Having to consider here the most rational co-ordination which
+is possible, I shall include, in the following sketch, only the calculus
+of indirect functions properly so called, reserving for the portion of
+this volume which relates to the philosophical study of _concrete_
+mathematics the general examination of its great geometrical and
+mechanical applications.
+
+
+_Two Cases: explicit and implicit Functions._ The fundamental division
+of the differential calculus, or of the general subject of
+differentiation, consists in distinguishing two cases, according as the
+analytical functions which are to be differentiated are _explicit_ or
+_implicit_; from which flow two parts ordinarily designated by the names
+of differentiation _of formulas_ and differentiation _of equations_. It
+is easy to understand, _à priori_, the importance of this
+classification. In fact, such a distinction would be illusory if the
+ordinary analysis was perfect; that is, if we knew how to resolve all
+equations algebraically, for then it would be possible to render every
+_implicit_ function _explicit_; and, by differentiating it in that
+state alone, the second part of the differential calculus would be
+immediately comprised in the first, without giving rise to any new
+difficulty. But the algebraical resolution of equations being, as we
+have seen, still almost in its infancy, and as yet impossible for most
+cases, it is plain that the case is very different, since we have,
+properly speaking, to differentiate a function without knowing it,
+although it is determinate. The differentiation of implicit functions
+constitutes then, by its nature, a question truly distinct from that
+presented by explicit functions, and necessarily more complicated. It is
+thus evident that we must commence with the differentiation of formulas,
+and reduce the differentiation of equations to this primary case by
+certain invariable analytical considerations, which need not be here
+mentioned.
+
+These two general cases of differentiation are also distinct in another
+point of view equally necessary, and too important to be left unnoticed.
+The relation which is obtained between the differentials is constantly
+more indirect, in comparison with that of the finite quantities, in the
+differentiation of implicit functions than in that of explicit
+functions. We know, in fact, from the considerations presented by
+Lagrange on the general formation of differential equations, that, on
+the one hand, the same primitive equation may give rise to a greater or
+less number of derived equations of very different forms, although at
+bottom equivalent, depending upon which of the arbitrary constants is
+eliminated, which is not the case in the differentiation of explicit
+formulas; and that, on the other hand, the unlimited system of the
+different primitive equations, which correspond to the same derived
+equation, presents a much more profound analytical variety than that of
+the different functions, which admit of one same explicit differential,
+and which are distinguished from each other only by a constant term.
+Implicit functions must therefore be regarded as being in reality still
+more modified by differentiation than explicit functions. We shall again
+meet with this consideration relatively to the integral calculus, where
+it acquires a preponderant importance.
+
+
+_Two Sub-cases: A single Variable or several Variables._ Each of the two
+fundamental parts of the Differential Calculus is subdivided into two
+very distinct theories, according as we are required to differentiate
+functions of a single variable or functions of several independent
+variables. This second case is, by its nature, quite distinct from the
+first, and evidently presents more complication, even in considering
+only explicit functions, and still more those which are implicit. As to
+the rest, one of these cases is deduced from the other in a general
+manner, by the aid of an invariable and very simple principle, which
+consists in regarding the total differential of a function which is
+produced by the simultaneous increments of the different independent
+variables which it contains, as the sum of the partial differentials
+which would be produced by the separate increment of each variable in
+turn, if all the others were constant. It is necessary, besides,
+carefully to remark, in connection with this subject, a new idea which
+is introduced by the distinction of functions into those of one variable
+and of several; it is the consideration of these different special
+derived functions, relating to each variable separately, and the number
+of which increases more and more in proportion as the order of the
+derivation becomes higher, and also when the variables become more
+numerous. It results from this that the differential relations belonging
+to functions of several variables are, by their nature, both much more
+indirect, and especially much more indeterminate, than those relating to
+functions of a single variable. This is most apparent in the case of
+implicit functions, in which, in the place of the simple arbitrary
+constants which elimination causes to disappear when we form the proper
+differential equations for functions of a single variable, it is the
+arbitrary functions of the proposed variables which are then eliminated;
+whence must result special difficulties when these equations come to be
+integrated.
+
+Finally, to complete this summary sketch of the different essential
+parts of the differential calculus proper, I should add, that in the
+differentiation of implicit functions, whether of a single variable or
+of several, it is necessary to make another distinction; that of the
+case in which it is required to differentiate at once different
+functions of this kind, _combined_ in certain primitive equations, from
+that in which all these functions are _separate_.
+
+The functions are evidently, in fact, still more implicit in the first
+case than in the second, if we consider that the same imperfection of
+ordinary analysis, which forbids our converting every implicit function
+into an equivalent explicit function, in like manner renders us unable
+to separate the functions which enter simultaneously into any system of
+equations. It is then necessary to differentiate, not only without
+knowing how to resolve the primitive equations, but even without being
+able to effect the proper eliminations among them, thus producing a new
+difficulty.
+
+
+_Reduction of the whole to the Differentiation of the ten elementary
+Functions._ Such, then, are the natural connection and the logical
+distribution of the different principal theories which compose the
+general system of differentiation. Since the differentiation of implicit
+functions is deduced from that of explicit functions by a single
+constant principle, and the differentiation of functions of several
+variables is reduced by another fixed principle to that of functions of
+a single variable, the whole of the differential calculus is finally
+found to rest upon the differentiation of explicit functions with a
+single variable, the only one which is ever executed directly. Now it is
+easy to understand that this first theory, the necessary basis of the
+entire system, consists simply in the differentiation of the ten simple
+functions, which are the uniform elements of all our analytical
+combinations, and the list of which has been given in the first chapter,
+on page 51; for the differentiation of compound functions is evidently
+deduced, in an immediate and necessary manner, from that of the simple
+functions which compose them. It is, then, to the knowledge of these ten
+fundamental differentials, and to that of the two general principles
+just mentioned, which bring under it all the other possible cases, that
+the whole system of differentiation is properly reduced. We see, by the
+combination of these different considerations, how simple and how
+perfect is the entire system of the differential calculus. It certainly
+constitutes, in its logical relations, the most interesting spectacle
+which mathematical analysis can present to our understanding.
+
+
+_Transformation of derived Functions for new Variables._ The general
+sketch which I have just summarily drawn would nevertheless present an
+important deficiency, if I did not here distinctly indicate a final
+theory, which forms, by its nature, the indispensable complement of the
+system of differentiation. It is that which has for its object the
+constant transformation of derived functions, as a result of determinate
+changes in the independent variables, whence results the possibility of
+referring to new variables all the general differential formulas
+primitively established for others. This question is now resolved in the
+most complete and the most simple manner, as are all those of which the
+differential calculus is composed. It is easy to conceive the general
+importance which it must have in any of the applications of the
+transcendental analysis, the fundamental resources of which it may be
+considered as augmenting, by permitting us to choose (in order to form
+the differential equations, in the first place, with more ease) that
+system of independent variables which may appear to be the most
+advantageous, although it is not to be finally retained. It is thus, for
+example, that most of the principal questions of geometry are resolved
+much more easily by referring the lines and surfaces to _rectilinear_
+co-ordinates, and that we may, nevertheless, have occasion to express
+these lines, etc., analytically by the aid of _polar_ co-ordinates, or
+in any other manner. We will then be able to commence the differential
+solution of the problem by employing the rectilinear system, but only as
+an intermediate step, from which, by the general theory here referred
+to, we can pass to the final system, which sometimes could not have been
+considered directly.
+
+
+_Different Orders of Differentiation._ In the logical classification of
+the differential calculus which has just been given, some may be
+inclined to suggest a serious omission, since I have not subdivided each
+of its four essential parts according to another general consideration,
+which seems at first view very important; namely, that of the higher or
+lower order of differentiation. But it is easy to understand that this
+distinction has no real influence in the differential calculus, inasmuch
+as it does not give rise to any new difficulty. If, indeed, the
+differential calculus was not rigorously complete, that is, if we did
+not know how to differentiate at will any function whatever, the
+differentiation to the second or higher order of each determinate
+function might engender special difficulties. But the perfect
+universality of the differential calculus plainly gives us the assurance
+of being able to differentiate, to any order whatever, all known
+functions whatever, the question reducing itself to a constantly
+repeated differentiation of the first order. This distinction,
+unimportant as it is for the differential calculus, acquires, however, a
+very great importance in the integral calculus, on account of the
+extreme imperfection of the latter.
+
+
+_Analytical Applications._ Finally, though this is not the place to
+consider the various applications of the differential calculus, yet an
+exception may be made for those which consist in the solution of
+questions which are purely analytical, which ought, indeed, to be
+logically treated in continuation of a system of differentiation,
+because of the evident homogeneity of the considerations involved. These
+questions may be reduced to three essential ones.
+
+Firstly, the _development into series_ of functions of one or more
+variables, or, more generally, the transformation of functions, which
+constitutes the most beautiful and the most important application of the
+differential calculus to general analysis, and which comprises, besides
+the fundamental series discovered by Taylor, the remarkable series
+discovered by Maclaurin, John Bernouilli, Lagrange, &c.:
+
+Secondly, the general _theory of maxima and minima_ values for any
+functions whatever, of one or more variables; one of the most
+interesting problems which analysis can present, however elementary it
+may now have become, and to the complete solution of which the
+differential calculus naturally applies:
+
+Thirdly, the general determination of the true value of functions which
+present themselves under an _indeterminate_ appearance for certain
+hypotheses made on the values of the corresponding variables; which is
+the least extensive and the least important of the three.
+
+The first question is certainly the principal one in all points of view;
+it is also the most susceptible of receiving a new extension hereafter,
+especially by conceiving, in a broader manner than has yet been done,
+the employment of the differential calculus in the transformation of
+functions, on which subject Lagrange has left some valuable hints.
+
+       *       *       *       *       *
+
+Having thus summarily, though perhaps too briefly, considered the chief
+points in the differential calculus, I now proceed to an equally rapid
+exposition of a systematic outline of the Integral Calculus, properly so
+called, that is, the abstract subject of integration.
+
+
+
+
+THE INTEGRAL CALCULUS.
+
+
+_Its Fundamental Division._ The fundamental division of the Integral
+Calculus is founded on the same principle as that of the Differential
+Calculus, in distinguishing the integration of _explicit_ differential
+formulas, and the integration of _implicit_ differentials or of
+differential equations. The separation of these two cases is even much
+more profound in relation to integration than to differentiation. In the
+differential calculus, in fact, this distinction rests, as we have seen,
+only on the extreme imperfection of ordinary analysis. But, on the other
+hand, it is easy to see that, even though all equations could be
+algebraically resolved, differential equations would none the less
+constitute a case of integration quite distinct from that presented by
+the explicit differential formulas; for, limiting ourselves, for the
+sake of simplicity, to the first order, and to a single function _y_ of
+a single variable _x_, if we suppose any differential equation between
+_x_, _y_, and _dy/dx_, to be resolved with reference to _dy/dx_, the
+expression of the derived function being then generally found to contain
+the primitive function itself, which is the object of the inquiry, the
+question of integration will not have at all changed its nature, and the
+solution will not really have made any other progress than that of
+having brought the proposed differential equation to be of only the
+first degree relatively to the derived function, which is in itself of
+little importance. The differential would not then be determined in a
+manner much less _implicit_ than before, as regards the integration,
+which would continue to present essentially the same characteristic
+difficulty. The algebraic resolution of equations could not make the
+case which we are considering come within the simple integration of
+explicit differentials, except in the special cases in which the
+proposed differential equation did not contain the primitive function
+itself, which would consequently permit us, by resolving it, to find
+_dy/dx_ in terms of _x_ only, and thus to reduce the question to the
+class of quadratures. Still greater difficulties would evidently be
+found in differential equations of higher orders, or containing
+simultaneously different functions of several independent variables.
+
+The integration of differential equations is then necessarily more
+complicated than that of explicit differentials, by the elaboration of
+which last the integral calculus has been created, and upon which the
+others have been made to depend as far as it has been possible. All the
+various analytical methods which have been proposed for integrating
+differential equations, whether it be the separation of the variables,
+the method of multipliers, &c., have in fact for their object to reduce
+these integrations to those of differential formulas, the only one
+which, by its nature, can be undertaken directly. Unfortunately,
+imperfect as is still this necessary base of the whole integral
+calculus, the art of reducing to it the integration of differential
+equations is still less advanced.
+
+
+_Subdivisions: one variable or several._ Each of these two fundamental
+branches of the integral calculus is next subdivided into two others (as
+in the differential calculus, and for precisely analogous reasons),
+according as we consider functions with a _single variable_, or
+functions with _several independent variables_.
+
+This distinction is, like the preceding one, still more important for
+integration than for differentiation. This is especially remarkable in
+reference to differential equations. Indeed, those which depend on
+several independent variables may evidently present this characteristic
+and much more serious difficulty, that the desired function may be
+differentially defined by a simple relation between its different
+special derivatives relative to the different variables taken
+separately. Hence results the most difficult and also the most extensive
+branch of the integral calculus, which is commonly named the _Integral
+Calculus of partial differences_, created by D'Alembert, and in which,
+according to the just appreciation of Lagrange, geometers ought to have
+seen a really new calculus, the philosophical character of which has not
+yet been determined with sufficient exactness. A very striking
+difference between this case and that of equations with a single
+independent variable consists, as has been already observed, in the
+arbitrary functions which take the place of the simple arbitrary
+constants, in order to give to the corresponding integrals all the
+proper generality.
+
+It is scarcely necessary to say that this higher branch of
+transcendental analysis is still entirely in its infancy, since, even in
+the most simple case, that of an equation of the first order between the
+partial derivatives of a single function with two independent variables,
+we are not yet completely able to reduce the integration to that of the
+ordinary differential equations. The integration of functions of several
+variables is much farther advanced in the case (infinitely more simple
+indeed) in which it has to do with only explicit differential formulas.
+We can then, in fact, when these formulas fulfil the necessary
+conditions of integrability, always reduce their integration to
+quadratures.
+
+
+_Other Subdivisions: different Orders of Differentiation._ A new general
+distinction, applicable as a subdivision to the integration of explicit
+or implicit differentials, with one variable or several, is drawn from
+the _higher or lower order of the differentials_: a distinction which,
+as we have above remarked, does not give rise to any special question in
+the differential calculus.
+
+Relatively to _explicit differentials_, whether of one variable or of
+several, the necessity of distinguishing their different orders belongs
+only to the extreme imperfection of the integral calculus. In fact, if
+we could always integrate every differential formula of the first order,
+the integration of a formula of the second order, or of any other, would
+evidently not form a new question, since, by integrating it at first in
+the first degree, we would arrive at the differential expression of the
+immediately preceding order, from which, by a suitable series of
+analogous integrations, we would be certain of finally arriving at the
+primitive function, the final object of these operations. But the little
+knowledge which we possess on integration of even the first order causes
+quite another state of affairs, so that a higher order of differentials
+produces new difficulties; for, having differential formulas of any
+order above the first, it may happen that we may be able to integrate
+them, either once, or several times in succession, and that we may still
+be unable to go back to the primitive functions, if these preliminary
+labours have produced, for the differentials of a lower order,
+expressions whose integrals are not known. This circumstance must occur
+so much the oftener (the number of known integrals being still very
+small), seeing that these successive integrals are generally very
+different functions from the derivatives which have produced them.
+
+With reference to _implicit differentials_, the distinction of orders is
+still more important; for, besides the preceding reason, the influence
+of which is evidently analogous in this case, and is even greater, it is
+easy to perceive that the higher order of the differential equations
+necessarily gives rise to questions of a new nature. In fact, even if we
+could integrate every equation of the first order relating to a single
+function, that would not be sufficient for obtaining the final integral
+of an equation of any order whatever, inasmuch as every differential
+equation is not reducible to that of an immediately inferior order.
+Thus, for example, if we have given any relation between _x_, _y_,
+_dx/dy_, and _d_²_y_/_dx_², to determine a function _y_ of a variable
+_x_, we shall not be able to deduce from it at once, after effecting a
+first integration, the corresponding differential relation between _x_,
+_y_, and _dy/dx_, from which, by a second integration, we could ascend
+to the primitive equations. This would not necessarily take place, at
+least without introducing new auxiliary functions, unless the proposed
+equation of the second order did not contain the required function _y_,
+together with its derivatives. As a general principle, differential
+equations will have to be regarded as presenting cases which are more
+and more _implicit_, as they are of a higher order, and which cannot be
+made to depend on one another except by special methods, the
+investigation of which consequently forms a new class of questions, with
+respect to which we as yet know scarcely any thing, even for functions
+of a single variable.[10]
+
+  [Footnote 10: The only important case of this class which has thus
+  far been completely treated is the general integration of _linear_
+  equations of any order whatever, with constant coefficients. Even
+  this case finally depends on the algebraic resolution of equations
+  of a degree equal to the order of differentiation.]
+
+_Another equivalent distinction._ Still farther, when we examine more
+profoundly this distinction of different orders of differential
+equations, we find that it can be always made to come under a final
+general distinction, relative to differential equations, which remains
+to be noticed. Differential equations with one or more independent
+variables may contain simply a single function, or (in a case evidently
+more complicated and more implicit, which corresponds to the
+differentiation of simultaneous implicit functions) we may have to
+determine at the same time several functions from the differential
+equations in which they are found united, together with their different
+derivatives. It is clear that such a state of the question necessarily
+presents a new special difficulty, that of separating the different
+functions desired, by forming for each, from the proposed differential
+equations, an isolated differential equation which does not contain the
+other functions or their derivatives. This preliminary labour, which is
+analogous to the elimination of algebra, is evidently indispensable
+before attempting any direct integration, since we cannot undertake
+generally (except by special artifices which are very rarely applicable)
+to determine directly several distinct functions at once.
+
+Now it is easy to establish the exact and necessary coincidence of this
+new distinction with the preceding one respecting the order of
+differential equations. We know, in fact, that the general method for
+isolating functions in simultaneous differential equations consists
+essentially in forming differential equations, separately in relation to
+each function, and of an order equal to the sum of all those of the
+different proposed equations. This transformation can always be
+effected. On the other hand, every differential equation of any order in
+relation to a single function might evidently always be reduced to the
+first order, by introducing a suitable number of auxiliary differential
+equations, containing at the same time the different anterior
+derivatives regarded as new functions to be determined. This method has,
+indeed, sometimes been actually employed with success, though it is not
+the natural one.
+
+Here, then, are two necessarily equivalent orders of conditions in the
+general theory of differential equations; the simultaneousness of a
+greater or smaller number of functions, and the higher or lower order of
+differentiation of a single function. By augmenting the order of the
+differential equations, we can isolate all the functions; and, by
+artificially multiplying the number of the functions, we can reduce all
+the equations to the first order. There is, consequently, in both cases,
+only one and the same difficulty from two different points of sight.
+But, however we may conceive it, this new difficulty is none the less
+real, and constitutes none the less, by its nature, a marked separation
+between the integration of equations of the first order and that of
+equations of a higher order. I prefer to indicate the distinction under
+this last form as being more simple, more general, and more logical.
+
+
+_Quadratures._ From the different considerations which have been
+indicated respecting the logical dependence of the various principal
+parts of the integral calculus, we see that the integration of explicit
+differential formulas of the first order and of a single variable is the
+necessary basis of all other integrations, which we never succeed in
+effecting but so far as we reduce them to this elementary case,
+evidently the only one which, by its nature, is capable of being treated
+directly. This simple fundamental integration is often designated by the
+convenient expression of _quadratures_, seeing that every integral of
+this kind, S_f_(_x_)_dx_, may, in fact, be regarded as representing the
+area of a curve, the equation of which in rectilinear co-ordinates would
+be _y_ = _f_(_x_). Such a class of questions corresponds, in the
+differential calculus, to the elementary case of the differentiation of
+explicit functions of a single variable. But the integral question is,
+by its nature, very differently complicated, and especially much more
+extensive than the differential question. This latter is, in fact,
+necessarily reduced, as we have seen, to the differentiation of the ten
+simple functions, the elements of all which are considered in analysis.
+On the other hand, the integration of compound functions does not
+necessarily follow from that of the simple functions, each combination
+of which may present special difficulties with respect to the integral
+calculus. Hence results the naturally indefinite extent, and the so
+varied complication of the question of _quadratures_, upon which, in
+spite of all the efforts of analysts, we still possess so little
+complete knowledge.
+
+In decomposing this question, as is natural, according to the different
+forms which may be assumed by the derivative function, we distinguish
+the case of _algebraic_ functions and that of _transcendental_
+functions.
+
+_Integration of Transcendental Functions._ The truly analytical
+integration of transcendental functions is as yet very little advanced,
+whether for _exponential_, or for _logarithmic_, or for _circular_
+functions. But a very small number of cases of these three different
+kinds have as yet been treated, and those chosen from among the
+simplest; and still the necessary calculations are in most cases
+extremely laborious. A circumstance which we ought particularly to
+remark in its philosophical connection is, that the different procedures
+of quadrature have no relation to any general view of integration, and
+consist of simple artifices very incoherent with each other, and very
+numerous, because of the very limited extent of each.
+
+One of these artifices should, however, here be noticed, which, without
+being really a method of integration, is nevertheless remarkable for its
+generality; it is the procedure invented by John Bernouilli, and known
+under the name of _integration by parts_, by means of which every
+integral may be reduced to another which is sometimes found to be more
+easy to be obtained. This ingenious relation deserves to be noticed for
+another reason, as having suggested the first idea of that
+transformation of integrals yet unknown, which has lately received a
+greater extension, and of which M. Fourier especially has made so new
+and important a use in the analytical questions produced by the theory
+of heat.
+
+_Integration of Algebraic Functions._ As to the integration of algebraic
+functions, it is farther advanced. However, we know scarcely any thing
+in relation to irrational functions, the integrals of which have been
+obtained only in extremely limited cases, and particularly by rendering
+them rational. The integration of rational functions is thus far the
+only theory of the integral calculus which has admitted of being treated
+in a truly complete manner; in a logical point of view, it forms, then,
+its most satisfactory part, but perhaps also the least important. It is
+even essential to remark, in order to have a just idea of the extreme
+imperfection of the integral calculus, that this case, limited as it is,
+is not entirely resolved except for what properly concerns integration
+viewed in an abstract manner; for, in the execution, the theory finds
+its progress most frequently quite stopped, independently of the
+complication of the calculations, by the imperfection of ordinary
+analysis, seeing that it makes the integration finally depend upon the
+algebraic resolution of equations, which greatly limits its use.
+
+To grasp in a general manner the spirit of the different procedures
+which are employed in quadratures, we must observe that, by their
+nature, they can be primitively founded only on the differentiation of
+the ten simple functions. The results of this, conversely considered,
+establish as many direct theorems of the integral calculus, the only
+ones which can be directly known. All the art of integration afterwards
+consists, as has been said in the beginning of this chapter, in reducing
+all the other quadratures, so far as is possible, to this small number
+of elementary ones, which unhappily we are in most cases unable to
+effect.
+
+_Singular Solutions._ In this systematic enumeration of the various
+essential parts of the integral calculus, considered in their logical
+relations, I have designedly neglected (in order not to break the chain
+of sequence) to consider a very important theory, which forms implicitly
+a portion of the general theory of the integration of differential
+equations, but which I ought here to notice separately, as being, so to
+speak, outside of the integral calculus, and being nevertheless of the
+greatest interest, both by its logical perfection and by the extent of
+its applications. I refer to what are called _Singular Solutions_ of
+differential equations, called sometimes, but improperly, _particular_
+solutions, which have been the subject of very remarkable investigations
+by Euler and Laplace, and of which Lagrange especially has presented
+such a beautiful and simple general theory. Clairaut, who first had
+occasion to remark their existence, saw in them a paradox of the
+integral calculus, since these solutions have the peculiarity of
+satisfying the differential equations without being comprised in the
+corresponding general integrals. Lagrange has since explained this
+paradox in the most ingenious and most satisfactory manner, by showing
+how such solutions are always derived from the general integral by the
+variation of the arbitrary constants. He was also the first to suitably
+appreciate the importance of this theory, and it is with good reason
+that he devoted to it so full a development in his "Calculus of
+Functions." In a logical point of view, this theory deserves all our
+attention by the character of perfect generality which it admits of,
+since Lagrange has given invariable and very simple procedures for
+finding the _singular_ solution of any differential equation which is
+susceptible of it; and, what is no less remarkable, these procedures
+require no integration, consisting only of differentiations, and are
+therefore always applicable. Differentiation has thus become, by a
+happy artifice, a means of compensating, in certain circumstances, for
+the imperfection of the integral calculus. Indeed, certain problems
+especially require, by their nature, the knowledge of these _singular_
+solutions; such, for example, in geometry, are all the questions in
+which a curve is to be determined from any property of its tangent or
+its osculating circle. In all cases of this kind, after having expressed
+this property by a differential equation, it will be, in its analytical
+relations, the _singular_ equation which will form the most important
+object of the inquiry, since it alone will represent the required curve;
+the general integral, which thenceforth it becomes unnecessary to know,
+designating only the system of the tangents, or of the osculating
+circles of this curve. We may hence easily understand all the importance
+of this theory, which seems to me to be not as yet sufficiently
+appreciated by most geometers.
+
+_Definite Integrals._ Finally, to complete our review of the vast
+collection of analytical researches of which is composed the integral
+calculus, properly so called, there remains to be mentioned one theory,
+very important in all the applications of the transcendental analysis,
+which I have had to leave outside of the system, as not being really
+destined for veritable integration, and proposing, on the contrary, to
+supply the place of the knowledge of truly analytical integrals, which
+are most generally unknown. I refer to the determination of _definite
+integrals_.
+
+The expression, always possible, of integrals in infinite series, may at
+first be viewed as a happy general means of compensating for the extreme
+imperfection of the integral calculus. But the employment of such
+series, because of their complication, and of the difficulty of
+discovering the law of their terms, is commonly of only moderate utility
+in the algebraic point of view, although sometimes very essential
+relations have been thence deduced. It is particularly in the
+arithmetical point of view that this procedure acquires a great
+importance, as a means of calculating what are called _definite
+integrals_, that is, the values of the required functions for certain
+determinate values of the corresponding variables.
+
+An inquiry of this nature exactly corresponds, in transcendental
+analysis, to the numerical resolution of equations in ordinary analysis.
+Being generally unable to obtain the veritable integral--named by
+opposition the _general_ or _indefinite_ integral; that is, the function
+which, differentiated, has produced the proposed differential
+formula--analysts have been obliged to employ themselves in determining
+at least, without knowing this function, the particular numerical values
+which it would take on assigning certain designated values to the
+variables. This is evidently resolving the arithmetical question without
+having previously resolved the corresponding algebraic one, which most
+generally is the most important one. Such an analysis is, then, by its
+nature, as imperfect as we have seen the numerical resolution of
+equations to be. It presents, like this last, a vicious confusion of
+arithmetical and algebraic considerations, whence result analogous
+inconveniences both in the purely logical point of view and in the
+applications. We need not here repeat the considerations suggested in
+our third chapter. But it will be understood that, unable as we almost
+always are to obtain the true integrals, it is of the highest importance
+to have been able to obtain this solution, incomplete and necessarily
+insufficient as it is. Now this has been fortunately attained at the
+present day for all cases, the determination of the value of definite
+integrals having been reduced to entirely general methods, which leave
+nothing to desire, in a great number of cases, but less complication in
+the calculations, an object towards which are at present directed all
+the special transformations of analysts. Regarding now this sort of
+_transcendental arithmetic_ as perfect, the difficulty in the
+applications is essentially reduced to making the proposed research
+depend, finally, on a simple determination of definite integrals, which
+evidently cannot always be possible, whatever analytical skill may be
+employed in effecting such a transformation.
+
+
+_Prospects of the Integral Calculus._ From the considerations indicated
+in this chapter, we see that, while the differential calculus
+constitutes by its nature a limited and perfect system, to which nothing
+essential remains to be added, the integral calculus, or the simple
+system of integration, presents necessarily an inexhaustible field for
+the activity of the human mind, independently of the indefinite
+applications of which the transcendental analysis is evidently
+susceptible. The general argument by which I have endeavoured, in the
+second chapter, to make apparent the impossibility of ever discovering
+the algebraic solution of equations of any degree and form whatsoever,
+has undoubtedly infinitely more force with regard to the search for a
+single method of integration, invariably applicable to all cases. "It
+is," says Lagrange, "one of those problems whose general solution we
+cannot hope for." The more we meditate on this subject, the more we
+will be convinced that such a research is utterly chimerical, as being
+far above the feeble reach of our intelligence; although the labours of
+geometers must certainly augment hereafter the amount of our knowledge
+respecting integration, and thus create methods of greater generality.
+The transcendental analysis is still too near its origin--there is
+especially too little time since it has been conceived in a truly
+rational manner--for us now to be able to have a correct idea of what it
+will hereafter become. But, whatever should be our legitimate hopes, let
+us not forget to consider, before all, the limits which are imposed by
+our intellectual constitution, and which, though not susceptible of a
+precise determination, have none the less an incontestable reality.
+
+I am induced to think that, when geometers shall have exhausted the most
+important applications of our present transcendental analysis, instead
+of striving to impress upon it, as now conceived, a chimerical
+perfection, they will rather create new resources by changing the mode
+of derivation of the auxiliary quantities introduced in order to
+facilitate the establishment of equations, and the formation of which
+might follow an infinity of other laws besides the very simple relation
+which has been chosen, according to the conception suggested in the
+first chapter. The resources of this nature appear to me susceptible of
+a much greater fecundity than those which would consist of merely
+pushing farther our present calculus of indirect functions. It is a
+suggestion which I submit to the geometers who have turned their
+thoughts towards the general philosophy of analysis.
+
+Finally, although, in the summary exposition which was the object of
+this chapter, I have had to exhibit the condition of extreme
+imperfection which still belongs to the integral calculus, the student
+would have a false idea of the general resources of the transcendental
+analysis if he gave that consideration too great an importance. It is
+with it, indeed, as with ordinary analysis, in which a very small amount
+of fundamental knowledge respecting the resolution of equations has been
+employed with an immense degree of utility. Little advanced as geometers
+really are as yet in the science of integrations, they have nevertheless
+obtained, from their scanty abstract conceptions, the solution of a
+multitude of questions of the first importance in geometry, in
+mechanics, in thermology, &c. The philosophical explanation of this
+double general fact results from the necessarily preponderating
+importance and grasp of _abstract_ branches of knowledge, the least of
+which is naturally found to correspond to a crowd of _concrete_
+researches, man having no other resource for the successive extension of
+his intellectual means than in the consideration of ideas more and more
+abstract, and still positive.
+
+       *       *       *       *       *
+
+In order to finish the complete exposition of the philosophical
+character of the transcendental analysis, there remains to be considered
+a final conception, by which the immortal Lagrange has rendered this
+analysis still better adapted to facilitate the establishment of
+equations in the most difficult problems, by considering a class of
+equations still more _indirect_ than the ordinary differential
+equations. It is the _Calculus_, or, rather, the _Method of Variations_;
+the general appreciation of which will be our next subject.
+
+
+
+
+CHAPTER V.
+
+THE CALCULUS OF VARIATIONS.
+
+
+In order to grasp with more ease the philosophical character of the
+_Method of Variations_, it will be well to begin by considering in a
+summary manner the special nature of the problems, the general
+resolution of which has rendered necessary the formation of this
+hyper-transcendental analysis. It is still too near its origin, and its
+applications have been too few, to allow us to obtain a sufficiently
+clear general idea of it from a purely abstract exposition of its
+fundamental theory.
+
+
+
+
+PROBLEMS GIVING RISE TO IT.
+
+
+The mathematical questions which have given birth to the _Calculus of
+Variations_ consist generally in the investigation of the _maxima_ and
+_minima_ of certain indeterminate integral formulas, which express the
+analytical law of such or such a phenomenon of geometry or mechanics,
+considered independently of any particular subject. Geometers for a long
+time designated all the questions of this character by the common name
+of _Isoperimetrical Problems_, which, however, is really suitable to
+only the smallest number of them.
+
+
+_Ordinary Questions of Maxima and Minima._ In the common theory of
+_maxima_ and _minima_, it is proposed to discover, with reference to a
+given function of one or more variables, what particular values must be
+assigned to these variables, in order that the corresponding value of
+the proposed function may be a _maximum_ or a _minimum_ with respect to
+those values which immediately precede and follow it; that is, properly
+speaking, we seek to know at what instant the function ceases to
+increase and commences to decrease, or reciprocally. The differential
+calculus is perfectly sufficient, as we know, for the general resolution
+of this class of questions, by showing that the values of the different
+variables, which suit either the maximum or minimum, must always reduce
+to zero the different first derivatives of the given function, taken
+separately with reference to each independent variable, and by
+indicating, moreover, a suitable characteristic for distinguishing the
+maximum from the minimum; consisting, in the case of a function of a
+single variable, for example, in the derived function of the second
+order taking a negative value for the maximum, and a positive value for
+the minimum. Such are the well-known fundamental conditions belonging to
+the greatest number of cases.
+
+
+_A new Class of Questions._ The construction of this general theory
+having necessarily destroyed the chief interest which questions of this
+kind had for geometers, they almost immediately rose to the
+consideration of a new order of problems, at once much more important
+and of much greater difficulty--those of _isoperimeters_. It is, then,
+no longer _the values of the variables_ belonging to the maximum or the
+minimum of a given function that it is required to determine. It is _the
+form of the function itself_ which is required to be discovered, from
+the condition of the maximum or of the minimum of a certain definite
+integral, merely indicated, which depends upon that function.
+
+
+_Solid of least Resistance._ The oldest question of this nature is that
+of _the solid of least resistance_, treated by Newton in the second book
+of the Principia, in which he determines what ought to be the meridian
+curve of a solid of revolution, in order that the resistance experienced
+by that body in the direction of its axis may be the least possible. But
+the course pursued by Newton, from the nature of his special method of
+transcendental analysis, had not a character sufficiently simple,
+sufficiently general, and especially sufficiently analytical, to attract
+geometers to this new order of problems. To effect this, the application
+of the infinitesimal method was needed; and this was done, in 1695, by
+John Bernouilli, in proposing the celebrated problem of the
+_Brachystochrone_.
+
+This problem, which afterwards suggested such a long series of analogous
+questions, consists in determining the curve which a heavy body must
+follow in order to descend from one point to another in the shortest
+possible time. Limiting the conditions to the simple fall in a vacuum,
+the only case which was at first considered, it is easily found that the
+required curve must be a reversed cycloid with a horizontal base, and
+with its origin at the highest point. But the question may become
+singularly complicated, either by taking into account the resistance of
+the medium, or the change in the intensity of gravity.
+
+
+_Isoperimeters._ Although this new class of problems was in the first
+place furnished by mechanics, it is in geometry that the principal
+investigations of this character were subsequently made. Thus it was
+proposed to discover which, among all the curves of the same contour
+traced between two given points, is that whose area is a maximum or
+minimum, whence has come the name of _Problem of Isoperimeters_; or it
+was required that the maximum or minimum should belong to the surface
+produced by the revolution of the required curve about an axis, or to
+the corresponding volume; in other cases, it was the vertical height of
+the center of gravity of the unknown curve, or of the surface and of the
+volume which it might generate, which was to become a maximum or
+minimum, &c. Finally, these problems were varied and complicated almost
+to infinity by the Bernouillis, by Taylor, and especially by Euler,
+before Lagrange reduced their solution to an abstract and entirely
+general method, the discovery of which has put a stop to the enthusiasm
+of geometers for such an order of inquiries. This is not the place for
+tracing the history of this subject. I have only enumerated some of the
+simplest principal questions, in order to render apparent the original
+general object of the method of variations.
+
+
+_Analytical Nature of these Problems._ We see that all these problems,
+considered in an analytical point of view, consist, by their nature, in
+determining what form a certain unknown function of one or more
+variables ought to have, in order that such or such an integral,
+dependent upon that function, shall have, within assigned limits, a
+value which is a maximum or a minimum with respect to all those which it
+would take if the required function had any other form whatever.
+
+Thus, for example, in the problem of the _brachystochrone_, it is well
+known that if _y_ = _f(z)_, _x_ = π(_z_), are the rectilinear equations
+of the required curve, supposing the axes of _x_ and of _y_ to be
+horizontal, and the axis of _z_ to be vertical, the time of the fall of
+a heavy body in that curve from the point whose ordinate is _z₁_, to
+that whose ordinate is _z₂_, is expressed in general terms by the
+definite integral
+
+  ∫_{_z₂_}^{_z₁_}√(1 + (_f'(z))²_ + (π'(_z_))²/(2_gz_))_dz._
+
+It is, then, necessary to find what the two unknown functions _f_ and π
+must be, in order that this integral may be a minimum.
+
+In the same way, to demand what is the curve among all plane
+isoperimetrical curves, which includes the greatest area, is the same
+thing as to propose to find, among all the functions _f(x)_ which can
+give a certain constant value to the integral
+
+  ∫_dx_√(1 + (_f'(x)_ )²),
+
+that one which renders the integral ∫_f(x)dx_, taken between the same
+limits, a maximum. It is evidently always so in other questions of this
+class.
+
+
+_Methods of the older Geometers._ In the solutions which geometers
+before Lagrange gave of these problems, they proposed, in substance, to
+reduce them to the ordinary theory of maxima and minima. But the means
+employed to effect this transformation consisted in special simple
+artifices peculiar to each case, and the discovery of which did not
+admit of invariable and certain rules, so that every really new question
+constantly reproduced analogous difficulties, without the solutions
+previously obtained being really of any essential aid, otherwise than by
+their discipline and training of the mind. In a word, this branch of
+mathematics presented, then, the necessary imperfection which always
+exists when the part common to all questions of the same class has not
+yet been distinctly grasped in order to be treated in an abstract and
+thenceforth general manner.
+
+
+
+
+METHOD OF LAGRANGE.
+
+
+Lagrange, in endeavouring to bring all the different problems of
+isoperimeters to depend upon a common analysis, organized into a
+distinct calculus, was led to conceive a new kind of differentiation, to
+which he has applied the characteristic δ, reserving the characteristic
+_d_ for the common differentials. These differentials of a new species,
+which he has designated under the name of _Variations_, consist of the
+infinitely small increments which the integrals receive, not by virtue
+of analogous increments on the part of the corresponding variables, as
+in the ordinary transcendental analysis, but by supposing that the
+_form_ of the function placed under the sign of integration undergoes an
+infinitely small change. This distinction is easily conceived with
+reference to curves, in which we see the ordinate, or any other variable
+of the curve, admit of two sorts of differentials, evidently very
+different, according as we pass from one point to another infinitely
+near it on the same curve, or to the corresponding point of the
+infinitely near curve produced by a certain determinate modification of
+the first curve.[11] It is moreover clear, that the relative
+_variations_ of different magnitudes connected with each other by any
+laws whatever are calculated, all but the characteristic, almost exactly
+in the same manner as the differentials. Finally, from the general
+notion of _variations_ are in like manner deduced the fundamental
+principles of the algorithm proper to this method, consisting simply in
+the evidently permissible liberty of transposing at will the
+characteristics specially appropriated to variations, before or after
+those which correspond to the ordinary differentials.
+
+  [Footnote 11: Leibnitz had already considered the comparison of one
+  curve with an other infinitely near to it, calling it
+  "_Differentiatio de curva in curvam_." But this comparison had no
+  analogy with the conception of Lagrange, the curves of Leibnitz
+  being embraced in the same general equation, from which they were
+  deduced by the simple change of an arbitrary constant.]
+
+This abstract conception having been once formed, Lagrange was able to
+reduce with ease, and in the most general manner, all the problems of
+_Isoperimeters_ to the simple ordinary theory of _maxima_ and _minima_.
+To obtain a clear idea of this great and happy transformation, we must
+previously consider an essential distinction which arises in the
+different questions of isoperimeters.
+
+
+_Two Classes of Questions._ These investigations must, in fact, be
+divided into two general classes, according as the maxima and minima
+demanded are _absolute_ or _relative_, to employ the abridged
+expressions of geometers.
+
+
+_Questions of the first Class._ The _first case_ is that in which the
+indeterminate definite integrals, the maximum or minimum of which is
+sought, are not subjected, by the nature of the problem, to any
+condition; as happens, for example, in the problem of the
+_brachystochrone_, in which the choice is to be made between all
+imaginable curves. The _second_ case takes place when, on the contrary,
+the variable integrals can vary only according to certain conditions,
+which usually consist in other definite integrals (which depend, in like
+manner, upon the required functions) always retaining the same given
+value; as, for example, in all the geometrical questions relating to
+real _isoperimetrical_ figures, and in which, by the nature of the
+problem, the integral relating to the length of the curve, or to the
+area of the surface, must remain constant during the variation of that
+integral which is the object of the proposed investigation.
+
+The _Calculus of Variations_ gives immediately the general solution of
+questions of the former class; for it evidently follows, from the
+ordinary theory of maxima and minima, that the required relation must
+reduce to zero the _variation_ of the proposed integral with reference
+to each independent variable; which gives the condition common to both
+the maximum and the minimum: and, as a characteristic for distinguishing
+the one from the other, that the variation of the second order of the
+same integral must be negative for the maximum and positive for the
+minimum. Thus, for example, in the problem of the brachystochrone, we
+will have, in order to determine the nature of the curve sought, the
+equation of condition
+
+  δ∫_{_z₂_}^{_z₁_}√([1 + (_f'(z)_)² + (π'(_z_))²]/(2_gz_))_dz_ = 0,
+
+which, being decomposed into two, with respect to the two unknown
+functions _f_ and π, which are independent of each other, will
+completely express the analytical definition of the required curve. The
+only difficulty peculiar to this new analysis consists in the
+elimination of the characteristic δ, for which the calculus of
+variations furnishes invariable and complete rules, founded, in general,
+on the method of "integration by parts," from which Lagrange has thus
+derived immense advantage. The constant object of this first analytical
+elaboration (which this is not the place for treating in detail) is to
+arrive at real differential equations, which can always be done; and
+thereby the question comes under the ordinary transcendental analysis,
+which furnishes the solution, at least so far as to reduce it to pure
+algebra if the integration can be effected. The general object of the
+method of variations is to effect this transformation, for which
+Lagrange has established rules, which are simple, invariable, and
+certain of success.
+
+
+_Equations of Limits._ Among the greatest special advantages of the
+method of variations, compared with the previous isolated solutions of
+isoperimetrical problems, is the important consideration of what
+Lagrange calls _Equations of Limits_, which were entirely neglected
+before him, though without them the greater part of the particular
+solutions remained necessarily incomplete. When the limits of the
+proposed integrals are to be fixed, their variations being zero, there
+is no occasion for noticing them. But it is no longer so when these
+limits, instead of being rigorously invariable, are only subjected to
+certain conditions; as, for example, if the two points between which the
+required curve is to be traced are not fixed, and have only to remain
+upon given lines or surfaces. Then it is necessary to pay attention to
+the variation of their co-ordinates, and to establish between them the
+relations which correspond to the equations of these lines or of these
+surfaces.
+
+
+_A more general consideration._ This essential consideration is only the
+final complement of a more general and more important consideration
+relative to the variations of different independent variables. If these
+variables are really independent of one another, as when we compare
+together all the imaginable curves susceptible of being traced between
+two points, it will be the same with their variations, and,
+consequently, the terms relating to each of these variations will have
+to be separately equal to zero in the general equation which expresses
+the maximum or the minimum. But if, on the contrary, we suppose the
+variables to be subjected to any fixed conditions, it will be necessary
+to take notice of the resulting relation between their variations, so
+that the number of the equations into which this general equation is
+then decomposed is always equal to only the number of the variables
+which remain truly independent. It is thus, for example, that instead of
+seeking for the shortest path between any two points, in choosing it
+from among all possible ones, it may be proposed to find only what is
+the shortest among all those which may be taken on any given surface; a
+question the general solution of which forms certainly one of the most
+beautiful applications of the method of variations.
+
+_Questions of the second Class._ Problems in which such modifying
+conditions are considered approach very nearly, in their nature, to the
+second general class of applications of the method of variations,
+characterized above as consisting in the investigation of _relative_
+maxima and minima. There is, however, this essential difference between
+the two cases, that in this last the modification is expressed by an
+integral which depends upon the function sought, while in the other it
+is designated by a finite equation which is immediately given. It is
+hence apparent that the investigation of _relative_ maxima and minima is
+constantly and necessarily more complicated than that of _absolute_
+maxima and minima. Luckily, a very important general theory, discovered
+by the genius of the great Euler before the invention of the Calculus of
+Variations, gives a uniform and very simple means of making one of
+these two classes of questions dependent on the other. It consists in
+this, that if we add to the integral which is to be a maximum or a
+minimum, a constant and indeterminate multiple of that one which, by the
+nature of the problem, is to remain constant, it will be sufficient to
+seek, by the general method of Lagrange above indicated, the _absolute_
+maximum or minimum of this whole expression. It can be easily conceived,
+indeed, that the part of the complete variation which would proceed from
+the last integral must be equal to zero (because of the constant
+character of this last) as well as the portion due to the first
+integral, which disappears by virtue of the maximum or minimum state.
+These two conditions evidently unite to produce, in that respect,
+effects exactly alike.
+
+Such is a sketch of the general manner in which the method of variation
+is applied to all the different questions which compose what is called
+the _Theory of Isoperimeters_. It will undoubtedly have been remarked in
+this summary exposition how much use has been made in this new analysis
+of the second fundamental property of the transcendental analysis
+noticed in the third chapter, namely, the generality of the
+infinitesimal expressions for the representation of the same geometrical
+or mechanical phenomenon, in whatever body it may be considered. Upon
+this generality, indeed, are founded, by their nature, all the solutions
+due to the method of variations. If a single formula could not express
+the length or the area of any curve whatever; if another fixed formula
+could not designate the time of the fall of a heavy body, according to
+whatever line it may descend, &c., how would it have been possible to
+resolve questions which unavoidably require, by their nature, the
+simultaneous consideration of all the cases which can be determined in
+each phenomenon by the different subjects which exhibit it.
+
+
+_Other Applications of this Method._ Notwithstanding the extreme
+importance of the theory of isoperimeters, and though the method of
+variations had at first no other object than the logical and general
+solution of this order of problems, we should still have but an
+incomplete idea of this beautiful analysis if we limited its destination
+to this. In fact, the abstract conception of two distinct natures of
+differentiation is evidently applicable not only to the cases for which
+it was created, but also to all those which present, for any reason
+whatever, two different manners of making the same magnitudes vary. It
+is in this way that Lagrange himself has made, in his "_Méchanique
+Analytique_," an extensive and important application of his calculus of
+variations, by employing it to distinguish the two sorts of changes
+which are naturally presented by the questions of rational mechanics for
+the different points which are considered, according as we compare the
+successive positions which are occupied, in virtue of its motion, by the
+same point of each body in two consecutive instants, or as we pass from
+one point of the body to another in the same instant. One of these
+comparisons produces ordinary differentials; the other gives rise to
+_variations_, which, there as every where, are only differentials taken
+under a new point of view. Such is the general acceptation in which we
+should conceive the Calculus of Variations, in order suitably to
+appreciate the importance of this admirable logical instrument, the
+most powerful that the human mind has as yet constructed.
+
+The method of variations being only an immense extension of the general
+transcendental analysis, I have no need of proving specially that it is
+susceptible of being considered under the different fundamental points
+of view which the calculus of indirect functions, considered as a whole,
+admits of. Lagrange invented the Calculus of Variations in accordance
+with the infinitesimal conception, and, indeed, long before he undertook
+the general reconstruction of the transcendental analysis. When he had
+executed this important reformation, he easily showed how it could also
+be applied to the Calculus of Variations, which he expounded with all
+the proper development, according to his theory of derivative functions.
+But the more that the use of the method of variations is difficult of
+comprehension, because of the higher degree of abstraction of the ideas
+considered, the more necessary is it, in its application, to economize
+the exertions of the mind, by adopting the most direct and rapid
+analytical conception, namely, that of Leibnitz. Accordingly, Lagrange
+himself has constantly preferred it in the important use which he has
+made of the Calculus of Variations in his "Analytical Mechanics." In
+fact, there does not exist the least hesitation in this respect among
+geometers.
+
+
+
+
+ITS RELATIONS TO THE ORDINARY CALCULUS.
+
+
+In order to make as clear as possible the philosophical character of the
+Calculus of Variations, I think that I should, in conclusion, briefly
+indicate a consideration which seems to me important, and by which I can
+approach it to the ordinary transcendental analysis in a higher degree
+than Lagrange seems to me to have done.[12]
+
+  [Footnote 12: I propose hereafter to develop this new
+  consideration, in a special work upon the _Calculus of Variations_,
+  intended to present this hyper-transcendental analysis in a new
+  point of view, which I think adapted to extend its general range.]
+
+We noticed in the preceding chapter the formation of the _calculus of
+partial differences_, created by D'Alembert, as having introduced into
+the transcendental analysis a new elementary idea; the notion of two
+kinds of increments, distinct and independent of one another, which a
+function of two variables may receive by virtue of the change of each
+variable separately. It is thus that the vertical ordinate of a surface,
+or any other magnitude which is referred to it, varies in two manners
+which are quite distinct, and which may follow the most different laws,
+according as we increase either the one or the other of the two
+horizontal co-ordinates. Now such a consideration seems to me very
+nearly allied, by its nature, to that which serves as the general basis
+of the method of variations. This last, indeed, has in reality done
+nothing but transfer to the independent variables themselves the
+peculiar conception which had been already adopted for the functions of
+these variables; a modification which has remarkably enlarged its use. I
+think, therefore, that so far as regards merely the fundamental
+conceptions, we may consider the calculus created by D'Alembert as
+having established a natural and necessary transition between the
+ordinary infinitesimal calculus and the calculus of variations; such a
+derivation of which seems to be adapted to make the general notion more
+clear and simple.
+
+According to the different considerations indicated in this chapter, the
+method of variations presents itself as the highest degree of perfection
+which the analysis of indirect functions has yet attained. In its
+primitive state, this last analysis presented itself as a powerful
+general means of facilitating the mathematical study of natural
+phenomena, by introducing, for the expression of their laws, the
+consideration of auxiliary magnitudes, chosen in such a manner that
+their relations are necessarily more simple and more easy to obtain than
+those of the direct magnitudes. But the formation of these differential
+equations was not supposed to admit of any general and abstract rules.
+Now the Analysis of Variations, considered in the most philosophical
+point of view, may be regarded as essentially destined, by its nature,
+to bring within the reach of the calculus the actual establishment of
+the differential equations; for, in a great number of important and
+difficult questions, such is the general effect of the _varied_
+equations, which, still more _indirect_ than the simple differential
+equations with respect to the special objects of the investigation, are
+also much more easy to form, and from which we may then, by invariable
+and complete analytical methods, the object of which is to eliminate the
+new order of auxiliary infinitesimals which have been introduced, deduce
+those ordinary differential equations which it would often have been
+impossible to establish directly. The method of variations forms, then,
+the most sublime part of that vast system of mathematical analysis,
+which, setting out from the most simple elements of algebra, organizes,
+by an uninterrupted succession of ideas, general methods more and more
+powerful, for the study of natural philosophy, and which, in its whole,
+presents the most incomparably imposing and unequivocal monument of the
+power of the human intellect.
+
+We must, however, also admit that the conceptions which are habitually
+considered in the method of variations being, by their nature, more
+indirect, more general, and especially more abstract than all others,
+the employment of such a method exacts necessarily and continuously the
+highest known degree of intellectual exertion, in order never to lose
+sight of the precise object of the investigation, in following
+reasonings which offer to the mind such uncertain resting-places, and in
+which signs are of scarcely any assistance. We must undoubtedly
+attribute in a great degree to this difficulty the little real use which
+geometers, with the exception of Lagrange, have as yet made of such an
+admirable conception.
+
+
+
+
+CHAPTER VI.
+
+THE CALCULUS OF FINITE DIFFERENCES.
+
+
+The different fundamental considerations indicated in the five preceding
+chapters constitute, in reality, all the essential bases of a complete
+exposition of mathematical analysis, regarded in the philosophical point
+of view. Nevertheless, in order not to neglect any truly important
+general conception relating to this analysis, I think that I should here
+very summarily explain the veritable character of a kind of calculus
+which is very extended, and which, though at bottom it really belongs to
+ordinary analysis, is still regarded as being of an essentially distinct
+nature. I refer to the _Calculus of Finite Differences_, which will be
+the special subject of this chapter.
+
+
+_Its general Character._ This calculus, created by Taylor, in his
+celebrated work entitled _Methodus Incrementorum_, consists essentially
+in the consideration of the finite increments which functions receive as
+a consequence of analogous increments on the part of the corresponding
+variables. These increments or _differences_, which take the
+characteristic Δ, to distinguish them from _differentials_, or
+infinitely small increments, may be in their turn regarded as new
+functions, and become the subject of a second similar consideration, and
+so on; from which results the notion of differences of various
+successive orders, analogous, at least in appearance, to the consecutive
+orders of differentials. Such a calculus evidently presents, like the
+calculus of indirect functions, two general classes of questions:
+
+1°. To determine the successive differences of all the various
+analytical functions of one or more variables, as the result of a
+definite manner of increase of the independent variables, which are
+generally supposed to augment in arithmetical progression.
+
+2°. Reciprocally, to start from these differences, or, more generally,
+from any equations established between them, and go back to the
+primitive functions themselves, or to their corresponding relations.
+
+Hence follows the decomposition of this calculus into two distinct ones,
+to which are usually given the names of the _Direct_, and the _Inverse
+Calculus of Finite Differences_, the latter being also sometimes called
+the _Integral Calculus of Finite Differences_. Each of these would,
+also, evidently admit of a logical distribution similar to that given in
+the fourth chapter for the differential and the integral calculus.
+
+
+_Its true Nature._ There is no doubt that Taylor thought that by such a
+conception he had founded a calculus of an entirely new nature,
+absolutely distinct from ordinary analysis, and more general than the
+calculus of Leibnitz, although resting on an analogous consideration. It
+is in this way, also, that almost all geometers have viewed the analysis
+of Taylor; but Lagrange, with his usual profundity, clearly perceived
+that these properties belonged much more to the forms and to the
+notations employed by Taylor than to the substance of his theory. In
+fact, that which constitutes the peculiar character of the analysis of
+Leibnitz, and makes of it a truly distinct and superior calculus, is the
+circumstance that the derived functions are in general of an entirely
+different nature from the primitive functions, so that they may give
+rise to more simple and more easily formed relations: whence result the
+admirable fundamental properties of the transcendental analysis, which
+have been already explained. But it is not so with the _differences_
+considered by Taylor; for these differences are, by their nature,
+functions essentially similar to those which have produced them, a
+circumstance which renders them unsuitable to facilitate the
+establishment of equations, and prevents their leading to more general
+relations. Every equation of finite differences is truly, at bottom, an
+equation directly relating to the very magnitudes whose successive
+states are compared. The scaffolding of new signs, which produce an
+illusion respecting the true character of these equations, disguises it,
+however, in a very imperfect manner, since it could always be easily
+made apparent by replacing the _differences_ by the equivalent
+combinations of the primitive magnitudes, of which they are really only
+the abridged designations. Thus the calculus of Taylor never has
+offered, and never can offer, in any question of geometry or of
+mechanics, that powerful general aid which we have seen to result
+necessarily from the analysis of Leibnitz. Lagrange has, moreover, very
+clearly proven that the pretended analogy observed between the calculus
+of differences and the infinitesimal calculus was radically vicious, in
+this way, that the formulas belonging to the former calculus can never
+furnish, as particular cases, those which belong to the latter, the
+nature of which is essentially distinct.
+
+From these considerations I am led to think that the calculus of finite
+differences is, in general, improperly classed with the transcendental
+analysis proper, that is, with the calculus of indirect functions. I
+consider it, on the contrary, in accordance with the views of Lagrange,
+to be only a very extensive and very important branch of ordinary
+analysis, that is to say, of that which I have named the calculus of
+direct functions, the equations which it considers being always, in
+spite of the notation, simple _direct_ equations.
+
+
+
+
+GENERAL THEORY OF SERIES.
+
+
+To sum up as briefly as possible the preceding explanation, the calculus
+of Taylor ought to be regarded as having constantly for its true object
+the general theory of _Series_, the most simple cases of which had alone
+been considered before that illustrious geometer. I ought, properly, to
+have mentioned this important theory in treating, in the second chapter,
+of Algebra proper, of which it is such an extensive branch. But, in
+order to avoid a double reference to it, I have preferred to notice it
+only in the consideration of the calculus of finite differences, which,
+reduced to its most simple general expression, is nothing but a complete
+logical study of questions relating to _series_.
+
+Every _Series_, or succession of numbers deduced from one another
+according to any constant law, necessarily gives rise to these two
+fundamental questions:
+
+1°. The law of the series being supposed known, to find the expression
+for its general term, so as to be able to calculate immediately any term
+whatever without being obliged to form successively all the preceding
+terms.
+
+2°. In the same circumstances, to determine the _sum_ of any number of
+terms of the series by means of their places, so that it can be known
+without the necessity of continually adding these terms together.
+
+These two fundamental questions being considered to be resolved, it may
+be proposed, reciprocally, to find the law of a series from the form of
+its general term, or the expression of the sum. Each of these different
+problems has so much the more extent and difficulty, as there can be
+conceived a greater number of different _laws_ for the series, according
+to the number of preceding terms on which each term directly depends,
+and according to the function which expresses that dependence. We may
+even consider series with several variable indices, as Laplace has done
+in his "Analytical Theory of Probabilities," by the analysis to which he
+has given the name of _Theory of Generating Functions_, although it is
+really only a new and higher branch of the calculus of finite
+differences or of the general theory of series.
+
+These general views which I have indicated give only an imperfect idea
+of the truly infinite extent and variety of the questions to which
+geometers have risen by means of this single consideration of series, so
+simple in appearance and so limited in its origin. It necessarily
+presents as many different cases as the algebraic resolution of
+equations, considered in its whole extent; and it is, by its nature,
+much more complicated, so much, indeed, that it always needs this last
+to conduct it to a complete solution. We may, therefore, anticipate what
+must still be its extreme imperfection, in spite of the successive
+labours of several geometers of the first order. We do not, indeed,
+possess as yet the complete and logical solution of any but the most
+simple questions of this nature.
+
+
+_Its identity with this Calculus._ It is now easy to conceive the
+necessary and perfect identity, which has been already announced,
+between the calculus of finite differences and the theory of series
+considered in all its bearings. In fact, every differentiation after the
+manner of Taylor evidently amounts to finding the _law_ of formation of
+a series with one or with several variable indices, from the expression
+of its general term; in the same way, every analogous integration may be
+regarded as having for its object the summation of a series, the general
+term of which would be expressed by the proposed difference. In this
+point of view, the various problems of the calculus of differences,
+direct or inverse, resolved by Taylor and his successors, have really a
+very great value, as treating of important questions relating to series.
+But it is very doubtful if the form and the notation introduced by
+Taylor really give any essential facility in the solution of questions
+of this kind. It would be, perhaps, more advantageous for most cases,
+and certainly more logical, to replace the _differences_ by the terms
+themselves, certain combinations of which they represent. As the
+calculus of Taylor does not rest on a truly distinct fundamental idea,
+and has nothing peculiar to it but its system of signs, there could
+never really be any important advantage in considering it as detached
+from ordinary analysis, of which it is, in reality, only an immense
+branch. This consideration of _differences_, most generally useless,
+even if it does not cause complication, seems to me to retain the
+character of an epoch in which, analytical ideas not being sufficiently
+familiar to geometers, they were naturally led to prefer the special
+forms suitable for simple numerical comparisons.
+
+
+
+
+PERIODIC OR DISCONTINUOUS FUNCTIONS.
+
+
+However that may be, I must not finish this general appreciation of the
+calculus of finite differences without noticing a new conception to
+which it has given birth, and which has since acquired a great
+importance. It is the consideration of those periodic or discontinuous
+functions which preserve the same value for an infinite series of values
+of the corresponding variables, subjected to a certain law, and which
+must be necessarily added to the integrals of the equations of finite
+differences in order to render them sufficiently general, as simple
+arbitrary constants are added to all quadratures in order to complete
+their generality. This idea, primitively introduced by Euler, has since
+been the subject of extended investigation by M. Fourier, who has made
+new and important applications of it in his mathematical theory of heat.
+
+
+
+
+APPLICATIONS OF THIS CALCULUS.
+
+
+_Series._ Among the principal general applications which have been made
+of the calculus of finite differences, it would be proper to place in
+the first rank, as the most extended and the most important, the
+solution of questions relating to series; if, as has been shown, the
+general theory of series ought not to be considered as constituting, by
+its nature, the actual foundation of the calculus of Taylor.
+
+
+_Interpolations._ This great class of problems being then set aside, the
+most essential of the veritable applications of the analysis of Taylor
+is, undoubtedly, thus far, the general method of _interpolations_, so
+frequently and so usefully employed in the investigation of the
+empirical laws of natural phenomena. The question consists, as is well
+known, in intercalating between certain given numbers other intermediate
+numbers, subjected to the same law which we suppose to exist between the
+first. We can abundantly verify, in this principal application of the
+calculus of Taylor, how truly foreign and often inconvenient is the
+consideration of _differences_ with respect to the questions which
+depend on that analysis. Indeed, Lagrange has replaced the formulas of
+interpolation, deduced from the ordinary algorithm of the calculus of
+finite differences, by much simpler general formulas, which are now
+almost always preferred, and which have been found directly, without
+making any use of the notion of _differences_, which only complicates
+the question.
+
+
+_Approximate Rectification, &c._ A last important class of applications
+of the calculus of finite differences, which deserves to be
+distinguished from the preceding, consists in the eminently useful
+employment made of it in geometry for determining by approximation the
+length and the area of any curve, and in the same way the cubature of a
+body of any form whatever. This procedure (which may besides be
+conceived abstractly as depending on the same analytical investigation
+as the question of interpolation) frequently offers a valuable
+supplement to the entirely logical geometrical methods which often lead
+to integrations, which we do not yet know how to effect, or to
+calculations of very complicated execution.
+
+       *       *       *       *       *
+
+Such are the various principal considerations to be noticed with respect
+to the calculus of finite differences. This examination completes the
+proposed philosophical outline of ABSTRACT MATHEMATICS.
+
+
+CONCRETE MATHEMATICS will now be the subject of a similar labour. In it
+we shall particularly devote ourselves to examining how it has been
+possible (supposing the general science of the calculus to be perfect),
+by invariable procedures, to reduce to pure questions of analysis all
+the problems which can be presented by _Geometry_ and _Mechanics_, and
+thus to impress on these two fundamental bases of natural philosophy a
+degree of precision and especially of unity; in a word, a character of
+high perfection, which could be communicated to them by such a course
+alone.
+
+
+
+
+BOOK II.
+
+GEOMETRY.
+
+
+
+
+BOOK II.
+
+GEOMETRY.
+
+
+
+
+CHAPTER I.
+
+GENERAL VIEW OF GEOMETRY.
+
+
+_Its true Nature._ After the general exposition of the philosophical
+character of concrete mathematics, compared with that of abstract
+mathematics, given in the introductory chapter, it need not here be
+shown in a special manner that geometry must be considered as a true
+natural science, only much more simple, and therefore much more perfect,
+than any other. This necessary perfection of geometry, obtained
+essentially by the application of mathematical analysis, which it so
+eminently admits, is apt to produce erroneous views of the real nature
+of this fundamental science, which most minds at present conceive to be
+a purely logical science quite independent of observation. It is
+nevertheless evident, to any one who examines with attention the
+character of geometrical reasonings, even in the present state of
+abstract geometry, that, although the facts which are considered in it
+are much more closely united than those relating to any other science,
+still there always exists, with respect to every body studied by
+geometers, a certain number of primitive phenomena, which, since they
+are not established by any reasoning, must be founded on observation
+alone, and which form the necessary basis of all the deductions.
+
+The scientific superiority of geometry arises from the phenomena which
+it considers being necessarily the most universal and the most simple of
+all. Not only may all the bodies of nature give rise to geometrical
+inquiries, as well as mechanical ones, but still farther, geometrical
+phenomena would still exist, even though all the parts of the universe
+should be considered as immovable. Geometry is then, by its nature, more
+general than mechanics. At the same time, its phenomena are more simple,
+for they are evidently independent of mechanical phenomena, while these
+latter are always complicated with the former. The same relations hold
+good in comparing geometry with abstract thermology.
+
+For these reasons, in our classification we have made geometry the first
+part of concrete mathematics; that part the study of which, in addition
+to its own importance, serves as the indispensable basis of all the
+rest.
+
+Before considering directly the philosophical study of the different
+orders of inquiries which constitute our present geometry, we should
+obtain a clear and exact idea of the general destination of that
+science, viewed in all its bearings. Such is the object of this chapter.
+
+
+_Definition._ Geometry is commonly defined in a very vague and entirely
+improper manner, as being _the science of extension_. An improvement on
+this would be to say that geometry has for its object the _measurement_
+of extension; but such an explanation would be very insufficient,
+although at bottom correct, and would be far from giving any idea of the
+true general character of geometrical science.
+
+To do this, I think that I should first explain _two fundamental ideas_,
+which, very simple in themselves, have been singularly obscured by the
+employment of metaphysical considerations.
+
+
+_The Idea of Space._ The first is that of _Space_. This conception
+properly consists simply in this, that, instead of considering extension
+in the bodies themselves, we view it in an indefinite medium, which we
+regard as containing all the bodies of the universe. This notion is
+naturally suggested to us by observation, when we think of the
+_impression_ which a body would leave in a fluid in which it had been
+placed. It is clear, in fact, that, as regards its geometrical
+relations, such an _impression_ may be substituted for the body itself,
+without altering the reasonings respecting it. As to the physical nature
+of this indefinite _space_, we are spontaneously led to represent it to
+ourselves, as being entirely analogous to the actual medium in which we
+live; so that if this medium was liquid instead of gaseous, our
+geometrical _space_ would undoubtedly be conceived as liquid also. This
+circumstance is, moreover, only very secondary, the essential object of
+such a conception being only to make us view extension separately from
+the bodies which manifest it to us. We can easily understand in advance
+the importance of this fundamental image, since it permits us to study
+geometrical phenomena in themselves, abstraction being made of all the
+other phenomena which constantly accompany them in real bodies, without,
+however, exerting any influence over them. The regular establishment of
+this general abstraction must be regarded as the first step which has
+been made in the rational study of geometry, which would have been
+impossible if it had been necessary to consider, together with the form
+and the magnitude of bodies, all their other physical properties. The
+use of such an hypothesis, which is perhaps the most ancient
+philosophical conception created by the human mind, has now become so
+familiar to us, that we have difficulty in exactly estimating its
+importance, by trying to appreciate the consequences which would result
+from its suppression.
+
+
+_Different Kinds of Extension._ The second preliminary geometrical
+conception which we have to examine is that of the different kinds of
+extension, designated by the words _volume_, _surface_, _line_, and even
+_point_, and of which the ordinary explanation is so unsatisfactory.[13]
+
+  [Footnote 13: Lacroix has justly criticised the expression of
+  _solid_, commonly used by geometers to designate a _volume_. It is
+  certain, in fact, that when we wish to consider separately a
+  certain portion of indefinite space, conceived as gaseous, we
+  mentally solidify its exterior envelope, so that a _line_ and a
+  _surface_ are habitually, to our minds, just as _solid_ as a
+  _volume_. It may also be remarked that most generally, in order
+  that bodies may penetrate one another with more facility, we are
+  obliged to imagine the interior of the _volumes_ to be hollow,
+  which renders still more sensible the impropriety of the word
+  _solid_.]
+
+Although it is evidently impossible to conceive any extension absolutely
+deprived of any one of the three fundamental dimensions, it is no less
+incontestable that, in a great number of occasions, even of immediate
+utility, geometrical questions depend on only two dimensions, considered
+separately from the third, or on a single dimension, considered
+separately from the two others. Again, independently of this direct
+motive, the study of extension with a single dimension, and afterwards
+with two, clearly presents itself as an indispensable preliminary for
+facilitating the study of complete bodies of three dimensions, the
+immediate theory of which would be too complicated. Such are the two
+general motives which oblige geometers to consider separately extension
+with regard to one or to two dimensions, as well as relatively to all
+three together.
+
+The general notions of _surface_ and of _line_ have been formed by the
+human mind, in order that it may be able to think, in a permanent
+manner, of extension in two directions, or in one only. The hyperbolical
+expressions habitually employed by geometers to define these notions
+tend to convey false ideas of them; but, examined in themselves, they
+have no other object than to permit us to reason with facility
+respecting these two kinds of extension, making complete abstraction of
+that which ought not to be taken into consideration. Now for this it is
+sufficient to conceive the dimension which we wish to eliminate as
+becoming gradually smaller and smaller, the two others remaining the
+same, until it arrives at such a degree of tenuity that it can no longer
+fix the attention. It is thus that we naturally acquire the real idea of
+a _surface_, and, by a second analogous operation, the idea of a _line_,
+by repeating for breadth what we had at first done for thickness.
+Finally, if we again repeat the same operation, we arrive at the idea of
+a _point_, or of an extension considered only with reference to its
+place, abstraction being made of all magnitude, and designed
+consequently to determine positions.
+
+_Surfaces_ evidently have, moreover, the general property of exactly
+circumscribing volumes; and in the same way, _lines_, in their turn,
+circumscribe _surfaces_ and are limited by _points_. But this
+consideration, to which too much importance is often given, is only a
+secondary one.
+
+Surfaces and lines are, then, in reality, always conceived with three
+dimensions; it would be, in fact, impossible to represent to one's self
+a surface otherwise than as an extremely thin plate, and a line
+otherwise than as an infinitely fine thread. It is even plain that the
+degree of tenuity attributed by each individual to the dimensions of
+which he wishes to make abstraction is not constantly identical, for it
+must depend on the degree of subtilty of his habitual geometrical
+observations. This want of uniformity has, besides, no real
+inconvenience, since it is sufficient, in order that the ideas of
+surface and of line should satisfy the essential condition of their
+destination, for each one to represent to himself the dimensions which
+are to be neglected as being smaller than all those whose magnitude his
+daily experience gives him occasion to appreciate.
+
+We hence see how devoid of all meaning are the fantastic discussions of
+metaphysicians upon the foundations of geometry. It should also be
+remarked that these primordial ideas are habitually presented by
+geometers in an unphilosophical manner, since, for example, they explain
+the notions of the different sorts of extent in an order absolutely the
+inverse of their natural dependence, which often produces the most
+serious inconveniences in elementary instruction.
+
+
+
+
+THE FINAL OBJECT OF GEOMETRY.
+
+
+These preliminaries being established, we can proceed directly to the
+general definition of geometry, continuing to conceive this science as
+having for its final object the _measurement_ of extension.
+
+It is necessary in this matter to go into a thorough explanation,
+founded on the distinction of the three kinds of extension, since the
+notion of _measurement_ is not exactly the same with reference to
+surfaces and volumes as to lines.
+
+
+_Nature of Geometrical Measurement._ If we take the word _measurement_
+in its direct and general mathematical acceptation, which signifies
+simply the determination of the value of the _ratios_ between any
+homogeneous magnitudes, we must consider, in geometry, that the
+_measurement_ of surfaces and of volumes, unlike that of lines, is never
+conceived, even in the most simple and the most favourable cases, as
+being effected directly. The comparison of two lines is regarded as
+direct; that of two surfaces or of two volumes is, on the contrary,
+always indirect. Thus we conceive that two lines may be superposed; but
+the superposition of two surfaces, or, still more so, of two volumes, is
+evidently impossible in most cases; and, even when it becomes rigorously
+practicable, such a comparison is never either convenient or exact. It
+is, then, very necessary to explain wherein properly consists the truly
+geometrical measurement of a surface or of a volume.
+
+
+_Measurement of Surfaces and of Volumes._ For this we must consider
+that, whatever may be the form of a body, there always exists a certain
+number of lines, more or less easy to be assigned, the length of which
+is sufficient to define exactly the magnitude of its surface or of its
+volume. Geometry, regarding these lines as alone susceptible of being
+directly measured, proposes to deduce, from the simple determination of
+them, the ratio of the surface or of the volume sought, to the unity of
+surface, or to the unity of volume. Thus the general object of
+geometry, with respect to surfaces and to volumes, is properly to reduce
+all comparisons of surfaces or of volumes to simple comparisons of
+lines.
+
+Besides the very great facility which such a transformation evidently
+offers for the measurement of volumes and of surfaces, there results
+from it, in considering it in a more extended and more scientific
+manner, the general possibility of reducing to questions of lines all
+questions relating to volumes and to surfaces, considered with reference
+to their magnitude. Such is often the most important use of the
+geometrical expressions which determine surfaces and volumes in
+functions of the corresponding lines.
+
+It is true that direct comparisons between surfaces or between volumes
+are sometimes employed; but such measurements are not regarded as
+geometrical, but only as a supplement sometimes necessary, although too
+rarely applicable, to the insufficiency or to the difficulty of truly
+rational methods. It is thus that we often determine the volume of a
+body, and in certain cases its surface, by means of its weight. In the
+same way, on other occasions, when we can substitute for the proposed
+volume an equivalent liquid volume, we establish directly the comparison
+of the two volumes, by profiting by the property possessed by liquid
+masses, of assuming any desired form. But all means of this nature are
+purely mechanical, and rational geometry necessarily rejects them.
+
+To render more sensible the difference between these modes of
+determination and true geometrical measurements, I will cite a single
+very remarkable example; the manner in which Galileo determined the
+ratio of the ordinary cycloid to that of the generating circle. The
+geometry of his time was as yet insufficient for the rational solution
+of such a problem. Galileo conceived the idea of discovering that ratio
+by a direct experiment. Having weighed as exactly as possible two plates
+of the same material and of equal thickness, one of them having the form
+of a circle and the other that of the generated cycloid, he found the
+weight of the latter always triple that of the former; whence he
+inferred that the area of the cycloid is triple that of the generating
+circle, a result agreeing with the veritable solution subsequently
+obtained by Pascal and Wallis. Such a success evidently depends on the
+extreme simplicity of the ratio sought; and we can understand the
+necessary insufficiency of such expedients, even when they are actually
+practicable.
+
+We see clearly, from what precedes, the nature of that part of geometry
+relating to _volumes_ and that relating to _surfaces_. But the character
+of the geometry of _lines_ is not so apparent, since, in order to
+simplify the exposition, we have considered the measurement of lines as
+being made directly. There is, therefore, needed a complementary
+explanation with respect to them.
+
+
+_Measurement of curved Lines._ For this purpose, it is sufficient to
+distinguish between the right line and curved lines, the measurement of
+the first being alone regarded as direct, and that of the other as
+always indirect. Although superposition is sometimes strictly
+practicable for curved lines, it is nevertheless evident that truly
+rational geometry must necessarily reject it, as not admitting of any
+precision, even when it is possible. The geometry of lines has, then,
+for its general object, to reduce in every case the measurement of
+curved lines to that of right lines; and consequently, in the most
+extended point of view, to reduce to simple questions of right lines all
+questions relating to the magnitude of any curves whatever. To
+understand the possibility of such a transformation, we must remark,
+that in every curve there always exist certain right lines, the length
+of which must be sufficient to determine that of the curve. Thus, in a
+circle, it is evident that from the length of the radius we must be able
+to deduce that of the circumference; in the same way, the length of an
+ellipse depends on that of its two axes; the length of a cycloid upon
+the diameter of the generating circle, &c.; and if, instead of
+considering the whole of each curve, we demand, more generally, the
+length of any arc, it will be sufficient to add to the different
+rectilinear parameters, which determine the whole curve, the chord of
+the proposed arc, or the co-ordinates of its extremities. To discover
+the relation which exists between the length of a curved line and that
+of similar right lines, is the general problem of the part of geometry
+which relates to the study of lines.
+
+Combining this consideration with those previously suggested with
+respect to volumes and to surfaces, we may form a very clear idea of the
+science of geometry, conceived in all its parts, by assigning to it, for
+its general object, the final reduction of the comparisons of all kinds
+of extent, volumes, surfaces, or lines, to simple comparisons of right
+lines, the only comparisons regarded as capable of being made directly,
+and which indeed could not be reduced to any others more easy to effect.
+Such a conception, at the same time, indicates clearly the veritable
+character of geometry, and seems suited to show at a single glance its
+utility and its perfection.
+
+
+_Measurement of right Lines._ In order to complete this fundamental
+explanation, I have yet to show how there can be, in geometry, a special
+section relating to the right line, which seems at first incompatible
+with the principle that the measurement of this class of lines must
+always be regarded as direct.
+
+It is so, in fact, as compared with that of curved lines, and of all the
+other objects which geometry considers. But it is evident that the
+estimation of a right line cannot be viewed as direct except so far as
+the linear unit can be applied to it. Now this often presents
+insurmountable difficulties, as I had occasion to show, for another
+reason, in the introductory chapter. We must, then, make the measurement
+of the proposed right line depend on other analogous measurements
+capable of being effected directly. There is, then, necessarily a
+primary distinct branch of geometry, exclusively devoted to the right
+line; its object is to determine certain right lines from others by
+means of the relations belonging to the figures resulting from their
+assemblage. This preliminary part of geometry, which is almost
+imperceptible in viewing the whole of the science, is nevertheless
+susceptible of a great development. It is evidently of especial
+importance, since all other geometrical measurements are referred to
+those of right lines, and if they could not be determined, the solution
+of every question would remain unfinished.
+
+Such, then, are the various fundamental parts of rational geometry,
+arranged according to their natural dependence; the geometry of _lines_
+being first considered, beginning with the right line; then the geometry
+of _surfaces_, and, finally, that of _solids_.
+
+
+
+
+INFINITE EXTENT OF ITS FIELD.
+
+
+Having determined with precision the general and final object of
+geometrical inquiries, the science must now be considered with respect
+to the field embraced by each of its three fundamental sections.
+
+Thus considered, geometry is evidently susceptible, by its nature, of an
+extension which is rigorously infinite; for the measurement of lines, of
+surfaces, or of volumes presents necessarily as many distinct questions
+as we can conceive different figures subjected to exact definitions; and
+their number is evidently infinite.
+
+Geometers limited themselves at first to consider the most simple
+figures which were directly furnished them by nature, or which were
+deduced from these primitive elements by the least complicated
+combinations. But they have perceived, since Descartes, that, in order
+to constitute the science in the most philosophical manner, it was
+necessary to make it apply to all imaginable figures. This abstract
+geometry will then inevitably comprehend as particular cases all the
+different real figures which the exterior world could present. It is
+then a fundamental principle in truly rational geometry to consider, as
+far as possible, all figures which can be rigorously conceived.
+
+The most superficial examination is enough to convince us that these
+figures present a variety which is quite infinite.
+
+
+_Infinity of Lines._ With respect to curved _lines_, regarding them as
+generated by the motion of a point governed by a certain law, it is
+plain that we shall have, in general, as many different curves as we
+conceive different laws for this motion, which may evidently be
+determined by an infinity of distinct conditions; although it may
+sometimes accidentally happen that new generations produce curves which
+have been already obtained. Thus, among plane curves, if a point moves
+so as to remain constantly at the same distance from a fixed point, it
+will generate a _circle_; if it is the sum or the difference of its
+distances from two fixed points which remains constant, the curve
+described will be an _ellipse_ or an _hyperbola_; if it is their
+product, we shall have an entirely different curve; if the point departs
+equally from a fixed point and from a fixed line, it will describe a
+_parabola_; if it revolves on a circle at the same time that this circle
+rolls along a straight line, we shall have a _cycloid_; if it advances
+along a straight line, while this line, fixed at one of its extremities,
+turns in any manner whatever, there will result what in general terms
+are called _spirals_, which of themselves evidently present as many
+perfectly distinct curves as we can suppose different relations between
+these two motions of translation and of rotation, &c. Each of these
+different curves may then furnish new ones, by the different general
+constructions which geometers have imagined, and which give rise to
+evolutes, to epicycloids, to caustics, &c. Finally, there exists a still
+greater variety among curves of double curvature.
+
+
+_Infinity of Surfaces._ As to _surfaces_, the figures are necessarily
+more different still, considering them as generated by the motion of
+lines. Indeed, the figure may then vary, not only in considering, as in
+curves, the different infinitely numerous laws to which the motion of
+the generating line may be subjected, but also in supposing that this
+line itself may change its nature; a circumstance which has nothing
+analogous in curves, since the points which describe them cannot have
+any distinct figure. Two classes of very different conditions may then
+cause the figures of surfaces to vary, while there exists only one for
+lines. It is useless to cite examples of this doubly infinite
+multiplicity of surfaces. It would be sufficient to consider the extreme
+variety of the single group of surfaces which may be generated by a
+right line, and which comprehends the whole family of cylindrical
+surfaces, that of conical surfaces, the most general class of
+developable surfaces, &c.
+
+
+_Infinity of Volumes._ With respect to _volumes_, there is no occasion
+for any special consideration, since they are distinguished from each
+other only by the surfaces which bound them.
+
+In order to complete this sketch, it should be added that surfaces
+themselves furnish a new general means of conceiving new curves, since
+every curve may be regarded as produced by the intersection of two
+surfaces. It is in this way, indeed, that the first lines which we may
+regard as having been truly invented by geometers were obtained, since
+nature gave directly the straight line and the circle. We know that the
+ellipse, the parabola, and the hyperbola, the only curves completely
+studied by the ancients, were in their origin conceived only as
+resulting from the intersection of a cone with circular base by a plane
+in different positions. It is evident that, by the combined employment
+of these different general means for the formation of lines and of
+surfaces, we could produce a rigorously infinitely series of distinct
+forms in starting from only a very small number of figures directly
+furnished by observation.
+
+
+_Analytical invention of Curves, &c._ Finally, all the various direct
+means for the invention of figures have scarcely any farther importance,
+since rational geometry has assumed its final character in the hands of
+Descartes. Indeed, as we shall see more fully in chapter iii., the
+invention of figures is now reduced to the invention of equations, so
+that nothing is more easy than to conceive new lines and new surfaces,
+by changing at will the functions introduced into the equations. This
+simple abstract procedure is, in this respect, infinitely more fruitful
+than all the direct resources of geometry, developed by the most
+powerful imagination, which should devote itself exclusively to that
+order of conceptions. It also explains, in the most general and the most
+striking manner, the necessarily infinite variety of geometrical forms,
+which thus corresponds to the diversity of analytical functions. Lastly,
+it shows no less clearly that the different forms of surfaces must be
+still more numerous than those of lines, since lines are represented
+analytically by equations with two variables, while surfaces give rise
+to equations with three variables, which necessarily present a greater
+diversity.
+
+The preceding considerations are sufficient to show clearly the
+rigorously infinite extent of each of the three general sections of
+geometry.
+
+
+
+
+EXPANSION OF ORIGINAL DEFINITION.
+
+
+To complete the formation of an exact and sufficiently extended idea of
+the nature of geometrical inquiries, it is now indispensable to return
+to the general definition above given, in order to present it under a
+new point of view, without which the complete science would be only very
+imperfectly conceived.
+
+When we assign as the object of geometry the _measurement_ of all sorts
+of lines, surfaces, and volumes, that is, as has been explained, the
+reduction of all geometrical comparisons to simple comparisons of right
+lines, we have evidently the advantage of indicating a general
+destination very precise and very easy to comprehend. But if we set
+aside every definition, and examine the actual composition of the
+science of geometry, we will at first be induced to regard the preceding
+definition as much too narrow; for it is certain that the greater part
+of the investigations which constitute our present geometry do not at
+all appear to have for their object the _measurement_ of extension. In
+spite of this fundamental objection, I will persist in retaining this
+definition; for, in fact, if, instead of confining ourselves to
+considering the different questions of geometry isolatedly, we endeavour
+to grasp the leading questions, in comparison with which all others,
+however important they may be, must be regarded as only secondary, we
+will finally recognize that the measurement of lines, of surfaces, and
+of volumes, is the invariable object, sometimes _direct_, though most
+often _indirect_, of all geometrical labours.
+
+This general proposition being fundamental, since it can alone give our
+definition all its value, it is indispensable to enter into some
+developments upon this subject.
+
+
+
+
+PROPERTIES OF LINES AND SURFACES.
+
+
+When we examine with attention the geometrical investigations which do
+not seem to relate to the _measurement_ of extent, we find that they
+consist essentially in the study of the different _properties_ of each
+line or of each surface; that is, in the knowledge of the different
+modes of generation, or at least of definition, peculiar to each figure
+considered. Now we can easily establish in the most general manner the
+necessary relation of such a study to the question of _measurement_, for
+which the most complete knowledge of the properties of each form is an
+indispensable preliminary. This is concurrently proven by two
+considerations, equally fundamental, although quite distinct in their
+nature.
+
+
+NECESSITY OF THEIR STUDY: 1. _To find the most suitable Property._ The
+_first_, purely scientific, consists in remarking that, if we did not
+know any other characteristic property of each line or surface than that
+one according to which geometers had first conceived it, in most cases
+it would be impossible to succeed in the solution of questions relating
+to its _measurement_. In fact, it is easy to understand that the
+different definitions which each figure admits of are not all equally
+suitable for such an object, and that they even present the most
+complete oppositions in that respect. Besides, since the primitive
+definition of each figure was evidently not chosen with this condition
+in view, it is clear that we must not expect, in general, to find it the
+most suitable; whence results the necessity of discovering others, that
+is, of studying as far as is possible the _properties_ of the proposed
+figure. Let us suppose, for example, that the circle is defined to be
+"the curve which, with the same contour, contains the greatest area."
+This is certainly a very characteristic property, but we would evidently
+find insurmountable difficulties in trying to deduce from such a
+starting point the solution of the fundamental questions relating to the
+rectification or to the quadrature of this curve. It is clear, in
+advance, that the property of having all its points equally distant from
+a fixed point must evidently be much better adapted to inquiries of this
+nature, even though it be not precisely the most suitable. In like
+manner, would Archimedes ever have been able to discover the quadrature
+of the parabola if he had known no other property of that curve than
+that it was the section of a cone with a circular base, by a plane
+parallel to its generatrix? The purely speculative labours of preceding
+geometers, in transforming this first definition, were evidently
+indispensable preliminaries to the direct solution of such a question.
+The same is true, in a still greater degree, with respect to surfaces.
+To form a just idea of this, we need only compare, as to the question of
+cubature or quadrature, the common definition of the sphere with that
+one, no less characteristic certainly, which would consist in regarding
+a spherical body, as that one which, with the same area, contains the
+greatest volume.
+
+No more examples are needed to show the necessity of knowing, so far as
+is possible, all the properties of each line or of each surface, in
+order to facilitate the investigation of rectifications, of quadratures,
+and of cubatures, which constitutes the final object of geometry. We may
+even say that the principal difficulty of questions of this kind
+consists in employing in each case the property which is best adapted
+to the nature of the proposed problem. Thus, while we continue to
+indicate, for more precision, the measurement of extension as the
+general destination of geometry, this first consideration, which goes to
+the very bottom of the subject, shows clearly the necessity of including
+in it the study, as thorough as possible, of the different generations
+or definitions belonging to the same form.
+
+
+2. _To pass from the Concrete to the Abstract._ A second consideration,
+of at least equal importance, consists in such a study being
+indispensable for organizing in a rational manner the relation of the
+abstract to the concrete in geometry.
+
+The science of geometry having to consider all imaginable figures which
+admit of an exact definition, it necessarily results from this, as we
+have remarked, that questions relating to any figures presented by
+nature are always implicitly comprised in this abstract geometry,
+supposed to have attained its perfection. But when it is necessary to
+actually pass to concrete geometry, we constantly meet with a
+fundamental difficulty, that of knowing to which of the different
+abstract types we are to refer, with sufficient approximation, the real
+lines or surfaces which we have to study. Now it is for the purpose of
+establishing such a relation that it is particularly indispensable to
+know the greatest possible number of properties of each figure
+considered in geometry.
+
+In fact, if we always confined ourselves to the single primitive
+definition of a line or of a surface, supposing even that we could then
+_measure_ it (which, according to the first order of considerations,
+would generally be impossible), this knowledge would remain almost
+necessarily barren in the application, since we should not ordinarily
+know how to recognize that figure in nature when it presented itself
+there; to ensure that, it would be necessary that the single
+characteristic, according to which geometers had conceived it, should be
+precisely that one whose verification external circumstances would
+admit: a coincidence which would be purely fortuitous, and on which we
+could not count, although it might sometimes take place. It is, then,
+only by multiplying as much as possible the characteristic properties of
+each abstract figure, that we can be assured, in advance, of recognizing
+it in the concrete state, and of thus turning to account all our
+rational labours, by verifying in each case the definition which is
+susceptible of being directly proven. This definition is almost always
+the only one in given circumstances, and varies, on the other hand, for
+the same figure, with different circumstances; a double reason for its
+previous determination.
+
+
+_Illustration: Orbits of the Planets._ The geometry of the heavens
+furnishes us with a very memorable example in this matter, well suited
+to show the general necessity of such a study. We know that the ellipse
+was discovered by Kepler to be the curve which the planets describe
+about the sun, and the satellites about their planets. Now would this
+fundamental discovery, which re-created astronomy, ever have been
+possible, if geometers had been always confined to conceiving the
+ellipse only as the oblique section of a circular cone by a plane? No
+such definition, it is evident, would admit of such a verification. The
+most general property of the ellipse, that the sum of the distances from
+any of its points to two fixed points is a constant quantity, is
+undoubtedly much more susceptible, by its nature, of causing the curve
+to be recognized in this case, but still is not directly suitable. The
+only characteristic which can here be immediately verified is that which
+is derived from the relation which exists in the ellipse between the
+length of the focal distances and their direction; the only relation
+which admits of an astronomical interpretation, as expressing the law
+which connects the distance from the planet to the sun, with the time
+elapsed since the beginning of its revolution. It was, then, necessary
+that the purely speculative labours of the Greek geometers on the
+properties of the conic sections should have previously presented their
+generation under a multitude of different points of view, before Kepler
+could thus pass from the abstract to the concrete, in choosing from
+among all these different characteristics that one which could be most
+easily proven for the planetary orbits.
+
+
+_Illustration: Figure of the Earth._ Another example of the same order,
+but relating to surfaces, occurs in considering the important question
+of the figure of the earth. If we had never known any other property of
+the sphere than its primitive character of having all its points equally
+distant from an interior point, how would we ever have been able to
+discover that the surface of the earth was spherical? For this, it was
+necessary previously to deduce from this definition of the sphere some
+properties capable of being verified by observations made upon the
+surface alone, such as the constant ratio which exists between the
+length of the path traversed in the direction of any meridian of a
+sphere going towards a pole, and the angular height of this pole above
+the horizon at each point. Another example, but involving a much longer
+series of preliminary speculations, is the subsequent proof that the
+earth is not rigorously spherical, but that its form is that of an
+ellipsoid of revolution.
+
+After such examples, it would be needless to give any others, which any
+one besides may easily multiply. All of them prove that, without a very
+extended knowledge of the different properties of each figure, the
+relation of the abstract to the concrete, in geometry, would be purely
+accidental, and that the science would consequently want one of its most
+essential foundations.
+
+Such, then, are two general considerations which fully demonstrate the
+necessity of introducing into geometry a great number of investigations
+which have not the _measurement_ of extension for their direct object;
+while we continue, however, to conceive such a measurement as being the
+final destination of all geometrical science. In this way we can retain
+the philosophical advantages of the clearness and precision of this
+definition, and still include in it, in a very logical though indirect
+manner, all known geometrical researches, in considering those which do
+not seem to relate to the measurement of extension, as intended either
+to prepare for the solution of the final questions, or to render
+possible the application of the solutions obtained.
+
+Having thus recognized, as a general principle, the close and necessary
+connexion of the study of the properties of lines and surfaces with
+those researches which constitute the final object of geometry, it is
+evident that geometers, in the progress of their labours, must by no
+means constrain themselves to keep such a connexion always in view.
+Knowing, once for all, how important it is to vary as much as possible
+the manner of conceiving each figure, they should pursue that study,
+without considering of what immediate use such or such a special
+property may be for rectifications, quadratures, and cubatures. They
+would uselessly fetter their inquiries by attaching a puerile importance
+to the continued establishment of that co-ordination.
+
+This general exposition of the general object of geometry is so much the
+more indispensable, since, by the very nature of the subject, this study
+of the different properties of each line and of each surface necessarily
+composes by far the greater part of the whole body of geometrical
+researches. Indeed, the questions directly relating to rectifications,
+to quadratures, and to cubatures, are evidently, by themselves, very few
+in number for each figure considered. On the other hand, the study of
+the properties of the same figure presents an unlimited field to the
+activity of the human mind, in which it may always hope to make new
+discoveries. Thus, although geometers have occupied themselves for
+twenty centuries, with more or less activity undoubtedly, but without
+any real interruption, in the study of the conic sections, they are far
+from regarding that so simple subject as being exhausted; and it is
+certain, indeed, that in continuing to devote themselves to it, they
+would not fail to find still unknown properties of those different
+curves. If labours of this kind have slackened considerably for a
+century past, it is not because they are completed, but only, as will be
+presently explained, because the philosophical revolution in geometry,
+brought about by Descartes, has singularly diminished the importance of
+such researches.
+
+It results from the preceding considerations that not only is the field
+of geometry necessarily infinite because of the variety of figures to
+be considered, but also in virtue of the diversity of the points of view
+under the same figure may be regarded. This last conception is, indeed,
+that which gives the broadest and most complete idea of the whole body
+of geometrical researches. We see that studies of this kind consist
+essentially, for each line or for each surface, in connecting all the
+geometrical phenomena which it can present, with a single fundamental
+phenomenon, regarded as the primitive definition.
+
+
+
+
+THE TWO GENERAL METHODS OF GEOMETRY.
+
+
+Having now explained in a general and yet precise manner the final
+object of geometry, and shown how the science, thus defined, comprehends
+a very extensive class of researches which did not at first appear
+necessarily to belong to it, there remains to be considered the _method_
+to be followed for the formation of this science. This discussion is
+indispensable to complete this first sketch of the philosophical
+character of geometry. I shall here confine myself to indicating the
+most general consideration in this matter, developing and summing up
+this important fundamental idea in the following chapters.
+
+Geometrical questions may be treated according to _two methods_ so
+different, that there result from them two sorts of geometry, so to say,
+the philosophical character of which does not seem to me to have yet
+been properly apprehended. The expressions of _Synthetical Geometry_ and
+_Analytical Geometry_, habitually employed to designate them, give a
+very false idea of them. I would much prefer the purely historical
+denominations of _Geometry of the Ancients_ and _Geometry of the
+Moderns_, which have at least the advantage of not causing their true
+character to be misunderstood. But I propose to employ henceforth the
+regular expressions of _Special Geometry_ and _General Geometry_, which
+seem to me suited to characterize with precision the veritable nature of
+the two methods.
+
+
+_Their fundamental Difference._ The fundamental difference between the
+manner in which we conceive Geometry since Descartes, and the manner in
+which the geometers of antiquity treated geometrical questions, is not
+the use of the Calculus (or Algebra), as is commonly thought to be the
+case. On the one hand, it is certain that the use of the calculus was
+not entirely unknown to the ancient geometers, since they used to make
+continual and very extensive applications of the theory of proportions,
+which was for them, as a means of deduction, a sort of real, though very
+imperfect and especially extremely limited equivalent for our present
+algebra. The calculus may even be employed in a much more complete
+manner than they have used it, in order to obtain certain geometrical
+solutions, which will still retain all the essential character of the
+ancient geometry; this occurs very frequently with respect to those
+problems of geometry of two or of three dimensions, which are commonly
+designated under the name of _determinate_. On the other hand, important
+as is the influence of the calculus in our modern geometry, various
+solutions obtained without algebra may sometimes manifest the peculiar
+character which distinguishes it from the ancient geometry, although
+analysis is generally indispensable. I will cite, as an example, the
+method of Roberval for tangents, the nature of which is essentially
+modern, and which, however, leads in certain cases to complete
+solutions, without any aid from the calculus. It is not, then, the
+instrument of deduction employed which is the principal distinction
+between the two courses which the human mind can take in geometry.
+
+The real fundamental difference, as yet imperfectly apprehended, seems
+to me to consist in the very nature of the questions considered. In
+truth, geometry, viewed as a whole, and supposed to have attained entire
+perfection, must, as we have seen on the one hand, embrace all
+imaginable figures, and, on the other, discover all the properties of
+each figure. It admits, from this double consideration, of being treated
+according to two essentially distinct plans; either, 1°, by grouping
+together all the questions, however different they may be, which relate
+to the same figure, and isolating those relating to different bodies,
+whatever analogy there may exist between them; or, 2°, on the contrary,
+by uniting under one point of view all similar inquiries, to whatever
+different figures they may relate, and separating the questions relating
+to the really different properties of the same body. In a word, the
+whole body of geometry may be essentially arranged either with reference
+to the _bodies_ studied or to the _phenomena_ to be considered. The
+first plan, which is the most natural, was that of the ancients; the
+second, infinitely more rational, is that of the moderns since
+Descartes.
+
+
+_Geometry of the Ancients._ Indeed, the principal characteristics of the
+ancient geometry is that they studied, one by one, the different lines
+and the different surfaces, not passing to the examination of a new
+figure till they thought they had exhausted all that there was
+interesting in the figures already known. In this way of proceeding,
+when they undertook the study of a new curve, the whole of the labour
+bestowed on the preceding ones could not offer directly any essential
+assistance, otherwise than by the geometrical practice to which it had
+trained the mind. Whatever might be the real similarity of the questions
+proposed as to two different figures, the complete knowledge acquired
+for the one could not at all dispense with taking up again the whole of
+the investigation for the other. Thus the progress of the mind was never
+assured; so that they could not be certain, in advance, of obtaining any
+solution whatever, however analogous the proposed problem might be to
+questions which had been already resolved. Thus, for example, the
+determination of the tangents to the three conic sections did not
+furnish any rational assistance for drawing the tangent to any other new
+curve, such as the conchoid, the cissoid, &c. In a word, the geometry of
+the ancients was, according to the expression proposed above,
+essentially special.
+
+
+_Geometry of the Moderns._ In the system of the moderns, geometry is, on
+the contrary, eminently _general_, that is to say, relating to any
+figures whatever. It is easy to understand, in the first place, that all
+geometrical expressions of any interest may be proposed with reference
+to all imaginable figures. This is seen directly in the fundamental
+problems--of rectifications, quadratures, and cubatures--which
+constitute, as has been shown, the final object of geometry. But this
+remark is no less incontestable, even for investigations which relate to
+the different _properties_ of lines and of surfaces, and of which the
+most essential, such as the question of tangents or of tangent planes,
+the theory of curvatures, &c., are evidently common to all figures
+whatever. The very few investigations which are truly peculiar to
+particular figures have only an extremely secondary importance. This
+being understood, modern geometry consists essentially in abstracting,
+in order to treat it by itself, in an entirely general manner, every
+question relating to the same geometrical phenomenon, in whatever bodies
+it may be considered. The application of the universal theories thus
+constructed to the special determination of the phenomenon which is
+treated of in each particular body, is now regarded as only a subaltern
+labour, to be executed according to invariable rules, and the success of
+which is certain in advance. This labour is, in a word, of the same
+character as the numerical calculation of an analytical formula. There
+can be no other merit in it than that of presenting in each case the
+solution which is necessarily furnished by the general method, with all
+the simplicity and elegance which the line or the surface considered can
+admit of. But no real importance is attached to any thing but the
+conception and the complete solution of a new question belonging to any
+figure whatever. Labours of this kind are alone regarded as producing
+any real advance in science. The attention of geometers, thus relieved
+from the examination of the peculiarities of different figures, and
+wholly directed towards general questions, has been thereby able to
+elevate itself to the consideration of new geometrical conceptions,
+which, applied to the curves studied by the ancients, have led to the
+discovery of important properties which they had not before even
+suspected. Such is geometry, since the radical revolution produced by
+Descartes in the general system of the science.
+
+
+_The Superiority of the modern Geometry._ The mere indication of the
+fundamental character of each of the two geometries is undoubtedly
+sufficient to make apparent the immense necessary superiority of modern
+geometry. We may even say that, before the great conception of
+Descartes, rational geometry was not truly constituted upon definitive
+bases, whether in its abstract or concrete relations. In fact, as
+regards science, considered speculatively, it is clear that, in
+continuing indefinitely to follow the course of the ancients, as did the
+moderns before Descartes, and even for a little while afterwards, by
+adding some new curves to the small number of those which they had
+studied, the progress thus made, however rapid it might have been, would
+still be found, after a long series of ages, to be very inconsiderable
+in comparison with the general system of geometry, seeing the infinite
+variety of the forms which would still have remained to be studied. On
+the contrary, at each question resolved according to the method of the
+moderns, the number of geometrical problems to be resolved is then, once
+for all, diminished by so much with respect to all possible bodies.
+Another consideration is, that it resulted, from their complete want of
+general methods, that the ancient geometers, in all their
+investigations, were entirely abandoned to their own strength, without
+ever having the certainty of obtaining, sooner or later, any solution
+whatever. Though this imperfection of the science was eminently suited
+to call forth all their admirable sagacity, it necessarily rendered
+their progress extremely slow; we can form some idea of this by the
+considerable time which they employed in the study of the conic
+sections. Modern geometry, making the progress of our mind certain,
+permits us, on the contrary, to make the greatest possible use of the
+forces of our intelligence, which the ancients were often obliged to
+waste on very unimportant questions.
+
+A no less important difference between the two systems appears when we
+come to consider geometry in the concrete point of view. Indeed, we have
+already remarked that the relation of the abstract to the concrete in
+geometry can be founded upon rational bases only so far as the
+investigations are made to bear directly upon all imaginable figures. In
+studying lines, only one by one, whatever may be the number, always
+necessarily very small, of those which we shall have considered, the
+application of such theories to figures really existing in nature will
+never have any other than an essentially accidental character, since
+there is nothing to assure us that these figures can really be brought
+under the abstract types considered by geometers.
+
+Thus, for example, there is certainly something fortuitous in the happy
+relation established between the speculations of the Greek geometers
+upon the conic sections and the determination of the true planetary
+orbits. In continuing geometrical researches upon the same plan, there
+was no good reason for hoping for similar coincidences; and it would
+have been possible, in these special studies, that the researches of
+geometers should have been directed to abstract figures entirely
+incapable of any application, while they neglected others, susceptible
+perhaps of an important and immediate application. It is clear, at
+least, that nothing positively guaranteed the necessary applicability of
+geometrical speculations. It is quite another thing in the modern
+geometry. From the single circumstance that in it we proceed by general
+questions relating to any figures whatever, we have in advance the
+evident certainty that the figures really existing in the external world
+could in no case escape the appropriate theory if the geometrical
+phenomenon which it considers presents itself in them.
+
+From these different considerations, we see that the ancient system of
+geometry wears essentially the character of the infancy of the science,
+which did not begin to become completely rational till after the
+philosophical resolution produced by Descartes. But it is evident, on
+the other hand, that geometry could not be at first conceived except in
+this _special_ manner. _General_ geometry would not have been possible,
+and its necessity could not even have been felt, if a long series of
+special labours on the most simple figures had not previously furnished
+bases for the conception of Descartes, and rendered apparent the
+impossibility of persisting indefinitely in the primitive geometrical
+philosophy.
+
+
+_The Ancient the Base of the Modern._ From this last consideration we
+must infer that, although the geometry which I have called _general_
+must be now regarded as the only true dogmatical geometry, and that to
+which we shall chiefly confine ourselves, the other having no longer
+much more than an historical interest, nevertheless it is not possible
+to entirely dispense with _special_ geometry in a rational exposition of
+the science. We undoubtedly need not borrow directly from ancient
+geometry all the results which it has furnished; but, from the very
+nature of the subject, it is necessarily impossible entirely to dispense
+with the ancient method, which will always serve as the preliminary
+basis of the science, dogmatically as well as historically. The reason
+of this is easy to understand. In fact, _general_ geometry being
+essentially founded, as we shall soon establish, upon the employment of
+the calculus in the transformation of geometrical into analytical
+considerations, such a manner of proceeding could not take possession of
+the subject immediately at its origin. We know that the application of
+mathematical analysis, from its nature, can never commence any science
+whatever, since evidently it cannot be employed until the science has
+already been sufficiently cultivated to establish, with respect to the
+phenomena considered, some _equations_ which can serve as starting
+points for the analytical operations. These fundamental equations being
+once discovered, analysis will enable us to deduce from them a multitude
+of consequences which it would have been previously impossible even to
+suspect; it will perfect the science to an immense degree, both with
+respect to the generality of its conceptions and to the complete
+co-ordination established between them. But mere mathematical analysis
+could never be sufficient to form the bases of any natural science, not
+even to demonstrate them anew when they have once been established.
+Nothing can dispense with the direct study of the subject, pursued up to
+the point of the discovery of precise relations.
+
+We thus see that the geometry of the ancients will always have, by its
+nature, a primary part, absolutely necessary and more or less extensive,
+in the complete system of geometrical knowledge. It forms a rigorously
+indispensable introduction to _general_ geometry. But it is to this that
+it must be limited in a completely dogmatic exposition. I will consider,
+then, directly, in the following chapter, this _special_ or
+_preliminary_ geometry restricted to exactly its necessary limits, in
+order to occupy myself thenceforth only with the philosophical
+examination of _general_ or _definitive_ geometry, the only one which is
+truly rational, and which at present essentially composes the science.
+
+
+
+
+CHAPTER II.
+
+ANCIENT OR SYNTHETIC GEOMETRY.
+
+
+The geometrical method of the ancients necessarily constituting a
+preliminary department in the dogmatical system of geometry, designed to
+furnish _general_ geometry with indispensable foundations, it is now
+proper to begin with determining wherein strictly consists this
+preliminary function of _special_ geometry, thus reduced to the
+narrowest possible limits.
+
+
+
+
+ITS PROPER EXTENT.
+
+
+_Lines; Polygons; Polyhedrons._ In considering it under this point of
+view, it is easy to recognize that we might restrict it to the study of
+the right line alone for what concerns the geometry of _lines_; to the
+_quadrature_ of rectilinear plane areas; and, lastly, to the _cubature_
+of bodies terminated by plane faces. The elementary propositions
+relating to these three fundamental questions form, in fact, the
+necessary starting point of all geometrical inquiries; they alone cannot
+be obtained except by a direct study of the subject; while, on the
+contrary, the complete theory of all other figures, even that of the
+circle, and of the surfaces and volumes which are connected with it, may
+at the present day be completely comprehended in the domain of _general_
+or _analytical_ geometry; these primitive elements at once furnishing
+_equations_ which are sufficient to allow of the application of the
+calculus to geometrical questions, which would not have been possible
+without this previous condition.
+
+It results from this consideration that, in common practice, we give to
+_elementary_ geometry more extent than would be rigorously necessary to
+it; since, besides the right line, polygons, and polyhedrons, we also
+include in it the circle and the "round" bodies; the study of which
+might, however, be as purely analytical as that, for example, of the
+conic sections. An unreflecting veneration for antiquity contributes to
+maintain this defect in method; but the best reason which can be given
+for it is the serious inconvenience for ordinary instruction which there
+would be in postponing, to so distant an epoch of mathematical
+education, the solution of several essential questions, which are
+susceptible of a direct and continual application to a great number of
+important uses. In fact, to proceed in the most rational manner, we
+should employ the integral calculus in obtaining the interesting results
+relating to the length or the area of the circle, or to the quadrature
+of the sphere, &c., which have been determined by the ancients from
+extremely simple considerations. This inconvenience would be of little
+importance with regard to the persons destined to study the whole of
+mathematical science, and the advantage of proceeding in a perfectly
+logical order would have a much greater comparative value. But the
+contrary case being the more frequent, theories so essential have
+necessarily been retained in elementary geometry. Perhaps the conic
+sections, the cycloid, &c., might be advantageously added in such cases.
+
+
+_Not to be farther restricted._ While this preliminary portion of
+geometry, which cannot be founded on the application of the calculus,
+is reduced by its nature to a very limited series of fundamental
+researches, relating to the right line, polygonal areas, and
+polyhedrons, it is certain, on the other hand, that we cannot restrict
+it any more; although, by a veritable abuse of the spirit of analysis,
+it has been recently attempted to present the establishment of the
+principal theorems of elementary geometry under an algebraical point of
+view. Thus some have pretended to demonstrate, by simple abstract
+considerations of mathematical analysis, the constant relation which
+exists between the three angles of a rectilinear triangle, the
+fundamental proposition of the theory of similar triangles, that of
+parallelopipedons, &c.; in a word, precisely the only geometrical
+propositions which cannot be obtained except by a direct study of the
+subject, without the calculus being susceptible of having any part in
+it. Such aberrations are the unreflecting exaggerations of that natural
+and philosophical tendency which leads us to extend farther and farther
+the influence of analysis in mathematical studies. In mechanics, the
+pretended analytical demonstrations of the parallelogram of forces are
+of similar character.
+
+The viciousness of such a manner of proceeding follows from the
+principles previously presented. We have already, in fact, recognized
+that, since the calculus is not, and cannot be, any thing but a means of
+deduction, it would indicate a radically false idea of it to wish to
+employ it in establishing the elementary foundations of any science
+whatever; for on what would the analytical reasonings in such an
+operation repose? A labour of this nature, very far from really
+perfecting the philosophical character of a science, would constitute a
+return towards the metaphysical age, in presenting real facts as mere
+logical abstractions.
+
+When we examine in themselves these pretended analytical demonstrations
+of the fundamental propositions of elementary geometry, we easily verify
+their necessary want of meaning. They are all founded on a vicious
+manner of conceiving the principle of _homogeneity_, the true general
+idea of which was explained in the second chapter of the preceding book.
+These demonstrations suppose that this principle does not allow us to
+admit the coexistence in the same equation of numbers obtained by
+different concrete comparisons, which is evidently false, and contrary
+to the constant practice of geometers. Thus it is easy to recognize
+that, by employing the law of homogeneity in this arbitrary and
+illegitimate acceptation, we could succeed in "demonstrating," with
+quite as much apparent rigour, propositions whose absurdity is manifest
+at the first glance. In examining attentively, for example, the
+procedure by the aid of which it has been attempted to prove
+analytically that the sum of the three angles of any rectilinear
+triangle is constantly equal to two right angles, we see that it is
+founded on this preliminary principle that, if two triangles have two of
+their angles respectively equal, the third angle of the one will
+necessarily be equal to the third angle of the other. This first point
+being granted, the proposed relation is immediately deduced from it in a
+very exact and simple manner. Now the analytical consideration by which
+this previous proposition has been attempted to be established, is of
+such a nature that, if it could be correct, we could rigorously deduce
+from it, in reproducing it conversely, this palpable absurdity, that two
+sides of a triangle are sufficient, without any angle, for the entire
+determination of the third side. We may make analogous remarks on all
+the demonstrations of this sort, the sophisms of which will be thus
+verified in a perfectly apparent manner.
+
+The more reason that we have here to consider geometry as being at the
+present day essentially analytical, the more necessary was it to guard
+against this abusive exaggeration of mathematical analysis, according to
+which all geometrical observation would be dispensed with, in
+establishing upon pure algebraical abstractions the very foundations of
+this natural science.
+
+
+_Attempted Demonstrations of Axioms, &c._ Another indication that
+geometers have too much overlooked the character of a natural science
+which is necessarily inherent in geometry, appears from their vain
+attempts, so long made, to _demonstrate_ rigorously, not by the aid of
+the calculus, but by means of certain constructions, several fundamental
+propositions of elementary geometry. Whatever may be effected, it will
+evidently be impossible to avoid sometimes recurring to simple and
+direct observation in geometry as a means of establishing various
+results. While, in this science, the phenomena which are considered are,
+by virtue of their extreme simplicity, much more closely connected with
+one another than those relating to any other physical science, some must
+still be found which cannot be deduced, and which, on the contrary,
+serve as starting points. It may be admitted that the greatest logical
+perfection of the science is to reduce these to the smallest number
+possible, but it would be absurd to pretend to make them completely
+disappear. I avow, moreover, that I find fewer real inconveniences in
+extending, a little beyond what would be strictly necessary, the number
+of these geometrical notions thus established by direct observation,
+provided they are sufficiently simple, than in making them the subjects
+of complicated and indirect demonstrations, even when these
+demonstrations may be logically irreproachable.
+
+The true dogmatic destination of the geometry of the ancients, reduced
+to its least possible indispensable developments, having thus been
+characterized as exactly as possible, it is proper to consider summarily
+each of the principal parts of which it must be composed. I think that I
+may here limit myself to considering the first and the most extensive of
+these parts, that which has for its object the study of _the right
+line_; the two other sections, namely, the _quadrature of polygons_ and
+the _cubature of polyhedrons_, from their limited extent, not being
+capable of giving rise to any philosophical consideration of any
+importance, distinct from those indicated in the preceding chapter with
+respect to the measure of areas and of volumes in general.
+
+
+
+
+GEOMETRY OF THE RIGHT LINE.
+
+
+The final question which we always have in view in the study of the
+right line, properly consists in determining, by means of one another,
+the different elements of any right-lined figure whatever; which enables
+us always to know indirectly the length and position of a right line, in
+whatever circumstances it may be placed. This fundamental problem is
+susceptible of two general solutions, the nature of which is quite
+distinct, the one _graphical_, the other _algebraic_. The first, though
+very imperfect, is that which must be first considered, because it is
+spontaneously derived from the direct study of the subject; the second,
+much more perfect in the most important respects, cannot be studied till
+afterwards, because it is founded upon the previous knowledge of the
+other.
+
+
+
+
+GRAPHICAL SOLUTIONS.
+
+
+The graphical solution consists in constructing at will the proposed
+figure, either with the same dimensions, or, more usually, with
+dimensions changed in any ratio whatever. The first mode need merely be
+mentioned as being the most simple and the one which would first occur
+to the mind, for it is evidently, by its nature, almost entirely
+incapable of application. The second is, on the contrary, susceptible of
+being most extensively and most usefully applied. We still make an
+important and continual use of it at the present day, not only to
+represent with exactness the forms of bodies and their relative
+positions, but even for the actual determination of geometrical
+magnitudes, when we do not need great precision. The ancients, in
+consequence of the imperfection of their geometrical knowledge, employed
+this procedure in a much more extensive manner, since it was for a long
+time the only one which they could apply, even in the most important
+precise determinations. It was thus, for example, that Aristarchus of
+Samos estimated the relative distance from the sun and from the moon to
+the earth, by making measurements on a triangle constructed as exactly
+as possible, so as to be similar to the right-angled triangle formed by
+the three bodies at the instant when the moon is in quadrature, and when
+an observation of the angle at the earth would consequently be
+sufficient to define the triangle. Archimedes himself, although he was
+the first to introduce calculated determinations into geometry, several
+times employed similar means. The formation of trigonometry did not
+cause this method to be entirely abandoned, although it greatly
+diminished its use; the Greeks and the Arabians continued to employ it
+for a great number of researches, in which we now regard the use of the
+calculus as indispensable.
+
+This exact reproduction of any figure whatever on a different scale
+cannot present any great theoretical difficulty when all the parts of
+the proposed figure lie in the same plane. But if we suppose, as most
+frequently happens, that they are situated in different planes, we see,
+then, a new order of geometrical considerations arise. The artificial
+figure, which is constantly plane, not being capable, in that case, of
+being a perfectly faithful image of the real figure, it is necessary
+previously to fix with precision the mode of representation, which gives
+rise to different systems of _Projection_.
+
+It then remains to be determined according to what laws the geometrical
+phenomena correspond in the two figures. This consideration generates a
+new series of geometrical investigations, the final object of which is
+properly to discover how we can replace constructions in relief by plane
+constructions. The ancients had to resolve several elementary questions
+of this kind for various cases in which we now employ spherical
+trigonometry, principally for different problems relating to the
+celestial sphere. Such was the object of their _analemmas_, and of the
+other plane figures which for a long time supplied the place of the
+calculus. We see by this that the ancients really knew the elements of
+what we now name _Descriptive Geometry_, although they did not conceive
+it in a distinct and general manner.
+
+I think it proper briefly to indicate in this place the true
+philosophical character of this "Descriptive Geometry;" although, being
+essentially a science of application, it ought not to be included within
+the proper domain of this work.
+
+
+
+
+DESCRIPTIVE GEOMETRY.
+
+
+All questions of geometry of three dimensions necessarily give rise,
+when we consider their graphical solution, to a common difficulty which
+is peculiar to them; that of substituting for the different
+constructions in relief, which are necessary to resolve them directly,
+and which it is almost always impossible to execute, simple equivalent
+plane constructions, by means of which we finally obtain the same
+results. Without this indispensable transformation, every solution of
+this kind would be evidently incomplete and really inapplicable in
+practice, although theoretically the constructions in space are usually
+preferable as being more direct. It was in order to furnish general
+means for always effecting such a transformation that _Descriptive
+Geometry_ was created, and formed into a distinct and homogeneous
+system, by the illustrious MONGE. He invented, in the first place, a
+uniform method of representing bodies by figures traced on a single
+plane, by the aid of _projections_ on two different planes, usually
+perpendicular to each other, and one of which is supposed to turn about
+their common intersection so as to coincide with the other produced; in
+this system, or in any other equivalent to it, it is sufficient to
+regard points and lines as being determined by their projections, and
+surfaces by the projections of their generating lines. This being
+established, Monge--analyzing with profound sagacity the various partial
+labours of this kind which had before been executed by a number of
+incongruous procedures, and considering also, in a general and direct
+manner, in what any questions of that nature must consist--found that
+they could always be reduced to a very small number of invariable
+abstract problems, capable of being resolved separately, once for all,
+by uniform operations, relating essentially some to the contacts and
+others to the intersections of surfaces. Simple and entirely general
+methods for the graphical solution of these two orders of problems
+having been formed, all the geometrical questions which may arise in any
+of the various arts of construction--stone-cutting, carpentry,
+perspective, dialling, fortification, &c.--can henceforth be treated as
+simple particular cases of a single theory, the invariable application
+of which will always necessarily lead to an exact solution, which may be
+facilitated in practice by profiting by the peculiar circumstances of
+each case.
+
+       *       *       *       *       *
+
+This important creation deserves in a remarkable degree to fix the
+attention of those philosophers who consider all that the human species
+has yet effected as a first step, and thus far the only really complete
+one, towards that general renovation of human labours, which must
+imprint upon all our arts a character of precision and of rationality,
+so necessary to their future progress. Such a revolution must, in fact,
+inevitably commence with that class of industrial labours, which is
+essentially connected with that science which is the most simple, the
+most perfect, and the most ancient. It cannot fail to extend hereafter,
+though with less facility, to all other practical operations. Indeed
+Monge himself, who conceived the true philosophy of the arts better than
+any one else, endeavoured to sketch out a corresponding system for the
+mechanical arts.
+
+Essential as the conception of descriptive geometry really is, it is
+very important not to deceive ourselves with respect to its true
+destination, as did those who, in the excitement of its first discovery,
+saw in it a means of enlarging the general and abstract domain of
+rational geometry. The result has in no way answered to these mistaken
+hopes. And, indeed, is it not evident that descriptive geometry has no
+special value except as a science of application, and as forming the
+true special theory of the geometrical arts? Considered in its abstract
+relations, it could not introduce any truly distinct order of
+geometrical speculations. We must not forget that, in order that a
+geometrical question should fall within the peculiar domain of
+descriptive geometry, it must necessarily have been previously resolved
+by speculative geometry, the solutions of which then, as we have seen,
+always need to be prepared for practice in such a way as to supply the
+place of constructions in relief by plane constructions; a substitution
+which really constitutes the only characteristic function of descriptive
+geometry.
+
+It is proper, however, to remark here, that, with regard to intellectual
+education, the study of descriptive geometry possesses an important
+philosophical peculiarity, quite independent of its high industrial
+utility. This is the advantage which it so pre-eminently offers--in
+habituating the mind to consider very complicated geometrical
+combinations in space, and to follow with precision their continual
+correspondence with the figures which are actually traced--of thus
+exercising to the utmost, in the most certain and precise manner, that
+important faculty of the human mind which is properly called
+"imagination," and which consists, in its elementary and positive
+acceptation, in representing to ourselves, clearly and easily, a vast
+and variable collection of ideal objects, as if they were really before
+us.
+
+Finally, to complete the indication of the general nature of descriptive
+geometry by determining its logical character, we have to observe that,
+while it belongs to the geometry of the ancients by the character of its
+solutions, on the other hand it approaches the geometry of the moderns
+by the nature of the questions which compose it. These questions are in
+fact eminently remarkable for that generality which, as we saw in the
+preceding chapter, constitutes the true fundamental character of modern
+geometry; for the methods used are always conceived as applicable to any
+figures whatever, the peculiarity of each having only a purely secondary
+influence. The solutions of descriptive geometry are then graphical,
+like most of those of the ancients, and at the same time general, like
+those of the moderns.
+
+       *       *       *       *       *
+
+After this important digression, we will pursue the philosophical
+examination of _special_ geometry, always considered as reduced to its
+least possible development, as an indispensable introduction to
+_general_ geometry. We have now sufficiently considered the _graphical_
+solution of the fundamental problem relating to the right line--that
+is, the determination of the different elements of any right-lined
+figure by means of one another--and have now to examine in a special
+manner the _algebraic_ solution.
+
+
+
+
+ALGEBRAIC SOLUTIONS.
+
+
+This kind of solution, the evident superiority of which need not here be
+dwelt upon, belongs necessarily, by the very nature of the question, to
+the system of the ancient geometry, although the logical method which is
+employed causes it to be generally, but very improperly, separated from
+it. We have thus an opportunity of verifying, in a very important
+respect, what was established generally in the preceding chapter, that
+it is not by the employment of the calculus that the modern geometry is
+essentially to be distinguished from the ancient. The ancients are in
+fact the true inventors of the present trigonometry, spherical as well
+as rectilinear; it being only much less perfect in their hands, on
+account of the extreme inferiority of their algebraical knowledge. It
+is, then, really in this chapter, and not, as it might at first be
+thought, in those which we shall afterwards devote to the philosophical
+examination of _general_ geometry, that it is proper to consider the
+character of this important preliminary theory, which is usually, though
+improperly, included in what is called _analytical geometry_, but which
+is really only a complement of _elementary geometry_ properly so called.
+
+Since all right-lined figures can be decomposed into triangles, it is
+evidently sufficient to know how to determine the different elements of
+a triangle by means of one another, which reduces _polygonometry_ to
+simple _trigonometry_.
+
+
+
+
+TRIGONOMETRY.
+
+
+The difficulty in resolving algebraically such a question as the above,
+consists essentially in forming, between the angles and the sides of a
+triangle, three distinct equations; which, when once obtained, will
+evidently reduce all trigonometrical problems to mere questions of
+analysis.
+
+
+_How to introduce Angles._ In considering the establishment of these
+equations in the most general manner, we immediately meet with a
+fundamental distinction with respect to the manner of introducing the
+angles into the calculation, according as they are made to enter
+_directly_, by themselves or by the circular arcs which are proportional
+to them; or _indirectly_, by the chords of these arcs, which are hence
+called their _trigonometrical lines_. Of these two systems of
+trigonometry the second was of necessity the only one originally
+adopted, as being the only practicable one, since the condition of
+geometry made it easy enough to find exact relations between the sides
+of the triangles and the trigonometrical lines which represent the
+angles, while it would have been absolutely impossible at that epoch to
+establish equations between the sides and the angles themselves.
+
+
+_Advantages of introducing Trigonometrical Lines._ At the present day,
+since the solution can be obtained by either system indifferently, that
+motive for preference no longer exists; but geometers have none the less
+persisted in following from choice the system primitively admitted from
+necessity; for, the same reason which enabled these trigonometrical
+equations to be obtained with much more facility, must, in like manner,
+as it is still more easy to conceive _à priori_, render these equations
+much more simple, since they then exist only between right lines,
+instead of being established between right lines and arcs of circles.
+Such a consideration has so much the more importance, as the question
+relates to formulas which are eminently elementary, and destined to be
+continually employed in all parts of mathematical science, as well as in
+all its various applications.
+
+It may be objected, however, that when an angle is given, it is, in
+reality, always given by itself, and not by its trigonometrical lines;
+and that when it is unknown, it is its angular value which is properly
+to be determined, and not that of any of its trigonometrical lines. It
+seems, according to this, that such lines are only useless
+intermediaries between the sides and the angles, which have to be
+finally eliminated, and the introduction of which does not appear
+capable of simplifying the proposed research. It is indeed important to
+explain, with more generality and precision than is customary, the great
+real utility of this manner of proceeding.
+
+
+_Division of Trigonometry into two Parts._ It consists in the fact that
+the introduction of these auxiliary magnitudes divides the entire
+question of trigonometry into two others essentially distinct, one of
+which has for its object to pass from the angles to their
+trigonometrical lines, or the converse, and the other of which proposes
+to determine the sides of the triangles by the trigonometrical lines of
+their angles, or the converse. Now the first of these two fundamental
+questions is evidently susceptible, by its nature, of being entirely
+treated and reduced to numerical tables once for all, in considering all
+possible angles, since it depends only upon those angles, and not at all
+upon the particular triangles in which they may enter in each case;
+while the solution of the second question must necessarily be renewed,
+at least in its arithmetical relations, for each new triangle which it
+is necessary to resolve. This is the reason why the first portion of the
+complete work, which would be precisely the most laborious, is no longer
+taken into the account, being always done in advance; while, if such a
+decomposition had not been performed, we would evidently have found
+ourselves under the obligation of recommencing the entire calculation in
+each particular case. Such is the essential property of the present
+trigonometrical system, which in fact would really present no actual
+advantage, if it was necessary to calculate continually the
+trigonometrical line of each angle to be considered, or the converse;
+the intermediate agency introduced would then be more troublesome than
+convenient.
+
+In order to clearly comprehend the true nature of this conception, it
+will be useful to compare it with a still more important one, designed
+to produce an analogous effect either in its algebraic, or, still more,
+in its arithmetical relations--the admirable theory of _logarithms_. In
+examining in a philosophical manner the influence of this theory, we see
+in fact that its general result is to decompose all imaginable
+arithmetical operations into two distinct parts. The first and most
+complicated of these is capable of being executed in advance once for
+all (since it depends only upon the numbers to be considered, and not at
+all upon the infinitely different combinations into which they can
+enter), and consists in considering all numbers as assignable powers of
+a constant number. The second part of the calculation, which must of
+necessity be recommenced for each new formula which is to have its
+value determined, is thenceforth reduced to executing upon these
+exponents correlative operations which are infinitely more simple. I
+confine myself here to merely indicating this resemblance, which any one
+can carry out for himself.
+
+We must besides observe, as a property (secondary at the present day,
+but all-important at its origin) of the trigonometrical system adopted,
+the very remarkable circumstance that the determination of angles by
+their trigonometrical lines, or the converse, admits of an arithmetical
+solution (the only one which is directly indispensable for the special
+destination of trigonometry) without the previous resolution of the
+corresponding algebraic question. It is doubtless to such a peculiarity
+that the ancients owed the possibility of knowing trigonometry. The
+investigation conceived in this way was so much the more easy, inasmuch
+as tables of chords (which the ancients naturally took as the
+trigonometrical lines) had been previously constructed for quite a
+different object, in the course of the labours of Archimedes on the
+rectification of the circle, from which resulted the actual
+determination of a certain series of chords; so that when Hipparchus
+subsequently invented trigonometry, he could confine himself to
+completing that operation by suitable intercalations; which shows
+clearly the connexion of ideas in that matter.
+
+
+_The Increase of such Trigonometrical Lines._ To complete this
+philosophical sketch of trigonometry, it is proper now to observe that
+the extension of the same considerations which lead us to replace angles
+or arcs of circles by straight lines, with the view of simplifying our
+equations, must also lead us to employ concurrently several
+trigonometrical lines, instead of confining ourselves to one only (as
+did the ancients), so as to perfect this system by choosing that one
+which will be algebraically the most convenient on each occasion. In
+this point of view, it is clear that the number of these lines is in
+itself no ways limited; provided that they are determined by the arc,
+and that they determine it, whatever may be the law according to which
+they are derived from it, they are suitable to be substituted for it in
+the equations. The Arabians, and subsequently the moderns, in confining
+themselves to the most simple constructions, have carried to four or
+five the number of _direct_ trigonometrical lines, which might be
+extended much farther.
+
+But instead of recurring to geometrical formations, which would finally
+become very complicated, we conceive with the utmost facility as many
+new trigonometrical lines as the analytical transformations may require,
+by means of a remarkable artifice, which is not usually apprehended in a
+sufficiently general manner. It consists in not directly multiplying the
+trigonometrical lines appropriate to each arc considered, but in
+introducing new ones, by considering this arc as indirectly determined
+by all lines relating to an arc which is a very simple function of the
+first. It is thus, for example, that, in order to calculate an angle
+with more facility, we will determine, instead of its sine, the sine of
+its half, or of its double, &c. Such a creation of _indirect_
+trigonometrical lines is evidently much more fruitful than all the
+direct geometrical methods for obtaining new ones. We may accordingly
+say that the number of trigonometrical lines actually employed at the
+present day by geometers is in reality unlimited, since at every
+instant, so to say, the transformations of analysis may lead us to
+augment it by the method which I have just indicated. Special names,
+however, have been given to those only of these _indirect_ lines which
+refer to the complement of the primitive arc, the others not occurring
+sufficiently often to render such denominations necessary; a
+circumstance which has caused a common misconception of the true extent
+of the system of trigonometry.
+
+
+_Study of their Mutual Relations._ This multiplicity of trigonometrical
+lines evidently gives rise to a third fundamental question in
+trigonometry, the study of the relations which exist between these
+different lines; since, without such a knowledge, we could not make use,
+for our analytical necessities, of this variety of auxiliary magnitudes,
+which, however, have no other destination. It is clear, besides, from
+the consideration just indicated, that this essential part of
+trigonometry, although simply preparatory, is, by its nature,
+susceptible of an indefinite extension when we view it in its entire
+generality, while the two others are circumscribed within rigorously
+defined limits.
+
+It is needless to add that these three principal parts of trigonometry
+have to be studied in precisely the inverse order from that in which we
+have seen them necessarily derived from the general nature of the
+subject; for the third is evidently independent of the two others, and
+the second, of that which was first presented--the resolution of
+triangles, properly so called--which must for that reason be treated in
+the last place; which rendered so much the more important the
+consideration of their natural succession and logical relations to one
+another.
+
+It is useless to consider here separately _spherical trigonometry_,
+which cannot give rise to any special philosophical consideration;
+since, essential as it is by the importance and the multiplicity of its
+uses, it can be treated at the present day only as a simple application
+of rectilinear trigonometry, which furnishes directly its fundamental
+equations, by substituting for the spherical triangle the corresponding
+trihedral angle.
+
+This summary exposition of the philosophy of trigonometry has been here
+given in order to render apparent, by an important example, that
+rigorous dependence and those successive ramifications which are
+presented by what are apparently the most simple questions of elementary
+geometry.
+
+       *       *       *       *       *
+
+Having thus examined the peculiar character of _special_ geometry
+reduced to its only dogmatic destination, that of furnishing to general
+geometry an indispensable preliminary basis, we have now to give all our
+attention to the true science of geometry, considered as a whole, in the
+most rational manner. For that purpose, it is necessary to carefully
+examine the great original idea of Descartes, upon which it is entirely
+founded. This will be the object of the following chapter.
+
+
+
+
+CHAPTER III.
+
+MODERN OR ANALYTICAL GEOMETRY.
+
+
+_General_ (or _Analytical_) geometry being entirely founded upon the
+transformation of geometrical considerations into equivalent analytical
+considerations, we must begin with examining directly and in a thorough
+manner the beautiful conception by which Descartes has established in a
+uniform manner the constant possibility of such a co-relation. Besides
+its own extreme importance as a means of highly perfecting geometrical
+science, or, rather, of establishing the whole of it on rational bases,
+the philosophical study of this admirable conception must have so much
+the greater interest in our eyes from its characterizing with perfect
+clearness the general method to be employed in organizing the relations
+of the abstract to the concrete in mathematics, by the analytical
+representation of natural phenomena. There is no conception, in the
+whole philosophy of mathematics which better deserves to fix all our
+attention.
+
+
+
+
+ANALYTICAL REPRESENTATION OF FIGURES.
+
+
+In order to succeed in expressing all imaginable geometrical phenomena
+by simple analytical relations, we must evidently, in the first place,
+establish a general method for representing analytically the subjects
+themselves in which these phenomena are found, that is, the lines or the
+surfaces to be considered. The _subject_ being thus habitually
+considered in a purely analytical point of view, we see how it is
+thenceforth possible to conceive in the same manner the various
+_accidents_ of which it is susceptible.
+
+In order to organize the representation of geometrical figures by
+analytical equations, we must previously surmount a fundamental
+difficulty; that of reducing the general elements of the various
+conceptions of geometry to simply numerical ideas; in a word, that of
+substituting in geometry pure considerations of _quantity_ for all
+considerations of _quality_.
+
+
+_Reduction of Figure to Position._ For this purpose let us observe, in
+the first place, that all geometrical ideas relate necessarily to these
+three universal categories: the _magnitude_, the _figure_, and the
+_position_ of the extensions to be considered. As to the first, there is
+evidently no difficulty; it enters at once into the ideas of numbers.
+With relation to the second, it must be remarked that it will always
+admit of being reduced to the third. For the figure of a body evidently
+results from the mutual position of the different points of which it is
+composed, so that the idea of position necessarily comprehends that of
+figure, and every circumstance of figure can be translated by a
+circumstance of position. It is in this way, in fact, that the human
+mind has proceeded in order to arrive at the analytical representation
+of geometrical figures, their conception relating directly only to
+positions. All the elementary difficulty is then properly reduced to
+that of referring ideas of situation to ideas of magnitude. Such is the
+direct destination of the preliminary conception upon which Descartes
+has established the general system of analytical geometry.
+
+His philosophical labour, in this relation, has consisted simply in the
+entire generalization of an elementary operation, which we may regard as
+natural to the human mind, since it is performed spontaneously, so to
+say, in all minds, even the most uncultivated. Thus, when we have to
+indicate the situation of an object without directly pointing it out,
+the method which we always adopt, and evidently the only one which can
+be employed, consists in referring that object to others which are
+known, by assigning the magnitude of the various geometrical elements,
+by which we conceive it connected with the known objects. These elements
+constitute what Descartes, and after him all geometers, have called the
+_co-ordinates_ of each point considered. They are necessarily two in
+number, if it is known in advance in what plane the point is situated;
+and three, if it may be found indifferently in any region of space. As
+many different constructions as can be imagined for determining the
+position of a point, whether on a plane or in space, so many distinct
+systems of co-ordinates may be conceived; they are consequently
+susceptible of being multiplied to infinity. But, whatever may be the
+system adopted, we shall always have reduced the ideas of situation to
+simple ideas of magnitude, so that we will consider the change in the
+position of a point as produced by mere numerical variations in the
+values of its co-ordinates.
+
+
+_Determination of the Position of a Point._ Considering at first only
+the least complicated case, that of _plane geometry_, it is in this way
+that we usually determine the position of a point on a plane, by its
+distances from two fixed right lines considered as known, which are
+called _axes_, and which are commonly supposed to be perpendicular to
+each other. This system is that most frequently adopted, because of its
+simplicity; but geometers employ occasionally an infinity of others.
+Thus the position of a point on a plane may be determined, 1°, by its
+distances from two fixed points; or, 2°, by its distance from a single
+fixed point, and the direction of that distance, estimated by the
+greater or less angle which it makes with a fixed right line, which
+constitutes the system of what are called _polar_ co-ordinates, the most
+frequently used after the system first mentioned; or, 3°, by the angles
+which the right lines drawn from the variable point to two fixed points
+make with the right line which joins these last; or, 4°, by the
+distances from that point to a fixed right line and a fixed point, &c.
+In a word, there is no geometrical figure whatever from which it is not
+possible to deduce a certain system of co-ordinates more or less
+susceptible of being employed.
+
+A general observation, which it is important to make in this connexion,
+is, that every system of co-ordinates is equivalent to determining a
+point, in plane geometry, by the intersection of two lines, each of
+which is subjected to certain fixed conditions of determination; a
+single one of these conditions remaining variable, sometimes the one,
+sometimes the other, according to the system considered. We could not,
+indeed, conceive any other means of constructing a point than to mark it
+by the meeting of two lines. Thus, in the most common system, that of
+_rectilinear co-ordinates_, properly so called, the point is determined
+by the intersection of two right lines, each of which remains constantly
+parallel to a fixed axis, at a greater or less distance from it; in the
+_polar_ system, the position of the point is marked by the meeting of a
+circle, of variable radius and fixed centre, with a movable right line
+compelled to turn about this centre: in other systems, the required
+point might be designated by the intersection of two circles, or of any
+other two lines, &c. In a word, to assign the value of one of the
+co-ordinates of a point in any system whatever, is always necessarily
+equivalent to determining a certain line on which that point must be
+situated. The geometers of antiquity had already made this essential
+remark, which served as the base of their method of geometrical _loci_,
+of which they made so happy a use to direct their researches in the
+resolution of _determinate_ problems, in considering separately the
+influence of each of the two conditions by which was defined each point
+constituting the object, direct or indirect, of the proposed question.
+It was the general systematization of this method which was the
+immediate motive of the labours of Descartes, which led him to create
+analytical geometry.
+
+After having clearly established this preliminary conception--by means
+of which ideas of position, and thence, implicitly, all elementary
+geometrical conceptions are capable of being reduced to simple numerical
+considerations--it is easy to form a direct conception, in its entire
+generality, of the great original idea of Descartes, relative to the
+analytical representation of geometrical figures: it is this which forms
+the special object of this chapter. I will continue to consider at
+first, for more facility, only geometry of two dimensions, which alone
+was treated by Descartes; and will afterwards examine separately, under
+the same point of view, the theory of surfaces and curves of double
+curvature.
+
+
+
+
+PLANE CURVES.
+
+
+_Expression of Lines by Equations._ In accordance with the manner of
+expressing analytically the position of a point on a plane, it can be
+easily established that, by whatever property any line may be defined,
+that definition always admits of being replaced by a corresponding
+equation between the two variable co-ordinates of the point which
+describes this line; an equation which will be thenceforth the
+analytical representation of the proposed line, every phenomenon of
+which will be translated by a certain algebraic modification of its
+equation. Thus, if we suppose that a point moves on a plane without its
+course being in any manner determined, we shall evidently have to regard
+its co-ordinates, to whatever system they may belong, as two variables
+entirely independent of one another. But if, on the contrary, this point
+is compelled to describe a certain line, we shall necessarily be
+compelled to conceive that its co-ordinates, in all the positions which
+it can take, retain a certain permanent and precise relation to each
+other, which is consequently susceptible of being expressed by a
+suitable equation; which will become the very clear and very rigorous
+analytical definition of the line under consideration, since it will
+express an algebraical property belonging exclusively to the
+co-ordinates of all the points of this line. It is clear, indeed, that
+when a point is not subjected to any condition, its situation is not
+determined except in giving at once its two co-ordinates, independently
+of each other; while, when the point must continue upon a defined line,
+a single co-ordinate is sufficient for completely fixing its position.
+The second co-ordinate is then a determinate _function_ of the first;
+or, in other words, there must exist between them a certain _equation_,
+of a nature corresponding to that of the line on which the point is
+compelled to remain. In a word, each of the co-ordinates of a point
+requiring it to be situated on a certain line, we conceive reciprocally
+that the condition, on the part of a point, of having to belong to a
+line defined in any manner whatever, is equivalent to assigning the
+value of one of the two co-ordinates; which is found in that case to be
+entirely dependent on the other. The analytical relation which expresses
+this dependence may be more or less difficult to discover, but it must
+evidently be always conceived to exist, even in the cases in which our
+present means may be insufficient to make it known. It is by this simple
+consideration that we may demonstrate, in an entirely general
+manner--independently of the particular verifications on which this
+fundamental conception is ordinarily established for each special
+definition of a line--the necessity of the analytical representation of
+lines by equations.
+
+
+_Expression of Equations by Lines._ Taking up again the same reflections
+in the inverse direction, we could show as easily the geometrical
+necessity of the representation of every equation of two variables, in a
+determinate system of co-ordinates, by a certain line; of which such a
+relation would be, in the absence of any other known property, a very
+characteristic definition, the scientific destination of which will be
+to fix the attention directly upon the general course of the solutions
+of the equation, which will thus be noted in the most striking and the
+most simple manner. This picturing of equations is one of the most
+important fundamental advantages of analytical geometry, which has
+thereby reacted in the highest degree upon the general perfecting of
+analysis itself; not only by assigning to purely abstract researches a
+clearly determined object and an inexhaustible career, but, in a still
+more direct relation, by furnishing a new philosophical medium for
+analytical meditation which could not be replaced by any other. In fact,
+the purely algebraic discussion of an equation undoubtedly makes known
+its solutions in the most precise manner, but in considering them only
+one by one, so that in this way no general view of them could be
+obtained, except as the final result of a long and laborious series of
+numerical comparisons. On the other hand, the geometrical _locus_ of the
+equation, being only designed to represent distinctly and with perfect
+clearness the summing up of all these comparisons, permits it to be
+directly considered, without paying any attention to the details which
+have furnished it. It can thereby suggest to our mind general analytical
+views, which we should have arrived at with much difficulty in any other
+manner, for want of a means of clearly characterizing their object. It
+is evident, for example, that the simple inspection of the logarithmic
+curve, or of the curve _y_ = sin. _x_, makes us perceive much more
+distinctly the general manner of the variations of logarithms with
+respect to their numbers, or of sines with respect to their arcs, than
+could the most attentive study of a table of logarithms or of natural
+sines. It is well known that this method has become entirely elementary
+at the present day, and that it is employed whenever it is desired to
+get a clear idea of the general character of the law which reigns in a
+series of precise observations of any kind whatever.
+
+
+_Any Change in the Line causes a Change in the Equation._ Returning to
+the representation of lines by equations, which is our principal object,
+we see that this representation is, by its nature, so faithful, that the
+line could not experience any modification, however slight it might be,
+without causing a corresponding change in the equation. This perfect
+exactitude even gives rise oftentimes to special difficulties; for
+since, in our system of analytical geometry, the mere displacements of
+lines affect the equations, as well as their real variations in
+magnitude or form, we should be liable to confound them with one another
+in our analytical expressions, if geometers had not discovered an
+ingenious method designed expressly to always distinguish them. This
+method is founded on this principle, that although it is impossible to
+change analytically at will the position of a line with respect to the
+axes of the co-ordinates, we can change in any manner whatever the
+situation of the axes themselves, which evidently amounts to the same;
+then, by the aid of the very simple general formula by which this
+transformation of the axes is produced, it becomes easy to discover
+whether two different equations are the analytical expressions of only
+the same line differently situated, or refer to truly distinct
+geometrical loci; since, in the former case, one of them will pass into
+the other by suitably changing the axes or the other constants of the
+system of co-ordinates employed. It must, moreover, be remarked on this
+subject, that general inconveniences of this nature seem to be
+absolutely inevitable in analytical geometry; for, since the ideas of
+position are, as we have seen, the only geometrical ideas immediately
+reducible to numerical considerations, and the conceptions of figure
+cannot be thus reduced, except by seeing in them relations of situation,
+it is impossible for analysis to escape confounding, at first, the
+phenomena of figure with simple phenomena of position, which alone are
+directly expressed by the equations.
+
+
+_Every Definition of a Line is an Equation._ In order to complete the
+philosophical explanation of the fundamental conception which serves as
+the base of analytical geometry, I think that I should here indicate a
+new general consideration, which seems to me particularly well adapted
+for putting in the clearest point of view this necessary representation
+of lines by equations with two variables. It consists in this, that not
+only, as we have shown, must every defined line necessarily give rise to
+a certain equation between the two co-ordinates of any one of its
+points, but, still farther, every definition of a line may be regarded
+as being already of itself an equation of that line in a suitable system
+of co-ordinates.
+
+It is easy to establish this principle, first making a preliminary
+logical distinction with respect to different kinds of definitions. The
+rigorously indispensable condition of every definition is that of
+distinguishing the object defined from all others, by assigning to it a
+property which belongs to it exclusively. But this end may be generally
+attained in two very different ways; either by a definition which is
+simply _characteristic_, that is, indicative of a property which,
+although truly exclusive, does not make known the mode of generation of
+the object; or by a definition which is really _explanatory_, that is,
+which characterizes the object by a property which expresses one of its
+modes of generation. For example, in considering the circle as the line,
+which, under the same contour, contains the greatest area, we have
+evidently a definition of the first kind; while in choosing the property
+of its having all its points equally distant from a fixed point, we have
+a definition of the second kind. It is, besides, evident, as a general
+principle, that even when any object whatever is known at first only by
+a _characteristic_ definition, we ought, nevertheless, to regard it as
+susceptible of _explanatory_ definitions, which the farther study of the
+object would necessarily lead us to discover.
+
+This being premised, it is clear that the general observation above
+made, which represents every definition of a line as being necessarily
+an equation of that line in a certain system of co-ordinates, cannot
+apply to definitions which are simply _characteristic_; it is to be
+understood only of definitions which are truly _explanatory_. But, in
+considering only this class, the principle is easy to prove. In fact, it
+is evidently impossible to define the generation of a line without
+specifying a certain relation between the two simple motions of
+translation or of rotation, into which the motion of the point which
+describes it will be decomposed at each instant. Now if we form the most
+general conception of what constitutes _a system of co-ordinates_, and
+admit all possible systems, it is clear that such a relation will be
+nothing else but the _equation_ of the proposed line, in a system of
+co-ordinates of a nature corresponding to that of the mode of generation
+considered. Thus, for example, the common definition of the _circle_ may
+evidently be regarded as being immediately the _polar equation_ of this
+curve, taking the centre of the circle for the pole. In the same way,
+the elementary definition of the _ellipse_ or of the _hyperbola_--as
+being the curve generated by a point which moves in such a manner that
+the sum or the difference of its distances from two fixed points remains
+constant--gives at once, for either the one or the other curve, the
+equation _y_ + _x_ = _c_, taking for the system of co-ordinates that in
+which the position of a point would be determined by its distances from
+two fixed points, and choosing for these poles the two given foci. In
+like manner, the common definition of any _cycloid_ would furnish
+directly, for that curve, the equation _y_ = _mx_; adopting as the
+co-ordinates of each point the arc which it marks upon a circle of
+invariable radius, measuring from the point of contact of that circle
+with a fixed line, and the rectilinear distance from that point of
+contact to a certain origin taken on that right line. We can make
+analogous and equally easy verifications with respect to the customary
+definitions of spirals, of epicycloids, &c. We shall constantly find
+that there exists a certain system of co-ordinates, in which we
+immediately obtain a very simple equation of the proposed line, by
+merely writing algebraically the condition imposed by the mode of
+generation considered.
+
+Besides its direct importance as a means of rendering perfectly apparent
+the necessary representation of every line by an equation, the preceding
+consideration seems to me to possess a true scientific utility, in
+characterizing with precision the principal general difficulty which
+occurs in the actual establishment of these equations, and in
+consequently furnishing an interesting indication with respect to the
+course to be pursued in inquiries of this kind, which, by their nature,
+could not admit of complete and invariable rules. In fact, since any
+definition whatever of a line, at least among those which indicate a
+mode of generation, furnishes directly the equation of that line in a
+certain system of co-ordinates, or, rather, of itself constitutes that
+equation, it follows that the difficulty which we often experience in
+discovering the equation of a curve, by means of certain of its
+characteristic properties, a difficulty which is sometimes very great,
+must proceed essentially only from the commonly imposed condition of
+expressing this curve analytically by the aid of a designated system of
+co-ordinates, instead of admitting indifferently all possible systems.
+These different systems cannot be regarded in analytical geometry as
+being all equally suitable; for various reasons, the most important of
+which will be hereafter discussed, geometers think that curves should
+almost always be referred, as far as is possible, to _rectilinear
+co-ordinates_, properly so called. Now we see, from what precedes, that
+in many cases these particular co-ordinates will not be those with
+reference to which the equation of the curve will be found to be
+directly established by the proposed definition. The principal
+difficulty presented by the formation of the equation of a line really
+consists, then, in general, in a certain transformation of co-ordinates.
+It is undoubtedly true that this consideration does not subject the
+establishment of these equations to a truly complete general method, the
+success of which is always certain; which, from the very nature of the
+subject, is evidently chimerical: but such a view may throw much useful
+light upon the course which it is proper to adopt, in order to arrive at
+the end proposed. Thus, after having in the first place formed the
+preparatory equation, which is spontaneously derived from the definition
+which we are considering, it will be necessary, in order to obtain the
+equation belonging to the system of co-ordinates which must be finally
+admitted, to endeavour to express in a function of these last
+co-ordinates those which naturally correspond to the given mode of
+generation. It is upon this last labour that it is evidently impossible
+to give invariable and precise precepts. We can only say that we shall
+have so many more resources in this matter as we shall know more of true
+analytical geometry, that is, as we shall know the algebraical
+expression of a greater number of different algebraical phenomena.
+
+
+
+
+CHOICE OF CO-ORDINATES.
+
+
+In order to complete the philosophical exposition of the conception
+which serves as the base of analytical geometry, I have yet to notice
+the considerations relating to the choice of the system of co-ordinates
+which is in general the most suitable. They will give the rational
+explanation of the preference unanimously accorded to the ordinary
+rectilinear system; a preference which has hitherto been rather the
+effect of an empirical sentiment of the superiority of this system, than
+the exact result of a direct and thorough analysis.
+
+
+_Two different Points of View._ In order to decide clearly between all
+the different systems of co-ordinates, it is indispensable to
+distinguish with care the two general points of view, the converse of
+one another, which belong to analytical geometry; namely, the relation
+of algebra to geometry, founded upon the representation of lines by
+equations; and, reciprocally, the relation of geometry to algebra,
+founded on the representation of equations by lines.
+
+It is evident that in every investigation of general geometry these two
+fundamental points of view are of necessity always found combined,
+since we have always to pass alternately, and at insensible intervals,
+so to say, from geometrical to analytical considerations, and from
+analytical to geometrical considerations. But the necessity of here
+temporarily separating them is none the less real; for the answer to the
+question of method which we are examining is, in fact, as we shall see
+presently, very far from being the same in both these relations, so that
+without this distinction we could not form any clear idea of it.
+
+
+1. _Representation of Lines by Equations._ Under _the first point of
+view_--the representation of lines by equations--the only reason which
+could lead us to prefer one system of co-ordinates to another would be
+the greater simplicity of the equation of each line, and greater
+facility in arriving at it. Now it is easy to see that there does not
+exist, and could not be expected to exist, any system of co-ordinates
+deserving in that respect a constant preference over all others. In
+fact, we have above remarked that for each geometrical definition
+proposed we can conceive a system of co-ordinates in which the equation
+of the line is obtained at once, and is necessarily found to be also
+very simple; and this system, moreover, inevitably varies with the
+nature of the characteristic property under consideration. The
+rectilinear system could not, therefore, be constantly the most
+advantageous for this object, although it may often be very favourable;
+there is probably no system which, in certain particular cases, should
+not be preferred to it, as well as to every other.
+
+
+2. _Representation of Equations by Lines._ It is by no means so,
+however, under the _second point of view_. We can, indeed, easily
+establish, as a general principle, that the ordinary rectilinear system
+must necessarily be better adapted than any other to the representation
+of equations by the corresponding geometrical _loci_; that is to say,
+that this representation is constantly more simple and more faithful in
+it than in any other.
+
+Let us consider, for this object, that, since every system of
+co-ordinates consists in determining a point by the intersection of two
+lines, the system adapted to furnish the most suitable geometrical
+_loci_ must be that in which these two lines are the simplest possible;
+a consideration which confines our choice to the _rectilinear_ system.
+In truth, there is evidently an infinite number of systems which deserve
+that name, that is to say, which employ only right lines to determine
+points, besides the ordinary system which assigns the distances from two
+fixed lines as co-ordinates; such, for example, would be that in which
+the co-ordinates of each point should be the two angles which the right
+lines, which go from that point to two fixed points, make with the right
+line, which joins these last points: so that this first consideration is
+not rigorously sufficient to explain the preference unanimously given to
+the common system. But in examining in a more thorough manner the nature
+of every system of co-ordinates, we also perceive that each of the two
+lines, whose meeting determines the point considered, must necessarily
+offer at every instant, among its different conditions of determination,
+a single variable condition, which gives rise to the corresponding
+co-ordinate, all the rest being fixed, and constituting the _axes_ of
+the system, taking this term in its most extended mathematical
+acceptation. The variation is indispensable, in order that we may be
+able to consider all possible positions; and the fixity is no less so,
+in order that there may exist means of comparison. Thus, in all
+_rectilinear_ systems, each of the two right lines will be subjected to
+a fixed condition, and the ordinate will result from the variable
+condition.
+
+
+_Superiority of rectilinear Co-ordinates._ From these considerations it
+is evident, as a general principle, that the most favourable system for
+the construction of geometrical _loci_ will necessarily be that in which
+the variable condition of each right line shall be the simplest
+possible; the fixed condition being left free to be made complex, if
+necessary to attain that object. Now, of all possible manners of
+determining two movable right lines, the easiest to follow geometrically
+is certainly that in which, the direction of each right line remaining
+invariable, it only approaches or recedes, more or less, to or from a
+constant axis. It would be, for example, evidently more difficult to
+figure to one's self clearly the changes of place of a point which is
+determined by the intersection of two right lines, which each turn
+around a fixed point, making a greater or smaller angle with a certain
+axis, as in the system of co-ordinates previously noticed. Such is the
+true general explanation of the fundamental property possessed by the
+common rectilinear system, of being better adapted than any other to the
+geometrical representation of equations, inasmuch as it is that one in
+which it is the easiest to conceive the change of place of a point
+resulting from the change in the value of its co-ordinates. In order to
+feel clearly all the force of this consideration, it would be sufficient
+to carefully compare this system with the polar system, in which this
+geometrical image, so simple and so easy to follow, of two right lines
+moving parallel, each one of them, to its corresponding axis, is
+replaced by the complicated picture of an infinite series of concentric
+circles, cut by a right line compelled to turn about a fixed point. It
+is, moreover, easy to conceive in advance what must be the extreme
+importance to analytical geometry of a property so profoundly
+elementary, which, for that reason, must be recurring at every instant,
+and take a progressively increasing value in all labours of this kind.
+
+
+_Perpendicularity of the Axes._ In pursuing farther the consideration
+which demonstrates the superiority of the ordinary system of
+co-ordinates over any other as to the representation of equations, we
+may also take notice of the utility for this object of the common usage
+of taking the two axes perpendicular to each other, whenever possible,
+rather than with any other inclination. As regards the representation of
+lines by equations, this secondary circumstance is no more universally
+proper than we have seen the general nature of the system to be; since,
+according to the particular occasion, any other inclination of the axes
+may deserve our preference in that respect. But, in the inverse point of
+view, it is easy to see that rectangular axes constantly permit us to
+represent equations in a more simple and even more faithful manner; for,
+with oblique axes, space being divided by them into regions which no
+longer have a perfect identity, it follows that, if the geometrical
+_locus_ of the equation extends into all these regions at once, there
+will be presented, by reason merely of this inequality of the angles,
+differences of figure which do not correspond to any analytical
+diversity, and will necessarily alter the rigorous exactness of the
+representation, by being confounded with the proper results of the
+algebraic comparisons. For example, an equation like: _x^m_ + _y^m_ =
+_c_, which, by its perfect symmetry, should evidently give a curve
+composed of four identical quarters, will be represented, on the
+contrary, if we take axes not rectangular, by a geometric _locus_, the
+four parts of which will be unequal. It is plain that the only means of
+avoiding all inconveniences of this kind is to suppose the angle of the
+two axes to be a right angle.
+
+The preceding discussion clearly shows that, although the ordinary
+system of rectilinear co-ordinates has no constant superiority over all
+others in one of the two fundamental points of view which are
+continually combined in analytical geometry, yet as, on the other hand,
+it is not constantly inferior, its necessary and absolute greater
+aptitude for the representation of equations must cause it to generally
+receive the preference; although it may evidently happen, in some
+particular cases, that the necessity of simplifying equations and of
+obtaining them more easily may determine geometers to adopt a less
+perfect system. The rectilinear system is, therefore, the one by means
+of which are ordinarily constructed the most essential theories of
+general geometry, intended to express analytically the most important
+geometrical phenomena. When it is thought necessary to choose some
+other, the polar system is almost always the one which is fixed upon,
+this system being of a nature sufficiently opposite to that of the
+rectilinear system to cause the equations, which are too complicated
+with respect to the latter, to become, in general, sufficiently simple
+with respect to the other. Polar co-ordinates, moreover, have often the
+advantage of admitting of a more direct and natural concrete
+signification; as is the case in mechanics, for the geometrical
+questions to which the theory of circular movement gives rise, and in
+almost all the cases of celestial geometry.
+
+       *       *       *       *       *
+
+In order to simplify the exposition, we have thus far considered the
+fundamental conception of analytical geometry only with respect to
+_plane curves_, the general study of which was the only object of the
+great philosophical renovation produced by Descartes. To complete this
+important explanation, we have now to show summarily how this elementary
+idea was extended by Clairaut, about a century afterwards, to the
+general study of _surfaces_ and _curves of double curvature_. The
+considerations which have been already given will permit me to limit
+myself on this subject to the rapid examination of what is strictly
+peculiar to this new case.
+
+
+
+
+SURFACES.
+
+
+_Determination of a Point in Space._ The complete analytical
+determination of a point in space evidently requires the values of three
+co-ordinates to be assigned; as, for example, in the system which is
+generally adopted, and which corresponds to the _rectilinear_ system of
+plane geometry, distances from the point to three fixed planes, usually
+perpendicular to one another; which presents the point as the
+intersection of three planes whose direction is invariable. We might
+also employ the distances from the movable point to three fixed points,
+which would determine it by the intersection of three spheres with a
+common centre. In like manner, the position of a point would be defined
+by giving its distance from a fixed point, and the direction of that
+distance, by means of the two angles which this right line makes with
+two invariable axes; this is the _polar_ system of geometry of three
+dimensions; the point is then constructed by the intersection of a
+sphere having a fixed centre, with two right cones with circular bases,
+whose axes and common summit do not change. In a word, there is
+evidently, in this case at least, the same infinite variety among the
+various possible systems of co-ordinates which we have already observed
+in geometry of two dimensions. In general, we have to conceive a point
+as being always determined by the intersection of any three surfaces
+whatever, as it was in the former case by that of two lines: each of
+these three surfaces has, in like manner, all its conditions of
+determination constant, excepting one, which gives rise to the
+corresponding co-ordinates, whose peculiar geometrical influence is thus
+to constrain the point to be situated upon that surface.
+
+This being premised, it is clear that if the three co-ordinates of a
+point are entirely independent of one another, that point can take
+successively all possible positions in space. But if the point is
+compelled to remain upon a certain surface defined in any manner
+whatever, then two co-ordinates are evidently sufficient for determining
+its situation at each instant, since the proposed surface will take the
+place of the condition imposed by the third co-ordinate. We must then,
+in this case, under the analytical point of view, necessarily conceive
+this last co-ordinate as a determinate function of the two others, these
+latter remaining perfectly independent of each other. Thus there will be
+a certain equation between the three variable co-ordinates, which will
+be permanent, and which will be the only one, in order to correspond to
+the precise degree of indetermination in the position of the point.
+
+
+_Expression of Surfaces by Equations._ This equation, more or less easy
+to be discovered, but always possible, will be the analytical definition
+of the proposed surface, since it must be verified for all the points of
+that surface, and for them alone. If the surface undergoes any change
+whatever, even a simple change of place, the equation must undergo a
+more or less serious corresponding modification. In a word, all
+geometrical phenomena relating to surfaces will admit of being
+translated by certain equivalent analytical conditions appropriate to
+equations of three variables; and in the establishment and
+interpretation of this general and necessary harmony will essentially
+consist the science of analytical geometry of three dimensions.
+
+
+_Expression of Equations by Surfaces._ Considering next this fundamental
+conception in the inverse point of view, we see in the same manner that
+every equation of three variables may, in general, be represented
+geometrically by a determinate surface, primitively defined by the very
+characteristic property, that the co-ordinates of all its points always
+retain the mutual relation enunciated in this equation. This geometrical
+locus will evidently change, for the same equation, according to the
+system of co-ordinates which may serve for the construction of this
+representation. In adopting, for example, the rectilinear system, it is
+clear that in the equation between the three variables, _x_, _y_, _z_,
+every particular value attributed to _z_ will give an equation between
+at _x_ and _y_, the geometrical locus of which will be a certain line
+situated in a plane parallel to the plane of _x_ and _y_, and at a
+distance from this last equal to the value of _z_; so that the complete
+geometrical locus will present itself as composed of an infinite series
+of lines superimposed in a series of parallel planes (excepting the
+interruptions which may exist), and will consequently form a veritable
+surface. It would be the same in considering any other system of
+co-ordinates, although the geometrical construction of the equation
+becomes more difficult to follow.
+
+Such is the elementary conception, the complement of the original idea
+of Descartes, on which is founded general geometry relative to surfaces.
+It would be useless to take up here directly the other considerations
+which have been above indicated, with respect to lines, and which any
+one can easily extend to surfaces; whether to show that every definition
+of a surface by any method of generation whatever is really a direct
+equation of that surface in a certain system of co-ordinates, or to
+determine among all the different systems of possible co-ordinates that
+one which is generally the most convenient. I will only add, on this
+last point, that the necessary superiority of the ordinary rectilinear
+system, as to the representation of equations, is evidently still more
+marked in analytical geometry of three dimensions than in that of two,
+because of the incomparably greater geometrical complication which would
+result from the choice of any other system. This can be verified in the
+most striking manner by considering the polar system in particular,
+which is the most employed after the ordinary rectilinear system, for
+surfaces as well as for plane curves, and for the same reasons.
+
+In order to complete the general exposition of the fundamental
+conception relative to the analytical study of surfaces, a philosophical
+examination should be made of a final improvement of the highest
+importance, which Monge has introduced into the very elements of this
+theory, for the classification of surfaces in natural families,
+established according to the mode of generation, and expressed
+algebraically by common differential equations, or by finite equations
+containing arbitrary functions.
+
+
+
+
+CURVES OF DOUBLE CURVATURE.
+
+
+Let us now consider the last elementary point of view of analytical
+geometry of three dimensions; that relating to the algebraic
+representation of curves considered in space, in the most general
+manner. In continuing to follow the principle which has been constantly
+employed, that of the degree of indetermination of the geometrical
+locus, corresponding to the degree of independence of the variables, it
+is evident, as a general principle, that when a point is required to be
+situated upon some certain curve, a single co-ordinate is enough for
+completely determining its position, by the intersection of this curve
+with the surface which results from this co-ordinate. Thus, in this
+case, the two other co-ordinates of the point must be conceived as
+functions necessarily determinate and distinct from the first. It
+follows that every line, considered in space, is then represented
+analytically, no longer by a single equation, but by the system of two
+equations between the three co-ordinates of any one of its points. It is
+clear, indeed, from another point of view, that since each of these
+equations, considered separately, expresses a certain surface, their
+combination presents the proposed line as the intersection of two
+determinate surfaces. Such is the most general manner of conceiving the
+algebraic representation of a line in analytical geometry of three
+dimensions. This conception is commonly considered in too restricted a
+manner, when we confine ourselves to considering a line as determined by
+the system of its two _projections_ upon two of the co-ordinate planes;
+a system characterized, analytically, by this peculiarity, that each of
+the two equations of the line then contains only two of the three
+co-ordinates, instead of simultaneously including the three variables.
+This consideration, which consists in regarding the line as the
+intersection of two cylindrical surfaces parallel to two of the three
+axes of the co-ordinates, besides the inconvenience of being confined to
+the ordinary rectilinear system, has the fault, if we strictly confine
+ourselves to it, of introducing useless difficulties into the analytical
+representation of lines, since the combination of these two cylinders
+would evidently not be always the most suitable for forming the
+equations of a line. Thus, considering this fundamental notion in its
+entire generality, it will be necessary in each case to choose, from
+among the infinite number of couples of surfaces, the intersection of
+which might produce the proposed curve, that one which will lend itself
+the best to the establishment of equations, as being composed of the
+best known surfaces. Thus, if the problem is to express analytically a
+circle in space, it will evidently be preferable to consider it as the
+intersection of a sphere and a plane, rather than as proceeding from any
+other combination of surfaces which could equally produce it.
+
+In truth, this manner of conceiving the representation of lines by
+equations, in analytical geometry of three dimensions, produces, by its
+nature, a necessary inconvenience, that of a certain analytical
+confusion, consisting in this: that the same line may thus be expressed,
+with the same system of co-ordinates, by an infinite number of different
+couples of equations, on account of the infinite number of couples of
+surfaces which can form it; a circumstance which may cause some
+difficulties in recognizing this line under all the algebraical
+disguises of which it admits. But there exists a very simple method for
+causing this inconvenience to disappear; it consists in giving up the
+facilities which result from this variety of geometrical constructions.
+It suffices, in fact, whatever may be the analytical system primitively
+established for a certain line, to be able to deduce from it the system
+corresponding to a single couple of surfaces uniformly generated; as,
+for example, to that of the two cylindrical surfaces which _project_ the
+proposed line upon two of the co-ordinate planes; surfaces which will
+evidently be always identical, in whatever manner the line may have been
+obtained, and which will not vary except when that line itself shall
+change. Now, in choosing this fixed system, which is actually the most
+simple, we shall generally be able to deduce from the primitive
+equations those which correspond to them in this special construction,
+by transforming them, by two successive eliminations, into two
+equations, each containing only two of the variable co-ordinates, and
+thereby corresponding to the two surfaces of projection. Such is really
+the principal destination of this sort of geometrical combination, which
+thus offers to us an invariable and certain means of recognizing the
+identity of lines in spite of the diversity of their equations, which is
+sometimes very great.
+
+
+
+
+IMPERFECTIONS OF ANALYTICAL GEOMETRY.
+
+
+Having now considered the fundamental conception of analytical geometry
+under its principal elementary aspects, it is proper, in order to make
+the sketch complete, to notice here the general imperfections yet
+presented by this conception with respect to both geometry and to
+analysis.
+
+_Relatively to geometry_, we must remark that the equations are as yet
+adapted to represent only entire geometrical loci, and not at all
+determinate portions of those loci. It would, however, be necessary, in
+some circumstances, to be able to express analytically a part of a line
+or of a surface, or even a _discontinuous_ line or surface, composed of
+a series of sections belonging to distinct geometrical figures, such as
+the contour of a polygon, or the surface of a polyhedron. Thermology,
+especially, often gives rise to such considerations, to which our
+present analytical geometry is necessarily inapplicable. The labours of
+M. Fourier on discontinuous functions have, however, begun to fill up
+this great gap, and have thereby introduced a new and essential
+improvement into the fundamental conception of Descartes. But this
+manner of representing heterogeneous or partial figures, being founded
+on the employment of trigonometrical series proceeding according to the
+sines of an infinite series of multiple arcs, or on the use of certain
+definite integrals equivalent to those series, and the general integral
+of which is unknown, presents as yet too much complication to admit of
+being immediately introduced into the system of analytical geometry.
+
+_Relatively to analysis_, we must begin by observing that our inability
+to conceive a geometrical representation of equations containing four,
+five, or more variables, analogous to those representations which all
+equations of two or of three variables admit, must not be viewed as an
+imperfection of our system of analytical geometry, for it evidently
+belongs to the very nature of the subject. Analysis being necessarily
+more general than geometry, since it relates to all possible phenomena,
+it would be very unphilosophical to desire always to find among
+geometrical phenomena alone a concrete representation of all the laws
+which analysis can express.
+
+There exists, however, another imperfection of less importance, which
+must really be viewed as proceeding from the manner in which we conceive
+analytical geometry. It consists in the evident incompleteness of our
+present representation of equations of two or of three variables by
+lines or surfaces, inasmuch as in the construction of the geometric
+locus we pay regard only to the _real_ solutions of equations, without
+at all noticing any _imaginary_ solutions. The general course of these
+last should, however, by its nature, be quite as susceptible as that of
+the others of a geometrical representation. It follows from this
+omission that the graphic picture of the equation is constantly
+imperfect, and sometimes even so much so that there is no geometric
+representation at all when the equation admits of only imaginary
+solutions. But, even in this last case, we evidently ought to be able to
+distinguish between equations as different in themselves as these, for
+example,
+
+  _x²_ + _y²_ + 1 = 0, _x⁶_ + _y⁴_ + 1 = 0, _y²_ + _e^x_ = 0.
+
+We know, moreover, that this principal imperfection often brings with
+it, in analytical geometry of two or of three dimensions, a number of
+secondary inconveniences, arising from several analytical modifications
+not corresponding to any geometrical phenomena.
+
+       *       *       *       *       *
+
+Our philosophical exposition of the fundamental conception of analytical
+geometry shows us clearly that this science consists essentially in
+determining what is the general analytical expression of such or such a
+geometrical phenomenon belonging to lines or to surfaces; and,
+reciprocally, in discovering the geometrical interpretation of such or
+such an analytical consideration. A detailed examination of the most
+important general questions would show us how geometers have succeeded
+in actually establishing this beautiful harmony, and in thus imprinting
+on geometrical science, regarded as a whole, its present eminently
+perfect character of rationality and of simplicity.
+
+     _Note._--The author devotes the two following chapters of his
+     course to the more detailed examination of Analytical Geometry of
+     two and of three dimensions; but his subsequent publication of a
+     separate work upon this branch of mathematics has been thought to
+     render unnecessary the reproduction of these two chapters in the
+     present volume.
+
+
+THE END.
+
+
+
+
+
+End of Project Gutenberg's The philosophy of mathematics, by Auguste Comte
+
+*** END OF THIS PROJECT GUTENBERG EBOOK THE PHILOSOPHY OF MATHEMATICS ***
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+<pre>
+
+Project Gutenberg's The philosophy of mathematics, by Auguste Comte
+
+This eBook is for the use of anyone anywhere at no cost and with
+almost no restrictions whatsoever.  You may copy it, give it away or
+re-use it under the terms of the Project Gutenberg License included
+with this eBook or online at www.gutenberg.org/license
+
+
+Title: The philosophy of mathematics
+
+Author: Auguste Comte
+
+Translator: W. M. Gillespie
+
+Release Date: May 15, 2012 [EBook #39702]
+
+Language: English
+
+Character set encoding: UTF-8
+
+*** START OF THIS PROJECT GUTENBERG EBOOK THE PHILOSOPHY OF MATHEMATICS ***
+
+
+
+
+Produced by Anna Hall, Albert László and the Online
+Distributed Proofreading Team at http://www.pgdp.net (This
+file was produced from images generously made available
+by The Internet Archive)
+
+
+
+
+
+
+</pre>
+
+<p><span class="pagenum"><a name="Page_1" id="Page_1">[Pg 1]</a></span></p>
+
+
+<h1><span class="small">THE</span><br />
+PHILOSOPHY<br />
+<span class="small">OF</span><br />
+MATHEMATICS.</h1>
+
+<p><span class="pagenum"><a name="Page_2" id="Page_2">[Pg 2]</a></span></p>
+
+
+<p><span class="pagenum"><a name="Page_3" id="Page_3">[Pg 3]</a></span></p>
+
+<div class="figcenter" style="width: 600px;">
+<a href="images/tree_b.jpg" class="fnanchor">
+<img src="images/tree_s.jpg" width="600" height="282" alt="The science of mathematics" />
+</a>
+</div>
+
+
+<p class="center">THE
+PHILOSOPHY
+OF
+MATHEMATICS;</p>
+
+<p class="center">TRANSLATED FROM THE<br />
+COURS DE PHILOSOPHIE POSITIVE<br />
+<span class="small">OF</span><br />
+<span class="big">AUGUSTE COMTE,</span></p>
+<p class="center"><span class="small">BY</span><br />
+W. M. GILLESPIE,<br />
+<span class="small">PROFESSOR OF CIVIL ENGINEERING &amp; ADJ. PROF. OF MATHEMATICS<br />
+IN UNION COLLEGE.</span></p>
+
+<p class="center">NEW YORK:<br />
+HARPER &amp; BROTHERS, PUBLISHERS,<br />
+82 CLIFF STREET<br />
+1851.</p>
+<p><span class="pagenum"><a name="Page_4" id="Page_4">[Pg 4]</a></span></p>
+
+
+<p class="center">
+Entered, according to Act of Congress, in the year one thousand<br />
+eight hundred and fifty-one, by<br />
+<br />
+<span class="smcap">Harper &amp; Brothers</span>.<br />
+<br />
+in the Clerk's Office of the District Court of the Southern District<br />
+of New York.</p>
+<p><span class="pagenum"><a name="Page_5" id="Page_5">[Pg 5]</a></span></p>
+
+
+
+
+<h2><a name="PREFACE" id="PREFACE">PREFACE.</a></h2>
+
+
+<p>The pleasure and profit which the translator
+has received from the great work here presented,
+have induced him to lay it before his fellow-teachers
+and students of Mathematics in a more accessible
+form than that in which it has hitherto appeared.
+The want of a comprehensive map of the
+wide region of mathematical science&mdash;a bird's-eye
+view of its leading features, and of the true bearings
+and relations of all its parts&mdash;is felt by every
+thoughtful student. He is like the visitor to a
+great city, who gets no just idea of its extent and
+situation till he has seen it from some commanding
+eminence. To have a panoramic view of the
+whole district&mdash;presenting at one glance all the
+parts in due co-ordination, and the darkest nooks
+clearly shown&mdash;is invaluable to either traveller or
+student. It is this which has been most perfectly
+accomplished for mathematical science by the
+author whose work is here presented.</p>
+
+<p>Clearness and depth, comprehensiveness and
+precision, have never, perhaps, been so remarkably
+united as in <span class="smcap">Auguste Comte</span>. He views his subject
+from an elevation which gives to each part of
+the complex whole its true position and value,
+while his telescopic glance loses none of the needful
+details, and not only itself pierces to the heart<span class="pagenum"><a name="Page_6" id="Page_6">[Pg 6]</a></span>
+of the matter, but converts its opaqueness into
+such transparent crystal, that other eyes are enabled
+to see as deeply into it as his own.</p>
+
+<p>Any mathematician who peruses this volume
+will need no other justification of the high opinion
+here expressed; but others may appreciate the
+following endorsements of well-known authorities.
+<i>Mill</i>, in his "Logic," calls the work of M. Comte
+"by far the greatest yet produced on the Philosophy
+of the sciences;" and adds, "of this admirable
+work, one of the most admirable portions is that
+in which he may truly be said to have created the
+Philosophy of the higher Mathematics:" <i>Morell</i>,
+in his "Speculative Philosophy of Europe," says,
+"The classification given of the sciences at large,
+and their regular order of development, is unquestionably
+a master-piece of scientific thinking, as
+simple as it is comprehensive;" and <i>Lewes</i>, in
+his "Biographical History of Philosophy," names
+Comte "the Bacon of the nineteenth century,"
+and says, "I unhesitatingly record my conviction
+that this is the greatest work of our age."</p>
+
+<p>The complete work of M. Comte&mdash;his "<i>Cours
+de Philosophie Positive</i>"&mdash;fills six large octavo volumes,
+of six or seven hundred pages each, two
+thirds of the first volume comprising the purely
+mathematical portion. The great bulk of the
+"Course" is the probable cause of the fewness of
+those to whom even this section of it is known.
+Its presentation in its present form is therefore felt
+by the translator to be a most useful contribution
+to mathematical progress in this country.<span class="pagenum"><a name="Page_7" id="Page_7">[Pg 7]</a></span>
+The comprehensiveness of the style of the author&mdash;grasping
+all possible forms of an idea in one
+Briarean sentence, armed at all points against
+leaving any opening for mistake or forgetfulness&mdash;occasionally
+verges upon cumbersomeness and
+formality. The translator has, therefore, sometimes
+taken the liberty of breaking up or condensing
+a long sentence, and omitting a few passages
+not absolutely necessary, or referring to the peculiar
+"Positive philosophy" of the author; but he
+has generally aimed at a conscientious fidelity to
+the original. It has often been difficult to retain
+its fine shades and subtile distinctions of meaning,
+and, at the same time, replace the peculiarly
+appropriate French idioms by corresponding English
+ones. The attempt, however, has always
+been made, though, when the best course has been
+at all doubtful, the language of the original has
+been followed as closely as possible, and, when
+necessary, smoothness and grace have been unhesitatingly
+sacrificed to the higher attributes of
+clearness and precision.</p>
+
+<p>Some forms of expression may strike the reader
+as unusual, but they have been retained because
+they were characteristic, not of the mere language
+of the original, but of its spirit. When a great
+thinker has clothed his conceptions in phrases
+which are singular even in his own tongue, he who
+professes to translate him is bound faithfully to
+preserve such forms of speech, as far as is practicable;
+and this has been here done with respect
+to such peculiarities of expression as belong to the<span class="pagenum"><a name="Page_8" id="Page_8">[Pg 8]</a></span>
+author, not as a foreigner, but as an individual&mdash;not
+because he writes in French, but because he
+is Auguste Comte.</p>
+
+<p>The young student of Mathematics should not
+attempt to read the whole of this volume at once,
+but should peruse each portion of it in connexion
+with the temporary subject of his special study:
+the first chapter of the first book, for example,
+while he is studying Algebra; the first chapter of
+the second book, when he has made some progress
+in Geometry; and so with the rest. Passages
+which are obscure at the first reading will brighten
+up at the second; and as his own studies cover
+a larger portion of the field of Mathematics, he
+will see more and more clearly their relations to
+one another, and to those which he is next to take
+up. For this end he is urgently recommended to
+obtain a perfect familiarity with the "Analytical
+Table of Contents," which maps out the whole
+subject, the grand divisions of which are also indicated
+in the Tabular View facing the title-page.
+Corresponding heads will be found in the body of
+the work, the principal divisions being in <span class="smcap">small
+capitals</span>, and the subdivisions in <i>Italics</i>. For
+these details the translator alone is responsible.</p><p><span class="pagenum"><a name="Page_9" id="Page_9">[Pg 9]</a></span></p>
+
+
+
+
+<h3>ANALYTICAL TABLE OF CONTENTS.</h3>
+
+<h5>INTRODUCTION.</h5>
+
+<ul class="toc">
+<li>&nbsp; &nbsp; <span class="label">Page</span></li>
+
+<li>GENERAL CONSIDERATIONS ON MATHEMATICAL SCIENCE&nbsp; &nbsp; <span class="label"><a href="#Page_17">17</a></span></li>
+<li><span class="smcap">The Object of Mathematics</span>&nbsp; &nbsp; <span class="label"><a href="#Page_18">18</a></span></li>
+<li><ol class="subtoc">
+<li>Measuring Magnitudes&nbsp; &nbsp; <span class="label"><a href="#Page_18">18</a></span></li>
+<li><ol class="subtoc">
+<li>Difficulties&nbsp; &nbsp; <span class="label"><a href="#Page_19">19</a></span></li>
+<li>General Method&nbsp; &nbsp; <span class="label"><a href="#Page_20">20</a></span></li>
+<li>Illustrations&nbsp; &nbsp; <span class="label"><a href="#Page_21">21</a></span></li>
+<li><ol class="subtoc">
+<li>1. Falling Bodies&nbsp; &nbsp; <span class="label"><a href="#Page_21">21</a></span></li>
+<li>2. Inaccessible Distances&nbsp; &nbsp; <span class="label"><a href="#Page_23">23</a></span></li>
+<li>3. Astronomical Facts&nbsp; &nbsp; <span class="label"><a href="#Page_24">24</a></span></li>
+</ol></li></ol></li></ol></li>
+<li><span class="smcap">True Definition of Mathematics</span>&nbsp; &nbsp; <span class="label"><a href="#Page_25">25</a></span></li>
+<li><ol class="subtoc">
+<li>A Science, not an Art&nbsp; &nbsp; <span class="label"><a href="#Page_25">25</a></span></li>
+</ol></li>
+<li><span class="smcap">Its Two Fundamental Divisions</span>&nbsp; &nbsp; <span class="label"><a href="#Page_26">26</a></span></li>
+<li><ol class="subtoc">
+<li>Their different Objects&nbsp; &nbsp; <span class="label"><a href="#Page_27">27</a></span></li>
+<li>Their different Natures&nbsp; &nbsp; <span class="label"><a href="#Page_29">29</a></span></li>
+<li><i>Concrete Mathematics</i>&nbsp; &nbsp; <span class="label"><a href="#Page_31">31</a></span></li>
+<li>Geometry and Mechanics&nbsp; &nbsp; <span class="label"><a href="#Page_32">32</a></span></li>
+<li><i>Abstract Mathematics</i>&nbsp; &nbsp; <span class="label"><a href="#Page_33">33</a></span></li>
+<li>The Calculus, or Analysis&nbsp; &nbsp; <span class="label"><a href="#Page_33">33</a></span></li>
+</ol></li>
+<li><span class="smcap">Extent of Its Field</span>&nbsp; &nbsp; <span class="label"><a href="#Page_35">35</a></span></li>
+<li><ol class="subtoc">
+<li>Its Universality&nbsp; &nbsp; <span class="label"><a href="#Page_36">36</a></span></li>
+<li>Its Limitations&nbsp; &nbsp; <span class="label"><a href="#Page_37">37</a></span></li>
+</ol></li>
+</ul>
+<p><span class="pagenum"><a name="Page_10" id="Page_10">[Pg 10]</a></span></p>
+
+
+
+<h4>BOOK I.<br />
+ANALYSIS.</h4>
+
+<h5>CHAPTER I.</h5>
+<ul class="toc">
+<li>&nbsp; &nbsp; <span class="label">Page</span></li>
+
+<li>GENERAL VIEW OF MATHEMATICAL ANALYSIS&nbsp; &nbsp; <span class="label"><a href="#Page_45">45</a></span></li>
+<li><span class="smcap">The True Idea of an Equation</span>&nbsp; &nbsp; <span class="label"><a href="#Page_46">46</a></span></li>
+<li><ol class="subtoc">
+<li>Division of Functions into Abstract and Concrete&nbsp; &nbsp; <span class="label"><a href="#Page_47">47</a></span></li>
+<li>Enumeration of Abstract Functions&nbsp; &nbsp; <span class="label"><a href="#Page_50">50</a></span></li>
+</ol></li>
+<li><span class="smcap">Divisions of the Calculus</span>&nbsp; &nbsp; <span class="label"><a href="#Page_53">53</a></span></li>
+<li><ol class="subtoc">
+<li><i>The Calculus of Values, or Arithmetic</i>&nbsp; &nbsp; <span class="label"><a href="#Page_57">57</a></span></li>
+<li>Its Extent&nbsp; &nbsp; <span class="label"><a href="#Page_57">57</a></span></li>
+<li>Its true Nature&nbsp; &nbsp; <span class="label"><a href="#Page_59">59</a></span></li>
+<li><i>The Calculus of Functions</i>&nbsp; &nbsp; <span class="label"><a href="#Page_61">61</a></span></li>
+<li>Two Modes of obtaining Equations&nbsp; &nbsp; <span class="label"><a href="#Page_61">61</a></span></li>
+<li><ol class="subtoc">
+<li>1. By the Relations between the given Quantities&nbsp; &nbsp; <span class="label"><a href="#Page_61">61</a></span></li>
+<li>2. By the Relations between auxiliary Quantities&nbsp; &nbsp; <span class="label"><a href="#Page_64">64</a></span></li>
+</ol></li>
+<li>Corresponding Divisions of the Calculus of Functions&nbsp; &nbsp; <span class="label"><a href="#Page_67">67</a></span></li>
+</ol></li>
+</ul>
+<h5>CHAPTER II.</h5>
+
+<ul class="toc">
+<li>ORDINARY ANALYSIS; OR, ALGEBRA.&nbsp; &nbsp; <span class="label"><a href="#Page_69">69</a></span></li>
+<li><ol class="subtoc">
+<li>Its Object&nbsp; &nbsp; <span class="label"><a href="#Page_69">69</a></span></li>
+<li>Classification of Equations&nbsp; &nbsp; <span class="label"><a href="#Page_70">70</a></span></li>
+</ol></li>
+<li><span class="smcap">Algebraic Equations</span>&nbsp; &nbsp; <span class="label"><a href="#Page_71">71</a></span></li>
+<li><ol class="subtoc">
+<li>Their Classification&nbsp; &nbsp; <span class="label"><a href="#Page_71">71</a></span></li>
+</ol></li>
+<li><span class="smcap">Algebraic Resolution of Equations</span>&nbsp; &nbsp; <span class="label"><a href="#Page_72">72</a></span></li>
+<li><ol class="subtoc">
+<li>Its Limits&nbsp; &nbsp; <span class="label"><a href="#Page_72">72</a></span></li>
+<li>General Solution&nbsp; &nbsp; <span class="label"><a href="#Page_72">72</a></span></li>
+<li>What we know in Algebra&nbsp; &nbsp; <span class="label"><a href="#Page_74">74</a></span></li>
+</ol></li>
+<li><span class="smcap">Numerical Resolution of Equations</span>&nbsp; &nbsp; <span class="label"><a href="#Page_75">75</a></span></li>
+<li><ol class="subtoc">
+<li>Its limited Usefulness&nbsp; &nbsp; <span class="label"><a href="#Page_76">76</a></span></li>
+</ol></li>
+<li>Different Divisions of the two Systems&nbsp; &nbsp; <span class="label"><a href="#Page_78">78</a></span></li>
+<li><span class="smcap">The Theory of Equations</span>&nbsp; &nbsp; <span class="label"><a href="#Page_79">79</a></span></li>
+<li><span class="smcap">The Method of Indeterminate Coefficients</span>&nbsp; &nbsp; <span class="label"><a href="#Page_80">80</a></span></li>
+<li><span class="smcap">Imaginary Quantities</span>&nbsp; &nbsp; <span class="label"><a href="#Page_81">81</a></span></li>
+<li><span class="smcap">Negative Quantities</span>&nbsp; &nbsp; <span class="label"><a href="#Page_81">81</a></span></li>
+<li><span class="smcap">The Principle of Homogeneity</span>&nbsp; &nbsp; <span class="label"><a href="#Page_84">84</a></span></li>
+</ul>
+<p><span class="pagenum"><a name="Page_11" id="Page_11">[Pg 11]</a></span></p>
+
+<h5>CHAPTER III.</h5>
+<ul class="toc">
+<li>TRANSCENDENTAL ANALYSIS: <span class="smcap">its different conceptions</span>&nbsp; &nbsp; <span class="label"><a href="#Page_88">88</a></span></li>
+<li><ol class="subtoc">
+<li>Preliminary Remarks&nbsp; &nbsp; <span class="label"><a href="#Page_88">88</a></span></li>
+<li>Its early History&nbsp; &nbsp; <span class="label"><a href="#Page_89">89</a></span></li>
+</ol></li>
+<li><span class="smcap">Method of Leibnitz</span>&nbsp; &nbsp; <span class="label"><a href="#Page_91">91</a></span></li>
+<li><ol class="subtoc">
+<li>Infinitely small Elements&nbsp; &nbsp; <span class="label"><a href="#Page_91">91</a></span></li>
+<li><i>Examples</i>:</li>
+<li><ol class="subtoc">
+<li>1. Tangents&nbsp; &nbsp; <span class="label"><a href="#Page_93">93</a></span></li>
+<li>2. Rectification of an Arc&nbsp; &nbsp; <span class="label"><a href="#Page_94">94</a></span></li>
+<li>3. Quadrature of a Curve&nbsp; &nbsp; <span class="label"><a href="#Page_95">95</a></span></li>
+<li>4. Velocity in variable Motion&nbsp; &nbsp; <span class="label"><a href="#Page_95">95</a></span></li>
+<li>5. Distribution of Heat&nbsp; &nbsp; <span class="label"><a href="#Page_96">96</a></span></li>
+</ol></li>
+<li>Generality of the Formulas&nbsp; &nbsp; <span class="label"><a href="#Page_97">97</a></span></li>
+<li>Demonstration of the Method&nbsp; &nbsp; <span class="label"><a href="#Page_98">98</a></span></li>
+<li><ol class="subtoc">
+<li>Illustration by Tangents&nbsp; &nbsp; <span class="label"><a href="#Page_102">102</a></span></li>
+</ol></li>
+</ol></li>
+<li><span class="smcap">Method of Newton</span>&nbsp; &nbsp; <span class="label"><a href="#Page_103">103</a></span></li>
+<li><ol class="subtoc">
+<li>Method of Limits&nbsp; &nbsp; <span class="label"><a href="#Page_103">103</a></span></li>
+<li><i>Examples</i>:</li>
+<li><ol class="subtoc">
+<li>1. Tangents&nbsp; &nbsp; <span class="label"><a href="#Page_104">104</a></span></li>
+<li>2. Rectifications&nbsp; &nbsp; <span class="label"><a href="#Page_105">105</a></span></li>
+</ol></li>
+<li>Fluxions and Fluents&nbsp; &nbsp; <span class="label"><a href="#Page_106">106</a></span></li>
+</ol></li>
+<li><span class="smcap">Method of Lagrange</span>&nbsp; &nbsp; <span class="label"><a href="#Page_108">108</a></span></li>
+<li><ol class="subtoc">
+<li>Derived Functions&nbsp; &nbsp; <span class="label"><a href="#Page_108">108</a></span></li>
+<li>An extension of ordinary Analysis&nbsp; &nbsp; <span class="label"><a href="#Page_108">108</a></span></li>
+<li><i>Example</i>: Tangents&nbsp; &nbsp; <span class="label"><a href="#Page_109">109</a></span></li>
+<li><i>Fundamental Identity of the three Methods</i>&nbsp; &nbsp; <span class="label"><a href="#Page_110">110</a></span></li>
+<li><i>Their comparative Value</i>&nbsp; &nbsp; <span class="label"><a href="#Page_113">113</a></span></li>
+<li>That of Leibnitz&nbsp; &nbsp; <span class="label"><a href="#Page_113">113</a></span></li>
+<li>That of Newton&nbsp; &nbsp; <span class="label"><a href="#Page_115">115</a></span></li>
+<li>That of Lagrange&nbsp; &nbsp; <span class="label"><a href="#Page_117">117</a></span></li>
+</ol></li>
+</ul>
+
+<p><span class="pagenum"><a name="Page_12" id="Page_12">[Pg 12]</a></span></p>
+
+<h5>CHAPTER IV.</h5>
+
+<ul class="toc">
+<li>THE DIFFERENTIAL AND INTEGRAL CALCULUS&nbsp; &nbsp; <span class="label"><a href="#Page_120">120</a></span></li>
+<li><span class="smcap">Its two fundamental Divisions</span>&nbsp; &nbsp; <span class="label"><a href="#Page_120">120</a></span></li>
+<li><span class="smcap">Their Relations to each Other</span>&nbsp; &nbsp; <span class="label"><a href="#Page_121">121</a></span></li>
+<li><ol class="subtoc">
+<li>1. Use of the Differential Calculus as preparatory to that of the Integral&nbsp; &nbsp; <span class="label"><a href="#Page_123">123</a></span></li>
+<li>2. Employment of the Differential Calculus alone&nbsp; &nbsp; <span class="label"><a href="#Page_125">125</a></span></li>
+<li>3. Employment of the Integral Calculus alone&nbsp; &nbsp; <span class="label"><a href="#Page_125">125</a></span></li>
+<li><ol class="subtoc">
+<li>Three Classes of Questions hence resulting&nbsp; &nbsp; <span class="label"><a href="#Page_126">126</a></span></li>
+</ol></li>
+</ol></li>
+<li><span class="smcap">The Differential Calculus</span>&nbsp; &nbsp; <span class="label"><a href="#Page_127">127</a></span></li>
+<li><ol class="subtoc">
+<li>Two Cases: Explicit and Implicit Functions&nbsp; &nbsp; <span class="label"><a href="#Page_127">127</a></span></li>
+<li><ol class="subtoc">
+<li>Two sub-Cases: a single Variable or several&nbsp; &nbsp; <span class="label"><a href="#Page_129">129</a></span></li>
+<li>Two other Cases: Functions separate or combined&nbsp; &nbsp; <span class="label"><a href="#Page_130">130</a></span></li>
+</ol></li>
+<li>Reduction of all to the Differentiation of the ten elementary Functions&nbsp; &nbsp; <span class="label"><a href="#Page_131">131</a></span></li>
+<li>Transformation of derived Functions for new Variables&nbsp; &nbsp; <span class="label"><a href="#Page_132">132</a></span></li>
+<li>Different Orders of Differentiation&nbsp; &nbsp; <span class="label"><a href="#Page_133">133</a></span></li>
+<li>Analytical Applications&nbsp; &nbsp; <span class="label"><a href="#Page_133">133</a></span></li>
+</ol></li>
+<li><span class="smcap">The Integral Calculus</span>&nbsp; &nbsp; <span class="label"><a href="#Page_135">135</a></span></li>
+<li><ol class="subtoc">
+<li>Its fundamental Division: Explicit and Implicit Functions&nbsp; &nbsp; <span class="label"><a href="#Page_135">135</a></span></li>
+<li>Subdivisions: a single Variable or several&nbsp; &nbsp; <span class="label"><a href="#Page_136">136</a></span></li>
+<li>Calculus of partial Differences&nbsp; &nbsp; <span class="label"><a href="#Page_137">137</a></span></li>
+<li>Another Subdivision: different Orders of Differentiation&nbsp; &nbsp; <span class="label"><a href="#Page_138">138</a></span></li>
+<li>Another equivalent Distinction&nbsp; &nbsp; <span class="label"><a href="#Page_140">140</a></span></li>
+<li><i>Quadratures</i>&nbsp; &nbsp; <span class="label"><a href="#Page_142">142</a></span></li>
+<li><ol class="subtoc">
+<li>Integration of Transcendental Functions&nbsp; &nbsp; <span class="label"><a href="#Page_143">143</a></span></li>
+<li>Integration by Parts&nbsp; &nbsp; <span class="label"><a href="#Page_143">143</a></span></li>
+<li>Integration of Algebraic Functions&nbsp; &nbsp; <span class="label"><a href="#Page_143">143</a></span></li>
+</ol></li>
+<li>Singular Solutions&nbsp; &nbsp; <span class="label"><a href="#Page_144">144</a></span></li>
+<li>Definite Integrals&nbsp; &nbsp; <span class="label"><a href="#Page_146">146</a></span></li>
+<li>Prospects of the Integral Calculus&nbsp; &nbsp; <span class="label"><a href="#Page_148">148</a></span></li>
+</ol></li>
+</ul>
+
+<p><span class="pagenum"><a name="Page_13" id="Page_13">[Pg 13]</a></span></p>
+
+<h5>CHAPTER V.</h5>
+
+<ul class="toc">
+<li>THE CALCULUS OF VARIATIONS&nbsp; &nbsp; <span class="label"><a href="#Page_151">151</a></span></li>
+<li><span class="smcap">Problems giving rise to it</span>&nbsp; &nbsp; <span class="label"><a href="#Page_151">151</a></span></li>
+<li><ol class="subtoc">
+<li>Ordinary Questions of Maxima and Minima&nbsp; &nbsp; <span class="label"><a href="#Page_151">151</a></span></li>
+<li>A new Class of Questions&nbsp; &nbsp; <span class="label"><a href="#Page_152">152</a></span></li>
+<li><ol class="subtoc">
+<li>Solid of least Resistance; Brachystochrone; Isoperimeters&nbsp; &nbsp; <span class="label"><a href="#Page_153">153</a></span></li>
+</ol></li>
+</ol></li>
+<li><span class="smcap">Analytical Nature of these Questions</span>&nbsp; &nbsp; <span class="label"><a href="#Page_154">154</a></span></li>
+<li><span class="smcap">Methods of the older Geometers</span>&nbsp; &nbsp; <span class="label"><a href="#Page_155">155</a></span></li>
+<li><span class="smcap">Method of Lagrange</span>&nbsp; &nbsp; <span class="label"><a href="#Page_156">156</a></span></li>
+<li><ol class="subtoc">
+<li>Two Classes of Questions&nbsp; &nbsp; <span class="label"><a href="#Page_157">157</a></span></li>
+<li><ol class="subtoc">
+<li>1. Absolute Maxima and Minima&nbsp; &nbsp; <span class="label"><a href="#Page_157">157</a></span></li>
+<li>Equations of Limits&nbsp; &nbsp; <span class="label"><a href="#Page_159">159</a></span></li>
+<li><ol class="subtoc">
+<li>A more general Consideration&nbsp; &nbsp; <span class="label"><a href="#Page_159">159</a></span></li>
+</ol></li>
+<li>2. Relative Maxima and Minima&nbsp; &nbsp; <span class="label"><a href="#Page_160">160</a></span></li>
+</ol></li>
+<li>Other Applications of the Method of Variations&nbsp; &nbsp; <span class="label"><a href="#Page_162">162</a></span></li>
+</ol></li>
+<li><span class="smcap">Its Relations to the ordinary Calculus</span>&nbsp; &nbsp; <span class="label"><a href="#Page_163">163</a></span></li>
+</ul>
+
+<h5>CHAPTER VI.</h5>
+
+<ul class="toc">
+<li>THE CALCULUS OF FINITE DIFFERENCES&nbsp; &nbsp; <span class="label"><a href="#Page_167">167</a></span></li>
+<li><ol class="subtoc">
+<li>Its general Character&nbsp; &nbsp; <span class="label"><a href="#Page_167">167</a></span></li>
+<li>Its true Nature&nbsp; &nbsp; <span class="label"><a href="#Page_168">168</a></span></li>
+</ol></li>
+<li><span class="smcap">General Theory of Series</span>&nbsp; &nbsp; <span class="label"><a href="#Page_170">170</a></span></li>
+<li><ol class="subtoc">
+<li>Its Identity with this Calculus&nbsp; &nbsp; <span class="label"><a href="#Page_172">172</a></span></li>
+</ol></li>
+<li><span class="smcap">Periodic or discontinuous Functions</span>&nbsp; &nbsp; <span class="label"><a href="#Page_173">173</a></span></li>
+<li><span class="smcap">Applications of this Calculus</span>&nbsp; &nbsp; <span class="label"><a href="#Page_173">173</a></span></li>
+<li><ol class="subtoc">
+<li>Series&nbsp; &nbsp; <span class="label"><a href="#Page_173">173</a></span></li>
+<li>Interpolation&nbsp; &nbsp; <span class="label"><a href="#Page_173">173</a></span></li>
+<li>Approximate Rectification, &amp;c.&nbsp; &nbsp; <span class="label"><a href="#Page_174">174</a></span></li>
+</ol></li>
+</ul>
+
+<p><span class="pagenum"><a name="Page_14" id="Page_14">[Pg 14]</a></span></p>
+
+
+<h4>BOOK II.<br />
+GEOMETRY.</h4>
+
+<h5>CHAPTER I.</h5>
+
+<ul class="toc">
+<li>A GENERAL VIEW OF GEOMETRY&nbsp; &nbsp; <span class="label"><a href="#Page_179">179</a></span></li>
+<li><ol class="subtoc">
+<li>The true Nature of Geometry&nbsp; &nbsp; <span class="label"><a href="#Page_179">179</a></span></li>
+<li>Two fundamental Ideas&nbsp; &nbsp; <span class="label"><a href="#Page_181">181</a></span></li>
+<li><ol class="subtoc">
+<li>1. The Idea of Space&nbsp; &nbsp; <span class="label"><a href="#Page_181">181</a></span></li>
+<li>2. Different kinds of Extension&nbsp; &nbsp; <span class="label"><a href="#Page_182">182</a></span></li>
+</ol></li>
+</ol></li>
+<li><span class="smcap">The final object of Geometry</span>&nbsp; &nbsp; <span class="label"><a href="#Page_184">184</a></span></li>
+<li><ol class="subtoc">
+<li>Nature of Geometrical Measurement&nbsp; &nbsp; <span class="label"><a href="#Page_185">185</a></span></li>
+<li><ol class="subtoc">
+<li>Of Surfaces and Volumes&nbsp; &nbsp; <span class="label"><a href="#Page_185">185</a></span></li>
+<li>Of curve Lines&nbsp; &nbsp; <span class="label"><a href="#Page_187">187</a></span></li>
+<li>Of right Lines&nbsp; &nbsp; <span class="label"><a href="#Page_189">189</a></span></li>
+</ol></li>
+</ol></li>
+<li><span class="smcap">The infinite extent of its Field</span>&nbsp; &nbsp; <span class="label"><a href="#Page_190">190</a></span></li>
+<li><ol class="subtoc">
+<li>Infinity of Lines&nbsp; &nbsp; <span class="label"><a href="#Page_190">190</a></span></li>
+<li>Infinity of Surfaces&nbsp; &nbsp; <span class="label"><a href="#Page_191">191</a></span></li>
+<li>Infinity of Volumes&nbsp; &nbsp; <span class="label"><a href="#Page_192">192</a></span></li>
+<li>Analytical Invention of Curves, &amp;c.&nbsp; &nbsp; <span class="label"><a href="#Page_193">193</a></span></li>
+</ol></li>
+<li><span class="smcap">Expansion of Original Definition</span>&nbsp; &nbsp; <span class="label"><a href="#Page_193">193</a></span></li>
+<li><ol class="subtoc">
+<li>Properties of Lines and Surfaces&nbsp; &nbsp; <span class="label"><a href="#Page_195">195</a></span></li>
+<li>Necessity of their Study&nbsp; &nbsp; <span class="label"><a href="#Page_195">195</a></span></li>
+<li><ol class="subtoc">
+<li>1. To find the most suitable Property&nbsp; &nbsp; <span class="label"><a href="#Page_195">195</a></span></li>
+<li>2. To pass from the Concrete to the Abstract&nbsp; &nbsp; <span class="label"><a href="#Page_197">197</a></span></li>
+</ol></li>
+<li>Illustrations:</li>
+<li><ol class="subtoc">
+<li>Orbits of the Planets&nbsp; &nbsp; <span class="label"><a href="#Page_198">198</a></span></li>
+<li>Figure of the Earth&nbsp; &nbsp; <span class="label"><a href="#Page_199">199</a></span></li>
+</ol></li>
+</ol></li>
+<li><span class="smcap">The two general Methods of Geometry</span>&nbsp; &nbsp; <span class="label"><a href="#Page_202">202</a></span></li>
+<li><ol class="subtoc">
+<li>Their fundamental Difference&nbsp; &nbsp; <span class="label"><a href="#Page_203">203</a></span></li>
+<li><ol class="subtoc">
+<li>1⁰. Different Questions with respect to the same Figure&nbsp; &nbsp; <span class="label"><a href="#Page_204">204</a></span></li>
+<li>2⁰. Similar Questions with respect to different Figures&nbsp; &nbsp; <span class="label"><a href="#Page_204">204</a></span></li>
+</ol></li>
+<li>Geometry of the Ancients&nbsp; &nbsp; <span class="label"><a href="#Page_204">204</a></span></li>
+<li>Geometry of the Moderns&nbsp; &nbsp; <span class="label"><a href="#Page_206">206</a></span></li>
+<li>Superiority of the Modern&nbsp; &nbsp; <span class="label"><a href="#Page_207">207</a></span></li>
+<li>The Ancient the base of the Modern&nbsp; &nbsp; <span class="label"><a href="#Page_209">209</a></span></li>
+</ol></li>
+</ul>
+
+<p><span class="pagenum"><a name="Page_15" id="Page_15">[Pg 15]</a></span></p>
+
+<h5>CHAPTER II.</h5>
+
+<ul class="toc">
+<li>ANCIENT OR SYNTHETIC GEOMETRY&nbsp; &nbsp; <span class="label"><a href="#Page_212">212</a></span></li>
+<li><span class="smcap">Its proper Extent</span>&nbsp; &nbsp; <span class="label"><a href="#Page_212">212</a></span></li>
+<li><ol class="subtoc">
+<li>Lines; Polygons; Polyhedrons&nbsp; &nbsp; <span class="label"><a href="#Page_212">212</a></span></li>
+<li>Not to be farther restricted&nbsp; &nbsp; <span class="label"><a href="#Page_213">213</a></span></li>
+<li>Improper Application of Analysis&nbsp; &nbsp; <span class="label"><a href="#Page_214">214</a></span></li>
+<li>Attempted Demonstrations of Axioms&nbsp; &nbsp; <span class="label"><a href="#Page_216">216</a></span></li>
+</ol></li>
+<li><span class="smcap">Geometry of the right Line</span>&nbsp; &nbsp; <span class="label"><a href="#Page_217">217</a></span></li>
+<li><span class="smcap">Graphical Solutions</span>&nbsp; &nbsp; <span class="label"><a href="#Page_218">218</a></span></li>
+<li><ol class="subtoc">
+<li><i>Descriptive Geometry</i>&nbsp; &nbsp; <span class="label"><a href="#Page_220">220</a></span></li>
+</ol></li>
+<li><span class="smcap">Algebraical Solutions</span>&nbsp; &nbsp; <span class="label"><a href="#Page_224">224</a></span></li>
+<li><ol class="subtoc">
+<li><i>Trigonometry</i>&nbsp; &nbsp; <span class="label"><a href="#Page_225">225</a></span></li>
+<li>Two Methods of introducing Angles&nbsp; &nbsp; <span class="label"><a href="#Page_226">226</a></span></li>
+<li><ol class="subtoc">
+<li>1. By Arcs&nbsp; &nbsp; <span class="label"><a href="#Page_226">226</a></span></li>
+<li>2. By trigonometrical Lines&nbsp; &nbsp; <span class="label"><a href="#Page_226">226</a></span></li>
+</ol></li>
+<li>Advantages of the latter&nbsp; &nbsp; <span class="label"><a href="#Page_226">226</a></span></li>
+<li>Its Division of trigonometrical Questions&nbsp; &nbsp; <span class="label"><a href="#Page_227">227</a></span></li>
+<li><ol class="subtoc">
+<li>1. Relations between Angles and trigonometrical Lines&nbsp; &nbsp; <span class="label"><a href="#Page_228">228</a></span></li>
+<li>2. Relations between trigonometrical Lines and Sides&nbsp; &nbsp; <span class="label"><a href="#Page_228">228</a></span></li>
+</ol></li>
+<li>Increase of trigonometrical Lines&nbsp; &nbsp; <span class="label"><a href="#Page_228">228</a></span></li>
+<li>Study of the Relations between them&nbsp; &nbsp; <span class="label"><a href="#Page_230">230</a></span></li>
+</ol></li>
+</ul>
+
+<p><span class="pagenum"><a name="Page_16" id="Page_16">[Pg 16]</a></span></p>
+
+<h5>CHAPTER III.</h5>
+
+<ul class="toc">
+<li>MODERN OR ANALYTICAL GEOMETRY&nbsp; &nbsp; <span class="label"><a href="#Page_232">232</a></span></li>
+<li><span class="smcap">The analytical Representation of Figures</span>&nbsp; &nbsp; <span class="label"><a href="#Page_232">232</a></span></li>
+<li><ol class="subtoc">
+<li>Reduction of Figure to Position&nbsp; &nbsp; <span class="label"><a href="#Page_233">233</a></span></li>
+<li>Determination of the position of a Point&nbsp; &nbsp; <span class="label"><a href="#Page_234">234</a></span></li>
+</ol></li>
+<li><span class="smcap">Plane Curves</span>&nbsp; &nbsp; <span class="label"><a href="#Page_237">237</a></span></li>
+<li><ol class="subtoc">
+<li>Expression of Lines by Equations&nbsp; &nbsp; <span class="label"><a href="#Page_237">237</a></span></li>
+<li>Expression of Equations by Lines&nbsp; &nbsp; <span class="label"><a href="#Page_238">238</a></span></li>
+<li>Any change in the Line changes the Equation&nbsp; &nbsp; <span class="label"><a href="#Page_240">240</a></span></li>
+<li>Every "Definition" of a Line is an Equation&nbsp; &nbsp; <span class="label"><a href="#Page_241">241</a></span></li>
+<li><i>Choice of Co-ordinates</i>&nbsp; &nbsp; <span class="label"><a href="#Page_245">245</a></span></li>
+<li>Two different points of View&nbsp; &nbsp; <span class="label"><a href="#Page_245">245</a></span></li>
+<li><ol class="subtoc">
+<li>1. Representation of Lines by Equations&nbsp; &nbsp; <span class="label"><a href="#Page_246">246</a></span></li>
+<li>2. Representation of Equations by Lines&nbsp; &nbsp; <span class="label"><a href="#Page_246">246</a></span></li>
+</ol></li>
+<li>Superiority of the rectilinear System&nbsp; &nbsp; <span class="label"><a href="#Page_248">248</a></span></li>
+<li><ol class="subtoc">
+<li>Advantages of perpendicular Axes&nbsp; &nbsp; <span class="label"><a href="#Page_249">249</a></span></li>
+</ol></li>
+</ol></li>
+<li><span class="smcap">Surfaces</span>&nbsp; &nbsp; <span class="label"><a href="#Page_251">251</a></span></li>
+<li><ol class="subtoc">
+<li>Determination of a Point in Space&nbsp; &nbsp; <span class="label"><a href="#Page_251">251</a></span></li>
+<li>Expression of Surfaces by Equations&nbsp; &nbsp; <span class="label"><a href="#Page_253">253</a></span></li>
+<li>Expression of Equations by Surfaces&nbsp; &nbsp; <span class="label"><a href="#Page_253">253</a></span></li>
+</ol></li>
+<li><span class="smcap">Curves in Space</span>&nbsp; &nbsp; <span class="label"><a href="#Page_255">255</a></span></li>
+<li>Imperfections of Analytical Geometry&nbsp; &nbsp; <span class="label"><a href="#Page_258">258</a></span></li>
+<li><ol class="subtoc">
+<li>Relatively to Geometry&nbsp; &nbsp; <span class="label"><a href="#Page_258">258</a></span></li>
+<li>Relatively to Analysis&nbsp; &nbsp; <span class="label"><a href="#Page_258">258</a></span></li>
+</ol></li>
+</ul>
+
+<p><span class="pagenum"><a name="Page_17" id="Page_17">[Pg 17]</a></span></p>
+
+
+
+
+<p class="center">THE<br />
+<span class="big">PHILOSOPHY OF MATHEMATICS.</span></p>
+
+<h2>INTRODUCTION.</h2>
+
+<p class="center">GENERAL CONSIDERATIONS.</p>
+
+
+<p>Although Mathematical Science is the most ancient
+and the most perfect of all, yet the general idea which
+we ought to form of it has not yet been clearly determined.
+Its definition and its principal divisions have
+remained till now vague and uncertain. Indeed the
+plural name&mdash;"The Mathematics"&mdash;by which we commonly
+designate it, would alone suffice to indicate the
+want of unity in the common conception of it.</p>
+
+<p>In truth, it was not till the commencement of the last
+century that the different fundamental conceptions which
+constitute this great science were each of them sufficiently
+developed to permit the true spirit of the whole
+to manifest itself with clearness. Since that epoch the
+attention of geometers has been too exclusively absorbed
+by the special perfecting of the different branches, and
+by the application which they have made of them to the
+most important laws of the universe, to allow them to
+give due attention to the general system of the science.</p>
+
+<p>But at the present time the progress of the special
+departments is no longer so rapid as to forbid the contemplation
+of the whole. The science of mathematics<span class="pagenum"><a name="Page_18" id="Page_18">[Pg 18]</a></span>
+is now sufficiently developed, both in itself and as to its
+most essential application, to have arrived at that state
+of consistency in which we ought to strive to arrange its
+different parts in a single system, in order to prepare for
+new advances. We may even observe that the last important
+improvements of the science have directly paved
+the way for this important philosophical operation, by impressing
+on its principal parts a character of unity which
+did not previously exist.</p>
+
+<p>To form a just idea of the object of mathematical science,
+we may start from the indefinite and meaningless
+definition of it usually given, in calling it "<i>The science
+of magnitudes</i>," or, which is more definite, "<i>The science
+which has for its object the measurement of magnitudes.</i>"
+Let us see how we can rise from this rough
+sketch (which is singularly deficient in precision and
+depth, though, at bottom, just) to a veritable definition,
+worthy of the importance, the extent, and the difficulty
+of the science.</p>
+
+
+<h3>THE OBJECT OF MATHEMATICS.</h3>
+
+<p><i>Measuring Magnitudes.</i> The question of <i>measuring</i>
+a magnitude in itself presents to the mind no other
+idea than that of the simple direct comparison of this
+magnitude with another similar magnitude, supposed to
+be known, which it takes for the <i>unit</i> of comparison
+among all others of the same kind. According to this
+definition, then, the science of mathematics&mdash;vast and
+profound as it is with reason reputed to be&mdash;instead of
+being an immense concatenation of prolonged mental labours,
+which offer inexhaustible occupation to our intellectual
+activity, would seem to consist of a simple<span class="pagenum"><a name="Page_19" id="Page_19">[Pg 19]</a></span>
+series of mechanical processes for obtaining directly the
+ratios of the quantities to be measured to those by which
+we wish to measure them, by the aid of operations of
+similar character to the superposition of lines, as practiced
+by the carpenter with his rule.</p>
+
+<p>The error of this definition consists in presenting as
+direct an object which is almost always, on the contrary,
+very indirect. The <i>direct</i> measurement of a magnitude,
+by superposition or any similar process, is most frequently
+an operation quite impossible for us to perform; so
+that if we had no other means for determining magnitudes
+than direct comparisons, we should be obliged to renounce
+the knowledge of most of those which interest us.</p>
+
+<p><i>Difficulties.</i> The force of this general observation
+will be understood if we limit ourselves to consider specially
+the particular case which evidently offers the most
+facility&mdash;that of the measurement of one straight line
+by another. This comparison, which is certainly the
+most simple which we can conceive, can nevertheless
+scarcely ever be effected directly. In reflecting on the
+whole of the conditions necessary to render a line susceptible
+of a direct measurement, we see that most frequently
+they cannot be all fulfilled at the same time.
+The first and the most palpable of these conditions&mdash;that
+of being able to pass over the line from one end of
+it to the other, in order to apply the unit of measurement
+to its whole length&mdash;evidently excludes at once by far the
+greater part of the distances which interest us the most;
+in the first place, all the distances between the celestial
+bodies, or from any one of them to the earth; and then,
+too, even the greater number of terrestrial distances, which
+are so frequently inaccessible. But even if this first condition<span class="pagenum"><a name="Page_20" id="Page_20">[Pg 20]</a></span>
+be found to be fulfilled, it is still farther necessary
+that the length be neither too great nor too small, which
+would render a direct measurement equally impossible.
+The line must also be suitably situated; for let it be one
+which we could measure with the greatest facility, if it
+were horizontal, but conceive it to be turned up vertically,
+and it becomes impossible to measure it.</p>
+
+<p>The difficulties which we have indicated in reference
+to measuring lines, exist in a very much greater degree
+in the measurement of surfaces, volumes, velocities, times,
+forces, &amp;c. It is this fact which makes necessary the
+formation of mathematical science, as we are going to
+see; for the human mind has been compelled to renounce,
+in almost all cases, the direct measurement of
+magnitudes, and to seek to determine them <i>indirectly</i>,
+and it is thus that it has been led to the creation of
+mathematics.</p>
+
+<p><i>General Method.</i> The general method which is constantly
+employed, and evidently the only one conceivable,
+to ascertain magnitudes which do not admit of a direct
+measurement, consists in connecting them with others
+which are susceptible of being determined immediately,
+and by means of which we succeed in discovering
+the first through the relations which subsist between
+the two. Such is the precise object of mathematical
+science viewed as a whole. In order to form a sufficiently
+extended idea of it, we must consider that this
+indirect determination of magnitudes may be indirect in
+very different degrees. In a great number of cases,
+which are often the most important, the magnitudes, by
+means of which the principal magnitudes sought are to
+be determined, cannot themselves be measured directly,<span class="pagenum"><a name="Page_21" id="Page_21">[Pg 21]</a></span>
+and must therefore, in their turn, become the subject of a
+similar question, and so on; so that on many occasions
+the human mind is obliged to establish a long series of
+intermediates between the system of unknown magnitudes
+which are the final objects of its researches, and
+the system of magnitudes susceptible of direct measurement,
+by whose means we finally determine the first,
+with which at first they appear to have no connexion.</p>
+
+<p><i>Illustrations.</i> Some examples will make clear any
+thing which may seem too abstract in the preceding
+generalities.</p>
+
+<p>1. <i>Falling Bodies.</i> Let us consider, in the first place,
+a natural phenomenon, very simple, indeed, but which
+may nevertheless give rise to a mathematical question,
+really existing, and susceptible of actual applications&mdash;the
+phenomenon of the vertical fall of heavy bodies.</p>
+
+<p>The mind the most unused to mathematical conceptions,
+in observing this phenomenon, perceives at once
+that the two <i>quantities</i> which it presents&mdash;namely, the
+<i>height</i> from which a body has fallen, and the <i>time</i> of its
+fall&mdash;are necessarily connected with each other, since they
+vary together, and simultaneously remain fixed; or, in
+the language of geometers, that they are "<i>functions</i>" of
+each other. The phenomenon, considered under this
+point of view, gives rise then to a mathematical question,
+which consists in substituting for the direct measurement
+of one of these two magnitudes, when it is impossible,
+the measurement of the other. It is thus, for
+example, that we may determine indirectly the depth of
+a precipice, by merely measuring the time that a heavy
+body would occupy in falling to its bottom, and by suitable
+procedures this inaccessible depth will be known<span class="pagenum"><a name="Page_22" id="Page_22">[Pg 22]</a></span>
+with as much precision as if it was a horizontal line
+placed in the most favourable circumstances for easy and
+exact measurement. On other occasions it is the height
+from which a body has fallen which it will be easy to ascertain,
+while the time of the fall could not be observed
+directly; then the same phenomenon would give rise to
+the inverse question, namely, to determine the time from
+the height; as, for example, if we wished to ascertain
+what would be the duration of the vertical fall of a body
+falling from the moon to the earth.</p>
+
+<p>In this example the mathematical question is very simple,
+at least when we do not pay attention to the variation
+in the intensity of gravity, or the resistance of the fluid
+which the body passes through in its fall. But, to extend
+the question, we have only to consider the same
+phenomenon in its greatest generality, in supposing the
+fall oblique, and in taking into the account all the principal
+circumstances. Then, instead of offering simply
+two variable quantities connected with each other by a
+relation easy to follow, the phenomenon will present a
+much greater number; namely, the space traversed,
+whether in a vertical or horizontal direction; the time
+employed in traversing it; the velocity of the body at
+each point of its course; even the intensity and the
+direction of its primitive impulse, which may also be
+viewed as variables; and finally, in certain cases (to
+take every thing into the account), the resistance of the
+medium and the intensity of gravity. All these different
+quantities will be connected with one another, in such a
+way that each in its turn may be indirectly determined
+by means of the others; and this will present as many
+distinct mathematical questions as there may be co-existing<span class="pagenum"><a name="Page_23" id="Page_23">[Pg 23]</a></span>
+magnitudes in the phenomenon under consideration.
+Such a very slight change in the physical conditions of
+a problem may cause (as in the above example) a mathematical
+research, at first very elementary, to be placed at
+once in the rank of the most difficult questions, whose
+complete and rigorous solution surpasses as yet the utmost
+power of the human intellect.</p>
+
+<p>2. <i>Inaccessible Distances.</i> Let us take a second example
+from geometrical phenomena. Let it be proposed
+to determine a distance which is not susceptible of direct
+measurement; it will be generally conceived as making
+part of a <i>figure</i>, or certain system of lines, chosen in
+such a way that all its other parts may be observed directly;
+thus, in the case which is most simple, and to
+which all the others may be finally reduced, the proposed
+distance will be considered as belonging to a triangle,
+in which we can determine directly either another
+side and two angles, or two sides and one angle. Thence-forward,
+the knowledge of the desired distance, instead
+of being obtained directly, will be the result of a mathematical
+calculation, which will consist in deducing it
+from the observed elements by means of the relation
+which connects it with them. This calculation will become
+successively more and more complicated, if the parts
+which we have supposed to be known cannot themselves
+be determined (as is most frequently the case) except in
+an indirect manner, by the aid of new auxiliary systems,
+the number of which, in great operations of this kind,
+finally becomes very considerable. The distance being
+once determined, the knowledge of it will frequently be
+sufficient for obtaining new quantities, which will become
+the subject of new mathematical questions. Thus, when<span class="pagenum"><a name="Page_24" id="Page_24">[Pg 24]</a></span>
+we know at what distance any object is situated, the
+simple observation of its apparent diameter will evidently
+permit us to determine indirectly its real dimensions,
+however inaccessible it may be, and, by a series of analogous
+investigations, its surface, its volume, even its
+weight, and a number of other properties, a knowledge
+of which seemed forbidden to us.</p>
+
+<p>3. <i>Astronomical Facts.</i> It is by such calculations
+that man has been able to ascertain, not only the distances
+from the planets to the earth, and, consequently,
+from each other, but their actual magnitude, their true
+figure, even to the inequalities of their surface; and, what
+seemed still more completely hidden from us, their respective
+masses, their mean densities, the principal circumstances
+of the fall of heavy bodies on the surface of
+each of them, &amp;c.</p>
+
+<p>By the power of mathematical theories, all these different
+results, and many others relative to the different
+classes of mathematical phenomena, have required no
+other direct measurements than those of a very small
+number of straight lines, suitably chosen, and of a greater
+number of angles. We may even say, with perfect
+truth, so as to indicate in a word the general range of
+the science, that if we did not fear to multiply calculations
+unnecessarily, and if we had not, in consequence,
+to reserve them for the determination of the quantities
+which could not be measured directly, the determination
+of all the magnitudes susceptible of precise estimation,
+which the various orders of phenomena can offer us,
+could be finally reduced to the direct measurement of a
+single straight line and of a suitable number of angles.</p><p><span class="pagenum"><a name="Page_25" id="Page_25">[Pg 25]</a></span></p>
+
+
+<h3>TRUE DEFINITION OF MATHEMATICS.</h3>
+
+<p>We are now able to define mathematical science with
+precision, by assigning to it as its object the <i>indirect</i>
+measurement of magnitudes, and by saying it constantly
+proposes <i>to determine certain magnitudes from others
+by means of the precise relations existing between them</i>.</p>
+
+<p>This enunciation, instead of giving the idea of only an
+<i>art</i>, as do all the ordinary definitions, characterizes immediately
+a true <i>science</i>, and shows it at once to be composed
+of an immense chain of intellectual operations,
+which may evidently become very complicated, because
+of the series of intermediate links which it will be necessary
+to establish between the unknown quantities and
+those which admit of a direct measurement; of the number
+of variables coexistent in the proposed question; and
+of the nature of the relations between all these different
+magnitudes furnished by the phenomena under consideration.
+According to such a definition, the spirit of
+mathematics consists in always regarding all the quantities
+which any phenomenon can present, as connected
+and interwoven with one another, with the view of deducing
+them from one another. Now there is evidently
+no phenomenon which cannot give rise to considerations
+of this kind; whence results the naturally indefinite extent
+and even the rigorous logical universality of mathematical
+science. We shall seek farther on to circumscribe
+as exactly as possible its real extension.</p>
+
+<p>The preceding explanations establish clearly the propriety
+of the name employed to designate the science
+which we are considering. This denomination, which
+has taken to-day so definite a meaning by itself signifies<span class="pagenum"><a name="Page_26" id="Page_26">[Pg 26]</a></span>
+simply <i>science</i> in general. Such a designation, rigorously
+exact for the Greeks, who had no other real science,
+could be retained by the moderns only to indicate
+the mathematics as <i>the</i> science, beyond all others&mdash;the
+science of sciences.</p>
+
+<p>Indeed, every true science has for its object the determination
+of certain phenomena by means of others, in
+accordance with the relations which exist between them.
+Every <i>science</i> consists in the co-ordination of facts; if
+the different observations were entirely isolated, there
+would be no science. We may even say, in general terms,
+that <i>science</i> is essentially destined to dispense, so far as
+the different phenomena permit it, with all direct observation,
+by enabling us to deduce from the smallest
+possible number of immediate data the greatest possible
+number of results. Is not this the real use, whether in
+speculation or in action, of the <i>laws</i> which we succeed
+in discovering among natural phenomena? Mathematical
+science, in this point of view, merely pushes to the
+highest possible degree the same kind of researches which
+are pursued, in degrees more or less inferior, by every
+real science in its respective sphere.</p>
+
+
+<h3>ITS TWO FUNDAMENTAL DIVISIONS.</h3>
+
+<p>We have thus far viewed mathematical science only
+as a whole, without paying any regard to its divisions.
+We must now, in order to complete this general view,
+and to form a just idea of the philosophical character of
+the science, consider its fundamental division. The secondary
+divisions will be examined in the following chapters.</p>
+
+<p>This principal division, which we are about to investigate,<span class="pagenum"><a name="Page_27" id="Page_27">[Pg 27]</a></span>
+can be truly rational, and derived from the real nature
+of the subject, only so far as it spontaneously presents
+itself to us, in making the exact analysis of a complete
+mathematical question. We will, therefore, having
+determined above what is the general object of mathematical
+labours, now characterize with precision the
+principal different orders of inquiries, of which they are
+constantly composed.</p>
+
+<p><i>Their different Objects.</i> The complete solution of
+every mathematical question divides itself necessarily
+into two parts, of natures essentially distinct, and with
+relations invariably determinate. We have seen that
+every mathematical inquiry has for its object to determine
+unknown magnitudes, according to the relations between
+them and known magnitudes. Now for this object,
+it is evidently necessary, in the first place, to ascertain
+with precision the relations which exist between
+the quantities which we are considering. This first
+branch of inquiries constitutes that which I call the <i>concrete</i>
+part of the solution. When it is finished, the question
+changes; it is now reduced to a pure question of
+numbers, consisting simply in determining unknown
+numbers, when we know what precise relations connect
+them with known numbers. This second branch of inquiries
+is what I call the <i>abstract</i> part of the solution.
+Hence follows the fundamental division of general mathematical
+science into <i>two</i> great sciences&mdash;<small>ABSTRACT MATHEMATICS</small>,
+and <small>CONCRETE MATHEMATICS</small>.</p>
+
+<p>This analysis may be observed in every complete
+mathematical question, however simple or complicated
+it may be. A single example will suffice to make it
+intelligible.</p><p><span class="pagenum"><a name="Page_28" id="Page_28">[Pg 28]</a></span></p>
+
+<p>Taking up again the phenomenon of the vertical fall
+of a heavy body, and considering the simplest case, we
+see that in order to succeed in determining, by means of
+one another, the height whence the body has fallen, and
+the duration of its fall, we must commence by discovering
+the exact relation of these two quantities, or, to use the
+language of geometers, the <i>equation</i> which exists between
+them. Before this first research is completed,
+every attempt to determine numerically the value of one
+of these two magnitudes from the other would evidently
+be premature, for it would have no basis. It is not enough
+to know vaguely that they depend on one another&mdash;which
+every one at once perceives&mdash;but it is necessary to determine
+in what this dependence consists. This inquiry
+may be very difficult, and in fact, in the present case,
+constitutes incomparably the greater part of the problem.
+The true scientific spirit is so modern, that no one, perhaps,
+before Galileo, had ever remarked the increase of
+velocity which a body experiences in its fall: a circumstance
+which excludes the hypothesis, towards which our
+mind (always involuntarily inclined to suppose in every
+phenomenon the most simple <i>functions</i>, without any other
+motive than its greater facility in conceiving them)
+would be naturally led, that the height was proportional
+to the time. In a word, this first inquiry terminated
+in the discovery of the law of Galileo.</p>
+
+<p>When this <i>concrete</i> part is completed, the inquiry becomes
+one of quite another nature. Knowing that the
+spaces passed through by the body in each successive second
+of its fall increase as the series of odd numbers, we
+have then a problem purely numerical and <i>abstract</i>; to
+deduce the height from the time, or the time from the<span class="pagenum"><a name="Page_29" id="Page_29">[Pg 29]</a></span>
+height; and this consists in finding that the first of these
+two quantities, according to the law which has been established,
+is a known multiple of the second power of the
+other; from which, finally, we have to calculate the value
+of the one when that of the other is given.</p>
+
+<p>In this example the concrete question is more difficult
+than the abstract one. The reverse would be the case
+if we considered the same phenomenon in its greatest
+generality, as I have done above for another object.
+According to the circumstances, sometimes the first,
+sometimes the second, of these two parts will constitute
+the principal difficulty of the whole question; for the
+mathematical law of the phenomenon may be very simple,
+but very difficult to obtain, or it may be easy to discover,
+but very complicated; so that the two great sections
+of mathematical science, when we compare them
+as wholes, must be regarded as exactly equivalent in extent
+and in difficulty, as well as in importance, as we
+shall show farther on, in considering each of them separately.</p>
+
+<p><i>Their different Natures.</i> These two parts, essentially
+distinct in their <i>object</i>, as we have just seen, are no less
+so with regard to the <i>nature</i> of the inquiries of which
+they are composed.</p>
+
+<p>The first should be called <i>concrete</i>, since it evidently
+depends on the character of the phenomena considered,
+and must necessarily vary when we examine new phenomena;
+while the second is completely independent of
+the nature of the objects examined, and is concerned with
+only the <i>numerical</i> relations which they present, for which
+reason it should be called <i>abstract</i>. The same relations
+may exist in a great number of different phenomena,<span class="pagenum"><a name="Page_30" id="Page_30">[Pg 30]</a></span>
+which, in spite of their extreme diversity, will be viewed
+by the geometer as offering an analytical question susceptible,
+when studied by itself, of being resolved once
+for all. Thus, for instance, the same law which exists
+between the space and the time of the vertical fall of a
+body in a vacuum, is found again in many other phenomena
+which offer no analogy with the first nor with
+each other; for it expresses the relation between the surface
+of a spherical body and the length of its diameter;
+it determines, in like manner, the decrease of the intensity
+of light or of heat in relation to the distance of the objects
+lighted or heated, &amp;c. The abstract part, common
+to these different mathematical questions, having
+been treated in reference to one of these, will thus have
+been treated for all; while the concrete part will have
+necessarily to be again taken up for each question separately,
+without the solution of any one of them being
+able to give any direct aid, in that connexion, for the solution
+of the rest.</p>
+
+<p>The abstract part of mathematics is, then, general in
+its nature; the concrete part, special.</p>
+
+<p>To present this comparison under a new point of view,
+we may say concrete mathematics has a philosophical
+character, which is essentially experimental, physical,
+phenomenal; while that of abstract mathematics is purely
+logical, rational. The concrete part of every mathematical
+question is necessarily founded on the consideration
+of the external world, and could never be resolved
+by a simple series of intellectual combinations. The abstract
+part, on the contrary, when it has been very completely
+separated, can consist only of a series of logical
+deductions, more or less prolonged; for if we have once<span class="pagenum"><a name="Page_31" id="Page_31">[Pg 31]</a></span>
+found the equations of a phenomenon, the determination
+of the quantities therein considered, by means of one another,
+is a matter for reasoning only, whatever the difficulties
+may be. It belongs to the understanding alone
+to deduce from these equations results which are evidently
+contained in them, although perhaps in a very involved
+manner, without there being occasion to consult
+anew the external world; the consideration of which,
+having become thenceforth foreign to the subject, ought
+even to be carefully set aside in order to reduce the labour
+to its true peculiar difficulty. The <i>abstract</i> part
+of mathematics is then purely instrumental, and is only
+an immense and admirable extension of natural logic to a
+certain class of deductions. On the other hand, geometry
+and mechanics, which, as we shall see presently, constitute
+the <i>concrete</i> part, must be viewed as real natural
+sciences, founded on observation, like all the rest,
+although the extreme simplicity of their phenomena permits
+an infinitely greater degree of systematization,
+which has sometimes caused a misconception of the experimental
+character of their first principles.</p>
+
+<p>We see, by this brief general comparison, how natural
+and profound is our fundamental division of mathematical
+science.</p>
+
+<p>We have now to circumscribe, as exactly as we can
+in this first sketch, each of these two great sections.</p>
+
+
+<h3>CONCRETE MATHEMATICS.</h3>
+
+<p><i>Concrete Mathematics</i> having for its object the discovery
+of the <i>equations</i> of phenomena, it would seem at
+first that it must be composed of as many distinct sciences
+as we find really distinct categories among natural<span class="pagenum"><a name="Page_32" id="Page_32">[Pg 32]</a></span>
+phenomena. But we are yet very far from having discovered
+mathematical laws in all kinds of phenomena;
+we shall even see, presently, that the greater part will
+very probably always hide themselves from our investigations.
+In reality, in the present condition of the human
+mind, there are directly but two great general classes of
+phenomena, whose equations we constantly know; these
+are, firstly, geometrical, and, secondly, mechanical phenomena.
+Thus, then, the concrete part of mathematics
+is composed of <span class="smcap">Geometry</span> and <span class="smcap">Rational Mechanics</span>.</p>
+
+<p>This is sufficient, it is true, to give to it a complete
+character of logical universality, when we consider all
+phenomena from the most elevated point of view of natural
+philosophy. In fact, if all the parts of the universe
+were conceived as immovable, we should evidently have
+only geometrical phenomena to observe, since all would
+be reduced to relations of form, magnitude, and position;
+then, having regard to the motions which take place in it,
+we would have also to consider mechanical phenomena.
+Hence the universe, in the statical point of view, presents
+only geometrical phenomena; and, considered dynamically,
+only mechanical phenomena. Thus geometry
+and mechanics constitute the two fundamental natural
+sciences, in this sense, that all natural effects may be conceived
+as simple necessary results, either of the laws of
+extension or of the laws of motion.</p>
+
+<p>But although this conception is always logically possible,
+the difficulty is to specialize it with the necessary
+precision, and to follow it exactly in each of the general
+cases offered to us by the study of nature; that is, to
+effectually reduce each principal question of natural philosophy,
+for a certain determinate order of phenomena, to<span class="pagenum"><a name="Page_33" id="Page_33">[Pg 33]</a></span>
+the question of geometry or mechanics, to which we might
+rationally suppose it should be brought. This transformation,
+which requires great progress to have been previously
+made in the study of each class of phenomena, has thus
+far been really executed only for those of astronomy, and
+for a part of those considered by terrestrial physics, properly
+so called. It is thus that astronomy, acoustics, optics,
+&amp;c., have finally become applications of mathematical
+science to certain orders of observations.<a name="FNanchor_1_1" id="FNanchor_1_1"></a><a href="#Footnote_1_1" class="fnanchor">[1]</a> But these
+applications not being by their nature rigorously circumscribed,
+to confound them with the science would be to
+assign to it a vague and indefinite domain; and this is
+done in the usual division, so faulty in so many other
+respects, of the mathematics into "Pure" and "Applied."</p>
+
+
+<h3>ABSTRACT MATHEMATICS.</h3>
+
+<p>The nature of abstract mathematics (the general division
+of which will be examined in the following chapter) is
+clearly and exactly determined. It is composed of what is
+called the <i>Calculus</i>,<a name="FNanchor_2_2" id="FNanchor_2_2"></a><a href="#Footnote_2_2" class="fnanchor">[2]</a> taking this word in its greatest extent,
+which reaches from the most simple numerical operations
+to the most sublime combinations of transcendental
+analysis. The <i>Calculus</i> has the solution of all questions
+<span class="pagenum"><a name="Page_34" id="Page_34">[Pg 34]</a></span>relating to numbers for its peculiar object. Its <i>starting
+point</i> is, constantly and necessarily, the knowledge of the
+precise relations, <i>i.e.</i>, of the <i>equations</i>, between the different
+magnitudes which are simultaneously considered;
+that which is, on the contrary, the <i>stopping point</i> of concrete
+mathematics. However complicated, or however indirect
+these relations may be, the final object of the calculus
+always is to obtain from them the values of the unknown
+quantities by means of those which are known.
+This <i>science</i>, although nearer perfection than any other,
+is really little advanced as yet, so that this object is rarely
+attained in a manner completely satisfactory.</p>
+
+<p>Mathematical analysis is, then, the true rational basis
+of the entire system of our actual knowledge. It constitutes
+the first and the most perfect of all the fundamental
+sciences. The ideas with which it occupies itself
+are the most universal, the most abstract, and the
+most simple which it is possible for us to conceive.</p>
+
+<p>This peculiar nature of mathematical analysis enables
+us easily to explain why, when it is properly employed,
+it is such a powerful instrument, not only to give more
+precision to our real knowledge, which is self-evident, but
+especially to establish an infinitely more perfect co-ordination
+in the study of the phenomena which admit of
+that application; for, our conceptions having been so
+generalized and simplified that a single analytical question,
+abstractly resolved, contains the <i>implicit</i> solution
+of a great number of diverse physical questions, the human
+mind must necessarily acquire by these means a
+greater facility in perceiving relations between phenomena
+which at first appeared entirely distinct from one
+another. We thus naturally see arise, through the medium<span class="pagenum"><a name="Page_35" id="Page_35">[Pg 35]</a></span>
+of analysis, the most frequent and the most unexpected
+approximations between problems which at first
+offered no apparent connection, and which we often end
+in viewing as identical. Could we, for example, without
+the aid of analysis, perceive the least resemblance
+between the determination of the direction of a curve at
+each of its points and that of the velocity acquired by a
+body at every instant of its variable motion? and yet
+these questions, however different they may be, compose
+but one in the eyes of the geometer.</p>
+
+<p>The high relative perfection of mathematical analysis
+is as easily perceptible. This perfection is not due, as
+some have thought, to the nature of the signs which are
+employed as instruments of reasoning, eminently concise
+and general as they are. In reality, all great analytical
+ideas have been formed without the algebraic signs having
+been of any essential aid, except for working them
+out after the mind had conceived them. The superior
+perfection of the science of the calculus is due principally
+to the extreme simplicity of the ideas which it considers,
+by whatever signs they may be expressed; so that
+there is not the least hope, by any artifice of scientific
+language, of perfecting to the same degree theories which
+refer to more complex subjects, and which are necessarily
+condemned by their nature to a greater or less logical inferiority.</p>
+
+
+<h3>THE EXTENT OF ITS FIELD.</h3>
+
+<p>Our examination of the philosophical character of mathematical
+science would remain incomplete, if, after having
+viewed its object and composition, we did not examine
+the real extent of its domain.</p><p><span class="pagenum"><a name="Page_36" id="Page_36">[Pg 36]</a></span></p>
+
+<p><i>Its Universality</i>. For this purpose it is indispensable
+to perceive, first of all, that, in the purely logical
+point of view, this science is by itself necessarily and
+rigorously universal; for there is no question whatever
+which may not be finally conceived as consisting in determining
+certain quantities from others by means of certain
+relations, and consequently as admitting of reduction,
+in final analysis, to a simple question of numbers.
+In all our researches, indeed, on whatever subject, our
+object is to arrive at numbers, at quantities, though often
+in a very imperfect manner and by very uncertain methods.
+Thus, taking an example in the class of subjects
+the least accessible to mathematics, the phenomena of
+living bodies, even when considered (to take the most
+complicated case) in the state of disease, is it not manifest
+that all the questions of therapeutics may be viewed
+as consisting in determining the <i>quantities</i> of the different
+agents which modify the organism, and which must
+act upon it to bring it to its normal state, admitting, for
+some of these quantities in certain cases, values which
+are equal to zero, or negative, or even contradictory?</p>
+
+<p>The fundamental idea of Descartes on the relation of
+the concrete to the abstract in mathematics, has proven,
+in opposition to the superficial distinction of metaphysics,
+that all ideas of quality may be reduced to those of
+quantity. This conception, established at first by its
+immortal author in relation to geometrical phenomena
+only, has since been effectually extended to mechanical
+phenomena, and in our days to those of heat. As a result
+of this gradual generalization, there are now no geometers
+who do not consider it, in a purely theoretical
+sense, as capable of being applied to all our real ideas of<span class="pagenum"><a name="Page_37" id="Page_37">[Pg 37]</a></span>
+every sort, so that every phenomenon is logically susceptible
+of being represented by an <i>equation</i>; as much so,
+indeed, as is a curve or a motion, excepting the difficulty
+of discovering it, and then of <i>resolving</i> it, which
+may be, and oftentimes are, superior to the greatest powers
+of the human mind.</p>
+
+<p><i>Its Limitations</i>. Important as it is to comprehend
+the rigorous universality, in a logical point of view, of
+mathematical science, it is no less indispensable to consider
+now the great real <i>limitations</i>, which, through the
+feebleness of our intellect, narrow in a remarkable degree
+its actual domain, in proportion as phenomena, in
+becoming special, become complicated.</p>
+
+<p>Every question may be conceived as capable of being
+reduced to a pure question of numbers; but the difficulty
+of effecting such a transformation increases so much
+with the complication of the phenomena of natural philosophy,
+that it soon becomes insurmountable.</p>
+
+<p>This will be easily seen, if we consider that to bring
+a question within the field of mathematical analysis, we
+must first have discovered the precise relations which exist
+between the quantities which are found in the phenomenon
+under examination, the establishment of these
+equations being the necessary starting point of all analytical
+labours. This must evidently be so much the
+more difficult as we have to do with phenomena which
+are more special, and therefore more complicated. We
+shall thus find that it is only in <i>inorganic physics</i>, at
+the most, that we can justly hope ever to obtain that
+high degree of scientific perfection.</p>
+
+<p>The <i>first</i> condition which is necessary in order that
+phenomena may admit of mathematical laws, susceptible<span class="pagenum"><a name="Page_38" id="Page_38">[Pg 38]</a></span>
+of being discovered, evidently is, that their different quantities
+should admit of being expressed by fixed numbers.
+We soon find that in this respect the whole of <i>organic
+physics</i>, and probably also the most complicated parts of
+inorganic physics, are necessarily inaccessible, by their
+nature, to our mathematical analysis, by reason of the
+extreme numerical variability of the corresponding phenomena.
+Every precise idea of fixed numbers is truly
+out of place in the phenomena of living bodies, when we
+wish to employ it otherwise than as a means of relieving
+the attention, and when we attach any importance to the
+exact relations of the values assigned.</p>
+
+<p>We ought not, however, on this account, to cease to
+conceive all phenomena as being necessarily subject to
+mathematical laws, which we are condemned to be ignorant
+of, only because of the too great complication of the
+phenomena. The most complex phenomena of living
+bodies are doubtless essentially of no other special nature
+than the simplest phenomena of unorganized matter. If
+it were possible to isolate rigorously each of the simple
+causes which concur in producing a single physiological
+phenomenon, every thing leads us to believe that it would
+show itself endowed, in determinate circumstances, with
+a kind of influence and with a quantity of action as exactly
+fixed as we see it in universal gravitation, a veritable
+type of the fundamental laws of nature.</p>
+
+<p>There is a <i>second</i> reason why we cannot bring complicated
+phenomena under the dominion of mathematical
+analysis. Even if we could ascertain the mathematical
+law which governs each agent, taken by itself, the combination
+of so great a number of conditions would render
+the corresponding mathematical problem so far above our<span class="pagenum"><a name="Page_39" id="Page_39">[Pg 39]</a></span>
+feeble means, that the question would remain in most
+cases incapable of solution.</p>
+
+<p>To appreciate this difficulty, let us consider how complicated
+mathematical questions become, even those relating
+to the most simple phenomena of unorganized bodies,
+when we desire to bring sufficiently near together the abstract
+and the concrete state, having regard to all the
+principal conditions which can exercise a real influence
+over the effect produced. We know, for example, that
+the very simple phenomenon of the flow of a fluid through
+a given orifice, by virtue of its gravity alone, has not as
+yet any complete mathematical solution, when we take
+into the account all the essential circumstances. It is
+the same even with the still more simple motion of a
+solid projectile in a resisting medium.</p>
+
+<p>Why has mathematical analysis been able to adapt itself
+with such admirable success to the most profound study
+of celestial phenomena? Because they are, in spite of
+popular appearances, much more simple than any others.
+The most complicated problem which they present, that
+of the modification produced in the motions of two bodies
+tending towards each other by virtue of their gravitation,
+by the influence of a third body acting on both of them
+in the same manner, is much less complex than the most
+simple terrestrial problem. And, nevertheless, even it
+presents difficulties so great that we yet possess only
+approximate solutions of it. It is even easy to see that
+the high perfection to which solar astronomy has been
+able to elevate itself by the employment of mathematical
+science is, besides, essentially due to our having skilfully
+profited by all the particular, and, so to say, accidental
+facilities presented by the peculiarly favourable constitution<span class="pagenum"><a name="Page_40" id="Page_40">[Pg 40]</a></span>
+of our planetary system. The planets which compose
+it are quite few in number, and their masses are in
+general very unequal, and much less than that of the
+sun; they are, besides, very distant from one another;
+they have forms almost spherical; their orbits are nearly
+circular, and only slightly inclined to each other, and so
+on. It results from all these circumstances that the perturbations
+are generally inconsiderable, and that to calculate
+them it is usually sufficient to take into the account,
+in connexion with the action of the sun on each
+particular planet, the influence of only one other planet,
+capable, by its size and its proximity, of causing perceptible
+derangements.</p>
+
+<p>If, however, instead of such a state of things, our solar
+system had been composed of a greater number of
+planets concentrated into a less space, and nearly equal
+in mass; if their orbits had presented very different inclinations,
+and considerable eccentricities; if these bodies
+had been of a more complicated form, such as very eccentric
+ellipsoids, it is certain that, supposing the same
+law of gravitation to exist, we should not yet have succeeded
+in subjecting the study of the celestial phenomena
+to our mathematical analysis, and probably we should
+not even have been able to disentangle the present principal
+law.</p>
+
+<p>These hypothetical conditions would find themselves
+exactly realized in the highest degree in <i>chemical</i> phenomena,
+if we attempted to calculate them by the theory
+of general gravitation.</p>
+
+<p>On properly weighing the preceding considerations,
+the reader will be convinced, I think, that in reducing
+the future extension of the great applications of mathematical<span class="pagenum"><a name="Page_41" id="Page_41">[Pg 41]</a></span>
+analysis, which are really possible, to the field
+comprised in the different departments of inorganic physics,
+I have rather exaggerated than contracted the extent
+of its actual domain. Important as it was to render
+apparent the rigorous logical universality of mathematical
+science, it was equally so to indicate the conditions
+which limit for us its real extension, so as not to
+contribute to lead the human mind astray from the true
+scientific direction in the study of the most complicated
+phenomena, by the chimerical search after an impossible
+perfection.</p>
+<p><span class="pagenum"><a name="Page_42" id="Page_42">[Pg 42]</a></span></p><p><span class="pagenum"><a name="Page_43" id="Page_43">[Pg 43]</a></span></p>
+<hr class="tb" />
+
+<p>Having thus exhibited the essential object and the
+principal composition of mathematical science, as well as
+its general relations with the whole body of natural philosophy,
+we have now to pass to the special examination
+of the great sciences of which it is composed.</p>
+
+<div class="blockquot"><p><i>Note.</i>&mdash;<span class="smcap">Analysis</span> and <span class="smcap">Geometry</span> are the two great heads under which
+the subject is about to be examined. To these <i>M. Comte</i> adds Rational
+<span class="smcap">Mechanics</span>; but as it is not comprised in the usual idea of Mathematics,
+and as its discussion would be of but limited utility and interest, it is not
+included in the present translation.</p></div>
+
+
+
+
+<p class="center big">BOOK I.</p>
+
+<p class="center">ANALYSIS.</p>
+
+<p><span class="pagenum"><a name="Page_44" id="Page_44">[Pg 44]</a><br /><a name="Page_45" id="Page_45">[Pg 45]</a></span></p>
+
+
+<p class="center big">BOOK I.</p>
+
+<h1>ANALYSIS.</h1>
+
+
+
+
+<h2><a name="CHAPTER_I" id="CHAPTER_I">CHAPTER I.</a></h2>
+
+<h3>GENERAL VIEW OF MATHEMATICAL ANALYSIS.</h3>
+
+
+<p>In the historical development of mathematical science
+since the time of Descartes, the advances of its abstract
+portion have always been determined by those of its concrete
+portion; but it is none the less necessary, in order
+to conceive the science in a manner truly logical, to
+consider the Calculus in all its principal branches before
+proceeding to the philosophical study of Geometry and
+Mechanics. Its analytical theories, more simple and
+more general than those of concrete mathematics, are in
+themselves essentially independent of the latter; while
+these, on the contrary, have, by their nature, a continual
+need of the former, without the aid of which they could
+make scarcely any progress. Although the principal
+conceptions of analysis retain at present some very perceptible
+traces of their geometrical or mechanical origin,
+they are now, however, mainly freed from that primitive
+character, which no longer manifests itself except in some
+secondary points; so that it is possible (especially since
+the labours of Lagrange) to present them in a dogmatic
+exposition, by a purely abstract method, in a single and<span class="pagenum"><a name="Page_46" id="Page_46">[Pg 46]</a></span>
+continuous system. It is this which will be undertaken
+in the present and the five following chapters, limiting our
+investigations to the most general considerations upon
+each principal branch of the science of the calculus.</p>
+
+<p>The definite object of our researches in concrete mathematics
+being the discovery of the <i>equations</i> which express
+the mathematical laws of the phenomenon under
+consideration, and these equations constituting the true
+starting point of the calculus, which has for its object
+to obtain from them the determination of certain quantities
+by means of others, I think it indispensable, before
+proceeding any farther, to go more deeply than has
+been customary into that fundamental idea of <i>equation</i>,
+the continual subject, either as end or as beginning, of
+all mathematical labours. Besides the advantage of circumscribing
+more definitely the true field of analysis,
+there will result from it the important consequence of
+tracing in a more exact manner the real line of demarcation
+between the concrete and the abstract part of
+mathematics, which will complete the general exposition
+of the fundamental division established in the introductory
+chapter.</p>
+
+
+
+
+<h3><a name="THE_TRUE_IDEA_OF_AN_EQUATION" id="THE_TRUE_IDEA_OF_AN_EQUATION">THE TRUE IDEA OF AN EQUATION.</a></h3>
+
+
+<p>We usually form much too vague an idea of what an
+<i>equation</i> is, when we give that name to every kind of
+relation of equality between <i>any</i> two functions of the
+magnitudes which we are considering. For, though every
+equation is evidently a relation of equality, it is far
+from being true that, reciprocally, every relation of equality
+is a veritable <i>equation</i>, of the kind of those to which,
+by their nature, the methods of analysis are applicable.</p><p><span class="pagenum"><a name="Page_47" id="Page_47">[Pg 47]</a></span></p>
+
+<p>This want of precision in the logical consideration of
+an idea which is so fundamental in mathematics, brings
+with it the serious inconvenience of rendering it almost
+impossible to explain, in general terms, the great and
+fundamental difficulty which we find in establishing the
+relation between the concrete and the abstract, and which
+stands out so prominently in each great mathematical
+question taken by itself. If the meaning of the word
+<i>equation</i> was truly as extended as we habitually suppose
+it to be in our definition of it, it is not apparent what
+great difficulty there could really be, in general, in establishing
+the equations of any problem whatsoever; for the
+whole would thus appear to consist in a simple question
+of form, which ought never even to exact any great intellectual
+efforts, seeing that we can hardly conceive of
+any precise relation which is not immediately a certain
+relation of equality, or which cannot be readily brought
+thereto by some very easy transformations.</p>
+
+<p>Thus, when we admit every species of <i>functions</i> into
+the definition of <i>equations</i>, we do not at all account for
+the extreme difficulty which we almost always experience
+in putting a problem into an equation, and which
+so often may be compared to the efforts required by the
+analytical elaboration of the equation when once obtained.
+In a word, the ordinary abstract and general idea
+of an <i>equation</i> does not at all correspond to the real
+meaning which geometers attach to that expression in
+the actual development of the science. Here, then, is a
+logical fault, a defect of correlation, which it is very important
+to rectify.</p>
+
+
+<p><i>Division of Functions into Abstract and Concrete.</i>
+To succeed in doing so, I begin by distinguishing two<span class="pagenum"><a name="Page_48" id="Page_48">[Pg 48]</a></span>
+sorts of <i>functions</i>, <i>abstract</i> or analytical functions, and
+<i>concrete</i> functions. The first alone can enter into veritable
+<i>equations</i>. We may, therefore, henceforth define
+every <i>equation</i>, in an exact and sufficiently profound manner,
+as a relation of equality between two <i>abstract</i> functions
+of the magnitudes under consideration. In order not
+to have to return again to this fundamental definition, I
+must add here, as an indispensable complement, without
+which the idea would not be sufficiently general, that
+these abstract functions may refer not only to the magnitudes
+which the problem presents of itself, but also to
+all the other auxiliary magnitudes which are connected
+with it, and which we will often be able to introduce,
+simply as a mathematical artifice, with the sole object
+of facilitating the discovery of the equations of the phenomena.
+I here anticipate summarily the result of a
+general discussion of the highest importance, which will
+be found at the end of this chapter. We will now return
+to the essential distinction of functions as abstract
+and concrete.</p>
+
+<p>This distinction may be established in two ways, essentially
+different, but complementary of each other, <i>à
+priori</i> and <i>à posteriori</i>; that is to say, by characterizing
+in a general manner the peculiar nature of each species
+of functions, and then by making the actual enumeration
+of all the abstract functions at present known,
+at least so far as relates to the elements of which they
+are composed.</p>
+
+<p><i>À priori</i>, the functions which I call <i>abstract</i> are those
+which express a manner of dependence between magnitudes,
+which can be conceived between numbers alone,
+without there being need of indicating any phenomenon<span class="pagenum"><a name="Page_49" id="Page_49">[Pg 49]</a></span>
+whatever in which it is realized. I name, on the other
+hand, <i>concrete</i> functions, those for which the mode of dependence
+expressed cannot be defined or conceived except
+by assigning a determinate case of physics, geometry, mechanics,
+&amp;c., in which it actually exists.</p>
+
+<p>Most functions in their origin, even those which are
+at present the most purely <i>abstract</i>, have begun by being
+<i>concrete</i>; so that it is easy to make the preceding
+distinction understood, by citing only the successive different
+points of view under which, in proportion as the
+science has become formed, geometers have considered
+the most simple analytical functions. I will indicate
+powers, for example, which have in general become abstract
+functions only since the labours of Vieta and Descartes.
+The functions <i>x<sup>2</sup></i>, <i>x<sup>3</sup></i>, which in our present analysis
+are so well conceived as simply <i>abstract</i>, were, for
+the geometers of antiquity, perfectly <i>concrete</i> functions,
+expressing the relation of the superficies of a square, or
+the volume of a cube to the length of their side. These
+had in their eyes such a character so exclusively, that
+it was only by means of the geometrical definitions that
+they discovered the elementary algebraic properties of
+these functions, relating to the decomposition of the
+variable into two parts, properties which were at that
+epoch only real theorems of geometry, to which a numerical
+meaning was not attached until long afterward.</p>
+
+<p>I shall have occasion to cite presently, for another reason,
+a new example, very suitable to make apparent the
+fundamental distinction which I have just exhibited; it
+is that of circular functions, both direct and inverse, which
+at the present time are still sometimes concrete, sometimes<span class="pagenum"><a name="Page_50" id="Page_50">[Pg 50]</a></span>
+abstract, according to the point of view under which
+they are regarded.</p>
+
+<p><i>À posteriori</i>, the general character which renders a
+function abstract or concrete having been established, the
+question as to whether a certain determinate function is
+veritably abstract, and therefore susceptible of entering
+into true analytical equations, becomes a simple question
+of fact, inasmuch as we are going to enumerate all the
+functions of this species.</p>
+
+
+<p><i>Enumeration of Abstract Functions.</i> At first view
+this enumeration seems impossible, the distinct analytical
+functions being infinite in number. But when we
+divide them into <i>simple</i> and <i>compound</i>, the difficulty disappears;
+for, though the number of the different functions
+considered in mathematical analysis is really infinite,
+they are, on the contrary, even at the present day,
+composed of a very small number of elementary functions,
+which can be easily assigned, and which are evidently
+sufficient for deciding the abstract or concrete character
+of any given function; which will be of the one or the
+other nature, according as it shall be composed exclusively
+of these simple abstract functions, or as it shall include
+others.</p>
+
+<p>We evidently have to consider, for this purpose, only
+the functions of a single variable, since those relative
+to several independent variables are constantly, by their
+nature, more or less <i>compound</i>.</p>
+
+<p>Let <i>x</i> be the independent variable, <i>y</i> the correlative
+variable which depends upon it. The different simple
+modes of abstract dependence, which we can now conceive
+between <i>y</i> and <i>x</i>, are expressed by the ten following elementary
+formulas, in which each function is coupled<span class="pagenum"><a name="Page_51" id="Page_51">[Pg 51]</a></span>
+with its <i>inverse</i>, that is, with that which would be obtained
+from the direct function by referring <i>x</i> to <i>y</i>, instead
+of referring <i>y</i> to <i>x</i>.</p>
+
+
+<div class="center">
+<table border="0" cellpadding="4" cellspacing="0" summary="functions">
+<tr><td>&nbsp;</td><td align="center">FUNCTION.</td><td align="center">ITS NAME.</td></tr>
+<tr><td align="left" rowspan="2">1st couple</td><td align="left">1° <i>y</i> = <i>a</i> + <i>x</i></td><td align="left"><i>Sum.</i></td></tr>
+<tr><td align="left">2° <i>y</i> = <i>a</i> - <i>x</i></td><td align="left"><i>Difference.</i></td></tr>
+<tr><td align="left" rowspan="2">2d couple</td><td align="left">1° <i>y</i> = <i>ax</i></td><td align="left"><i>Product.</i></td></tr>
+<tr><td align="left">2° <i>y</i> = <i>a/x</i></td><td align="left"><i>Quotient.</i></td></tr>
+<tr><td align="left" rowspan="2">3d couple</td><td align="left">1° <i>y</i> = <i>x^a</i></td><td align="left"><i>Power.</i></td></tr>
+<tr><td align="left">2° <i>y</i> = <i>[aroot]x</i></td><td align="left"><i>Root.</i></td></tr>
+<tr><td align="left" rowspan="2">4th couple</td><td align="left">1° <i>y</i> = <i>a^x</i></td><td align="left"><i>Exponential.</i></td></tr>
+<tr><td align="left">2° <i>y</i> = <i>[log a]x</i></td><td align="left"><i>Logarithmic.</i></td></tr>
+<tr><td align="left" rowspan="2">5th couple</td><td align="left">1° <i>y</i> = sin. <i>x</i></td><td align="left"><i>Direct Circular.</i></td></tr>
+<tr><td align="left">2° <i>y</i> = arc(sin. = <i>x</i>).</td><td align="left"><i>Inverse Circular.</i><a name="FNanchor_3_3" id="FNanchor_3_3"></a><a href="#Footnote_3_3" class="fnanchor">[3]</a></td></tr>
+</table></div>
+
+<p>Such are the elements, very few in number, which directly
+compose all the abstract functions known at the
+present day. Few as they are, they are evidently sufficient
+to give rise to an infinite number of analytical
+combinations.</p>
+<p><span class="pagenum"><a name="Page_52" id="Page_52">[Pg 52]</a></span></p>
+<p>No rational consideration rigorously circumscribes, <i>à
+priori</i>, the preceding table, which is only the actual expression
+of the present state of the science. Our analytical
+elements are at the present day more numerous
+than they were for Descartes, and even for Newton and
+Leibnitz: it is only a century since the last two couples
+have been introduced into analysis by the labours of John
+Bernouilli and Euler. Doubtless new ones will be hereafter
+admitted; but, as I shall show towards the end of
+this chapter, we cannot hope that they will ever be greatly
+multiplied, their real augmentation giving rise to very
+great difficulties.</p>
+
+<p>We can now form a definite, and, at the same time,
+sufficiently extended idea of what geometers understand
+by a veritable <i>equation</i>. This explanation is especially
+suited to make us understand how difficult it must be
+really to establish the <i>equations</i> of phenomena, since we
+have effectually succeeded in so doing only when we
+have been able to conceive the mathematical laws of
+these phenomena by the aid of functions entirely composed
+of only the mathematical elements which I have
+just enumerated. It is clear, in fact, that it is then
+only that the problem becomes truly abstract, and is reduced
+to a pure question of numbers, these functions
+being the only simple relations which we can conceive
+between numbers, considered by themselves. Up to this
+period of the solution, whatever the appearances may be,
+the question is still essentially concrete, and does not come
+within the domain of the <i>calculus</i>. Now the fundamental
+difficulty of this passage from the <i>concrete</i> to the <i>abstract</i>
+in general consists especially in the insufficiency
+of this very small number of analytical elements which<span class="pagenum"><a name="Page_53" id="Page_53">[Pg 53]</a></span>
+we possess, and by means of which, nevertheless, in spite
+of the little real variety which they offer us, we must
+succeed in representing all the precise relations which
+all the different natural phenomena can manifest to us.
+Considering the infinite diversity which must necessarily
+exist in this respect in the external world, we easily
+understand how far below the true difficulty our conceptions
+must frequently be found, especially if we add
+that as these elements of our analysis have been in the
+first place furnished to us by the mathematical consideration
+of the simplest phenomena, we have, <i>à priori</i>, no
+rational guarantee of their necessary suitableness to represent
+the mathematical law of every other class of phenomena.
+I will explain presently the general artifice, so
+profoundly ingenious, by which the human mind has succeeded
+in diminishing, in a remarkable degree, this fundamental
+difficulty which is presented by the relation of
+the concrete to the abstract in mathematics, without,
+however, its having been necessary to multiply the number
+of these analytical elements.</p>
+
+
+
+
+<h3><a name="THE_TWO_PRINCIPAL_DIVISIONS_OF_THE_CALCULUS" id="THE_TWO_PRINCIPAL_DIVISIONS_OF_THE_CALCULUS">THE TWO PRINCIPAL DIVISIONS OF THE CALCULUS.</a></h3>
+
+
+<p>The preceding explanations determine with precision
+the true object and the real field of abstract mathematics.
+I must now pass to the examination of its principal
+divisions, for thus far we have considered the calculus
+as a whole.</p>
+
+<p>The first direct consideration to be presented on the
+composition of the science of the <i>calculus</i> consists in dividing
+it, in the first place, into two principal branches,
+to which, for want of more suitable denominations, I will
+give the names of <i>Algebraic calculus</i>, or <i>Algebra</i>, and of<span class="pagenum"><a name="Page_54" id="Page_54">[Pg 54]</a></span>
+<i>Arithmetical calculus</i>, or <i>Arithmetic</i>; but with the caution
+to take these two expressions in their most extended
+logical acceptation, in the place of the by far too restricted
+meaning which is usually attached to them.</p>
+
+<p>The complete solution of every question of the <i>calculus</i>,
+from the most elementary up to the most transcendental,
+is necessarily composed of two successive parts,
+whose nature is essentially distinct. In the first, the object
+is to transform the proposed equations, so as to make
+apparent the manner in which the unknown quantities
+are formed by the known ones: it is this which constitutes
+the <i>algebraic</i> question. In the second, our object
+is to <i>find the values</i> of the formulas thus obtained; that
+is, to determine directly the values of the numbers sought,
+which are already represented by certain explicit functions
+of given numbers: this is the <i>arithmetical</i> question.<a name="FNanchor_4_4" id="FNanchor_4_4"></a><a href="#Footnote_4_4" class="fnanchor">[4]</a>
+It is apparent that, in every solution which is
+<span class="pagenum"><a name="Page_55" id="Page_55">[Pg 55]</a></span>truly rational, it necessarily follows the algebraical question,
+of which it forms the indispensable complement,
+since it is evidently necessary to know the mode of generation
+of the numbers sought for before determining their
+actual values for each particular case. Thus the stopping-place
+of the algebraic part of the solution becomes
+the starting point of the arithmetical part.</p>
+
+<p>We thus see that the <i>algebraic</i> calculus and the <i>arithmetical</i>
+calculus differ essentially in their object. They
+differ no less in the point of view under which they regard
+quantities; which are considered in the first as to their
+<i>relations</i>, and in the second as to their <i>values</i>. The
+true spirit of the calculus, in general, requires this distinction
+to be maintained with the most severe exactitude,
+and the line of demarcation between the two periods
+of the solution to be rendered as clear and distinct
+as the proposed question permits. The attentive observation
+of this precept, which is too much neglected, may
+be of much assistance, in each particular question, in directing
+the efforts of our mind, at any moment of the
+solution, towards the real corresponding difficulty. In
+truth, the imperfection of the science of the calculus
+obliges us very often (as will be explained in the next
+chapter) to intermingle algebraic and arithmetical considerations
+in the solution of the same question. But, however
+impossible it may be to separate clearly the two parts
+of the labour, yet the preceding indications will always
+enable us to avoid confounding them.</p>
+
+<p>In endeavouring to sum up as succinctly as possible
+the distinction just established, we see that <span class="smcap">Algebra</span>
+may be defined, in general, as having for its object the
+<i>resolution of equations</i>; taking this expression in its<span class="pagenum"><a name="Page_56" id="Page_56">[Pg 56]</a></span>
+full logical meaning, which signifies the transformation
+of <i>implicit</i> functions into equivalent <i>explicit</i> ones. In
+the same way, <span class="smcap">Arithmetic</span> may be defined as destined
+to <i>the determination of the values of functions</i>. Henceforth,
+therefore, we will briefly say that <span class="smcap">Algebra</span> is the
+<i>Calculus of Functions</i>, and <span class="smcap">Arithmetic</span> the <i>Calculus of
+Values</i>.</p>
+
+<p>We can now perceive how insufficient and even erroneous
+are the ordinary definitions. Most generally, the
+exaggerated importance attributed to Signs has led to the
+distinguishing the two fundamental branches of the science
+of the Calculus by the manner of designating in
+each the subjects of discussion, an idea which is evidently
+absurd in principle and false in fact. Even the celebrated
+definition given by Newton, characterizing <i>Algebra</i>
+as <i>Universal Arithmetic</i>, gives certainly a very false
+idea of the nature of algebra and of that of arithmetic.<a name="FNanchor_5_5" id="FNanchor_5_5"></a><a href="#Footnote_5_5" class="fnanchor">[5]</a></p>
+
+<p>Having thus established the fundamental division of
+the calculus into two principal branches, I have now to
+compare in general terms the extent, the importance, and
+the difficulty of these two sorts of calculus, so as to have
+hereafter to consider only the <i>Calculus of Functions</i>,
+which is to be the principal subject of our study.</p>
+<p><span class="pagenum"><a name="Page_57" id="Page_57">[Pg 57]</a></span></p>
+
+
+
+<h3><a name="THE_CALCULUS_OF_VALUES_OR_ARITHMETIC" id="THE_CALCULUS_OF_VALUES_OR_ARITHMETIC">THE CALCULUS OF VALUES, OR ARITHMETIC.</a></h3>
+
+
+<p><i>Its Extent.</i> The <i>Calculus of Values, or Arithmetic</i>,
+would appear, at first view, to present a field as vast as
+that of <i>algebra</i>, since it would seem to admit as many
+distinct questions as we can conceive different algebraic
+formulas whose values are to be determined. But a very
+simple reflection will show the difference. Dividing functions
+into <i>simple</i> and <i>compound</i>, it is evident that when
+we know how to determine the <i>value</i> of simple functions,
+the consideration of compound functions will no longer
+present any difficulty. In the algebraic point of view,
+a compound function plays a very different part from that
+of the elementary functions of which it consists, and from
+this, indeed, proceed all the principal difficulties of analysis.
+But it is very different with the Arithmetical Calculus.
+Thus the number of truly distinct arithmetical
+operations is only that determined by the number of the
+elementary abstract functions, the very limited list of
+which has been given above. The determination of the values
+of these ten functions necessarily gives that of all
+the functions, infinite in number, which are considered
+in the whole of mathematical analysis, such at least as
+it exists at present. There can be no new arithmetical
+operations without the creation of really new analytical
+elements, the number of which must always be extremely
+small. The field of <i>arithmetic</i> is, then, by its nature,
+exceedingly restricted, while that of algebra is rigorously
+indefinite.</p>
+
+<p>It is, however, important to remark, that the domain
+of the <i>calculus of values</i> is, in reality, much more extensive
+than it is commonly represented; for several questions<span class="pagenum"><a name="Page_58" id="Page_58">[Pg 58]</a></span>
+truly <i>arithmetical</i>, since they consist of determinations
+of values, are not ordinarily classed as such, because
+we are accustomed to treat them only as incidental
+in the midst of a body of analytical researches
+more or less elevated, the too high opinion commonly
+formed of the influence of signs being again the principal
+cause of this confusion of ideas. Thus not only the
+construction of a table of logarithms, but also the calculation
+of trigonometrical tables, are true arithmetical operations
+of a higher kind. We may also cite as being
+in the same class, although in a very distinct and more
+elevated order, all the methods by which we determine
+directly the value of any function for each particular system
+of values attributed to the quantities on which it depends,
+when we cannot express in general terms the explicit
+form of that function. In this point of view the
+<i>numerical</i> solution of questions which we cannot resolve
+algebraically, and even the calculation of "Definite Integrals,"
+whose general integrals we do not know, really
+make a part, in spite of all appearances, of the domain
+of <i>arithmetic</i>, in which we must necessarily comprise all
+that which has for its object the <i>determination of the
+values of functions</i>. The considerations relative to this
+object are, in fact, constantly homogeneous, whatever the
+<i>determinations</i> in question, and are always very distinct
+from truly <i>algebraic</i> considerations.</p>
+
+<p>To complete a just idea of the real extent of the calculus
+of values, we must include in it likewise that part
+of the general science of the calculus which now bears
+the name of the <i>Theory of Numbers</i>, and which is yet
+so little advanced. This branch, very extensive by its
+nature, but whose importance in the general system of<span class="pagenum"><a name="Page_59" id="Page_59">[Pg 59]</a></span>
+science is not very great, has for its object the discovery
+of the properties inherent in different numbers by virtue
+of their values, and independent of any particular system
+of numeration. It forms, then, a sort of <i>transcendental
+arithmetic</i>; and to it would really apply the definition
+proposed by Newton for algebra.</p>
+
+<p>The entire domain of arithmetic is, then, much more
+extended than is commonly supposed; but this <i>calculus
+of values</i> will still never be more than a point, so to
+speak, in comparison with the <i>calculus of functions</i>, of
+which mathematical science essentially consists. This
+comparative estimate will be still more apparent from
+some considerations which I have now to indicate respecting
+the true nature of arithmetical questions in general,
+when they are more profoundly examined.</p>
+
+
+<p><i>Its true Nature.</i> In seeking to determine with precision
+in what <i>determinations of values</i> properly consist,
+we easily recognize that they are nothing else but veritable
+<i>transformations</i> of the functions to be valued;
+transformations which, in spite of their special end, are
+none the less essentially of the same nature as all those
+taught by analysis. In this point of view, the <i>calculus
+of values</i> might be simply conceived as an appendix, and
+a particular application of the <i>calculus of functions</i>, so
+that <i>arithmetic</i> would disappear, so to say, as a distinct
+section in the whole body of abstract mathematics.</p>
+
+<p>In order thoroughly to comprehend this consideration,
+we must observe that, when we propose to determine the
+<i>value</i> of an unknown number whose mode of formation is
+given, it is, by the mere enunciation of the arithmetical
+question, already defined and expressed under a certain
+form; and that in <i>determining its value</i> we only put its<span class="pagenum"><a name="Page_60" id="Page_60">[Pg 60]</a></span>
+expression under another determinate form, to which we
+are accustomed to refer the exact notion of each particular
+number by making it re-enter into the regular system
+of <i>numeration</i>. The determination of values consists
+so completely of a simple <i>transformation</i>, that when the
+primitive expression of the number is found to be already
+conformed to the regular system of numeration, there
+is no longer any determination of value, properly speaking,
+or, rather, the question is answered by the question
+itself. Let the question be to add the two numbers <i>one</i>
+and <i>twenty</i>, we answer it by merely repeating the enunciation
+of the question,<a name="FNanchor_6_6" id="FNanchor_6_6"></a><a href="#Footnote_6_6" class="fnanchor">[6]</a> and nevertheless we think that
+we have <i>determined the value</i> of the sum. This signifies
+that in this case the first expression of the function
+had no need of being transformed, while it would not be
+thus in adding twenty-three and fourteen, for then the
+sum would not be immediately expressed in a manner
+conformed to the rank which it occupies in the fixed and
+general scale of numeration.</p>
+
+<p>To sum up as comprehensively as possible the preceding
+views, we may say, that to determine the <i>value</i> of
+a number is nothing else than putting its primitive expression
+under the form</p>
+
+<p>
+<i>a</i> + <i>bz</i> + <i>cz<sup>2</sup></i> + <i>dz<sup>3</sup></i> + <i>ez<sup>4</sup></i> . . . . . + <i>pz<sup>m</sup></i>,<br />
+</p>
+
+<p><i>z</i> being generally equal to 10, and the coefficients <i>a</i>, <i>b</i>,
+<i>c</i>, <i>d</i>, &amp;c., being subjected to the conditions of being whole
+numbers less than <i>z</i>; capable of becoming equal to zero;
+but never negative. Every arithmetical question may
+thus be stated as consisting in putting under such a form
+<span class="pagenum"><a name="Page_61" id="Page_61">[Pg 61]</a></span>any abstract function whatever of different quantities,
+which are supposed to have themselves a similar form
+already. We might then see in the different operations
+of arithmetic only simple particular cases of certain algebraic
+transformations, excepting the special difficulties
+belonging to conditions relating to the nature of the coefficients.</p>
+
+<p>It clearly follows that abstract mathematics is essentially
+composed of the <i>Calculus of Functions</i>, which had
+been already seen to be its most important, most extended,
+and most difficult part. It will henceforth be the exclusive
+subject of our analytical investigations. I will
+therefore no longer delay on the <i>Calculus of Values</i>, but
+pass immediately to the examination of the fundamental
+division of the <i>Calculus of Functions</i>.</p>
+
+
+
+
+<h3><a name="THE_CALCULUS_OF_FUNCTIONS_OR_ALGEBRA" id="THE_CALCULUS_OF_FUNCTIONS_OR_ALGEBRA">THE CALCULUS OF FUNCTIONS, OR ALGEBRA.</a></h3>
+
+
+<p><i>Principle of its Fundamental Division.</i> We have
+determined, at the beginning of this chapter, wherein
+properly consists the difficulty which we experience in
+putting mathematical questions into <i>equations</i>. It is essentially
+because of the insufficiency of the very small
+number of analytical elements which we possess, that
+the relation of the concrete to the abstract is usually so
+difficult to establish. Let us endeavour now to appreciate
+in a philosophical manner the general process by
+which the human mind has succeeded, in so great a number
+of important cases, in overcoming this fundamental
+obstacle to <i>The establishment of Equations</i>.</p>
+
+
+<p>1. <i>By the Creation of new Functions.</i> In looking at
+this important question from the most general point of
+view, we are led at once to the conception of one means of<span class="pagenum"><a name="Page_62" id="Page_62">[Pg 62]</a></span>
+facilitating the establishment of the equations of phenomena.
+Since the principal obstacle in this matter comes
+from the too small number of our analytical elements, the
+whole question would seem to be reduced to creating
+new ones. But this means, though natural, is really
+illusory; and though it might be useful, it is certainly
+insufficient.</p>
+
+<p>In fact, the creation of an elementary abstract function,
+which shall be veritably new, presents in itself the
+greatest difficulties. There is even something contradictory
+in such an idea; for a new analytical element
+would evidently not fulfil its essential and appropriate
+conditions, if we could not immediately <i>determine its
+value</i>. Now, on the other hand, how are we to <i>determine
+the value</i> of a new function which is truly <i>simple</i>,
+that is, which is not formed by a combination of those
+already known? That appears almost impossible. The
+introduction into analysis of another elementary abstract
+function, or rather of another couple of functions (for each
+would be always accompanied by its <i>inverse</i>), supposes
+then, of necessity, the simultaneous creation of a new
+arithmetical operation, which is certainly very difficult.</p>
+
+<p>If we endeavour to obtain an idea of the means which
+the human mind employs for inventing new analytical
+elements, by the examination of the procedures by the
+aid of which it has actually conceived those which we
+already possess, our observations leave us in that respect
+in an entire uncertainty, for the artifices which it has
+already made use of for that purpose are evidently exhausted.
+To convince ourselves of it, let us consider
+the last couple of simple functions which has been introduced
+into analysis, and at the formation of which we<span class="pagenum"><a name="Page_63" id="Page_63">[Pg 63]</a></span>
+have been present, so to speak, namely, the fourth couple;
+for, as I have explained, the fifth couple does not strictly
+give veritable new analytical elements. The function
+<i>a<sup>x</sup></i>, and, consequently, its inverse, have been formed by
+conceiving, under a new point of view, a function which
+had been a long time known, namely, powers&mdash;when the
+idea of them had become sufficiently generalized. The
+consideration of a power relatively to the variation of its
+exponent, instead of to the variation of its base, was sufficient
+to give rise to a truly novel simple function, the
+variation following then an entirely different route. But
+this artifice, as simple as ingenious, can furnish nothing
+more; for, in turning over in the same manner all our
+present analytical elements, we end in only making them
+return into one another.</p>
+
+<p>We have, then, no idea as to how we could proceed to
+the creation of new elementary abstract functions which
+would properly satisfy all the necessary conditions. This
+is not to say, however, that we have at present attained
+the effectual limit established in that respect by the
+bounds of our intelligence. It is even certain that the
+last special improvements in mathematical analysis have
+contributed to extend our resources in that respect, by
+introducing within the domain of the calculus certain definite
+integrals, which in some respects supply the place
+of new simple functions, although they are far from fulfilling
+all the necessary conditions, which has prevented
+me from inserting them in the table of true analytical
+elements. But, on the whole, I think it unquestionable
+that the number of these elements cannot increase except
+with extreme slowness. It is therefore not from
+these sources that the human mind has drawn its most<span class="pagenum"><a name="Page_64" id="Page_64">[Pg 64]</a></span>
+powerful means of facilitating, as much as is possible,
+the establishment of equations.</p>
+
+
+<p>2. <i>By the Conception of Equations between certain
+auxiliary Quantities.</i> This first method being set aside,
+there remains evidently but one other: it is, seeing the
+impossibility of finding directly the equations between
+the quantities under consideration, to seek for corresponding
+ones between other auxiliary quantities, connected
+with the first according to a certain determinate law,
+and from the relation between which we may return to
+that between the primitive magnitudes. Such is, in
+substance, the eminently fruitful conception, which the
+human mind has succeeded in establishing, and which
+constitutes its most admirable instrument for the mathematical
+explanation of natural phenomena; the <i>analysis</i>,
+called <i>transcendental</i>.</p>
+
+<p>As a general philosophical principle, the auxiliary
+quantities, which are introduced in the place of the primitive
+magnitudes, or concurrently with them, in order to
+facilitate the establishment of equations, might be derived
+according to any law whatever from the immediate
+elements of the question. This conception has thus a
+much more extensive reach than has been commonly attributed
+to it by even the most profound geometers. It
+is extremely important for us to view it in its whole logical
+extent, for it will perhaps be by establishing a general
+mode of <i>derivation</i> different from that to which we
+have thus far confined ourselves (although it is evidently
+very far from being the only possible one) that we shall
+one day succeed in essentially perfecting mathematical
+analysis as a whole, and consequently in establishing
+more powerful means of investigating the laws of nature<span class="pagenum"><a name="Page_65" id="Page_65">[Pg 65]</a></span>
+than our present processes, which are unquestionably susceptible
+of becoming exhausted.</p>
+
+<p>But, regarding merely the present constitution of the
+science, the only auxiliary quantities habitually introduced
+in the place of the primitive quantities in the
+<i>Transcendental Analysis</i> are what are called, 1<sup>o</sup>, <i>infinitely
+small</i> elements, the <i>differentials</i> (of different orders)
+of those quantities, if we regard this analysis in the
+manner of <span class="smcap">Leibnitz</span>; or, 2<sup>o</sup>, the <i>fluxions</i>, the limits of
+the ratios of the simultaneous increments of the primitive
+quantities compared with one another, or, more
+briefly, the <i>prime and ultimate ratios</i> of these increments,
+if we adopt the conception of <span class="smcap">Newton</span>; or, 3<sup>o</sup>,
+the <i>derivatives</i>, properly so called, of those quantities,
+that is, the coefficients of the different terms of their respective
+increments, according to the conception of <span class="smcap">Lagrange</span>.</p>
+
+<p>These three principal methods of viewing our present
+transcendental analysis, and all the other less distinctly
+characterized ones which have been successively proposed,
+are, by their nature, necessarily identical, whether
+in the calculation or in the application, as will be explained
+in a general manner in the third chapter. As to
+their relative value, we shall there see that the conception
+of Leibnitz has thus far, in practice, an incontestable
+superiority, but that its logical character is exceedingly
+vicious; while that the conception of Lagrange,
+admirable by its simplicity, by its logical perfection, by
+the philosophical unity which it has established in mathematical
+analysis (till then separated into two almost entirely
+independent worlds), presents, as yet, serious inconveniences
+in the applications, by retarding the progress<span class="pagenum"><a name="Page_66" id="Page_66">[Pg 66]</a></span>
+of the mind. The conception of Newton occupies nearly
+middle ground in these various relations, being less rapid,
+but more rational than that of Leibnitz; less philosophical,
+but more applicable than that of Lagrange.</p>
+
+<p>This is not the place to explain the advantages of the
+introduction of this kind of auxiliary quantities in the
+place of the primitive magnitudes. The third chapter
+is devoted to this subject. At present I limit myself to
+consider this conception in the most general manner, in
+order to deduce therefrom the fundamental division of
+the <i>calculus of functions</i> into two systems essentially
+distinct, whose dependence, for the complete solution of
+any one mathematical question, is invariably determinate.</p>
+
+<p>In this connexion, and in the logical order of ideas,
+the transcendental analysis presents itself as being necessarily
+the first, since its general object is to facilitate
+the establishment of equations, an operation which must
+evidently precede the <i>resolution</i> of those equations, which
+is the object of the ordinary analysis. But though it is
+exceedingly important to conceive in this way the true
+relations of these two systems of analysis, it is none the
+less proper, in conformity with the regular usage, to
+study the transcendental analysis after ordinary analysis;
+for though the former is, at bottom, by itself logically
+independent of the latter, or, at least, may be essentially
+disengaged from it, yet it is clear that, since
+its employment in the solution of questions has always
+more or less need of being completed by the use of the
+ordinary analysis, we would be constrained to leave the
+questions in suspense if this latter had not been previously
+studied.</p><p><span class="pagenum"><a name="Page_67" id="Page_67">[Pg 67]</a></span></p>
+
+
+<p><i>Corresponding Divisions of the Calculus of Functions.</i>
+It follows from the preceding considerations that
+the <i>Calculus of Functions</i>, or <i>Algebra</i> (taking this word
+in its most extended meaning), is composed of two distinct
+fundamental branches, one of which has for its immediate
+object the <i>resolution</i> of equations, when they
+are directly established between the magnitudes themselves
+which are under consideration; and the other,
+starting from equations (generally much easier to form)
+between quantities indirectly connected with those of
+the problem, has for its peculiar and constant destination
+the deduction, by invariable analytical methods, of
+the corresponding equations between the direct magnitudes
+which we are considering; which brings the question
+within the domain of the preceding calculus.</p>
+
+<p>The former calculus bears most frequently the name
+of <i>Ordinary Analysis</i>, or of <i>Algebra</i>, properly so called.
+The second constitutes what is called the <i>Transcendental
+Analysis</i>, which has been designated by the different
+denominations of <i>Infinitesimal Calculus</i>, <i>Calculus of
+Fluxions and of Fluents</i>, <i>Calculus of Vanishing Quantities</i>,
+the <i>Differential and Integral Calculus</i>, &amp;c., according
+to the point of view in which it has been conceived.</p>
+
+<p>In order to remove every foreign consideration, I will
+propose to name it <span class="smcap">Calculus of Indirect Functions</span>, giving
+to ordinary analysis the title of <span class="smcap">Calculus of Direct
+Functions</span>. These expressions, which I form essentially
+by generalizing and epitomizing the ideas of Lagrange,
+are simply intended to indicate with precision the true
+general character belonging to each of these two forms
+of analysis.</p><p><span class="pagenum"><a name="Page_68" id="Page_68">[Pg 68]</a></span></p>
+
+<p>Having now established the fundamental division of
+mathematical analysis, I have next to consider separately
+each of its two parts, commencing with the <i>Calculus
+of Direct Functions</i>, and reserving more extended developments
+for the different branches of the <i>Calculus of
+Indirect Functions</i>.</p><hr class="chap" /><p><span class="pagenum"><a name="Page_69" id="Page_69">[Pg 69]</a></span></p>
+
+
+
+
+<h2><a name="CHAPTER_II" id="CHAPTER_II">CHAPTER II.</a></h2>
+
+<h3>ORDINARY ANALYSIS, OR ALGEBRA.</h3>
+
+
+<p>The <i>Calculus of direct Functions</i>, or <i>Algebra</i>, is (as
+was shown at the end of the preceding chapter) entirely
+sufficient for the solution of mathematical questions, when
+they are so simple that we can form directly the equations
+between the magnitudes themselves which we are
+considering, without its being necessary to introduce in
+their place, or conjointly with them, any system of auxiliary
+quantities <i>derived</i> from the first. It is true that
+in the greatest number of important cases its use requires
+to be preceded and prepared by that of the <i>Calculus
+of indirect Functions</i>, which is intended to facilitate
+the establishment of equations. But, although algebra
+has then only a secondary office to perform, it has
+none the less a necessary part in the complete solution
+of the question, so that the <i>Calculus of direct Functions</i>
+must continue to be, by its nature, the fundamental base
+of all mathematical analysis. We must therefore, before
+going any further, consider in a general manner the logical
+composition of this calculus, and the degree of development
+to which it has at the present day arrived.</p>
+
+
+<p><i>Its Object.</i> The final object of this calculus being the
+<i>resolution</i> (properly so called) of <i>equations</i>, that is, the
+discovery of the manner in which the unknown quantities
+are formed from the known quantities, in accordance
+with the <i>equations</i> which exist between them, it
+naturally presents as many different departments as we<span class="pagenum"><a name="Page_70" id="Page_70">[Pg 70]</a></span>
+can conceive truly distinct classes of equations. Its appropriate
+extent is consequently rigorously indefinite, the
+number of analytical functions susceptible of entering
+into equations being in itself quite unlimited, although
+they are composed of only a very small number of primitive
+elements.</p>
+
+
+<p><i>Classification of Equations.</i> The rational classification
+of equations must evidently be determined by the
+nature of the analytical elements of which their numbers
+are composed; every other classification would be essentially
+arbitrary. Accordingly, analysts begin by dividing
+equations with one or more variables into two principal
+classes, according as they contain functions of only
+the first three couples (see the table in chapter i., page
+51), or as they include also exponential or circular functions.
+The names of <i>Algebraic</i> functions and <i>Transcendental</i>
+functions, commonly given to these two principal
+groups of analytical elements, are undoubtedly very inappropriate.
+But the universally established division between
+the corresponding equations is none the less very
+real in this sense, that the resolution of equations containing
+the functions called <i>transcendental</i> necessarily
+presents more difficulties than those of the equations
+called <i>algebraic</i>. Hence the study of the former is as
+yet exceedingly imperfect, so that frequently the resolution
+of the most simple of them is still unknown to us,<a name="FNanchor_7_7" id="FNanchor_7_7"></a><a href="#Footnote_7_7" class="fnanchor">[7]</a>
+and our analytical methods have almost exclusive reference
+to the elaboration of the latter.</p>
+<p><span class="pagenum"><a name="Page_71" id="Page_71">[Pg 71]</a></span></p>
+
+
+
+<h3><a name="ALGEBRAIC_EQUATIONS" id="ALGEBRAIC_EQUATIONS">ALGEBRAIC EQUATIONS.</a></h3>
+
+
+<p>Considering now only these <i>Algebraic</i> equations, we
+must observe, in the first place, that although they may
+often contain <i>irrational</i> functions of the unknown quantities
+as well as <i>rational</i> functions, we can always, by
+more or less easy transformations, make the first case
+come under the second, so that it is with this last that
+analysts have had to occupy themselves exclusively in
+order to resolve all sorts of <i>algebraic</i> equations.</p>
+
+
+<p><i>Their Classification.</i> In the infancy of algebra, these
+equations were classed according to the number of their
+terms. But this classification was evidently faulty, since
+it separated cases which were really similar, and brought
+together others which had nothing in common besides this
+unimportant characteristic.<a name="FNanchor_8_8" id="FNanchor_8_8"></a><a href="#Footnote_8_8" class="fnanchor">[8]</a> It has been retained only
+for equations with two terms, which are, in fact, capable
+of being resolved in a manner peculiar to themselves.</p>
+
+<p>The classification of equations by what is called their
+<i>degrees</i>, is, on the other hand, eminently natural, for this
+distinction rigorously determines the greater or less difficulty
+of their <i>resolution</i>. This gradation is apparent
+in the cases of all the equations which can be resolved;
+but it may be indicated in a general manner independently
+of the fact of the resolution. We need only consider
+that the most general equation of each degree necessarily
+comprehends all those of the different inferior degrees,
+as must also the formula which determines the unknown
+quantity. Consequently, however slight we may
+suppose the difficulty peculiar to the <i>degree</i> which we
+<span class="pagenum"><a name="Page_72" id="Page_72">[Pg 72]</a></span>are considering, since it is inevitably complicated in the
+execution with those presented by all the preceding degrees,
+the resolution really offers more and more obstacles,
+in proportion as the degree of the equation is elevated.</p>
+
+
+
+
+<h3><a name="ALGEBRAIC_RESOLUTION_OF_EQUATIONS" id="ALGEBRAIC_RESOLUTION_OF_EQUATIONS">ALGEBRAIC RESOLUTION OF EQUATIONS.</a></h3>
+
+
+<p><i>Its Limits.</i> The resolution of algebraic equations is
+as yet known to us only in the four first degrees, such
+is the increase of difficulty noticed above. In this respect,
+algebra has made no considerable progress since
+the labours of Descartes and the Italian analysts of the
+sixteenth century, although in the last two centuries
+there has been perhaps scarcely a single geometer who
+has not busied himself in trying to advance the resolution
+of equations. The general equation of the fifth degree
+itself has thus far resisted all attacks.</p>
+
+<p>The constantly increasing complication which the
+formulas for resolving equations must necessarily present,
+in proportion as the degree increases (the difficulty
+of using the formula of the fourth degree rendering it almost
+inapplicable), has determined analysts to renounce,
+by a tacit agreement, the pursuit of such researches, although
+they are far from regarding it as impossible to
+obtain the resolution of equations of the fifth degree, and
+of several other higher ones.</p>
+
+
+<p><i>General Solution.</i> The only question of this kind
+which would be really of great importance, at least in
+its logical relations, would be the general resolution of
+algebraic equations of any degree whatsoever. Now,
+the more we meditate on this subject, the more we are
+led to think, with Lagrange, that it really surpasses the
+scope of our intelligence. We must besides observe that<span class="pagenum"><a name="Page_73" id="Page_73">[Pg 73]</a></span>
+the formula which would express the <i>root</i> of an equation
+of the <i>m<sup>th</sup></i> degree would necessarily include radicals of
+the <i>m<sup>th</sup></i> order (or functions of an equivalent multiplicity),
+because of the <i>m</i> determinations which it must admit.
+Since we have seen, besides, that this formula
+must also embrace, as a particular case, that formula
+which corresponds to every lower degree, it follows that
+it would inevitably also contain radicals of the next
+lower degree, the next lower to that, &amp;c., so that, even
+if it were possible to discover it, it would almost always
+present too great a complication to be capable of being
+usefully employed, unless we could succeed in simplifying
+it, at the same time retaining all its generality, by
+the introduction of a new class of analytical elements of
+which we yet have no idea. We have, then, reason to
+believe that, without having already here arrived at the
+limits imposed by the feeble extent of our intelligence,
+we should not be long in reaching them if we actively
+and earnestly prolonged this series of investigations.</p>
+
+<p>It is, besides, important to observe that, even supposing
+we had obtained the resolution of <i>algebraic</i> equations
+of any degree whatever, we would still have treated
+only a very small part of <i>algebra</i>, properly so called,
+that is, of the calculus of direct functions, including the
+resolution of all the equations which can be formed by
+the known analytical functions.</p>
+
+<p>Finally, we must remember that, by an undeniable
+law of human nature, our means for conceiving new
+questions being much more powerful than our resources
+for resolving them, or, in other words, the human mind
+being much more ready to inquire than to reason, we
+shall necessarily always remain <i>below</i> the difficulty, no<span class="pagenum"><a name="Page_74" id="Page_74">[Pg 74]</a></span>
+matter to what degree of development our intellectual
+labour may arrive. Thus, even though we should some
+day discover the complete resolution of all the analytical
+equations at present known, chimerical as the supposition
+is, there can be no doubt that, before attaining this
+end, and probably even as a subsidiary means, we would
+have already overcome the difficulty (a much smaller one,
+though still very great) of conceiving new analytical elements,
+the introduction of which would give rise to classes
+of equations of which, at present, we are completely
+ignorant; so that a similar imperfection in algebraic science
+would be continually reproduced, in spite of the real
+and very important increase of the absolute mass of our
+knowledge.</p>
+
+
+<p><i>What we know in Algebra.</i> In the present condition
+of algebra, the complete resolution of the equations
+of the first four degrees, of any binomial equations, of
+certain particular equations of the higher degrees, and of
+a very small number of exponential, logarithmic, or circular
+equations, constitute the fundamental methods
+which are presented by the calculus of direct functions
+for the solution of mathematical problems. But, limited
+as these elements are, geometers have nevertheless succeeded
+in treating, in a truly admirable manner, a very
+great number of important questions, as we shall find in
+the course of the volume. The general improvements
+introduced within a century into the total system of
+mathematical analysis, have had for their principal object
+to make immeasurably useful this little knowledge
+which we have, instead of tending to increase it. This
+result has been so fully obtained, that most frequently
+this calculus has no real share in the complete solution<span class="pagenum"><a name="Page_75" id="Page_75">[Pg 75]</a></span>
+of the question, except by its most simple parts; those
+which have reference to equations of the two first degrees,
+with one or more variables.</p>
+
+
+
+
+<h3><a name="NUMERICAL_RESOLUTION_OF_EQUATIONS" id="NUMERICAL_RESOLUTION_OF_EQUATIONS">NUMERICAL RESOLUTION OF EQUATIONS.</a></h3>
+
+
+<p>The extreme imperfection of algebra, with respect to
+the resolution of equations, has led analysts to occupy
+themselves with a new class of questions, whose true
+character should be here noted. They have busied themselves
+in filling up the immense gap in the resolution of
+algebraic equations of the higher degrees, by what they
+have named the <i>numerical resolution</i> of equations. Not
+being able to obtain, in general, the <i>formula</i> which expresses
+what explicit function of the given quantities the
+unknown one is, they have sought (in the absence of this
+kind of resolution, the only one really <i>algebraic</i>) to determine,
+independently of that formula, at least the <i>value</i>
+of each unknown quantity, for various designated systems
+of particular values attributed to the given quantities.
+By the successive labours of analysts, this incomplete
+and illegitimate operation, which presents an
+intimate mixture of truly algebraic questions with others
+which are purely arithmetical, has been rendered possible
+in all cases for equations of any degree and even of
+any form. The methods for this which we now possess
+are sufficiently general, although the calculations to which
+they lead are often so complicated as to render it almost
+impossible to execute them. We have nothing else to
+do, then, in this part of algebra, but to simplify the methods
+sufficiently to render them regularly applicable, which
+we may hope hereafter to effect. In this condition of
+the calculus of direct functions, we endeavour, in its application,<span class="pagenum"><a name="Page_76" id="Page_76">[Pg 76]</a></span>
+so to dispose the proposed questions as finally to
+require only this numerical resolution of the equations.</p>
+
+
+<p><i>Its limited Usefulness.</i> Valuable as is such a resource
+in the absence of the veritable solution, it is essential
+not to misconceive the true character of these
+methods, which analysts rightly regard as a very imperfect
+algebra. In fact, we are far from being always able
+to reduce our mathematical questions to depend finally
+upon only the <i>numerical</i> resolution of equations; that
+can be done only for questions quite isolated or truly
+final, that is, for the smallest number. Most questions,
+in fact, are only preparatory, and intended to serve as an
+indispensable preparation for the solution of other questions.
+Now, for such an object, it is evident that it is
+not the actual <i>value</i> of the unknown quantity which it
+is important to discover, but the <i>formula</i>, which shows
+how it is derived from the other quantities under consideration.
+It is this which happens, for example, in a
+very extensive class of cases, whenever a certain question
+includes at the same time several unknown quantities.
+We have then, first of all, to separate them. By
+suitably employing the simple and general method so
+happily invented by analysts, and which consists in referring
+all the other unknown quantities to one of them,
+the difficulty would always disappear if we knew how to
+obtain the algebraic resolution of the equations under
+consideration, while the <i>numerical</i> solution would then
+be perfectly useless. It is only for want of knowing the
+<i>algebraic</i> resolution of equations with a single unknown
+quantity, that we are obliged to treat <i>Elimination</i> as a
+distinct question, which forms one of the greatest special
+difficulties of common algebra. Laborious as are the<span class="pagenum"><a name="Page_77" id="Page_77">[Pg 77]</a></span>
+methods by the aid of which we overcome this difficulty,
+they are not even applicable, in an entirely general manner,
+to the elimination of one unknown quantity between
+two equations of any form whatever.</p>
+
+<p>In the most simple questions, and when we have really
+to resolve only a single equation with a single unknown
+quantity, this <i>numerical</i> resolution is none the less a
+very imperfect method, even when it is strictly sufficient.
+It presents, in fact, this serious inconvenience of obliging
+us to repeat the whole series of operations for the slightest
+change which may take place in a single one of the
+quantities considered, although their relations to one another
+remain unchanged; the calculations made for one
+case not enabling us to dispense with any of those which
+relate to a case very slightly different. This happens because
+of our inability to abstract and treat separately
+that purely algebraic part of the question which is common
+to all the cases which result from the mere variation
+of the given numbers.</p>
+
+<p>According to the preceding considerations, the calculus
+of direct functions, viewed in its present state, divides
+into two very distinct branches, according as its
+subject is the <i>algebraic</i> resolution of equations or their
+<i>numerical</i> resolution. The first department, the only
+one truly satisfactory, is unhappily very limited, and will
+probably always remain so; the second, too often insufficient,
+has, at least, the advantage of a much greater
+generality. The necessity of clearly distinguishing these
+two parts is evident, because of the essentially different
+object proposed in each, and consequently the peculiar
+point of view under which quantities are therein considered.</p><p><span class="pagenum"><a name="Page_78" id="Page_78">[Pg 78]</a></span></p>
+
+
+<p><i>Different Divisions of the two Methods of Resolution.</i>
+If, moreover, we consider these parts with reference
+to the different methods of which each is composed,
+we find in their logical distribution an entirely different
+arrangement. In fact, the first part must be divided
+according to the nature of the equations which we are
+able to resolve, and independently of every consideration
+relative to the <i>values</i> of the unknown quantities. In
+the second part, on the contrary, it is not according to
+the <i>degrees</i> of the equations that the methods are naturally
+distinguished, since they are applicable to equations
+of any degree whatever; it is according to the numerical
+character of the <i>values</i> of the unknown quantities;
+for, in calculating these numbers directly, without deducing
+them from general formulas, different means would
+evidently be employed when the numbers are not susceptible
+of having their values determined otherwise than
+by a series of approximations, always incomplete, or when
+they can be obtained with entire exactness. This distinction
+of <i>incommensurable</i> and of <i>commensurable</i> roots,
+which require quite different principles for their determination,
+important as it is in the numerical resolution of
+equations, is entirely insignificant in the algebraic resolution,
+in which the <i>rational</i> or <i>irrational</i> nature of the
+numbers which are obtained is a mere accident of the
+calculation, which cannot exercise any influence over the
+methods employed; it is, in a word, a simple arithmetical
+consideration. We may say as much, though in a less
+degree, of the division of the commensurable roots themselves
+into <i>entire</i> and <i>fractional</i>. In fine, the case is
+the same, in a still greater degree, with the most general
+classification of roots, as <i>real</i> and <i>imaginary</i>. All<span class="pagenum"><a name="Page_79" id="Page_79">[Pg 79]</a></span>
+these different considerations, which are preponderant as
+to the numerical resolution of equations, and which are
+of no importance in their algebraic resolution, render more
+and more sensible the essentially distinct nature of these
+two principal parts of algebra.</p>
+
+
+
+
+<h3><a name="THE_THEORY_OF_EQUATIONS" id="THE_THEORY_OF_EQUATIONS">THE THEORY OF EQUATIONS.</a></h3>
+
+
+<p>These two departments, which constitute the immediate
+object of the calculus of direct functions, are subordinate
+to a third one, purely speculative, from which both
+of them borrow their most powerful resources, and which
+has been very exactly designated by the general name
+of <i>Theory of Equations</i>, although it as yet relates only
+to <i>Algebraic</i> equations. The numerical resolution of
+equations, because of its generality, has special need of
+this rational foundation.</p>
+
+<p>This last and important branch of algebra is naturally
+divided into two orders of questions, viz., those which refer
+to the <i>composition</i> of equations, and those which concern
+their <i>transformation</i>; these latter having for their
+object to modify the roots of an equation without knowing
+them, in accordance with any given law, providing
+that this law is uniform in relation to all the parts.<a name="FNanchor_9_9" id="FNanchor_9_9"></a><a href="#Footnote_9_9" class="fnanchor">[9]</a></p>
+<p><span class="pagenum"><a name="Page_80" id="Page_80">[Pg 80]</a></span></p>
+
+
+
+<h3><a name="THE_METHOD_OF_INDETERMINATE_COEFFICIENTS" id="THE_METHOD_OF_INDETERMINATE_COEFFICIENTS">THE METHOD OF INDETERMINATE COEFFICIENTS.</a></h3>
+
+
+<p>To complete this rapid general enumeration of the different
+essential parts of the calculus of direct functions,
+I must, lastly, mention expressly one of the most fruitful
+and important theories of algebra proper, that relating
+to the transformation of functions into series by the aid
+of what is called the <i>Method of indeterminate Coefficients</i>.
+This method, so eminently analytical, and which
+must be regarded as one of the most remarkable discoveries
+of Descartes, has undoubtedly lost some of its importance
+since the invention and the development of the
+infinitesimal calculus, the place of which it might so happily
+take in some particular respects. But the increasing
+extension of the transcendental analysis, although it
+has rendered this method much less necessary, has, on
+the other hand, multiplied its applications and enlarged
+its resources; so that by the useful combination between
+the two theories, which has finally been effected, the use
+of the method of indeterminate coefficients has become
+at present much more extensive than it was even before
+the formation of the calculus of indirect functions.</p>
+<p><span class="pagenum"><a name="Page_81" id="Page_81">[Pg 81]</a></span></p>
+<hr class="tb" />
+
+<p>Having thus sketched the general outlines of algebra
+proper, I have now to offer some considerations on several
+leading points in the calculus of direct functions, our
+ideas of which may be advantageously made more clear
+by a philosophical examination.</p>
+
+
+
+
+<h3><a name="IMAGINARY_QUANTITIES" id="IMAGINARY_QUANTITIES">IMAGINARY QUANTITIES.</a></h3>
+
+
+<p>The difficulties connected with several peculiar symbols
+to which algebraic calculations sometimes lead, and
+especially to the expressions called <i>imaginary</i>, have been,
+I think, much exaggerated through purely metaphysical
+considerations, which have been forced upon them, in the
+place of regarding these abnormal results in their true
+point of view as simple analytical facts. Viewing them
+thus, we readily see that, since the spirit of mathematical
+analysis consists in considering magnitudes in reference
+to their relations only, and without any regard to
+their determinate value, analysts are obliged to admit indifferently
+every kind of expression which can be engendered
+by algebraic combinations. The interdiction of
+even one expression because of its apparent singularity
+would destroy the generality of their conceptions. The
+common embarrassment on this subject seems to me to
+proceed essentially from an unconscious confusion between
+the idea of <i>function</i> and the idea of <i>value</i>, or, what
+comes to the same thing, between the <i>algebraic</i> and the
+<i>arithmetical</i> point of view. A thorough examination
+would show mathematical analysis to be much more clear
+in its nature than even mathematicians commonly suppose.</p>
+
+
+
+
+<h3><a name="NEGATIVE_QUANTITIES" id="NEGATIVE_QUANTITIES">NEGATIVE QUANTITIES.</a></h3>
+
+
+<p>As to negative quantities, which have given rise to so
+many misplaced discussions, as irrational as useless, we
+must distinguish between their <i>abstract</i> signification and
+their <i>concrete</i> interpretation, which have been almost always
+confounded up to the present day. Under the first<span class="pagenum"><a name="Page_82" id="Page_82">[Pg 82]</a></span>
+point of view, the theory of negative quantities can be
+established in a complete manner by a single algebraical
+consideration. The necessity of admitting such expressions
+is the same as for imaginary quantities, as above
+indicated; and their employment as an analytical artifice,
+to render the formulas more comprehensive, is a
+mechanism of calculation which cannot really give rise
+to any serious difficulty. We may therefore regard the
+abstract theory of negative quantities as leaving nothing
+essential to desire; it presents no obstacles but those inappropriately
+introduced by sophistical considerations.</p>
+
+<p>It is far from being so, however, with their concrete
+theory. This consists essentially in that admirable property
+of the signs + and-, of representing analytically
+the oppositions of directions of which certain magnitudes
+are susceptible. This <i>general theorem</i> on the relation
+of the concrete to the abstract in mathematics is one of
+the most beautiful discoveries which we owe to the genius
+of Descartes, who obtained it as a simple result of properly
+directed philosophical observation. A great number
+of geometers have since striven to establish directly
+its general demonstration, but thus far their efforts have
+been illusory. Their vain metaphysical considerations
+and heterogeneous minglings of the abstract and the
+concrete have so confused the subject, that it becomes
+necessary to here distinctly enunciate the general fact.
+It consists in this: if, in any equation whatever, expressing
+the relation of certain quantities which are susceptible
+of opposition of directions, one or more of those quantities
+come to be reckoned in a direction contrary to that
+which belonged to them when the equation was first established,
+it will not be necessary to form directly a new<span class="pagenum"><a name="Page_83" id="Page_83">[Pg 83]</a></span>
+equation for this second state of the phenomena; it will
+suffice to change, in the first equation, the sign of each
+of the quantities which shall have changed its direction;
+and the equation, thus modified, will always rigorously
+coincide with that which we would have arrived at in
+recommencing to investigate, for this new case, the analytical
+law of the phenomenon. The general theorem
+consists in this constant and necessary coincidence. Now,
+as yet, no one has succeeded in directly proving this; we
+have assured ourselves of it only by a great number of
+geometrical and mechanical verifications, which are, it
+is true, sufficiently multiplied, and especially sufficiently
+varied, to prevent any clear mind from having the least
+doubt of the exactitude and the generality of this essential
+property, but which, in a philosophical point of view,
+do not at all dispense with the research for so important
+an explanation. The extreme extent of the theorem must
+make us comprehend both the fundamental difficulties of
+this research and the high utility for the perfecting of
+mathematical science which would belong to the general
+conception of this great truth. This imperfection of theory,
+however, has not prevented geometers from making
+the most extensive and the most important use of this
+property in all parts of concrete mathematics.</p>
+
+<p>It follows from the above general enunciation of the
+fact, independently of any demonstration, that the property
+of which we speak must never be applied to magnitudes
+whose directions are continually varying, without
+giving rise to a simple opposition of direction; in
+that case, the sign with which every result of calculation
+is necessarily affected is not susceptible of any concrete
+interpretation, and the attempts sometimes made to establish<span class="pagenum"><a name="Page_84" id="Page_84">[Pg 84]</a></span>
+one are erroneous. This circumstance occurs,
+among other occasions, in the case of a radius vector in
+geometry, and diverging forces in mechanics.</p>
+
+
+
+
+<h3><a name="PRINCIPLE_OF_HOMOGENEITY" id="PRINCIPLE_OF_HOMOGENEITY">PRINCIPLE OF HOMOGENEITY.</a></h3>
+
+
+<p>A second general theorem on the relation of the concrete
+to the abstract is that which is ordinarily designated
+under the name of <i>Principle of Homogeneity</i>. It
+is undoubtedly much less important in its applications
+than the preceding, but it particularly merits our attention
+as having, by its nature, a still greater extent,
+since it is applicable to all phenomena without distinction,
+and because of the real utility which it often possesses
+for the verification of their analytical laws. I
+can, moreover, exhibit a direct and general demonstration
+of it which seems to me very simple. It is founded
+on this single observation, which is self-evident, that the
+exactitude of every relation between any concrete magnitudes
+whatsoever is independent of the value of the
+<i>units</i> to which they are referred for the purpose of expressing
+them in numbers. For example, the relation
+which exists between the three sides of a right-angled
+triangle is the same, whether they are measured by yards,
+or by miles, or by inches.</p>
+
+<p>It follows from this general consideration, that every
+equation which expresses the analytical law of any phenomenon
+must possess this property of being in no way
+altered, when all the quantities which are found in it
+are made to undergo simultaneously the change corresponding
+to that which their respective units would
+experience. Now this change evidently consists in all
+the quantities of each sort becoming at once <i>m</i> times<span class="pagenum"><a name="Page_85" id="Page_85">[Pg 85]</a></span>
+smaller, if the unit which corresponds to them becomes
+<i>m</i> times greater, or reciprocally. Thus every equation
+which represents any concrete relation whatever must
+possess this characteristic of remaining the same, when
+we make <i>m</i> times greater all the quantities which it contains,
+and which express the magnitudes between which
+the relation exists; excepting always the numbers which
+designate simply the mutual <i>ratios</i> of these different
+magnitudes, and which therefore remain invariable during
+the change of the units. It is this property which
+constitutes the law of Homogeneity in its most extended
+signification, that is, of whatever analytical functions the
+equations may be composed.</p>
+
+<p>But most frequently we consider only the cases in
+which the functions are such as are called <i>algebraic</i>,
+and to which the idea of <i>degree</i> is applicable. In this
+case we can give more precision to the general proposition
+by determining the analytical character which must
+be necessarily presented by the equation, in order that
+this property may be verified. It is easy to see, then,
+that, by the modification just explained, all the <i>terms</i> of
+the first degree, whatever may be their form, rational or
+irrational, entire or fractional, will become <i>m</i> times greater;
+all those of the second degree, <i>m<sup>2</sup></i> times; those of
+the third, <i>m<sup>3</sup></i> times, &amp;c. Thus the terms of the same degree,
+however different may be their composition, varying
+in the same manner, and the terms of different degrees
+varying in an unequal proportion, whatever similarity
+there may be in their composition, it will be necessary,
+to prevent the equation from being disturbed,
+that all the terms which it contains should be of the same
+degree. It is in this that properly consists the ordinary<span class="pagenum"><a name="Page_86" id="Page_86">[Pg 86]</a></span>
+theorem of <i>Homogeneity</i>, and it is from this circumstance
+that the general law has derived its name, which,
+however, ceases to be exactly proper for all other functions.</p>
+
+<p>In order to treat this subject in its whole extent, it is
+important to observe an essential condition, to which attention
+must be paid in applying this property when the
+phenomenon expressed by the equation presents magnitudes
+of different natures. Thus it may happen that
+the respective units are completely independent of each
+other, and then the theorem of Homogeneity will hold
+good, either with reference to all the corresponding classes
+of quantities, or with regard to only a single one or more
+of them. But it will happen on other occasions that the
+different units will have fixed relations to one another,
+determined by the nature of the question; then it will
+be necessary to pay attention to this subordination of
+the units in verifying the homogeneity, which will not
+exist any longer in a purely algebraic sense, and the
+precise form of which will vary according to the nature
+of the phenomena. Thus, for example, to fix our ideas,
+when, in the analytical expression of geometrical phenomena,
+we are considering at once lines, areas, and volumes,
+it will be necessary to observe that the three corresponding
+units are necessarily so connected with each
+other that, according to the subordination generally established
+in that respect, when the first becomes <i>m</i> times
+greater, the second becomes <i>m<sup>2</sup></i> times, and the third <i>m<sup>3</sup></i>
+times. It is with such a modification that homogeneity
+will exist in the equations, in which, if they are <i>algebraic</i>,
+we will have to estimate the degree of each term
+by doubling the exponents of the factors which correspond<span class="pagenum"><a name="Page_87" id="Page_87">[Pg 87]</a></span>
+to areas, and tripling those of the factors relating
+to volumes.</p>
+<p><span class="pagenum"><a name="Page_88" id="Page_88">[Pg 88]</a></span></p>
+<hr class="tb" />
+
+<p>Such are the principal general considerations relating
+to the <i>Calculus of Direct Functions</i>. We have now to
+pass to the philosophical examination of the <i>Calculus of
+Indirect Functions</i>, the much superior importance and
+extent of which claim a fuller development.</p>
+
+
+
+
+<h2><a name="CHAPTER_III" id="CHAPTER_III">CHAPTER III.</a></h2>
+
+<h3>TRANSCENDENTAL ANALYSIS:<br />
+DIFFERENT MODES OF VIEWING IT.</h3>
+
+
+<p>We determined, in the second chapter, the philosophical
+character of the transcendental analysis, in whatever
+manner it may be conceived, considering only the general
+nature of its actual destination as a part of mathematical
+science. This analysis has been presented by
+geometers under several points of view, really distinct,
+although necessarily equivalent, and leading always to
+identical results. They may be reduced to three principal
+ones; those of <span class="smcap">Leibnitz</span>, of <span class="smcap">Newton</span>, and of <span class="smcap">Lagrange</span>,
+of which all the others are only secondary modifications.
+In the present state of science, each of these
+three general conceptions offers essential advantages which
+pertain to it exclusively, without our having yet succeeded
+in constructing a single method uniting all these
+different characteristic qualities. This combination will
+probably be hereafter effected by some method founded
+upon the conception of Lagrange when that important
+philosophical labour shall have been accomplished,
+the study of the other conceptions will have only a historic
+interest; but, until then, the science must be considered
+as in only a provisional state, which requires the
+simultaneous consideration of all the various modes of
+viewing this calculus. Illogical as may appear this multiplicity
+of conceptions of one identical subject, still,
+without them all, we could form but a very insufficient<span class="pagenum"><a name="Page_89" id="Page_89">[Pg 89]</a></span>
+idea of this analysis, whether in itself, or more especially
+in relation to its applications. This want of system
+in the most important part of mathematical analysis will
+not appear strange if we consider, on the one hand, its
+great extent and its superior difficulty, and, on the other,
+its recent formation.</p>
+
+
+
+
+<h3><a name="ITS_EARLY_HISTORY" id="ITS_EARLY_HISTORY">ITS EARLY HISTORY.</a></h3>
+
+
+<p>If we had to trace here the systematic history of the
+successive formation of the transcendental analysis, it
+would be necessary previously to distinguish carefully
+from the <i>calculus of indirect functions</i>, properly so called,
+the original idea of the <i>infinitesimal method</i>, which
+can be conceived by itself, independently of any <i>calculus</i>.
+We should see that the first germ of this idea is found
+in the procedure constantly employed by the Greek geometers,
+under the name of the <i>Method of Exhaustions</i>,
+as a means of passing from the properties of straight lines
+to those of curves, and consisting essentially in substituting
+for the curve the auxiliary consideration of an inscribed
+or circumscribed polygon, by means of which they
+rose to the curve itself, taking in a suitable manner the
+limits of the primitive ratios. Incontestable as is this
+filiation of ideas, it would be giving it a greatly exaggerated
+importance to see in this method of exhaustions
+the real equivalent of our modern methods, as some geometers
+have done; for the ancients had no logical and
+general means for the determination of these limits, and
+this was commonly the greatest difficulty of the question;
+so that their solutions were not subjected to abstract
+and invariable rules, the uniform application of
+which would lead with certainty to the knowledge sought;<span class="pagenum"><a name="Page_90" id="Page_90">[Pg 90]</a></span>
+which is, on the contrary, the principal characteristic of
+our transcendental analysis. In a word, there still remained
+the task of generalizing the conceptions used by
+the ancients, and, more especially, by considering it in a
+manner purely abstract, of reducing it to a complete system
+of calculation, which to them was impossible.</p>
+
+<p>The first idea which was produced in this new direction
+goes back to the great geometer Fermat, whom Lagrange
+has justly presented as having blocked out the
+direct formation of the transcendental analysis by his
+method for the determination of <i>maxima</i> and <i>minima</i>,
+and for the finding of <i>tangents</i>, which consisted essentially
+in introducing the auxiliary consideration of the
+correlative increments of the proposed variables, increments
+afterward suppressed as equal to zero when the
+equations had undergone certain suitable transformations.
+But, although Fermat was the first to conceive
+this analysis in a truly abstract manner, it was yet far
+from being regularly formed into a general and distinct
+calculus having its own notation, and especially freed
+from the superfluous consideration of terms which, in the
+analysis of Fermat, were finally not taken into the account,
+after having nevertheless greatly complicated all
+the operations by their presence. This is what Leibnitz
+so happily executed, half a century later, after some intermediate
+modifications of the ideas of Fermat introduced
+by Wallis, and still more by Barrow; and he has
+thus been the true creator of the transcendental analysis,
+such as we now employ it. This admirable discovery
+was so ripe (like all the great conceptions of the
+human intellect at the moment of their manifestation),
+that Newton, on his side, had arrived, at the same time,<span class="pagenum"><a name="Page_91" id="Page_91">[Pg 91]</a></span>
+or a little earlier, at a method exactly equivalent, by
+considering this analysis under a very different point of
+view, which, although more logical in itself, is really
+less adapted to give to the common fundamental method
+all the extent and the facility which have been imparted
+to it by the ideas of Leibnitz. Finally, Lagrange, putting
+aside the heterogeneous considerations which had
+guided Leibnitz and Newton, has succeeded in reducing
+the transcendental analysis, in its greatest perfection, to
+a purely algebraic system, which only wants more aptitude
+for its practical applications.</p>
+
+<p>After this summary glance at the general history of
+the transcendental analysis, we will proceed to the dogmatic
+exposition of the three principal conceptions, in order
+to appreciate exactly their characteristic properties,
+and to show the necessary identity of the methods which
+are thence derived. Let us begin with that of Leibnitz.</p>
+
+
+
+
+<h3><a name="METHOD_OF_LEIBNITZ" id="METHOD_OF_LEIBNITZ">METHOD OF LEIBNITZ.</a></h3>
+
+
+<p><i>Infinitely small Elements.</i> This consists in introducing
+into the calculus, in order to facilitate the establishment
+of equations, the infinitely small elements of which
+all the quantities, the relations between which are sought,
+are considered to be composed. These elements or <i>differentials</i>
+will have certain relations to one another,
+which are constantly and necessarily more simple and
+easy to discover than those of the primitive quantities, and
+by means of which we will be enabled (by a special calculus
+having for its peculiar object the elimination of these
+auxiliary infinitesimals) to go back to the desired equations,
+which it would have been most frequently impossible
+to obtain directly. This indirect analysis may have<span class="pagenum"><a name="Page_92" id="Page_92">[Pg 92]</a></span>
+different degrees of indirectness; for, when there is too
+much difficulty in forming immediately the equation between
+the differentials of the magnitudes under consideration,
+a second application of the same general artifice
+will have to be made, and these differentials be treated,
+in their turn, as new primitive quantities, and a relation
+be sought between their infinitely small elements (which,
+with reference to the final objects of the question, will be
+<i>second differentials</i>), and so on; the same transformation
+admitting of being repeated any number of times,
+on the condition of finally eliminating the constantly increasing
+number of infinitesimal quantities introduced as
+auxiliaries.</p>
+
+<p>A person not yet familiar with these considerations
+does not perceive at once how the employment of these
+auxiliary quantities can facilitate the discovery of the
+analytical laws of phenomena; for the infinitely small
+increments of the proposed magnitudes being of the same
+species with them, it would seem that their relations
+should not be obtained with more ease, inasmuch as the
+greater or less value of a quantity cannot, in fact, exercise
+any influence on an inquiry which is necessarily independent,
+by its nature, of every idea of value. But
+it is easy, nevertheless, to explain very clearly, and in a
+quite general manner, how far the question must be simplified
+by such an artifice. For this purpose, it is necessary
+to begin by distinguishing <i>different orders</i> of infinitely
+small quantities, a very precise idea of which
+may be obtained by considering them as being either the
+successive powers of the same primitive infinitely small
+quantity, or as being quantities which may be regarded
+as having finite ratios with these powers; so that, to<span class="pagenum"><a name="Page_93" id="Page_93">[Pg 93]</a></span>
+take an example, the second, third, &amp;c., differentials of
+any one variable are classed as infinitely small quantities
+of the second order, the third, &amp;c., because it is
+easy to discover in them finite multiples of the second,
+third, &amp;c., powers of a certain first differential. These
+preliminary ideas being established, the spirit of the infinitesimal
+analysis consists in constantly neglecting the
+infinitely small quantities in comparison with finite quantities,
+and generally the infinitely small quantities of any
+order whatever in comparison with all those of an inferior
+order. It is at once apparent how much such a
+liberty must facilitate the formation of equations between
+the differentials of quantities, since, in the place of these
+differentials, we can substitute such other elements as we
+may choose, and as will be more simple to consider, only
+taking care to conform to this single condition, that the
+new elements differ from the preceding ones only by quantities
+infinitely small in comparison with them. It is
+thus that it will be possible, in geometry, to treat curved
+lines as composed of an infinity of rectilinear elements,
+curved surfaces as formed of plane elements, and, in mechanics,
+variable motions as an infinite series of uniform
+motions, succeeding one another at infinitely small intervals
+of time.</p>
+
+
+<p><span class="smcap">Examples.</span> Considering the importance of this admirable
+conception, I think that I ought here to complete
+the illustration of its fundamental character by the summary
+indication of some leading examples.</p>
+
+
+<p>1. <i>Tangents.</i> Let it be required to determine, for
+each point of a plane curve, the equation of which is
+given, the direction of its tangent; a question whose
+general solution was the primitive object of the inventors<span class="pagenum"><a name="Page_94" id="Page_94">[Pg 94]</a></span>
+of the transcendental analysis. We will consider the
+tangent as a secant joining two points infinitely near to
+each other; and then, designating by <i>dy</i> and <i>dx</i> the infinitely
+small differences of the co-ordinates of those two
+points, the elementary principles of geometry will immediately
+give the equation <i>t</i> = <i>dy</i>/<i>dx</i> for the trigonometrical
+tangent of the angle which is made with the axis of the
+abscissas by the desired tangent, this being the most simple
+way of fixing its position in a system of rectilinear
+co-ordinates. This equation, common to all curves, being
+established, the question is reduced to a simple analytical
+problem, which will consist in eliminating the infinitesimals
+<i>dx</i> and <i>dy</i>, which were introduced as auxiliaries, by
+determining in each particular case, by means of the equation
+of the proposed curve, the ratio of <i>dy</i> to <i>dx</i>, which will
+be constantly done by uniform and very simple methods.</p>
+
+
+<p>2. <i>Rectification of an Arc.</i> In the second place, suppose
+that we wish to know the length of the arc of any
+curve, considered as a function of the co-ordinates of its extremities.
+It would be impossible to establish directly the
+equation between this arc s and these co-ordinates, while
+it is easy to find the corresponding relation between the
+differentials of these different magnitudes. The most simple
+theorems of elementary geometry will in fact give at
+once, considering the infinitely small arc <i>ds</i> as a right
+line, the equations</p>
+
+<p>
+<i>ds<sup>2</sup></i> = <i>dy<sup>2</sup></i> + <i>dx<sup>2</sup></i>, or <i>ds<sup>2</sup></i> = <i>dx<sup>2</sup></i> + <i>dy<sup>2</sup></i> + <i>dz<sup>2</sup></i>,<br />
+</p>
+
+<p>according as the curve is of single or double curvature.
+In either case, the question is now entirely within the
+domain of analysis, which, by the elimination of the differentials
+(which is the peculiar object of the calculus of<span class="pagenum"><a name="Page_95" id="Page_95">[Pg 95]</a></span>
+indirect functions), will carry us back from this relation
+to that which exists between the finite quantities themselves
+under examination.</p>
+
+
+<p>3. <i>Quadrature of a Curve.</i> It would be the same
+with the quadrature of curvilinear areas. If the curve is
+a plane one, and referred to rectilinear co-ordinates, we
+will conceive the area A comprised between this curve,
+the axis of the abscissas, and two extreme co-ordinates,
+to increase by an infinitely small quantity <i>d</i>A, as the result
+of a corresponding increment of the abscissa. The
+relation between these two differentials can be immediately
+obtained with the greatest facility by substituting for
+the curvilinear element of the proposed area the rectangle
+formed by the extreme ordinate and the element of the
+abscissa, from which it evidently differs only by an infinitely
+small quantity of the second order. This will at
+once give, whatever may be the curve, the very simple
+differential equation</p>
+
+<p>
+<i>d</i>A = <i>ydx</i>,<br />
+</p>
+
+<p>from which, when the curve is defined, the calculus of
+indirect functions will show how to deduce the finite
+equation, which is the immediate object of the problem.</p>
+
+
+<p>4. <i>Velocity in Variable Motion.</i> In like manner, in
+Dynamics, when we desire to know the expression for
+the velocity acquired at each instant by a body impressed
+with a motion varying according to any law, we will
+consider the motion as being uniform during an infinitely
+small element of the time <i>t</i>, and we will thus immediately
+form the differential equation <i>de</i> = <i>vdt</i>, in which
+<i>v</i> designates the velocity acquired when the body has
+passed over the space <i>e</i>; and thence it will be easy to
+deduce, by simple and invariable analytical procedures,<span class="pagenum"><a name="Page_96" id="Page_96">[Pg 96]</a></span>
+the formula which would give the velocity in each particular
+motion, in accordance with the corresponding relation
+between the time and the space; or, reciprocally,
+what this relation would be if the mode of variation of
+the velocity was supposed to be known, whether with respect
+to the space or to the time.</p>
+
+
+<p>5. <i>Distribution of Heat.</i> Lastly, to indicate another
+kind of questions, it is by similar steps that we are able,
+in the study of thermological phenomena, according to
+the happy conception of M. Fourier, to form in a very
+simple manner the general differential equation which
+expresses the variable distribution of heat in any body
+whatever, subjected to any influences, by means of the
+single and easily-obtained relation, which represents the
+uniform distribution of heat in a right-angled parallelopipedon,
+considering (geometrically) every other body as
+decomposed into infinitely small elements of a similar
+form, and (thermologically) the flow of heat as constant
+during an infinitely small element of time. Henceforth,
+all the questions which can be presented by abstract thermology
+will be reduced, as in geometry and mechanics,
+to mere difficulties of analysis, which will always consist
+in the elimination of the differentials introduced as auxiliaries
+to facilitate the establishment of the equations.</p>
+
+<p>Examples of such different natures are more than sufficient
+to give a clear general idea of the immense scope
+of the fundamental conception of the transcendental analysis
+as formed by Leibnitz, constituting, as it undoubtedly
+does, the most lofty thought to which the human
+mind has as yet attained.</p>
+
+<p>It is evident that this conception was indispensable to
+complete the foundation of mathematical science, by enabling<span class="pagenum"><a name="Page_97" id="Page_97">[Pg 97]</a></span>
+us to establish, in a broad and fruitful manner,
+the relation of the concrete to the abstract. In this respect
+it must be regarded as the necessary complement
+of the great fundamental idea of Descartes on the general
+analytical representation of natural phenomena: an
+idea which did not begin to be worthily appreciated and
+suitably employed till after the formation of the infinitesimal
+analysis, without which it could not produce,
+even in geometry, very important results.</p>
+
+
+<p><i>Generality of the Formulas.</i> Besides the admirable
+facility which is given by the transcendental analysis for
+the investigation of the mathematical laws of all phenomena,
+a second fundamental and inherent property, perhaps
+as important as the first, is the extreme generality of
+the differential formulas, which express in a single equation
+each determinate phenomenon, however varied the
+subjects in relation to which it is considered. Thus we
+see, in the preceding examples, that a single differential
+equation gives the tangents of all curves, another their
+rectifications, a third their quadratures; and in the same
+way, one invariable formula expresses the mathematical
+law of every variable motion; and, finally, a single equation
+constantly represents the distribution of heat in any
+body and for any case. This generality, which is so exceedingly
+remarkable, and which is for geometers the
+basis of the most elevated considerations, is a fortunate
+and necessary consequence of the very spirit of the transcendental
+analysis, especially in the conception of Leibnitz.
+Thus the infinitesimal analysis has not only furnished
+a general method for indirectly forming equations
+which it would have been impossible to discover in a direct
+manner, but it has also permitted us to consider, for<span class="pagenum"><a name="Page_98" id="Page_98">[Pg 98]</a></span>
+the mathematical study of natural phenomena, a new
+order of more general laws, which nevertheless present a
+clear and precise signification to every mind habituated
+to their interpretation. By virtue of this second characteristic
+property, the entire system of an immense science,
+such as geometry or mechanics, has been condensed
+into a small number of analytical formulas, from which
+the human mind can deduce, by certain and invariable
+rules, the solution of all particular problems.</p>
+
+
+<p><i>Demonstration of the Method.</i> To complete the general
+exposition of the conception of Leibnitz, there remains
+to be considered the demonstration of the logical
+procedure to which it leads, and this, unfortunately, is
+the most imperfect part of this beautiful method.</p>
+
+<p>In the beginning of the infinitesimal analysis, the
+most celebrated geometers rightly attached more importance
+to extending the immortal discovery of Leibnitz
+and multiplying its applications than to rigorously establishing
+the logical bases of its operations. They contented
+themselves for a long time by answering the objections
+of second-rate geometers by the unhoped-for solution
+of the most difficult problems; doubtless persuaded
+that in mathematical science, much more than in any
+other, we may boldly welcome new methods, even when
+their rational explanation is imperfect, provided they are
+fruitful in results, inasmuch as its much easier and more
+numerous verifications would not permit any error to remain
+long undiscovered. But this state of things could
+not long exist, and it was necessary to go back to the
+very foundations of the analysis of Leibnitz in order to
+prove, in a perfectly general manner, the rigorous exactitude
+of the procedures employed in this method, in spite<span class="pagenum"><a name="Page_99" id="Page_99">[Pg 99]</a></span>
+of the apparent infractions of the ordinary rules of reasoning
+which it permitted.</p>
+
+<p>Leibnitz, urged to answer, had presented an explanation
+entirely erroneous, saying that he treated infinitely
+small quantities as <i>incomparables</i>, and that he neglected
+them in comparison with finite quantities, "like grains
+of sand in comparison with the sea:" a view which would
+have completely changed the nature of his analysis, by
+reducing it to a mere approximative calculus, which, under
+this point of view, would be radically vicious, since
+it would be impossible to foresee, in general, to what degree
+the successive operations might increase these first
+errors, which could thus evidently attain any amount.
+Leibnitz, then, did not see, except in a very confused
+manner, the true logical foundations of the analysis which
+he had created. His earliest successors limited themselves,
+at first, to verifying its exactitude by showing the
+conformity of its results, in particular applications, to
+those obtained by ordinary algebra or the geometry of the
+ancients; reproducing, according to the ancient methods,
+so far as they were able, the solutions of some problems after
+they had been once obtained by the new method, which
+alone was capable of discovering them in the first place.</p>
+
+<p>When this great question was considered in a more
+general manner, geometers, instead of directly attacking
+the difficulty, preferred to elude it in some way, as Euler
+and D'Alembert, for example, have done, by demonstrating
+the necessary and constant conformity of the
+conception of Leibnitz, viewed in all its applications,
+with other fundamental conceptions of the transcendental
+analysis, that of Newton especially, the exactitude of
+which was free from any objection. Such a general verification<span class="pagenum"><a name="Page_100" id="Page_100">[Pg 100]</a></span>
+is undoubtedly strictly sufficient to dissipate any
+uncertainty as to the legitimate employment of the analysis
+of Leibnitz. But the infinitesimal method is so important&mdash;it
+offers still, in almost all its applications, such
+a practical superiority over the other general conceptions
+which have been successively proposed&mdash;that there
+would be a real imperfection in the philosophical character
+of the science if it could not justify itself, and needed
+to be logically founded on considerations of another order,
+which would then cease to be employed.</p>
+
+<p>It was, then, of real importance to establish directly
+and in a general manner the necessary rationality of the
+infinitesimal method. After various attempts more or
+less imperfect, a distinguished geometer, Carnot, presented
+at last the true direct logical explanation of the method
+of Leibnitz, by showing it to be founded on the principle
+of the necessary compensation of errors, this being,
+in fact, the precise and luminous manifestation of what
+Leibnitz had vaguely and confusedly perceived. Carnot
+has thus rendered the science an essential service, although,
+as we shall see towards the end of this chapter,
+all this logical scaffolding of the infinitesimal method,
+properly so called, is very probably susceptible of only a
+provisional existence, inasmuch as it is radically vicious
+in its nature. Still, we should not fail to notice the
+general system of reasoning proposed by Carnot, in order
+to directly legitimate the analysis of Leibnitz. Here is
+the substance of it:</p>
+
+<p>In establishing the differential equation of a phenomenon,
+we substitute, for the immediate elements of the different
+quantities considered, other simpler infinitesimals,
+which differ from them infinitely little in comparison<span class="pagenum"><a name="Page_101" id="Page_101">[Pg 101]</a></span>
+with them; and this substitution constitutes the principal
+artifice of the method of Leibnitz, which without it
+would possess no real facility for the formation of equations.
+Carnot regards such an hypothesis as really producing
+an error in the equation thus obtained, and which
+for this reason he calls <i>imperfect</i>; only, it is clear that
+this error must be infinitely small. Now, on the other
+hand, all the analytical operations, whether of differentiation
+or of integration, which are performed upon these
+differential equations, in order to raise them to finite
+equations by eliminating all the infinitesimals which
+have been introduced as auxiliaries, produce as constantly,
+by their nature, as is easily seen, other analogous errors,
+so that an exact compensation takes place, and the
+final equations, in the words of Carnot, become <i>perfect</i>.
+Carnot views, as a certain and invariable indication of
+the actual establishment of this necessary compensation,
+the complete elimination of the various infinitely small
+quantities, which is always, in fact, the final object of
+all the operations of the transcendental analysis; for if
+we have committed no other infractions of the general
+rules of reasoning than those thus exacted by the very
+nature of the infinitesimal method, the infinitely small
+errors thus produced cannot have engendered other than
+infinitely small errors in all the equations, and the relations
+are necessarily of a rigorous exactitude as soon as
+they exist between finite quantities alone, since the only
+errors then possible must be finite ones, while none such
+can have entered. All this general reasoning is founded
+on the conception of infinitesimal quantities, regarded as
+indefinitely decreasing, while those from which they are
+derived are regarded as fixed.</p><p><span class="pagenum"><a name="Page_102" id="Page_102">[Pg 102]</a></span></p>
+
+
+<p><i>Illustration by Tangents.</i> Thus, to illustrate this abstract
+exposition by a single example, let us take up again
+the question of <i>tangents</i>, which is the most easy to analyze
+completely. We will regard the equation <i>t</i> = <i>dy/dx</i>,
+obtained above, as being affected with an infinitely small
+error, since it would be perfectly rigorous only for the
+secant. Now let us complete the solution by seeking,
+according to the equation of each curve, the ratio between
+the differentials of the co-ordinates. If we suppose
+this equation to be <i>y</i> = <i>ax<sup>2</sup></i>, we shall evidently have</p>
+
+<p>
+<i>dy</i> = 2<i>axdx</i> + <i>adx<sup>2</sup></i>.<br />
+</p>
+
+<p>In this formula we shall have to neglect the term <i>dx<sup>2</sup></i>
+as an infinitely small quantity of the second order. Then
+the combination of the two <i>imperfect</i> equations.</p>
+
+<p>
+<i>t</i> = <i>dy/dx</i>, <i>dy</i> = 2<i>ax(dx)</i>,<br />
+</p>
+
+<p>being sufficient to eliminate entirely the infinitesimals,
+the finite result, <i>t</i> = 2<i>ax</i>, will necessarily be rigorously correct,
+from the effect of the exact compensation of the two
+errors committed; since, by its finite nature, it cannot be
+affected by an infinitely small error, and this is, nevertheless,
+the only one which it could have, according to
+the spirit of the operations which have been executed.</p>
+
+<p>It would be easy to reproduce in a uniform manner
+the same reasoning with reference to all the other general
+applications of the analysis of Leibnitz.</p>
+
+<p>This ingenious theory is undoubtedly more subtile than
+solid, when we examine it more profoundly; but it has
+really no other radical logical fault than that of the infinitesimal
+method itself, of which it is, it seems to me,
+the natural development and the general explanation, so<span class="pagenum"><a name="Page_103" id="Page_103">[Pg 103]</a></span>
+that it must be adopted for as long a time as it shall be
+thought proper to employ this method directly.</p>
+
+<hr class="tb" />
+
+<p>I pass now to the general exposition of the two other
+fundamental conceptions of the transcendental analysis,
+limiting myself in each to its principal idea, the philosophical
+character of the analysis having been sufficiently
+determined above in the examination of the conception
+of Leibnitz, which I have specially dwelt upon because
+it admits of being most easily grasped as a whole, and
+most rapidly described.</p>
+
+
+
+
+<h3><a name="METHOD_OF_NEWTON" id="METHOD_OF_NEWTON">METHOD OF NEWTON.</a></h3>
+
+
+<p>Newton has successively presented his own method of
+conceiving the transcendental analysis under several different
+forms. That which is at present the most commonly
+adopted was designated by Newton, sometimes under
+the name of the <i>Method of prime and ultimate Ratios</i>,
+sometimes under that of the <i>Method of Limits</i>.</p>
+
+
+<p><i>Method of Limits.</i> The general spirit of the transcendental
+analysis, from this point of view, consists in
+introducing as auxiliaries, in the place of the primitive
+quantities, or concurrently with them, in order to facilitate
+the establishment of equations, the <i>limits of the ratios</i>
+of the simultaneous increments of these quantities;
+or, in other words, the <i>final ratios</i> of these increments;
+limits or final ratios which can be easily shown to have
+a determinate and finite value. A special calculus, which
+is the equivalent of the infinitesimal calculus, is then
+employed to pass from the equations between these limits
+to the corresponding equations between the primitive
+quantities themselves.</p><p><span class="pagenum"><a name="Page_104" id="Page_104">[Pg 104]</a></span></p>
+
+<p>The power which is given by such an analysis, of expressing
+with more ease the mathematical laws of phenomena,
+depends in general on this, that since the calculus
+applies, not to the increments themselves of the proposed
+quantities, but to the limits of the ratios of those
+increments, we can always substitute for each increment
+any other magnitude more easy to consider, provided that
+their final ratio is the ratio of equality, or, in other words,
+that the limit of their ratio is unity. It is clear, indeed,
+that the calculus of limits would be in no way affected
+by this substitution. Starting from this principle, we
+find nearly the equivalent of the facilities offered by the
+analysis of Leibnitz, which are then merely conceived under
+another point of view. Thus curves will be regarded
+as the <i>limits</i> of a series of rectilinear polygons, variable
+motions as the <i>limits</i> of a collection of uniform motions
+of constantly diminishing durations, and so on.</p>
+
+
+<p><span class="smcap">Examples.</span> 1. <i>Tangents.</i> Suppose, for example, that
+we wish to determine the direction of the tangent to a
+curve; we will regard it as the limit towards which would
+tend a secant, which should turn about the given point
+so that its second point of intersection should indefinitely
+approach the first. Representing the differences of the co-ordinates
+of the two points by &#916;<i>y</i> and &#916;<i>x</i>, we would have
+at each instant, for the trigonometrical tangent of the angle
+which the secant makes with the axis of abscissas,</p>
+
+<p>
+<i>t</i> = &#916;<i>y</i>/&#916;<i>x</i>;<br />
+</p>
+
+<p>from which, taking the limits, we will obtain, relatively
+to the tangent itself, this general formula of transcendental
+analysis,</p>
+
+<p>
+<i>t</i> = <i>L</i>(&#916;<i>y</i>/&#916;<i>x</i>),<br />
+</p><p><span class="pagenum"><a name="Page_105" id="Page_105">[Pg 105]</a></span></p>
+
+<p>the characteristic <i>L</i> being employed to designate the limit.
+The calculus of indirect functions will show how to deduce
+from this formula in each particular case, when the
+equation of the curve is given, the relation between <i>t</i> and
+<i>x</i>, by eliminating the auxiliary quantities which have
+been introduced. If we suppose, in order to complete the
+solution, that the equation of the proposed curve is <i>y</i> = <i>ax<sup>2</sup></i>,
+we shall evidently have</p>
+
+<p>
+&#916;<i>y</i> = 2<i>ax</i>&#916;<i>x</i> + <i>a</i>(&#916;<i>x</i>)<sup>2</sup>,<br />
+</p>
+
+<p>from which we shall obtain</p>
+
+<p>
+&#916;<i>y</i>/&#916;<i>x</i> = 2<i>ax</i> + <i>a</i>&#916;<i>x</i>.<br />
+</p>
+
+<p>Now it is clear that the <i>limit</i> towards which the second
+number tends, in proportion as &#916;<i>x</i> diminishes, is 2<i>ax</i>.
+We shall therefore find, by this method, <i>t</i> = 2<i>ax</i>, as we
+obtained it for the same case by the method of Leibnitz.</p>
+
+<p>2. <i>Rectifications.</i> In like manner, when the rectification
+of a curve is desired, we must substitute for the increment
+of the arc s the chord of this increment, which
+evidently has such a connexion with it that the limit
+of their ratio is unity; and then we find (pursuing in
+other respects the same plan as with the method of Leibnitz)
+this general equation of rectifications:</p>
+
+<p>
+(<i>L</i>&#916;<i>s</i>/&#916;<i>x</i>)² = 1 + (<i>L</i>&#916;<i>y</i>/&#916;<i>x</i>)²,<br />
+or (<i>L</i>&#916;<i>s</i>/&#916;<i>x</i>)<sup>2</sup> = 1 + (<i>L</i>&#916;<i>y</i>/&#916;<i>x</i>)<sup>2</sup> + (<i>L</i>&#916;<i>z</i>/&#916;<i>x</i>)<sup>2</sup>,<br />
+</p>
+
+<p>according as the curve is plane or of double curvature.
+It will now be necessary, for each particular curve, to
+pass from this equation to that between the arc and the
+abscissa, which depends on the transcendental calculus
+properly so called.</p><p><span class="pagenum"><a name="Page_106" id="Page_106">[Pg 106]</a></span></p>
+
+<p>We could take up, with the same facility, by the
+method of limits, all the other general questions, the solution
+of which has been already indicated according to the
+infinitesimal method.</p>
+
+<p>Such is, in substance, the conception which Newton
+formed for the transcendental analysis, or, more precisely,
+that which Maclaurin and D'Alembert have presented
+as the most rational basis of that analysis, in seeking to
+fix and to arrange the ideas of Newton upon that subject.</p>
+
+
+<p><i>Fluxions and Fluents.</i> Another distinct form under
+which Newton has presented this same method should be
+here noticed, and deserves particularly to fix our attention,
+as much by its ingenious clearness in some cases
+as by its having furnished the notation best suited to this
+manner of viewing the transcendental analysis, and, moreover,
+as having been till lately the special form of the calculus
+of indirect functions commonly adopted by the English
+geometers. I refer to the calculus of <i>fluxions</i> and
+of <i>fluents</i>, founded on the general idea of <i>velocities</i>.</p>
+
+<p>To facilitate the conception of the fundamental idea,
+let us consider every curve as generated by a point impressed
+with a motion varying according to any law whatever.
+The different quantities which the curve can present,
+the abscissa, the ordinate, the arc, the area, &amp;c.,
+will be regarded as simultaneously produced by successive
+degrees during this motion. The <i>velocity</i> with which
+each shall have been described will be called the <i>fluxion</i>
+of that quantity, which will be inversely named its <i>fluent</i>.
+Henceforth the transcendental analysis will consist,
+according to this conception, in forming directly the
+equations between the fluxions of the proposed quantities,
+in order to deduce therefrom, by a special calculus,<span class="pagenum"><a name="Page_107" id="Page_107">[Pg 107]</a></span>
+the equations between the fluents themselves. What
+has been stated respecting curves may, moreover, evidently
+be applied to any magnitudes whatever, regarded,
+by the aid of suitable images, as produced by motion.</p>
+
+<p>It is easy to understand the general and necessary
+identity of this method with that of limits complicated
+with the foreign idea of motion. In fact, resuming the
+case of the curve, if we suppose, as we evidently always
+may, that the motion of the describing point is uniform
+in a certain direction, that of the abscissa, for example,
+then the fluxion of the abscissa will be constant, like the
+element of the time; for all the other quantities generated,
+the motion cannot be conceived to be uniform, except
+for an infinitely small time. Now the velocity being
+in general according to its mechanical conception, the
+ratio of each space to the time employed in traversing it,
+and this time being here proportional to the increment of
+the abscissa, it follows that the fluxions of the ordinate,
+of the arc, of the area, &amp;c., are really nothing else (rejecting
+the intermediate consideration of time) than the
+final ratios of the increments of these different quantities
+to the increment of the abscissa. This method of fluxions
+and fluents is, then, in reality, only a manner of
+representing, by a comparison borrowed from mechanics,
+the method of prime and ultimate ratios, which alone can
+be reduced to a calculus. It evidently, then, offers the
+same general advantages in the various principal applications
+of the transcendental analysis, without its being
+necessary to present special proofs of this.</p><hr class="chap" /><p><span class="pagenum"><a name="Page_108" id="Page_108">[Pg 108]</a></span></p>
+
+
+
+
+<h3><a name="METHOD_OF_LAGRANGE" id="METHOD_OF_LAGRANGE">METHOD OF LAGRANGE.</a></h3>
+
+
+<p><i>Derived Functions.</i> The conception of Lagrange,
+in its admirable simplicity, consists in representing the
+transcendental analysis as a great algebraic artifice, by
+which, in order to facilitate the establishment of equations,
+we introduce, in the place of the primitive functions,
+or concurrently with them, their <i>derived</i> functions;
+that is, according to the definition of Lagrange,
+the coefficient of the first term of the increment of each
+function, arranged according to the ascending powers of
+the increment of its variable. The special calculus of
+indirect functions has for its constant object, here as
+well as in the conceptions of Leibnitz and of Newton, to
+eliminate these <i>derivatives</i> which have been thus employed
+as auxiliaries, in order to deduce from their relations
+the corresponding equations between the primitive
+magnitudes.</p>
+
+
+<p><i>An Extension of ordinary Analysis.</i> The transcendental
+analysis is, then, nothing but a simple though very
+considerable extension of ordinary analysis. Geometers
+have long been accustomed to introduce in analytical investigations,
+in the place of the magnitudes themselves
+which they wished to study, their different powers, or
+their logarithms, or their sines, &amp;c., in order to simplify
+the equations, and even to obtain them more easily.
+This successive <i>derivation</i> is an artifice of the same
+nature, only of greater extent, and procuring, in consequence,
+much more important resources for this common
+object.</p>
+
+<p>But, although we can readily conceive, <i>à priori</i>, that
+the auxiliary consideration of these derivatives <i>may</i> facilitate<span class="pagenum"><a name="Page_109" id="Page_109">[Pg 109]</a></span>
+the establishment of equations, it is not easy to
+explain why this <i>must</i> necessarily follow from this mode
+of derivation rather than from any other transformation.
+Such is the weak point of the great idea of Lagrange.
+The precise advantages of this analysis cannot as yet be
+grasped in an abstract manner, but only shown by considering
+separately each principal question, so that the
+verification is often exceedingly laborious.</p>
+
+
+<p><span class="smcap">Example.</span> <i>Tangents.</i> This manner of conceiving the
+transcendental analysis may be best illustrated by its application
+to the most simple of the problems above examined&mdash;that
+of tangents.</p>
+
+<p>Instead of conceiving the tangent as the prolongation
+of the infinitely small element of the curve, according to
+the notion of Leibnitz&mdash;or as the limit of the secants, according
+to the ideas of Newton&mdash;Lagrange considers it,
+according to its simple geometrical character, analogous
+to the definitions of the ancients, to be a right line such
+that no other right line can pass through the point of
+contact between it and the curve. Then, to determine
+its direction, we must seek the general expression of its
+distance from the curve, measured in any direction whatever&mdash;in
+that of the ordinate, for example&mdash;and dispose
+of the arbitrary constant relating to the inclination of the
+right line, which will necessarily enter into that expression,
+in such a way as to diminish that separation as much
+as possible. Now this distance, being evidently equal
+to the difference of the two ordinates of the curve and of
+the right line, which correspond to the same new abscissa
+<i>x</i> + <i>h</i>, will be represented by the formula</p>
+
+<p>
+(<i>f'</i>(<i>x</i>) - <i>t</i>)<i>h</i> + <i>qh<sup>2</sup></i> + <i>rh<sup>3</sup></i> + etc.,<br />
+</p>
+
+<p>in which <i>t</i> designates, as above, the unknown trigonometrical<span class="pagenum"><a name="Page_110" id="Page_110">[Pg 110]</a></span>
+tangent of the angle which the required line makes
+with the axis of abscissas, and <i>f'</i>(<i>x</i>) the derived function
+of the ordinate <i>f</i>(<i>x</i>). This being understood, it is easy
+to see that, by disposing of <i>t</i> so as to make the first term
+of the preceding formula equal to zero, we will render the
+interval between the two lines the least possible, so that
+any other line for which <i>t</i> did not have the value thus
+determined would necessarily depart farther from the proposed
+curve. We have, then, for the direction of the tangent
+sought, the general expression <i>t</i> = <i>f'</i>(<i>x</i>), a result exactly
+equivalent to those furnished by the Infinitesimal
+Method and the Method of Limits. We have yet to find
+<i>f'</i>(<i>x</i>) in each particular curve, which is a mere question
+of analysis, quite identical with those which are presented,
+at this stage of the operations, by the other methods.</p>
+
+<p>After these considerations upon the principal general
+conceptions, we need not stop to examine some other theories
+proposed, such as Euler's <i>Calculus of Vanishing
+Quantities</i>, which are really modifications&mdash;more or less
+important, and, moreover, no longer used&mdash;of the preceding
+methods.</p>
+
+<p>I have now to establish the comparison and the appreciation
+of these three fundamental methods. Their <i>perfect
+and necessary conformity</i> is first to be proven in a
+general manner.</p>
+
+
+
+
+<h3><a name="FUNDAMENTAL_IDENTITY_OF_THE_THREE_METHODS" id="FUNDAMENTAL_IDENTITY_OF_THE_THREE_METHODS">FUNDAMENTAL IDENTITY OF THE THREE METHODS.</a></h3>
+
+
+<p>It is, in the first place, evident from what precedes,
+considering these three methods as to their actual destination,
+independently of their preliminary ideas, that
+they all consist in the same general logical artifice, which
+has been characterized in the first chapter; to wit, the<span class="pagenum"><a name="Page_111" id="Page_111">[Pg 111]</a></span>
+introduction of a certain system of auxiliary magnitudes,
+having uniform relations to those which are the special
+objects of the inquiry, and substituted for them expressly
+to facilitate the analytical expression of the mathematical
+laws of the phenomena, although they have finally to
+be eliminated by the aid of a special calculus. It is
+this which has determined me to regularly define the
+transcendental analysis as <i>the calculus of indirect functions</i>,
+in order to mark its true philosophical character,
+at the same time avoiding any discussion upon the best
+manner of conceiving and applying it. The general effect
+of this analysis, whatever the method employed, is,
+then, to bring every mathematical question much more
+promptly within the power of the <i>calculus</i>, and thus to
+diminish considerably the serious difficulty which is usually
+presented by the passage from the concrete to the abstract.
+Whatever progress we may make, we can never
+hope that the calculus will ever be able to grasp every
+question of natural philosophy, geometrical, or mechanical,
+or thermological, &amp;c., immediately upon its birth,
+which would evidently involve a contradiction. Every
+problem will constantly require a certain preliminary labour
+to be performed, in which the calculus can be of no
+assistance, and which, by its nature, cannot be subjected
+to abstract and invariable rules; it is that which has
+for its special object the establishment of equations, which
+form the indispensable starting point of all analytical researches.
+But this preliminary labour has been remarkably
+simplified by the creation of the transcendental analysis,
+which has thus hastened the moment at which the
+solution admits of the uniform and precise application of
+general and abstract methods; by reducing, in each case,<span class="pagenum"><a name="Page_112" id="Page_112">[Pg 112]</a></span>
+this special labour to the investigation of equations between
+the auxiliary magnitudes; from which the calculus
+then leads to equations directly referring to the proposed
+magnitudes, which, before this admirable conception, it
+had been necessary to establish directly and separately.
+Whether these indirect equations are <i>differential</i> equations,
+according to the idea of Leibnitz, or equations of
+<i>limits</i>, conformably to the conception of Newton, or, lastly,
+<i>derived</i> equations, according to the theory of Lagrange,
+the general procedure is evidently always the same.</p>
+
+<p>But the coincidence of these three principal methods
+is not limited to the common effect which they produce;
+it exists, besides, in the very manner of obtaining it. In
+fact, not only do all three consider, in the place of the
+primitive magnitudes, certain auxiliary ones, but, still
+farther, the quantities thus introduced as subsidiary are
+exactly identical in the three methods, which consequently
+differ only in the manner of viewing them. This
+can be easily shown by taking for the general term of
+comparison any one of the three conceptions, especially
+that of Lagrange, which is the most suitable to serve as
+a type, as being the freest from foreign considerations.
+Is it not evident, by the very definition of <i>derived functions</i>,
+that they are nothing else than what Leibnitz calls
+<i>differential coefficients</i>, or the ratios of the differential
+of each function to that of the corresponding variable,
+since, in determining the first differential, we will be
+obliged, by the very nature of the infinitesimal method,
+to limit ourselves to taking the only term of the increment
+of the function which contains the first power of
+the infinitely small increment of the variable? In the
+same way, is not the derived function, by its nature,<span class="pagenum"><a name="Page_113" id="Page_113">[Pg 113]</a></span>
+likewise the necessary <i>limit</i> towards which tends the ratio
+between the increment of the primitive function and
+that of its variable, in proportion as this last indefinitely
+diminishes, since it evidently expresses what that ratio
+becomes when we suppose the increment of the variable
+to equal zero? That which is designated by <i>dx</i>/<i>dy</i> in the
+method of Leibnitz; that which ought to be noted as
+<i>L</i>(&#916;<i>y</i>/&#916;<i>x</i>) in that of Newton; and that which Lagrange has
+indicated by <i>f'</i>(<i>x</i>), is constantly one same function, seen
+from three different points of view, the considerations
+of Leibnitz and Newton properly consisting in making
+known two general necessary properties of the derived
+function. The transcendental analysis, examined abstractedly
+and in its principle, is then always the same,
+whatever may be the conception which is adopted, and
+the procedures of the calculus of indirect functions are
+necessarily identical in these different methods, which in
+like manner must, for any application whatever, lead constantly
+to rigorously uniform results.</p>
+
+
+
+
+<h3><a name="COMPARATIVE_VALUE_OF_THE_THREE_METHODS" id="COMPARATIVE_VALUE_OF_THE_THREE_METHODS">COMPARATIVE VALUE OF THE THREE METHODS.</a></h3>
+
+
+<p>If now we endeavour to estimate the comparative value
+of these three equivalent conceptions, we shall find in
+each advantages and inconveniences which are peculiar
+to it, and which still prevent geometers from confining
+themselves to any one of them, considered as final.</p>
+
+
+<p><i>That of Leibnitz.</i> The conception of Leibnitz presents
+incontestably, in all its applications, a very marked
+superiority, by leading in a much more rapid manner,
+and with much less mental effort, to the formation of<span class="pagenum"><a name="Page_114" id="Page_114">[Pg 114]</a></span>
+equations between the auxiliary magnitudes. It is to its
+use that we owe the high perfection which has been acquired
+by all the general theories of geometry and mechanics.
+Whatever may be the different speculative
+opinions of geometers with respect to the infinitesimal
+method, in an abstract point of view, all tacitly agree in
+employing it by preference, as soon as they have to treat
+a new question, in order not to complicate the necessary
+difficulty by this purely artificial obstacle proceeding from
+a misplaced obstinacy in adopting a less expeditious course.
+Lagrange himself, after having reconstructed the transcendental
+analysis on new foundations, has (with that
+noble frankness which so well suited his genius) rendered
+a striking and decisive homage to the characteristic properties
+of the conception of Leibnitz, by following it exclusively
+in the entire system of his <i>Méchanique Analytique</i>.
+Such a fact renders any comments unnecessary.</p>
+
+<p>But when we consider the conception of Leibnitz in
+itself and in its logical relations, we cannot escape admitting,
+with Lagrange, that it is radically vicious in
+this, that, adopting its own expressions, the notion of infinitely
+small quantities is a <i>false idea</i>, of which it is in
+fact impossible to obtain a clear conception, however we
+may deceive ourselves in that matter. Even if we adopt
+the ingenious idea of the compensation of errors, as above
+explained, this involves the radical inconvenience of being
+obliged to distinguish in mathematics two classes of reasonings,
+those which are perfectly rigorous, and those in
+which we designedly commit errors which subsequently
+have to be compensated. A conception which leads to
+such strange consequences is undoubtedly very unsatisfactory
+in a logical point of view.</p><p><span class="pagenum"><a name="Page_115" id="Page_115">[Pg 115]</a></span></p>
+
+<p>To say, as do some geometers, that it is possible in
+every case to reduce the infinitesimal method to that of
+limits, the logical character of which is irreproachable,
+would evidently be to elude the difficulty rather than to
+remove it; besides, such a transformation almost entirely
+strips the conception of Leibnitz of its essential advantages
+of facility and rapidity.</p>
+
+<p>Finally, even disregarding the preceding important
+considerations, the infinitesimal method would no less
+evidently present by its nature the very serious defect of
+breaking the unity of abstract mathematics, by creating
+a transcendental analysis founded on principles so different
+from those which form the basis of the ordinary analysis.
+This division of analysis into two worlds almost
+entirely independent of each other, tends to hinder the
+formation of truly general analytical conceptions. To
+fully appreciate the consequences of this, we should have
+to go back to the state of the science before Lagrange
+had established a general and complete harmony between
+these two great sections.</p>
+
+
+<p><i>That of Newton.</i> Passing now to the conception of
+Newton, it is evident that by its nature it is not exposed
+to the fundamental logical objections which are called
+forth by the method of Leibnitz. The notion of <i>limits</i>
+is, in fact, remarkable for its simplicity and its precision.
+In the transcendental analysis presented in this manner,
+the equations are regarded as exact from their very origin,
+and the general rules of reasoning are as constantly
+observed as in ordinary analysis. But, on the other
+hand, it is very far from offering such powerful resources
+for the solution of problems as the infinitesimal method.
+The obligation which it imposes, of never considering<span class="pagenum"><a name="Page_116" id="Page_116">[Pg 116]</a></span>
+the increments of magnitudes separately and by themselves,
+nor even in their ratios, but only in the limits of
+those ratios, retards considerably the operations of the
+mind in the formation of auxiliary equations. We may
+even say that it greatly embarrasses the purely analytical
+transformations. Thus the transcendental analysis,
+considered separately from its applications, is far from presenting
+in this method the extent and the generality which
+have been imprinted upon it by the conception of Leibnitz.
+It is very difficult, for example, to extend the theory
+of Newton to functions of several independent variables.
+But it is especially with reference to its applications
+that the relative inferiority of this theory is most
+strongly marked.</p>
+
+<p>Several Continental geometers, in adopting the method
+of Newton as the more logical basis of the transcendental
+analysis, have partially disguised this inferiority by a serious
+inconsistency, which consists in applying to this method
+the notation invented by Leibnitz for the infinitesimal
+method, and which is really appropriate to it alone.
+In designating by <i>dy</i>/<i>dx</i> that which logically ought, in the
+theory of limits, to be denoted by <i>L</i>(&#916;<i>y</i>/&#916;<i>x</i>), and in extending
+to all the other analytical conceptions this displacement
+of signs, they intended, undoubtedly, to combine the special
+advantages of the two methods; but, in reality, they
+have only succeeded in causing a vicious confusion between
+them, a familiarity with which hinders the formation
+of clear and exact ideas of either. It would certainly
+be singular, considering this usage in itself, that,
+by the mere means of signs, it could be possible to effect<span class="pagenum"><a name="Page_117" id="Page_117">[Pg 117]</a></span>
+a veritable combination between two theories so distinct
+as those under consideration.</p>
+
+<p>Finally, the method of limits presents also, though in
+a less degree, the greater inconvenience, which I have
+above noted in reference to the infinitesimal method, of
+establishing a total separation between the ordinary and
+the transcendental analysis; for the idea of <i>limits</i>, though
+clear and rigorous, is none the less in itself, as Lagrange
+has remarked, a foreign idea, upon which analytical theories
+ought not to be dependent.</p>
+
+
+<p><i>That of Lagrange.</i> This perfect unity of analysis,
+and this purely abstract character of its fundamental notions,
+are found in the highest degree in the conception
+of Lagrange, and are found there alone; it is, for this
+reason, the most rational and the most philosophical of
+all. Carefully removing every heterogeneous consideration,
+Lagrange has reduced the transcendental analysis
+to its true peculiar character, that of presenting a very
+extensive class of analytical transformations, which facilitate
+in a remarkable degree the expression of the conditions
+of various problems. At the same time, this analysis
+is thus necessarily presented as a simple extension
+of ordinary analysis; it is only a higher algebra. All the
+different parts of abstract mathematics, previously so incoherent,
+have from that moment admitted of being conceived
+as forming a single system.</p>
+
+<p>Unhappily, this conception, which possesses such fundamental
+properties, independently of its so simple and
+so lucid notation, and which is undoubtedly destined to
+become the final theory of transcendental analysis, because
+of its high philosophical superiority over all the
+other methods proposed, presents in its present state too<span class="pagenum"><a name="Page_118" id="Page_118">[Pg 118]</a></span>
+many difficulties in its applications, as compared with the
+conception of Newton, and still more with that of Leibnitz,
+to be as yet exclusively adopted. Lagrange himself
+has succeeded only with great difficulty in rediscovering,
+by his method, the principal results already obtained
+by the infinitesimal method for the solution of the general
+questions of geometry and mechanics; we may judge
+from that what obstacles would be found in treating in
+the same manner questions which were truly new and
+important. It is true that Lagrange, on several occasions,
+has shown that difficulties call forth, from men of
+genius, superior efforts, capable of leading to the greatest
+results. It was thus that, in trying to adapt his method
+to the examination of the curvature of lines, which seemed
+so far from admitting its application, he arrived at that
+beautiful theory of contacts which has so greatly perfected
+that important part of geometry. But, in spite
+of such happy exceptions, the conception of Lagrange has
+nevertheless remained, as a whole, essentially unsuited
+to applications.</p>
+
+<p>The final result of the general comparison which I
+have too briefly sketched, is, then, as already suggested,
+that, in order to really understand the transcendental analysis,
+we should not only consider it in its principles according
+to the three fundamental conceptions of Leibnitz,
+of Newton, and of Lagrange, but should besides accustom
+ourselves to carry out almost indifferently, according
+to these three principal methods, and especially
+according to the first and the last, the solution of all important
+questions, whether of the pure calculus of indirect
+functions or of its applications. This is a course which
+I could not too strongly recommend to all those who desire<span class="pagenum"><a name="Page_119" id="Page_119">[Pg 119]</a></span>
+to judge philosophically of this admirable creation of
+the human mind, as well as to those who wish to learn
+to make use of this powerful instrument with success and
+with facility. In all the other parts of mathematical science,
+the consideration of different methods for a single
+class of questions may be useful, even independently of
+its historical interest, but it is not indispensable; here,
+on the contrary, it is strictly necessary.</p>
+
+<p>Having determined with precision, in this chapter, the
+philosophical character of the calculus of indirect functions,
+according to the principal fundamental conceptions
+of which it admits, we have next to consider, in the following
+chapter, the logical division and the general composition
+of this calculus.</p><hr class="chap" /><p><span class="pagenum"><a name="Page_120" id="Page_120">[Pg 120]</a></span></p>
+
+
+
+
+<h2><a name="CHAPTER_IV" id="CHAPTER_IV">CHAPTER IV.</a></h2>
+
+
+<h3>THE DIFFERENTIAL AND INTEGRAL CALCULUS.<br />
+ITS TWO FUNDAMENTAL DIVISIONS.</h3>
+
+
+<p>The <i>calculus of indirect functions</i>, in accordance with
+the considerations explained in the preceding chapter, is
+necessarily divided into two parts (or, more properly, is
+decomposed into two different <i>calculi</i> entirely distinct,
+although intimately connected by their nature), according
+as it is proposed to find the relations between the
+auxiliary magnitudes (the introduction of which constitutes
+the general spirit of this calculus) by means of the
+relations between the corresponding primitive magnitudes;
+or, conversely, to try to discover these direct
+equations by means of the indirect equations originally
+established. Such is, in fact, constantly the double object
+of the transcendental analysis.</p>
+
+<p>These two systems have received different names, according
+to the point of view under which this analysis
+has been regarded. The infinitesimal method, properly
+so called, having been the most generally employed for
+the reasons which have been given, almost all geometers
+employ habitually the denominations of <i>Differential
+Calculus</i> and of <i>Integral Calculus</i>, established by
+Leibnitz, and which are, in fact, very rational consequences
+of his conception. Newton, in accordance with
+his method, named the first the <i>Calculus of Fluxions</i>,
+and the second the <i>Calculus of Fluents</i>, expressions which
+were commonly employed in England. Finally, following<span class="pagenum"><a name="Page_121" id="Page_121">[Pg 121]</a></span>
+the eminently philosophical theory founded by Lagrange,
+one would be called the <i>Calculus of Derived
+Functions</i>, and the other the <i>Calculus of Primitive
+Functions</i>. I will continue to make use of the terms of
+Leibnitz, as being more convenient for the formation of
+secondary expressions, although I ought, in accordance
+with the suggestions made in the preceding chapter, to
+employ concurrently all the different conceptions, approaching
+as nearly as possible to that of Lagrange.</p>
+
+
+
+
+<h3><a name="THEIR_RELATIONS_TO_EACH_OTHER" id="THEIR_RELATIONS_TO_EACH_OTHER">THEIR RELATIONS TO EACH OTHER.</a></h3>
+
+
+<p>The differential calculus is evidently the logical basis
+of the integral calculus; for we do not and cannot
+know how to integrate directly any other differential expressions
+than those produced by the differentiation of
+the ten simple functions which constitute the general elements
+of our analysis. The art of integration consists,
+then, essentially in bringing all the other cases, as far as
+is possible, to finally depend on only this small number
+of fundamental integrations.</p>
+
+<p>In considering the whole body of the transcendental
+analysis, as I have characterized it in the preceding chapter,
+it is not at first apparent what can be the peculiar
+utility of the differential calculus, independently of this
+necessary relation with the integral calculus, which seems
+as if it must be, by itself, the only one directly indispensable.
+In fact, the elimination of the <i>infinitesimals</i> or
+of the <i>derivatives</i>, introduced as auxiliaries to facilitate
+the establishment of equations, constituting, as we have
+seen, the final and invariable object of the calculus of indirect
+functions, it is natural to think that the calculus
+which teaches how to deduce from the equations between<span class="pagenum"><a name="Page_122" id="Page_122">[Pg 122]</a></span>
+these auxiliary magnitudes, those which exist between the
+primitive magnitudes themselves, ought strictly to suffice
+for the general wants of the transcendental analysis without
+our perceiving, at the first glance, what special and
+constant part the solution of the inverse question can
+have in such an analysis. It would be a real error, though
+a common one, to assign to the differential calculus, in order
+to explain its peculiar, direct, and necessary influence,
+the destination of forming the differential equations, from
+which the integral calculus then enables us to arrive at
+the finite equations; for the primitive formation of differential
+equations is not and cannot be, properly speaking,
+the object of any calculus, since, on the contrary, it
+forms by its nature the indispensable starting point of any
+calculus whatever. How, in particular, could the differential
+calculus, which in itself is reduced to teaching the
+means of <i>differentiating</i> the different equations, be a
+general procedure for establishing them? That which
+in every application of the transcendental analysis really
+facilitates the formation of equations, is the infinitesimal
+<i>method</i>, and not the infinitesimal <i>calculus</i>, which is perfectly
+distinct from it, although it is its indispensable complement.
+Such a consideration would, then, give a false
+idea of the special destination which characterizes the differential
+calculus in the general system of the transcendental
+analysis.</p>
+
+<p>But we should nevertheless very imperfectly conceive
+the real peculiar importance of this first branch of the
+calculus of indirect functions, if we saw in it only a simple
+preliminary labour, having no other general and essential
+object than to prepare indispensable foundations
+for the integral calculus. As the ideas on this matter<span class="pagenum"><a name="Page_123" id="Page_123">[Pg 123]</a></span>
+are generally confused, I think that I ought here to explain
+in a summary manner this important relation as I
+view it, and to show that in every application of the
+transcendental analysis a primary, direct, and necessary
+part is constantly assigned to the differential calculus.</p>
+
+
+<p>1. <i>Use of the Differential Calculus as preparatory
+to that of the Integral.</i> In forming the differential equations
+of any phenomenon whatever, it is very seldom that
+we limit ourselves to introduce differentially only those
+magnitudes whose relations are sought. To impose that
+condition would be to uselessly diminish the resources
+presented by the transcendental analysis for the expression
+of the mathematical laws of phenomena. Most frequently
+we introduce into the primitive equations, through
+their differentials, other magnitudes whose relations are
+already known or supposed to be so, and without the
+consideration of which it would be frequently impossible
+to establish equations. Thus, for example, in the general
+problem of the rectification of curves, the differential
+equation,</p>
+
+<p>
+<i>ds</i><sup>2</sup> = <i>dy</i><sup>2</sup> + <i>dx</i><sup>2</sup>, or <i>ds</i><sup>2</sup> = <i>dx</i><sup>2</sup> + <i>dy</i><sup>2</sup> + <i>dz</i><sup>2</sup>,<br />
+</p>
+
+<p>is not only established between the desired function s and
+the independent variable <i>x</i>, to which it is referred, but, at
+the same time, there have been introduced, as indispensable
+intermediaries, the differentials of one or two other
+functions, <i>y</i> and <i>z</i>, which are among the data of the
+problem; it would not have been possible to form directly
+the equation between <i>ds</i> and <i>dx</i>, which would, besides,
+be peculiar to each curve considered. It is the same for
+most questions. Now in these cases it is evident that
+the differential equation is not immediately suitable for
+integration. It is previously necessary that the differentials<span class="pagenum"><a name="Page_124" id="Page_124">[Pg 124]</a></span>
+of the functions supposed to be known, which
+have been employed as intermediaries, should be entirely
+eliminated, in order that equations may be obtained between
+the differentials of the functions which alone are
+sought and those of the really independent variables, after
+which the question depends on only the integral calculus.
+Now this preparatory elimination of certain differentials,
+in order to reduce the infinitesimals to the
+smallest number possible, belongs simply to the differential
+calculus; for it must evidently be done by determining,
+by means of the equations between the functions
+supposed to be known, taken as intermediaries, the
+relations of their differentials, which is merely a question
+of differentiation. Thus, for example, in the case of rectifications,
+it will be first necessary to calculate <i>dy</i>, or <i>dy</i>
+and <i>dz</i>, by differentiating the equation or the equations
+of each curve proposed; after eliminating these expressions,
+the general differential formula above enunciated
+will then contain only <i>ds</i> and <i>dx</i>; having arrived at this
+point, the elimination of the infinitesimals can be completed
+only by the integral calculus.</p>
+
+<p>Such is, then, the general office necessarily belonging
+to the differential calculus in the complete solution of the
+questions which exact the employment of the transcendental
+analysis; to produce, as far as is possible, the elimination
+of the infinitesimals, that is, to reduce in each
+case the primitive differential equations so that they shall
+contain only the differentials of the really independent
+variables, and those of the functions sought, by causing
+to disappear, by elimination, the differentials of all the
+other known functions which may have been taken as intermediaries
+at the time of the formation of the differential<span class="pagenum"><a name="Page_125" id="Page_125">[Pg 125]</a></span>
+equations of the problem which is under consideration.</p>
+
+
+<p>2. <i>Employment of the Differential Calculus alone.</i>
+For certain questions, which, although few in number,
+have none the less, as we shall see hereafter, a very great
+importance, the magnitudes which are sought enter directly,
+and not by their differentials, into the primitive
+differential equations, which then contain differentially
+only the different known functions employed as intermediaries,
+in accordance with the preceding explanation.
+These cases are the most favourable of all; for it is evident
+that the differential calculus is then entirely sufficient
+for the complete elimination of the infinitesimals,
+without the question giving rise to any integration. This
+is what occurs, for example, in the problem of <i>tangents</i>
+in geometry; in that of <i>velocities</i> in mechanics, &amp;c.</p>
+
+
+<p>3. <i>Employment of the Integral Calculus alone.</i> Finally,
+some other questions, the number of which is also
+very small, but the importance of which is no less great,
+present a second exceptional case, which is in its nature
+exactly the converse of the preceding. They are those
+in which the differential equations are found to be immediately
+ready for integration, because they contain, at
+their first formation, only the infinitesimals which relate
+to the functions sought, or to the really independent variables,
+without its being necessary to introduce, differentially,
+other functions as intermediaries. If in these
+new cases we introduce these last functions, since, by hypothesis,
+they will enter directly and not by their differentials,
+ordinary algebra will suffice to eliminate them,
+and to bring the question to depend on only the integral
+calculus. The differential calculus will then have no<span class="pagenum"><a name="Page_126" id="Page_126">[Pg 126]</a></span>
+special part in the complete solution of the problem, which
+will depend entirely upon the integral calculus. The
+general question of <i>quadratures</i> offers an important example
+of this, for the differential equation being then
+<i>dA = ydx</i>, will become immediately fit for integration as
+soon as we shall have eliminated, by means of the equation
+of the proposed curve, the intermediary function <i>y</i>,
+which does not enter into it differentially. The same
+circumstances exist in the problem of <i>cubatures</i>, and in
+some others equally important.</p>
+
+
+<p><i>Three classes of Questions hence resulting.</i> As a
+general result of the previous considerations, it is then
+necessary to divide into three classes the mathematical
+questions which require the use of the transcendental
+analysis; the <i>first</i> class comprises the problems susceptible
+of being entirely resolved by means of the differential
+calculus alone, without any need of the integral calculus;
+the <i>second</i>, those which are, on the contrary, entirely
+dependent upon the integral calculus, without the
+differential calculus having any part in their solution;
+lastly, in the <i>third</i> and the most extensive, which constitutes
+the normal case, the two others being only exceptional,
+the differential and the integral calculus have
+each in their turn a distinct and necessary part in the
+complete solution of the problem, the former making the
+primitive differential equations undergo a preparation
+which is indispensable for the application of the latter.
+Such are exactly their general relations, of which too
+indefinite and inexact ideas are generally formed.</p>
+<p><span class="pagenum"><a name="Page_127" id="Page_127">[Pg 127]</a></span></p>
+<hr class="tb" />
+
+<p>Let us now take a general survey of the logical composition
+of each calculus, beginning with the differential.</p>
+
+
+
+
+<h3><a name="THE_DIFFERENTIAL_CALCULUS" id="THE_DIFFERENTIAL_CALCULUS">THE DIFFERENTIAL CALCULUS.</a></h3>
+
+
+<p>In the exposition of the transcendental analysis, it is
+customary to intermingle with the purely analytical part
+(which reduces itself to the treatment of the abstract
+principles of differentiation and integration) the study of
+its different principal applications, especially those which
+concern geometry. This confusion of ideas, which is a
+consequence of the actual manner in which the science
+has been developed, presents, in the dogmatic point of
+view, serious inconveniences in this respect, that it makes
+it difficult properly to conceive either analysis or geometry.
+Having to consider here the most rational co-ordination
+which is possible, I shall include, in the following
+sketch, only the calculus of indirect functions properly
+so called, reserving for the portion of this volume
+which relates to the philosophical study of <i>concrete</i> mathematics
+the general examination of its great geometrical
+and mechanical applications.</p>
+
+
+<p><i>Two Cases: explicit and implicit Functions.</i> The
+fundamental division of the differential calculus, or of
+the general subject of differentiation, consists in distinguishing
+two cases, according as the analytical functions
+which are to be differentiated are <i>explicit</i> or <i>implicit</i>;
+from which flow two parts ordinarily designated by the
+names of differentiation <i>of formulas</i> and differentiation
+<i>of equations</i>. It is easy to understand, <i>à priori</i>, the
+importance of this classification. In fact, such a distinction
+would be illusory if the ordinary analysis was
+perfect; that is, if we knew how to resolve all equations
+algebraically, for then it would be possible to render
+every <i>implicit</i> function <i>explicit</i>; and, by differentiating<span class="pagenum"><a name="Page_128" id="Page_128">[Pg 128]</a></span>
+it in that state alone, the second part of the differential
+calculus would be immediately comprised in the first,
+without giving rise to any new difficulty. But the algebraical
+resolution of equations being, as we have seen,
+still almost in its infancy, and as yet impossible for most
+cases, it is plain that the case is very different, since
+we have, properly speaking, to differentiate a function
+without knowing it, although it is determinate. The
+differentiation of implicit functions constitutes then, by
+its nature, a question truly distinct from that presented
+by explicit functions, and necessarily more complicated.
+It is thus evident that we must commence with the differentiation
+of formulas, and reduce the differentiation
+of equations to this primary case by certain invariable
+analytical considerations, which need not be here mentioned.</p>
+
+<p>These two general cases of differentiation are also distinct
+in another point of view equally necessary, and too
+important to be left unnoticed. The relation which is
+obtained between the differentials is constantly more indirect,
+in comparison with that of the finite quantities,
+in the differentiation of implicit functions than in that
+of explicit functions. We know, in fact, from the considerations
+presented by Lagrange on the general formation
+of differential equations, that, on the one hand, the
+same primitive equation may give rise to a greater or
+less number of derived equations of very different forms,
+although at bottom equivalent, depending upon which of
+the arbitrary constants is eliminated, which is not the
+case in the differentiation of explicit formulas; and
+that, on the other hand, the unlimited system of the
+different primitive equations, which correspond to the<span class="pagenum"><a name="Page_129" id="Page_129">[Pg 129]</a></span>
+same derived equation, presents a much more profound
+analytical variety than that of the different functions,
+which admit of one same explicit differential, and which
+are distinguished from each other only by a constant
+term. Implicit functions must therefore be regarded as
+being in reality still more modified by differentiation
+than explicit functions. We shall again meet with this
+consideration relatively to the integral calculus, where
+it acquires a preponderant importance.</p>
+
+
+<p><i>Two Sub-cases: A single Variable or several Variables.</i>
+Each of the two fundamental parts of the Differential
+Calculus is subdivided into two very distinct theories,
+according as we are required to differentiate functions
+of a single variable or functions of several independent
+variables. This second case is, by its nature,
+quite distinct from the first, and evidently presents more
+complication, even in considering only explicit functions,
+and still more those which are implicit. As to the rest,
+one of these cases is deduced from the other in a general
+manner, by the aid of an invariable and very simple
+principle, which consists in regarding the total differential
+of a function which is produced by the simultaneous
+increments of the different independent variables which
+it contains, as the sum of the partial differentials which
+would be produced by the separate increment of each
+variable in turn, if all the others were constant. It is
+necessary, besides, carefully to remark, in connection
+with this subject, a new idea which is introduced by
+the distinction of functions into those of one variable
+and of several; it is the consideration of these different
+special derived functions, relating to each variable separately,
+and the number of which increases more and<span class="pagenum"><a name="Page_130" id="Page_130">[Pg 130]</a></span>
+more in proportion as the order of the derivation becomes
+higher, and also when the variables become more numerous.
+It results from this that the differential relations
+belonging to functions of several variables are, by
+their nature, both much more indirect, and especially
+much more indeterminate, than those relating to functions
+of a single variable. This is most apparent in the
+case of implicit functions, in which, in the place of the
+simple arbitrary constants which elimination causes to
+disappear when we form the proper differential equations
+for functions of a single variable, it is the arbitrary functions
+of the proposed variables which are then eliminated;
+whence must result special difficulties when these
+equations come to be integrated.</p>
+
+<p>Finally, to complete this summary sketch of the different
+essential parts of the differential calculus proper,
+I should add, that in the differentiation of implicit functions,
+whether of a single variable or of several, it is necessary
+to make another distinction; that of the case in
+which it is required to differentiate at once different
+functions of this kind, <i>combined</i> in certain primitive
+equations, from that in which all these functions are
+<i>separate</i>.</p>
+
+<p>The functions are evidently, in fact, still more implicit
+in the first case than in the second, if we consider
+that the same imperfection of ordinary analysis, which
+forbids our converting every implicit function into an
+equivalent explicit function, in like manner renders us
+unable to separate the functions which enter simultaneously
+into any system of equations. It is then necessary
+to differentiate, not only without knowing how
+to resolve the primitive equations, but even without being<span class="pagenum"><a name="Page_131" id="Page_131">[Pg 131]</a></span>
+able to effect the proper eliminations among them,
+thus producing a new difficulty.</p>
+
+
+<p><i>Reduction of the whole to the Differentiation of the
+ten elementary Functions.</i> Such, then, are the natural
+connection and the logical distribution of the different
+principal theories which compose the general system of
+differentiation. Since the differentiation of implicit
+functions is deduced from that of explicit functions by
+a single constant principle, and the differentiation of
+functions of several variables is reduced by another fixed
+principle to that of functions of a single variable, the
+whole of the differential calculus is finally found to rest
+upon the differentiation of explicit functions with a single
+variable, the only one which is ever executed directly.
+Now it is easy to understand that this first theory,
+the necessary basis of the entire system, consists simply
+in the differentiation of the ten simple functions, which
+are the uniform elements of all our analytical combinations,
+and the list of which has been given in the first
+chapter, on page 51; for the differentiation of compound
+functions is evidently deduced, in an immediate and necessary
+manner, from that of the simple functions which
+compose them. It is, then, to the knowledge of these
+ten fundamental differentials, and to that of the two general
+principles just mentioned, which bring under it all
+the other possible cases, that the whole system of differentiation
+is properly reduced. We see, by the combination
+of these different considerations, how simple and
+how perfect is the entire system of the differential calculus.
+It certainly constitutes, in its logical relations,
+the most interesting spectacle which mathematical analysis
+can present to our understanding.</p><p><span class="pagenum"><a name="Page_132" id="Page_132">[Pg 132]</a></span></p>
+
+
+<p><i>Transformation of derived Functions for new Variables.</i>
+The general sketch which I have just summarily
+drawn would nevertheless present an important deficiency,
+if I did not here distinctly indicate a final theory,
+which forms, by its nature, the indispensable complement
+of the system of differentiation. It is that which has
+for its object the constant transformation of derived functions,
+as a result of determinate changes in the independent
+variables, whence results the possibility of referring
+to new variables all the general differential formulas
+primitively established for others. This question
+is now resolved in the most complete and the most simple
+manner, as are all those of which the differential
+calculus is composed. It is easy to conceive the general
+importance which it must have in any of the applications
+of the transcendental analysis, the fundamental
+resources of which it may be considered as augmenting,
+by permitting us to choose (in order to form the differential
+equations, in the first place, with more ease) that
+system of independent variables which may appear to
+be the most advantageous, although it is not to be finally
+retained. It is thus, for example, that most of the
+principal questions of geometry are resolved much more
+easily by referring the lines and surfaces to <i>rectilinear</i>
+co-ordinates, and that we may, nevertheless, have occasion
+to express these lines, etc., analytically by the aid
+of <i>polar</i> co-ordinates, or in any other manner. We will
+then be able to commence the differential solution of the
+problem by employing the rectilinear system, but only
+as an intermediate step, from which, by the general theory
+here referred to, we can pass to the final system,
+which sometimes could not have been considered directly.</p><p><span class="pagenum"><a name="Page_133" id="Page_133">[Pg 133]</a></span></p>
+
+
+<p><i>Different Orders of Differentiation.</i> In the logical
+classification of the differential calculus which has just
+been given, some may be inclined to suggest a serious
+omission, since I have not subdivided each of its four
+essential parts according to another general consideration,
+which seems at first view very important; namely,
+that of the higher or lower order of differentiation. But
+it is easy to understand that this distinction has no real
+influence in the differential calculus, inasmuch as it does
+not give rise to any new difficulty. If, indeed, the differential
+calculus was not rigorously complete, that is,
+if we did not know how to differentiate at will any function
+whatever, the differentiation to the second or higher
+order of each determinate function might engender special
+difficulties. But the perfect universality of the differential
+calculus plainly gives us the assurance of being
+able to differentiate, to any order whatever, all known
+functions whatever, the question reducing itself to a constantly
+repeated differentiation of the first order. This
+distinction, unimportant as it is for the differential calculus,
+acquires, however, a very great importance in the
+integral calculus, on account of the extreme imperfection
+of the latter.</p>
+
+
+<p><i>Analytical Applications.</i> Finally, though this is not
+the place to consider the various applications of the differential
+calculus, yet an exception may be made for
+those which consist in the solution of questions which are
+purely analytical, which ought, indeed, to be logically
+treated in continuation of a system of differentiation, because
+of the evident homogeneity of the considerations
+involved. These questions may be reduced to three essential
+ones.</p><p><span class="pagenum"><a name="Page_134" id="Page_134">[Pg 134]</a></span></p>
+
+<p>Firstly, the <i>development into series</i> of functions of
+one or more variables, or, more generally, the transformation
+of functions, which constitutes the most beautiful
+and the most important application of the differential calculus
+to general analysis, and which comprises, besides
+the fundamental series discovered by Taylor, the remarkable
+series discovered by Maclaurin, John Bernouilli, Lagrange,
+&amp;c.:</p>
+
+<p>Secondly, the general <i>theory of maxima and minima</i>
+values for any functions whatever, of one or more variables;
+one of the most interesting problems which analysis
+can present, however elementary it may now have
+become, and to the complete solution of which the differential
+calculus naturally applies:</p>
+
+<p>Thirdly, the general determination of the true value
+of functions which present themselves under an <i>indeterminate</i>
+appearance for certain hypotheses made on the
+values of the corresponding variables; which is the least
+extensive and the least important of the three.</p>
+
+<p>The first question is certainly the principal one in all
+points of view; it is also the most susceptible of receiving
+a new extension hereafter, especially by conceiving,
+in a broader manner than has yet been done, the employment
+of the differential calculus in the transformation
+of functions, on which subject Lagrange has left
+some valuable hints.</p>
+<p><span class="pagenum"><a name="Page_135" id="Page_135">[Pg 135]</a></span></p>
+<hr class="tb" />
+
+<p>Having thus summarily, though perhaps too briefly,
+considered the chief points in the differential calculus, I
+now proceed to an equally rapid exposition of a systematic
+outline of the Integral Calculus, properly so called,
+that is, the abstract subject of integration.</p>
+
+
+
+
+<h3><a name="THE_INTEGRAL_CALCULUS" id="THE_INTEGRAL_CALCULUS">THE INTEGRAL CALCULUS.</a></h3>
+
+
+<p><i>Its Fundamental Division.</i> The fundamental division
+of the Integral Calculus is founded on the same principle
+as that of the Differential Calculus, in distinguishing
+the integration of <i>explicit</i> differential formulas, and the
+integration of <i>implicit</i> differentials or of differential equations.
+The separation of these two cases is even much
+more profound in relation to integration than to differentiation.
+In the differential calculus, in fact, this distinction
+rests, as we have seen, only on the extreme imperfection
+of ordinary analysis. But, on the other hand,
+it is easy to see that, even though all equations could be
+algebraically resolved, differential equations would none
+the less constitute a case of integration quite distinct
+from that presented by the explicit differential formulas;
+for, limiting ourselves, for the sake of simplicity, to the
+first order, and to a single function <i>y</i> of a single variable
+<i>x</i>, if we suppose any differential equation between <i>x</i>, <i>y</i>,
+and <i>dy/dx</i>, to be resolved with reference to <i>dy/dx</i>, the expression
+of the derived function being then generally found
+to contain the primitive function itself, which is the object
+of the inquiry, the question of integration will not
+have at all changed its nature, and the solution will not
+really have made any other progress than that of having
+brought the proposed differential equation to be of only
+the first degree relatively to the derived function, which
+is in itself of little importance. The differential would
+not then be determined in a manner much less <i>implicit</i>
+than before, as regards the integration, which would continue
+to present essentially the same characteristic difficulty.<span class="pagenum"><a name="Page_136" id="Page_136">[Pg 136]</a></span>
+The algebraic resolution of equations could not
+make the case which we are considering come within the
+simple integration of explicit differentials, except in the
+special cases in which the proposed differential equation
+did not contain the primitive function itself, which would
+consequently permit us, by resolving it, to find <i>dy/dx</i> in
+terms of <i>x</i> only, and thus to reduce the question to the
+class of quadratures. Still greater difficulties would evidently
+be found in differential equations of higher orders,
+or containing simultaneously different functions of several
+independent variables.</p>
+
+<p>The integration of differential equations is then necessarily
+more complicated than that of explicit differentials,
+by the elaboration of which last the integral calculus
+has been created, and upon which the others have been
+made to depend as far as it has been possible. All the
+various analytical methods which have been proposed for
+integrating differential equations, whether it be the separation
+of the variables, the method of multipliers, &amp;c.,
+have in fact for their object to reduce these integrations
+to those of differential formulas, the only one which, by its
+nature, can be undertaken directly. Unfortunately, imperfect
+as is still this necessary base of the whole integral
+calculus, the art of reducing to it the integration of differential
+equations is still less advanced.</p>
+
+
+<p><i>Subdivisions: one variable or several.</i> Each of these
+two fundamental branches of the integral calculus is next
+subdivided into two others (as in the differential calculus,
+and for precisely analogous reasons), according as we
+consider functions with a <i>single variable</i>, or functions
+with <i>several independent variables</i>.</p><p><span class="pagenum"><a name="Page_137" id="Page_137">[Pg 137]</a></span></p>
+
+<p>This distinction is, like the preceding one, still more
+important for integration than for differentiation. This
+is especially remarkable in reference to differential equations.
+Indeed, those which depend on several independent
+variables may evidently present this characteristic
+and much more serious difficulty, that the desired function
+may be differentially defined by a simple relation between
+its different special derivatives relative to the different
+variables taken separately. Hence results the
+most difficult and also the most extensive branch of the
+integral calculus, which is commonly named the <i>Integral
+Calculus of partial differences</i>, created by D'Alembert,
+and in which, according to the just appreciation of
+Lagrange, geometers ought to have seen a really new
+calculus, the philosophical character of which has not yet
+been determined with sufficient exactness. A very striking
+difference between this case and that of equations
+with a single independent variable consists, as has been
+already observed, in the arbitrary functions which take
+the place of the simple arbitrary constants, in order to give
+to the corresponding integrals all the proper generality.</p>
+
+<p>It is scarcely necessary to say that this higher branch
+of transcendental analysis is still entirely in its infancy,
+since, even in the most simple case, that of an equation
+of the first order between the partial derivatives of a single
+function with two independent variables, we are not
+yet completely able to reduce the integration to that of
+the ordinary differential equations. The integration of
+functions of several variables is much farther advanced
+in the case (infinitely more simple indeed) in which it
+has to do with only explicit differential formulas. We
+can then, in fact, when these formulas fulfil the necessary<span class="pagenum"><a name="Page_138" id="Page_138">[Pg 138]</a></span>
+conditions of integrability, always reduce their integration
+to quadratures.</p>
+
+
+<p><i>Other Subdivisions: different Orders of Differentiation.</i>
+A new general distinction, applicable as a subdivision
+to the integration of explicit or implicit differentials,
+with one variable or several, is drawn from the <i>higher
+or lower order of the differentials</i>: a distinction which,
+as we have above remarked, does not give rise to any
+special question in the differential calculus.</p>
+
+<p>Relatively to <i>explicit differentials</i>, whether of one variable
+or of several, the necessity of distinguishing their
+different orders belongs only to the extreme imperfection
+of the integral calculus. In fact, if we could always integrate
+every differential formula of the first order, the
+integration of a formula of the second order, or of any
+other, would evidently not form a new question, since, by
+integrating it at first in the first degree, we would arrive
+at the differential expression of the immediately preceding
+order, from which, by a suitable series of analogous
+integrations, we would be certain of finally arriving at
+the primitive function, the final object of these operations.
+But the little knowledge which we possess on integration
+of even the first order causes quite another state
+of affairs, so that a higher order of differentials produces
+new difficulties; for, having differential formulas of any
+order above the first, it may happen that we may be able
+to integrate them, either once, or several times in succession,
+and that we may still be unable to go back to
+the primitive functions, if these preliminary labours have
+produced, for the differentials of a lower order, expressions
+whose integrals are not known. This circumstance
+must occur so much the oftener (the number of known<span class="pagenum"><a name="Page_139" id="Page_139">[Pg 139]</a></span>
+integrals being still very small), seeing that these successive
+integrals are generally very different functions
+from the derivatives which have produced them.</p>
+
+<p>With reference to <i>implicit differentials</i>, the distinction
+of orders is still more important; for, besides the
+preceding reason, the influence of which is evidently
+analogous in this case, and is even greater, it is easy to
+perceive that the higher order of the differential equations
+necessarily gives rise to questions of a new nature.
+In fact, even if we could integrate every equation of the
+first order relating to a single function, that would not
+be sufficient for obtaining the final integral of an equation
+of any order whatever, inasmuch as every differential
+equation is not reducible to that of an immediately inferior
+order. Thus, for example, if we have given any
+relation between <i>x</i>, <i>y</i>, <i>dx/dy</i>, and <i>d</i><sup>2</sup><i>y</i>/<i>dx</i><sup>2</sup>, to determine a function
+<i>y</i> of a variable <i>x</i>, we shall not be able to deduce
+from it at once, after effecting a first integration, the
+corresponding differential relation between <i>x</i>, <i>y</i>, and <i>dy/dx</i>,
+from which, by a second integration, we could ascend
+to the primitive equations. This would not necessarily
+take place, at least without introducing new auxiliary
+functions, unless the proposed equation of the second order
+did not contain the required function <i>y</i>, together with
+its derivatives. As a general principle, differential equations
+will have to be regarded as presenting cases which
+are more and more <i>implicit</i>, as they are of a higher order,
+and which cannot be made to depend on one another
+except by special methods, the investigation of which
+consequently forms a new class of questions, with respect<span class="pagenum"><a name="Page_140" id="Page_140">[Pg 140]</a></span>
+to which we as yet know scarcely any thing, even
+for functions of a single variable.<a name="FNanchor_10_10" id="FNanchor_10_10"></a><a href="#Footnote_10_10" class="fnanchor">[10]</a></p>
+
+<p><i>Another equivalent distinction.</i> Still farther, when
+we examine more profoundly this distinction of different
+orders of differential equations, we find that it can be
+always made to come under a final general distinction,
+relative to differential equations, which remains to be
+noticed. Differential equations with one or more independent
+variables may contain simply a single function,
+or (in a case evidently more complicated and more implicit,
+which corresponds to the differentiation of simultaneous
+implicit functions) we may have to determine
+at the same time several functions from the differential
+equations in which they are found united, together with
+their different derivatives. It is clear that such a state
+of the question necessarily presents a new special difficulty,
+that of separating the different functions desired,
+by forming for each, from the proposed differential equations,
+an isolated differential equation which does not
+contain the other functions or their derivatives. This
+preliminary labour, which is analogous to the elimination
+of algebra, is evidently indispensable before attempting
+any direct integration, since we cannot undertake
+generally (except by special artifices which are very
+rarely applicable) to determine directly several distinct
+functions at once.</p>
+
+<p>Now it is easy to establish the exact and necessary
+coincidence of this new distinction with the preceding
+<span class="pagenum"><a name="Page_141" id="Page_141">[Pg 141]</a></span>one respecting the order of differential equations. We
+know, in fact, that the general method for isolating functions
+in simultaneous differential equations consists essentially
+in forming differential equations, separately in
+relation to each function, and of an order equal to the
+sum of all those of the different proposed equations.
+This transformation can always be effected. On the
+other hand, every differential equation of any order in
+relation to a single function might evidently always be
+reduced to the first order, by introducing a suitable number
+of auxiliary differential equations, containing at the
+same time the different anterior derivatives regarded as
+new functions to be determined. This method has, indeed,
+sometimes been actually employed with success,
+though it is not the natural one.</p>
+
+<p>Here, then, are two necessarily equivalent orders of
+conditions in the general theory of differential equations;
+the simultaneousness of a greater or smaller number of
+functions, and the higher or lower order of differentiation
+of a single function. By augmenting the order of
+the differential equations, we can isolate all the functions;
+and, by artificially multiplying the number of
+the functions, we can reduce all the equations to the
+first order. There is, consequently, in both cases, only
+one and the same difficulty from two different points of
+sight. But, however we may conceive it, this new difficulty
+is none the less real, and constitutes none the
+less, by its nature, a marked separation between the integration
+of equations of the first order and that of equations
+of a higher order. I prefer to indicate the distinction
+under this last form as being more simple, more
+general, and more logical.</p><p><span class="pagenum"><a name="Page_142" id="Page_142">[Pg 142]</a></span></p>
+
+
+<p><i>Quadratures.</i> From the different considerations
+which have been indicated respecting the logical dependence
+of the various principal parts of the integral calculus,
+we see that the integration of explicit differential
+formulas of the first order and of a single variable is the
+necessary basis of all other integrations, which we never
+succeed in effecting but so far as we reduce them to this
+elementary case, evidently the only one which, by its
+nature, is capable of being treated directly. This simple
+fundamental integration is often designated by the
+convenient expression of <i>quadratures</i>, seeing that every
+integral of this kind, S<i>f</i>(<i>x</i>)<i>dx</i>, may, in fact, be regarded
+as representing the area of a curve, the equation of which
+in rectilinear co-ordinates would be <i>y</i> = <i>f</i>(<i>x</i>). Such a
+class of questions corresponds, in the differential calculus,
+to the elementary case of the differentiation of explicit
+functions of a single variable. But the integral question
+is, by its nature, very differently complicated, and
+especially much more extensive than the differential
+question. This latter is, in fact, necessarily reduced, as
+we have seen, to the differentiation of the ten simple
+functions, the elements of all which are considered in
+analysis. On the other hand, the integration of compound
+functions does not necessarily follow from that of
+the simple functions, each combination of which may
+present special difficulties with respect to the integral
+calculus. Hence results the naturally indefinite extent,
+and the so varied complication of the question of <i>quadratures</i>,
+upon which, in spite of all the efforts of analysts,
+we still possess so little complete knowledge.</p>
+
+<p>In decomposing this question, as is natural, according
+to the different forms which may be assumed by the<span class="pagenum"><a name="Page_143" id="Page_143">[Pg 143]</a></span>
+derivative function, we distinguish the case of <i>algebraic</i>
+functions and that of <i>transcendental</i> functions.</p>
+
+<p><i>Integration of Transcendental Functions.</i> The truly
+analytical integration of transcendental functions is as
+yet very little advanced, whether for <i>exponential</i>, or for
+<i>logarithmic</i>, or for <i>circular</i> functions. But a very small
+number of cases of these three different kinds have as
+yet been treated, and those chosen from among the simplest;
+and still the necessary calculations are in most
+cases extremely laborious. A circumstance which we
+ought particularly to remark in its philosophical connection
+is, that the different procedures of quadrature
+have no relation to any general view of integration, and
+consist of simple artifices very incoherent with each other,
+and very numerous, because of the very limited extent
+of each.</p>
+
+<p>One of these artifices should, however, here be noticed,
+which, without being really a method of integration,
+is nevertheless remarkable for its generality; it is
+the procedure invented by John Bernouilli, and known
+under the name of <i>integration by parts</i>, by means of
+which every integral may be reduced to another which
+is sometimes found to be more easy to be obtained.
+This ingenious relation deserves to be noticed for another
+reason, as having suggested the first idea of that transformation
+of integrals yet unknown, which has lately
+received a greater extension, and of which M. Fourier
+especially has made so new and important a use in the
+analytical questions produced by the theory of heat.</p>
+
+<p><i>Integration of Algebraic Functions.</i> As to the integration
+of algebraic functions, it is farther advanced.
+However, we know scarcely any thing in relation to irrational<span class="pagenum"><a name="Page_144" id="Page_144">[Pg 144]</a></span>
+functions, the integrals of which have been obtained
+only in extremely limited cases, and particularly by
+rendering them rational. The integration of rational
+functions is thus far the only theory of the integral calculus
+which has admitted of being treated in a truly complete
+manner; in a logical point of view, it forms, then,
+its most satisfactory part, but perhaps also the least important.
+It is even essential to remark, in order to have
+a just idea of the extreme imperfection of the integral
+calculus, that this case, limited as it is, is not entirely
+resolved except for what properly concerns integration
+viewed in an abstract manner; for, in the execution, the
+theory finds its progress most frequently quite stopped,
+independently of the complication of the calculations, by
+the imperfection of ordinary analysis, seeing that it
+makes the integration finally depend upon the algebraic
+resolution of equations, which greatly limits its use.</p>
+
+<p>To grasp in a general manner the spirit of the different
+procedures which are employed in quadratures, we
+must observe that, by their nature, they can be primitively
+founded only on the differentiation of the ten simple
+functions. The results of this, conversely considered,
+establish as many direct theorems of the integral calculus,
+the only ones which can be directly known. All the
+art of integration afterwards consists, as has been said
+in the beginning of this chapter, in reducing all the other
+quadratures, so far as is possible, to this small number
+of elementary ones, which unhappily we are in most
+cases unable to effect.</p>
+
+<p><i>Singular Solutions.</i> In this systematic enumeration
+of the various essential parts of the integral calculus, considered
+in their logical relations, I have designedly neglected<span class="pagenum"><a name="Page_145" id="Page_145">[Pg 145]</a></span>
+(in order not to break the chain of sequence) to
+consider a very important theory, which forms implicitly
+a portion of the general theory of the integration of differential
+equations, but which I ought here to notice separately,
+as being, so to speak, outside of the integral calculus,
+and being nevertheless of the greatest interest, both
+by its logical perfection and by the extent of its applications.
+I refer to what are called <i>Singular Solutions</i>
+of differential equations, called sometimes, but improperly,
+<i>particular</i> solutions, which have been the subject
+of very remarkable investigations by Euler and Laplace,
+and of which Lagrange especially has presented such a
+beautiful and simple general theory. Clairaut, who first
+had occasion to remark their existence, saw in them a
+paradox of the integral calculus, since these solutions
+have the peculiarity of satisfying the differential equations
+without being comprised in the corresponding general
+integrals. Lagrange has since explained this paradox
+in the most ingenious and most satisfactory manner,
+by showing how such solutions are always derived
+from the general integral by the variation of the arbitrary
+constants. He was also the first to suitably appreciate
+the importance of this theory, and it is with
+good reason that he devoted to it so full a development
+in his "Calculus of Functions." In a logical point of
+view, this theory deserves all our attention by the character
+of perfect generality which it admits of, since Lagrange
+has given invariable and very simple procedures
+for finding the <i>singular</i> solution of any differential equation
+which is susceptible of it; and, what is no less remarkable,
+these procedures require no integration, consisting
+only of differentiations, and are therefore always<span class="pagenum"><a name="Page_146" id="Page_146">[Pg 146]</a></span>
+applicable. Differentiation has thus become, by a happy
+artifice, a means of compensating, in certain circumstances,
+for the imperfection of the integral calculus.
+Indeed, certain problems especially require, by their nature,
+the knowledge of these <i>singular</i> solutions; such,
+for example, in geometry, are all the questions in which
+a curve is to be determined from any property of its tangent
+or its osculating circle. In all cases of this kind,
+after having expressed this property by a differential
+equation, it will be, in its analytical relations, the <i>singular</i>
+equation which will form the most important object
+of the inquiry, since it alone will represent the required
+curve; the general integral, which thenceforth it
+becomes unnecessary to know, designating only the system
+of the tangents, or of the osculating circles of this
+curve. We may hence easily understand all the importance
+of this theory, which seems to me to be not as yet
+sufficiently appreciated by most geometers.</p>
+
+<p><i>Definite Integrals.</i> Finally, to complete our review
+of the vast collection of analytical researches of which is
+composed the integral calculus, properly so called, there
+remains to be mentioned one theory, very important in
+all the applications of the transcendental analysis, which
+I have had to leave outside of the system, as not being
+really destined for veritable integration, and proposing, on
+the contrary, to supply the place of the knowledge of truly
+analytical integrals, which are most generally unknown.
+I refer to the determination of <i>definite integrals</i>.</p>
+
+<p>The expression, always possible, of integrals in infinite
+series, may at first be viewed as a happy general
+means of compensating for the extreme imperfection of
+the integral calculus. But the employment of such series,<span class="pagenum"><a name="Page_147" id="Page_147">[Pg 147]</a></span>
+because of their complication, and of the difficulty
+of discovering the law of their terms, is commonly of only
+moderate utility in the algebraic point of view, although
+sometimes very essential relations have been thence deduced.
+It is particularly in the arithmetical point of
+view that this procedure acquires a great importance, as
+a means of calculating what are called <i>definite integrals</i>,
+that is, the values of the required functions for certain
+determinate values of the corresponding variables.</p>
+
+<p>An inquiry of this nature exactly corresponds, in transcendental
+analysis, to the numerical resolution of equations
+in ordinary analysis. Being generally unable to
+obtain the veritable integral&mdash;named by opposition the
+<i>general</i> or <i>indefinite</i> integral; that is, the function which,
+differentiated, has produced the proposed differential formula&mdash;analysts
+have been obliged to employ themselves
+in determining at least, without knowing this function,
+the particular numerical values which it would take on
+assigning certain designated values to the variables.
+This is evidently resolving the arithmetical question
+without having previously resolved the corresponding algebraic
+one, which most generally is the most important
+one. Such an analysis is, then, by its nature, as
+imperfect as we have seen the numerical resolution of
+equations to be. It presents, like this last, a vicious
+confusion of arithmetical and algebraic considerations,
+whence result analogous inconveniences both in the
+purely logical point of view and in the applications.
+We need not here repeat the considerations suggested in
+our third chapter. But it will be understood that, unable
+as we almost always are to obtain the true integrals,
+it is of the highest importance to have been able<span class="pagenum"><a name="Page_148" id="Page_148">[Pg 148]</a></span>
+to obtain this solution, incomplete and necessarily insufficient
+as it is. Now this has been fortunately attained
+at the present day for all cases, the determination of
+the value of definite integrals having been reduced to
+entirely general methods, which leave nothing to desire,
+in a great number of cases, but less complication in the
+calculations, an object towards which are at present directed
+all the special transformations of analysts. Regarding
+now this sort of <i>transcendental arithmetic</i> as
+perfect, the difficulty in the applications is essentially
+reduced to making the proposed research depend, finally,
+on a simple determination of definite integrals, which
+evidently cannot always be possible, whatever analytical
+skill may be employed in effecting such a transformation.</p>
+
+
+<p><i>Prospects of the Integral Calculus.</i> From the considerations
+indicated in this chapter, we see that, while
+the differential calculus constitutes by its nature a limited
+and perfect system, to which nothing essential remains
+to be added, the integral calculus, or the simple system
+of integration, presents necessarily an inexhaustible field
+for the activity of the human mind, independently of
+the indefinite applications of which the transcendental
+analysis is evidently susceptible. The general argument
+by which I have endeavoured, in the second chapter,
+to make apparent the impossibility of ever discovering
+the algebraic solution of equations of any degree and
+form whatsoever, has undoubtedly infinitely more force
+with regard to the search for a single method of integration,
+invariably applicable to all cases. "It is," says
+Lagrange, "one of those problems whose general solution
+we cannot hope for." The more we meditate on<span class="pagenum"><a name="Page_149" id="Page_149">[Pg 149]</a></span>
+this subject, the more we will be convinced that such a
+research is utterly chimerical, as being far above the feeble
+reach of our intelligence; although the labours of
+geometers must certainly augment hereafter the amount
+of our knowledge respecting integration, and thus create
+methods of greater generality. The transcendental analysis
+is still too near its origin&mdash;there is especially too
+little time since it has been conceived in a truly rational
+manner&mdash;for us now to be able to have a correct idea of
+what it will hereafter become. But, whatever should be
+our legitimate hopes, let us not forget to consider, before
+all, the limits which are imposed by our intellectual constitution,
+and which, though not susceptible of a precise
+determination, have none the less an incontestable reality.</p>
+
+<p>I am induced to think that, when geometers shall have
+exhausted the most important applications of our present
+transcendental analysis, instead of striving to impress
+upon it, as now conceived, a chimerical perfection, they
+will rather create new resources by changing the mode
+of derivation of the auxiliary quantities introduced in
+order to facilitate the establishment of equations, and
+the formation of which might follow an infinity of other
+laws besides the very simple relation which has been
+chosen, according to the conception suggested in the first
+chapter. The resources of this nature appear to me susceptible
+of a much greater fecundity than those which
+would consist of merely pushing farther our present calculus
+of indirect functions. It is a suggestion which I
+submit to the geometers who have turned their thoughts
+towards the general philosophy of analysis.</p>
+
+<p>Finally, although, in the summary exposition which
+was the object of this chapter, I have had to exhibit the<span class="pagenum"><a name="Page_150" id="Page_150">[Pg 150]</a></span>
+condition of extreme imperfection which still belongs to
+the integral calculus, the student would have a false idea
+of the general resources of the transcendental analysis if
+he gave that consideration too great an importance. It
+is with it, indeed, as with ordinary analysis, in which a
+very small amount of fundamental knowledge respecting
+the resolution of equations has been employed with an
+immense degree of utility. Little advanced as geometers
+really are as yet in the science of integrations, they
+have nevertheless obtained, from their scanty abstract
+conceptions, the solution of a multitude of questions of
+the first importance in geometry, in mechanics, in thermology,
+&amp;c. The philosophical explanation of this
+double general fact results from the necessarily preponderating
+importance and grasp of <i>abstract</i> branches of
+knowledge, the least of which is naturally found to correspond
+to a crowd of <i>concrete</i> researches, man having
+no other resource for the successive extension of his intellectual
+means than in the consideration of ideas more
+and more abstract, and still positive.</p>
+<p><span class="pagenum"><a name="Page_151" id="Page_151">[Pg 151]</a></span></p>
+<hr class="tb" />
+
+<p>In order to finish the complete exposition of the philosophical
+character of the transcendental analysis, there
+remains to be considered a final conception, by which
+the immortal Lagrange has rendered this analysis still
+better adapted to facilitate the establishment of equations
+in the most difficult problems, by considering a class of
+equations still more <i>indirect</i> than the ordinary differential
+equations. It is the <i>Calculus</i>, or, rather, the <i>Method
+of Variations</i>; the general appreciation of which will be
+our next subject.</p>
+
+
+
+
+<h2><a name="CHAPTER_V" id="CHAPTER_V">CHAPTER V.</a></h2>
+
+<h3>THE CALCULUS OF VARIATIONS.</h3>
+
+
+<p>In order to grasp with more ease the philosophical
+character of the <i>Method of Variations</i>, it will be well to
+begin by considering in a summary manner the special
+nature of the problems, the general resolution of which
+has rendered necessary the formation of this hyper-transcendental
+analysis. It is still too near its origin, and
+its applications have been too few, to allow us to obtain
+a sufficiently clear general idea of it from a purely abstract
+exposition of its fundamental theory.</p>
+
+
+
+
+<h3><a name="PROBLEMS_GIVING_RISE_TO_IT" id="PROBLEMS_GIVING_RISE_TO_IT">PROBLEMS GIVING RISE TO IT.</a></h3>
+
+
+<p>The mathematical questions which have given birth
+to the <i>Calculus of Variations</i> consist generally in the
+investigation of the <i>maxima</i> and <i>minima</i> of certain indeterminate
+integral formulas, which express the analytical
+law of such or such a phenomenon of geometry
+or mechanics, considered independently of any particular
+subject. Geometers for a long time designated all the
+questions of this character by the common name of <i>Isoperimetrical
+Problems</i>, which, however, is really suitable
+to only the smallest number of them.</p>
+
+
+<p><i>Ordinary Questions of Maxima and Minima.</i> In
+the common theory of <i>maxima</i> and <i>minima</i>, it is proposed
+to discover, with reference to a given function of
+one or more variables, what particular values must be
+assigned to these variables, in order that the corresponding<span class="pagenum"><a name="Page_152" id="Page_152">[Pg 152]</a></span>
+value of the proposed function may be a <i>maximum</i>
+or a <i>minimum</i> with respect to those values which immediately
+precede and follow it; that is, properly speaking,
+we seek to know at what instant the function ceases
+to increase and commences to decrease, or reciprocally.
+The differential calculus is perfectly sufficient, as we
+know, for the general resolution of this class of questions,
+by showing that the values of the different variables,
+which suit either the maximum or minimum, must
+always reduce to zero the different first derivatives of
+the given function, taken separately with reference to
+each independent variable, and by indicating, moreover,
+a suitable characteristic for distinguishing the maximum
+from the minimum; consisting, in the case of a function
+of a single variable, for example, in the derived function
+of the second order taking a negative value for the maximum,
+and a positive value for the minimum. Such
+are the well-known fundamental conditions belonging to
+the greatest number of cases.</p>
+
+
+<p><i>A new Class of Questions.</i> The construction of this
+general theory having necessarily destroyed the chief
+interest which questions of this kind had for geometers,
+they almost immediately rose to the consideration of a
+new order of problems, at once much more important and
+of much greater difficulty&mdash;those of <i>isoperimeters</i>. It
+is, then, no longer <i>the values of the variables</i> belonging
+to the maximum or the minimum of a given function
+that it is required to determine. It is <i>the form of the
+function itself</i> which is required to be discovered, from
+the condition of the maximum or of the minimum of a
+certain definite integral, merely indicated, which depends
+upon that function.</p><p><span class="pagenum"><a name="Page_153" id="Page_153">[Pg 153]</a></span></p>
+
+
+<p><i>Solid of least Resistance.</i> The oldest question of
+this nature is that of <i>the solid of least resistance</i>, treated
+by Newton in the second book of the Principia, in
+which he determines what ought to be the meridian
+curve of a solid of revolution, in order that the resistance
+experienced by that body in the direction of its axis
+may be the least possible. But the course pursued by
+Newton, from the nature of his special method of transcendental
+analysis, had not a character sufficiently simple,
+sufficiently general, and especially sufficiently analytical,
+to attract geometers to this new order of problems.
+To effect this, the application of the infinitesimal
+method was needed; and this was done, in 1695, by
+John Bernouilli, in proposing the celebrated problem of
+the <i>Brachystochrone</i>.</p>
+
+<p>This problem, which afterwards suggested such a long
+series of analogous questions, consists in determining
+the curve which a heavy body must follow in order to
+descend from one point to another in the shortest possible
+time. Limiting the conditions to the simple fall
+in a vacuum, the only case which was at first considered,
+it is easily found that the required curve must be
+a reversed cycloid with a horizontal base, and with its
+origin at the highest point. But the question may become
+singularly complicated, either by taking into account
+the resistance of the medium, or the change in the
+intensity of gravity.</p>
+
+
+<p><i>Isoperimeters.</i> Although this new class of problems
+was in the first place furnished by mechanics, it is in
+geometry that the principal investigations of this character
+were subsequently made. Thus it was proposed
+to discover which, among all the curves of the same contour<span class="pagenum"><a name="Page_154" id="Page_154">[Pg 154]</a></span>
+traced between two given points, is that whose area
+is a maximum or minimum, whence has come the name
+of <i>Problem of Isoperimeters</i>; or it was required that
+the maximum or minimum should belong to the surface
+produced by the revolution of the required curve about
+an axis, or to the corresponding volume; in other cases,
+it was the vertical height of the center of gravity of the
+unknown curve, or of the surface and of the volume
+which it might generate, which was to become a maximum
+or minimum, &amp;c. Finally, these problems were
+varied and complicated almost to infinity by the Bernouillis,
+by Taylor, and especially by Euler, before Lagrange
+reduced their solution to an abstract and entirely
+general method, the discovery of which has put a
+stop to the enthusiasm of geometers for such an order of
+inquiries. This is not the place for tracing the history
+of this subject. I have only enumerated some of the
+simplest principal questions, in order to render apparent
+the original general object of the method of variations.</p>
+
+
+<p><i>Analytical Nature of these Problems.</i> We see that
+all these problems, considered in an analytical point of
+view, consist, by their nature, in determining what form
+a certain unknown function of one or more variables
+ought to have, in order that such or such an integral,
+dependent upon that function, shall have, within assigned
+limits, a value which is a maximum or a minimum
+with respect to all those which it would take if the required
+function had any other form whatever.</p>
+
+<p>Thus, for example, in the problem of the <i>brachystochrone</i>,
+it is well known that if <i>y</i> = <i>f(z)</i>, <i>x</i> = &#960;(<i>z</i>), are the
+rectilinear equations of the required curve, supposing
+the axes of <i>x</i> and of <i>y</i> to be horizontal, and the axis of<span class="pagenum"><a name="Page_155" id="Page_155">[Pg 155]</a></span>
+<i>z</i> to be vertical, the time of the fall of a heavy body in
+that curve from the point whose ordinate is <i>z<sub>1</sub></i>, to that
+whose ordinate is <i>z<sub>2</sub></i>, is expressed in general terms by
+the definite integral</p>
+
+<p>
+&#8747;_{<i>z_{2}</i>}<sup><i>z_{1</i></sup>}&#8730;(1 + (<i>f'(z))<sup>2</sup></i> + (&#960;'(<i>z</i>))<sup>2</sup>/(2<i>gz</i>))<i>dz.</i><br />
+</p>
+
+<p>It is, then, necessary to find what the two unknown
+functions <i>f</i> and &#960; must be, in order that this integral
+may be a minimum.</p>
+
+<p>In the same way, to demand what is the curve among
+all plane isoperimetrical curves, which includes the greatest
+area, is the same thing as to propose to find, among
+all the functions <i>f(x)</i> which can give a certain constant
+value to the integral</p>
+
+<p>
+&#8747;<i>dx</i>&#8730;(1 + (<i>f'(x)</i> )<sup>2</sup>),<br />
+</p>
+
+<p>that one which renders the integral &#8747;<i>f(x)dx</i>, taken between
+the same limits, a maximum. It is evidently always
+so in other questions of this class.</p>
+
+
+<p><i>Methods of the older Geometers.</i> In the solutions
+which geometers before Lagrange gave of these problems,
+they proposed, in substance, to reduce them to the
+ordinary theory of maxima and minima. But the means
+employed to effect this transformation consisted in special
+simple artifices peculiar to each case, and the discovery
+of which did not admit of invariable and certain
+rules, so that every really new question constantly reproduced
+analogous difficulties, without the solutions previously
+obtained being really of any essential aid, otherwise
+than by their discipline and training of the mind.
+In a word, this branch of mathematics presented, then,
+the necessary imperfection which always exists when the
+part common to all questions of the same class has not<span class="pagenum"><a name="Page_156" id="Page_156">[Pg 156]</a></span>
+yet been distinctly grasped in order to be treated in an
+abstract and thenceforth general manner.</p>
+
+
+
+
+<h3><a name="METHOD_OF_LAGRANGE2" id="METHOD_OF_LAGRANGE2">METHOD OF LAGRANGE.</a></h3>
+
+
+<p>Lagrange, in endeavouring to bring all the different
+problems of isoperimeters to depend upon a common analysis,
+organized into a distinct calculus, was led to conceive
+a new kind of differentiation, to which he has applied
+the characteristic &#948;, reserving the characteristic <i>d</i>
+for the common differentials. These differentials of a
+new species, which he has designated under the name of
+<i>Variations</i>, consist of the infinitely small increments
+which the integrals receive, not by virtue of analogous
+increments on the part of the corresponding variables, as
+in the ordinary transcendental analysis, but by supposing
+that the <i>form</i> of the function placed under the sign of
+integration undergoes an infinitely small change. This
+distinction is easily conceived with reference to curves,
+in which we see the ordinate, or any other variable of
+the curve, admit of two sorts of differentials, evidently
+very different, according as we pass from one point to another
+infinitely near it on the same curve, or to the corresponding
+point of the infinitely near curve produced by
+a certain determinate modification of the first curve.<a name="FNanchor_11_11" id="FNanchor_11_11"></a><a href="#Footnote_11_11" class="fnanchor">[11]</a> It
+is moreover clear, that the relative <i>variations</i> of different
+magnitudes connected with each other by any laws
+whatever are calculated, all but the characteristic, almost
+exactly in the same manner as the differentials. Finally,
+<span class="pagenum"><a name="Page_157" id="Page_157">[Pg 157]</a></span>from the general notion of <i>variations</i> are in like manner
+deduced the fundamental principles of the algorithm
+proper to this method, consisting simply in the evidently
+permissible liberty of transposing at will the characteristics
+specially appropriated to variations, before or after
+those which correspond to the ordinary differentials.</p>
+
+<p>This abstract conception having been once formed, Lagrange
+was able to reduce with ease, and in the most
+general manner, all the problems of <i>Isoperimeters</i> to the
+simple ordinary theory of <i>maxima</i> and <i>minima</i>. To obtain
+a clear idea of this great and happy transformation,
+we must previously consider an essential distinction which
+arises in the different questions of isoperimeters.</p>
+
+
+<p><i>Two Classes of Questions.</i> These investigations
+must, in fact, be divided into two general classes, according
+as the maxima and minima demanded are <i>absolute</i>
+or <i>relative</i>, to employ the abridged expressions of
+geometers.</p>
+
+
+<p><i>Questions of the first Class.</i> The <i>first case</i> is that
+in which the indeterminate definite integrals, the maximum
+or minimum of which is sought, are not subjected,
+by the nature of the problem, to any condition; as happens,
+for example, in the problem of the <i>brachystochrone</i>,
+in which the choice is to be made between all imaginable
+curves. The <i>second</i> case takes place when, on the
+contrary, the variable integrals can vary only according
+to certain conditions, which usually consist in other definite
+integrals (which depend, in like manner, upon the
+required functions) always retaining the same given value;
+as, for example, in all the geometrical questions relating
+to real <i>isoperimetrical</i> figures, and in which, by
+the nature of the problem, the integral relating to the<span class="pagenum"><a name="Page_158" id="Page_158">[Pg 158]</a></span>
+length of the curve, or to the area of the surface, must
+remain constant during the variation of that integral
+which is the object of the proposed investigation.</p>
+
+<p>The <i>Calculus of Variations</i> gives immediately the
+general solution of questions of the former class; for it
+evidently follows, from the ordinary theory of maxima
+and minima, that the required relation must reduce to
+zero the <i>variation</i> of the proposed integral with reference
+to each independent variable; which gives the condition
+common to both the maximum and the minimum: and,
+as a characteristic for distinguishing the one from the
+other, that the variation of the second order of the same
+integral must be negative for the maximum and positive
+for the minimum. Thus, for example, in the problem
+of the brachystochrone, we will have, in order to determine
+the nature of the curve sought, the equation of
+condition</p>
+
+<p>
+&#948;&#8747;_{<i>z_{2}</i>}<sup><i>z_{1</i></sup>}&#8730;([1 + (<i>f'</i>(<i>z</i>))<sup>2</sup> + (&#960;'(<i>z</i>))<sup>2</sup>]/(2<i>gz</i>))<i>dz</i> = 0,<br />
+</p>
+
+<p>which, being decomposed into two, with respect to the
+two unknown functions <i>f</i> and &#960;, which are independent
+of each other, will completely express the analytical
+definition of the required curve. The only difficulty
+peculiar to this new analysis consists in the elimination
+of the characteristic &#948;, for which the calculus of variations
+furnishes invariable and complete rules, founded, in
+general, on the method of "integration by parts," from
+which Lagrange has thus derived immense advantage.
+The constant object of this first analytical elaboration
+(which this is not the place for treating in detail) is to
+arrive at real differential equations, which can always
+be done; and thereby the question comes under the ordinary<span class="pagenum"><a name="Page_159" id="Page_159">[Pg 159]</a></span>
+transcendental analysis, which furnishes the solution,
+at least so far as to reduce it to pure algebra if
+the integration can be effected. The general object of
+the method of variations is to effect this transformation,
+for which Lagrange has established rules, which are simple,
+invariable, and certain of success.</p>
+
+
+<p><i>Equations of Limits.</i> Among the greatest special
+advantages of the method of variations, compared with
+the previous isolated solutions of isoperimetrical problems,
+is the important consideration of what Lagrange
+calls <i>Equations of Limits</i>, which were entirely neglected
+before him, though without them the greater part of
+the particular solutions remained necessarily incomplete.
+When the limits of the proposed integrals are to be fixed,
+their variations being zero, there is no occasion for
+noticing them. But it is no longer so when these limits,
+instead of being rigorously invariable, are only subjected
+to certain conditions; as, for example, if the two points
+between which the required curve is to be traced are
+not fixed, and have only to remain upon given lines or
+surfaces. Then it is necessary to pay attention to the
+variation of their co-ordinates, and to establish between
+them the relations which correspond to the equations of
+these lines or of these surfaces.</p>
+
+
+<p><i>A more general consideration.</i> This essential consideration
+is only the final complement of a more general
+and more important consideration relative to the
+variations of different independent variables. If these
+variables are really independent of one another, as when
+we compare together all the imaginable curves susceptible
+of being traced between two points, it will be the
+same with their variations, and, consequently, the terms<span class="pagenum"><a name="Page_160" id="Page_160">[Pg 160]</a></span>
+relating to each of these variations will have to be separately
+equal to zero in the general equation which expresses
+the maximum or the minimum. But if, on the
+contrary, we suppose the variables to be subjected to any
+fixed conditions, it will be necessary to take notice of the
+resulting relation between their variations, so that the
+number of the equations into which this general equation
+is then decomposed is always equal to only the
+number of the variables which remain truly independent.
+It is thus, for example, that instead of seeking
+for the shortest path between any two points, in choosing
+it from among all possible ones, it may be proposed to
+find only what is the shortest among all those which
+may be taken on any given surface; a question the general
+solution of which forms certainly one of the most
+beautiful applications of the method of variations.</p>
+
+<p><i>Questions of the second Class.</i> Problems in which
+such modifying conditions are considered approach very
+nearly, in their nature, to the second general class of
+applications of the method of variations, characterized
+above as consisting in the investigation of <i>relative</i> maxima
+and minima. There is, however, this essential difference
+between the two cases, that in this last the
+modification is expressed by an integral which depends
+upon the function sought, while in the other it is designated
+by a finite equation which is immediately given.
+It is hence apparent that the investigation of <i>relative</i>
+maxima and minima is constantly and necessarily more
+complicated than that of <i>absolute</i> maxima and minima.
+Luckily, a very important general theory, discovered by
+the genius of the great Euler before the invention of
+the Calculus of Variations, gives a uniform and very<span class="pagenum"><a name="Page_161" id="Page_161">[Pg 161]</a></span>
+simple means of making one of these two classes of
+questions dependent on the other. It consists in this,
+that if we add to the integral which is to be a maximum
+or a minimum, a constant and indeterminate multiple
+of that one which, by the nature of the problem, is to
+remain constant, it will be sufficient to seek, by the general
+method of Lagrange above indicated, the <i>absolute</i>
+maximum or minimum of this whole expression. It
+can be easily conceived, indeed, that the part of the complete
+variation which would proceed from the last integral
+must be equal to zero (because of the constant
+character of this last) as well as the portion due to the
+first integral, which disappears by virtue of the maximum
+or minimum state. These two conditions evidently
+unite to produce, in that respect, effects exactly
+alike.</p>
+
+<p>Such is a sketch of the general manner in which the
+method of variation is applied to all the different questions
+which compose what is called the <i>Theory of Isoperimeters</i>.
+It will undoubtedly have been remarked in
+this summary exposition how much use has been made
+in this new analysis of the second fundamental property
+of the transcendental analysis noticed in the third chapter,
+namely, the generality of the infinitesimal expressions
+for the representation of the same geometrical or
+mechanical phenomenon, in whatever body it may be
+considered. Upon this generality, indeed, are founded,
+by their nature, all the solutions due to the method of
+variations. If a single formula could not express the
+length or the area of any curve whatever; if another
+fixed formula could not designate the time of the fall of
+a heavy body, according to whatever line it may descend,<span class="pagenum"><a name="Page_162" id="Page_162">[Pg 162]</a></span>
+&amp;c., how would it have been possible to resolve
+questions which unavoidably require, by their nature, the
+simultaneous consideration of all the cases which can be
+determined in each phenomenon by the different subjects
+which exhibit it.</p>
+
+
+<p><i>Other Applications of this Method.</i> Notwithstanding
+the extreme importance of the theory of isoperimeters,
+and though the method of variations had at first no
+other object than the logical and general solution of this
+order of problems, we should still have but an incomplete
+idea of this beautiful analysis if we limited its
+destination to this. In fact, the abstract conception of
+two distinct natures of differentiation is evidently applicable
+not only to the cases for which it was created, but
+also to all those which present, for any reason whatever,
+two different manners of making the same magnitudes
+vary. It is in this way that Lagrange himself has made,
+in his "<i>Méchanique Analytique</i>," an extensive and important
+application of his calculus of variations, by employing
+it to distinguish the two sorts of changes which
+are naturally presented by the questions of rational mechanics
+for the different points which are considered, according
+as we compare the successive positions which
+are occupied, in virtue of its motion, by the same point
+of each body in two consecutive instants, or as we pass
+from one point of the body to another in the same instant.
+One of these comparisons produces ordinary differentials;
+the other gives rise to <i>variations</i>, which, there as every
+where, are only differentials taken under a new point of
+view. Such is the general acceptation in which we
+should conceive the Calculus of Variations, in order suitably
+to appreciate the importance of this admirable logical<span class="pagenum"><a name="Page_163" id="Page_163">[Pg 163]</a></span>
+instrument, the most powerful that the human mind
+has as yet constructed.</p>
+
+<p>The method of variations being only an immense extension
+of the general transcendental analysis, I have no
+need of proving specially that it is susceptible of being
+considered under the different fundamental points of view
+which the calculus of indirect functions, considered as a
+whole, admits of. Lagrange invented the Calculus of
+Variations in accordance with the infinitesimal conception,
+and, indeed, long before he undertook the general reconstruction
+of the transcendental analysis. When he
+had executed this important reformation, he easily showed
+how it could also be applied to the Calculus of Variations,
+which he expounded with all the proper development,
+according to his theory of derivative functions.
+But the more that the use of the method of variations is
+difficult of comprehension, because of the higher degree
+of abstraction of the ideas considered, the more necessary
+is it, in its application, to economize the exertions of the
+mind, by adopting the most direct and rapid analytical
+conception, namely, that of Leibnitz. Accordingly, Lagrange
+himself has constantly preferred it in the important
+use which he has made of the Calculus of Variations
+in his "Analytical Mechanics." In fact, there does
+not exist the least hesitation in this respect among geometers.</p>
+
+
+
+
+<h3><a name="ITS_RELATIONS_TO_THE_ORDINARY_CALCULUS" id="ITS_RELATIONS_TO_THE_ORDINARY_CALCULUS">ITS RELATIONS TO THE ORDINARY CALCULUS.</a></h3>
+
+
+<p>In order to make as clear as possible the philosophical
+character of the Calculus of Variations, I think that I
+should, in conclusion, briefly indicate a consideration
+which seems to me important, and by which I can approach<span class="pagenum"><a name="Page_164" id="Page_164">[Pg 164]</a></span>
+it to the ordinary transcendental analysis in a
+higher degree than Lagrange seems to me to have done.<a name="FNanchor_12_12" id="FNanchor_12_12"></a><a href="#Footnote_12_12" class="fnanchor">[12]</a></p>
+
+<p>We noticed in the preceding chapter the formation of
+the <i>calculus of partial differences</i>, created by D'Alembert,
+as having introduced into the transcendental analysis
+a new elementary idea; the notion of two kinds of
+increments, distinct and independent of one another,
+which a function of two variables may receive by virtue
+of the change of each variable separately. It is thus
+that the vertical ordinate of a surface, or any other magnitude
+which is referred to it, varies in two manners
+which are quite distinct, and which may follow the most
+different laws, according as we increase either the one
+or the other of the two horizontal co-ordinates. Now
+such a consideration seems to me very nearly allied, by
+its nature, to that which serves as the general basis of
+the method of variations. This last, indeed, has in reality
+done nothing but transfer to the independent variables
+themselves the peculiar conception which had been
+already adopted for the functions of these variables; a
+modification which has remarkably enlarged its use. I
+think, therefore, that so far as regards merely the fundamental
+conceptions, we may consider the calculus created
+by D'Alembert as having established a natural and necessary
+transition between the ordinary infinitesimal calculus
+and the calculus of variations; such a derivation
+of which seems to be adapted to make the general notion
+more clear and simple.</p>
+<p><span class="pagenum"><a name="Page_165" id="Page_165">[Pg 165]</a></span></p>
+<p>According to the different considerations indicated in
+this chapter, the method of variations presents itself as
+the highest degree of perfection which the analysis of indirect
+functions has yet attained. In its primitive state,
+this last analysis presented itself as a powerful general
+means of facilitating the mathematical study of natural
+phenomena, by introducing, for the expression of their
+laws, the consideration of auxiliary magnitudes, chosen
+in such a manner that their relations are necessarily more
+simple and more easy to obtain than those of the direct
+magnitudes. But the formation of these differential
+equations was not supposed to admit of any general and
+abstract rules. Now the Analysis of Variations, considered
+in the most philosophical point of view, may be
+regarded as essentially destined, by its nature, to bring
+within the reach of the calculus the actual establishment
+of the differential equations; for, in a great number of
+important and difficult questions, such is the general effect
+of the <i>varied</i> equations, which, still more <i>indirect</i>
+than the simple differential equations with respect to the
+special objects of the investigation, are also much more
+easy to form, and from which we may then, by invariable
+and complete analytical methods, the object of which
+is to eliminate the new order of auxiliary infinitesimals
+which have been introduced, deduce those ordinary differential
+equations which it would often have been impossible
+to establish directly. The method of variations
+forms, then, the most sublime part of that vast system
+of mathematical analysis, which, setting out from the
+most simple elements of algebra, organizes, by an uninterrupted
+succession of ideas, general methods more and
+more powerful, for the study of natural philosophy, and<span class="pagenum"><a name="Page_166" id="Page_166">[Pg 166]</a></span>
+which, in its whole, presents the most incomparably imposing
+and unequivocal monument of the power of the
+human intellect.</p>
+
+<p>We must, however, also admit that the conceptions
+which are habitually considered in the method of variations
+being, by their nature, more indirect, more general,
+and especially more abstract than all others, the
+employment of such a method exacts necessarily and
+continuously the highest known degree of intellectual
+exertion, in order never to lose sight of the precise object
+of the investigation, in following reasonings which
+offer to the mind such uncertain resting-places, and in
+which signs are of scarcely any assistance. We must
+undoubtedly attribute in a great degree to this difficulty
+the little real use which geometers, with the exception
+of Lagrange, have as yet made of such an admirable
+conception.</p><hr class="chap" /><p><span class="pagenum"><a name="Page_167" id="Page_167">[Pg 167]</a></span></p>
+
+
+
+
+<h2><a name="CHAPTER_VI" id="CHAPTER_VI">CHAPTER VI.</a></h2>
+
+<h3>THE CALCULUS OF FINITE DIFFERENCES.</h3>
+
+
+<p>The different fundamental considerations indicated in
+the five preceding chapters constitute, in reality, all the
+essential bases of a complete exposition of mathematical
+analysis, regarded in the philosophical point of view.
+Nevertheless, in order not to neglect any truly important
+general conception relating to this analysis, I think
+that I should here very summarily explain the veritable
+character of a kind of calculus which is very extended,
+and which, though at bottom it really belongs to ordinary
+analysis, is still regarded as being of an essentially
+distinct nature. I refer to the <i>Calculus of Finite Differences</i>,
+which will be the special subject of this chapter.</p>
+
+
+<p><i>Its general Character.</i> This calculus, created by
+Taylor, in his celebrated work entitled <i>Methodus Incrementorum</i>,
+consists essentially in the consideration of the
+finite increments which functions receive as a consequence
+of analogous increments on the part of the corresponding
+variables. These increments or <i>differences</i>,
+which take the characteristic &#916;, to distinguish them from
+<i>differentials</i>, or infinitely small increments, may be in
+their turn regarded as new functions, and become the
+subject of a second similar consideration, and so on; from
+which results the notion of differences of various successive
+orders, analogous, at least in appearance, to the
+consecutive orders of differentials. Such a calculus evidently<span class="pagenum"><a name="Page_168" id="Page_168">[Pg 168]</a></span>
+presents, like the calculus of indirect functions,
+two general classes of questions:</p>
+
+<p>1°. To determine the successive differences of all the
+various analytical functions of one or more variables, as
+the result of a definite manner of increase of the independent
+variables, which are generally supposed to augment
+in arithmetical progression.</p>
+
+<p>2°. Reciprocally, to start from these differences, or,
+more generally, from any equations established between
+them, and go back to the primitive functions themselves,
+or to their corresponding relations.</p>
+
+<p>Hence follows the decomposition of this calculus into
+two distinct ones, to which are usually given the names
+of the <i>Direct</i>, and the <i>Inverse Calculus of Finite Differences</i>,
+the latter being also sometimes called the <i>Integral
+Calculus of Finite Differences</i>. Each of these would,
+also, evidently admit of a logical distribution similar to
+that given in the fourth chapter for the differential and
+the integral calculus.</p>
+
+
+<p><i>Its true Nature.</i> There is no doubt that Taylor
+thought that by such a conception he had founded a calculus
+of an entirely new nature, absolutely distinct from
+ordinary analysis, and more general than the calculus of
+Leibnitz, although resting on an analogous consideration.
+It is in this way, also, that almost all geometers have
+viewed the analysis of Taylor; but Lagrange, with his
+usual profundity, clearly perceived that these properties
+belonged much more to the forms and to the notations
+employed by Taylor than to the substance of his theory.
+In fact, that which constitutes the peculiar character of
+the analysis of Leibnitz, and makes of it a truly distinct
+and superior calculus, is the circumstance that the derived<span class="pagenum"><a name="Page_169" id="Page_169">[Pg 169]</a></span>
+functions are in general of an entirely different nature
+from the primitive functions, so that they may give
+rise to more simple and more easily formed relations:
+whence result the admirable fundamental properties of
+the transcendental analysis, which have been already explained.
+But it is not so with the <i>differences</i> considered
+by Taylor; for these differences are, by their nature,
+functions essentially similar to those which have produced
+them, a circumstance which renders them unsuitable
+to facilitate the establishment of equations, and
+prevents their leading to more general relations. Every
+equation of finite differences is truly, at bottom, an equation
+directly relating to the very magnitudes whose successive
+states are compared. The scaffolding of new
+signs, which produce an illusion respecting the true character
+of these equations, disguises it, however, in a very
+imperfect manner, since it could always be easily made
+apparent by replacing the <i>differences</i> by the equivalent
+combinations of the primitive magnitudes, of which they
+are really only the abridged designations. Thus the calculus
+of Taylor never has offered, and never can offer, in
+any question of geometry or of mechanics, that powerful
+general aid which we have seen to result necessarily
+from the analysis of Leibnitz. Lagrange has, moreover,
+very clearly proven that the pretended analogy observed
+between the calculus of differences and the infinitesimal
+calculus was radically vicious, in this way, that the formulas
+belonging to the former calculus can never furnish,
+as particular cases, those which belong to the latter,
+the nature of which is essentially distinct.</p>
+
+<p>From these considerations I am led to think that the
+calculus of finite differences is, in general, improperly<span class="pagenum"><a name="Page_170" id="Page_170">[Pg 170]</a></span>
+classed with the transcendental analysis proper, that is,
+with the calculus of indirect functions. I consider it, on
+the contrary, in accordance with the views of Lagrange,
+to be only a very extensive and very important branch
+of ordinary analysis, that is to say, of that which I
+have named the calculus of direct functions, the equations
+which it considers being always, in spite of the
+notation, simple <i>direct</i> equations.</p>
+
+
+
+
+<h3><a name="GENERAL_THEORY_OF_SERIES" id="GENERAL_THEORY_OF_SERIES">GENERAL THEORY OF SERIES.</a></h3>
+
+
+<p>To sum up as briefly as possible the preceding explanation,
+the calculus of Taylor ought to be regarded
+as having constantly for its true object the general theory
+of <i>Series</i>, the most simple cases of which had alone
+been considered before that illustrious geometer. I
+ought, properly, to have mentioned this important theory
+in treating, in the second chapter, of Algebra proper,
+of which it is such an extensive branch. But, in order
+to avoid a double reference to it, I have preferred to notice
+it only in the consideration of the calculus of finite
+differences, which, reduced to its most simple general
+expression, is nothing but a complete logical study of
+questions relating to <i>series</i>.</p>
+
+<p>Every <i>Series</i>, or succession of numbers deduced from
+one another according to any constant law, necessarily
+gives rise to these two fundamental questions:</p>
+
+<p>1°. The law of the series being supposed known, to
+find the expression for its general term, so as to be able
+to calculate immediately any term whatever without being
+obliged to form successively all the preceding terms.</p>
+
+<p>2°. In the same circumstances, to determine the <i>sum</i>
+of any number of terms of the series by means of their<span class="pagenum"><a name="Page_171" id="Page_171">[Pg 171]</a></span>
+places, so that it can be known without the necessity
+of continually adding these terms together.</p>
+
+<p>These two fundamental questions being considered to
+be resolved, it may be proposed, reciprocally, to find the
+law of a series from the form of its general term, or the
+expression of the sum. Each of these different problems
+has so much the more extent and difficulty, as there
+can be conceived a greater number of different <i>laws</i> for
+the series, according to the number of preceding terms
+on which each term directly depends, and according to
+the function which expresses that dependence. We may
+even consider series with several variable indices, as Laplace
+has done in his "Analytical Theory of Probabilities,"
+by the analysis to which he has given the name
+of <i>Theory of Generating Functions</i>, although it is really
+only a new and higher branch of the calculus of finite
+differences or of the general theory of series.</p>
+
+<p>These general views which I have indicated give only
+an imperfect idea of the truly infinite extent and variety
+of the questions to which geometers have risen by means
+of this single consideration of series, so simple in appearance
+and so limited in its origin. It necessarily
+presents as many different cases as the algebraic resolution
+of equations, considered in its whole extent; and it
+is, by its nature, much more complicated, so much, indeed,
+that it always needs this last to conduct it to a complete
+solution. We may, therefore, anticipate what must
+still be its extreme imperfection, in spite of the successive
+labours of several geometers of the first order. We do
+not, indeed, possess as yet the complete and logical solution
+of any but the most simple questions of this nature.</p><p><span class="pagenum"><a name="Page_172" id="Page_172">[Pg 172]</a></span></p>
+
+
+<p><i>Its identity with this Calculus.</i> It is now easy to
+conceive the necessary and perfect identity, which has
+been already announced, between the calculus of finite
+differences and the theory of series considered in all its
+bearings. In fact, every differentiation after the manner
+of Taylor evidently amounts to finding the <i>law</i> of
+formation of a series with one or with several variable
+indices, from the expression of its general term; in the
+same way, every analogous integration may be regarded
+as having for its object the summation of a series, the
+general term of which would be expressed by the proposed
+difference. In this point of view, the various problems
+of the calculus of differences, direct or inverse, resolved
+by Taylor and his successors, have really a very
+great value, as treating of important questions relating
+to series. But it is very doubtful if the form and the
+notation introduced by Taylor really give any essential
+facility in the solution of questions of this kind. It
+would be, perhaps, more advantageous for most cases, and
+certainly more logical, to replace the <i>differences</i> by the
+terms themselves, certain combinations of which they
+represent. As the calculus of Taylor does not rest on
+a truly distinct fundamental idea, and has nothing peculiar
+to it but its system of signs, there could never really
+be any important advantage in considering it as detached
+from ordinary analysis, of which it is, in reality, only an
+immense branch. This consideration of <i>differences</i>, most
+generally useless, even if it does not cause complication,
+seems to me to retain the character of an epoch in which,
+analytical ideas not being sufficiently familiar to geometers,
+they were naturally led to prefer the special forms
+suitable for simple numerical comparisons.</p><hr class="chap" /><p><span class="pagenum"><a name="Page_173" id="Page_173">[Pg 173]</a></span></p>
+
+
+
+
+<h3><a name="PERIODIC_OR_DISCONTINUOUS_FUNCTIONS" id="PERIODIC_OR_DISCONTINUOUS_FUNCTIONS">PERIODIC OR DISCONTINUOUS FUNCTIONS.</a></h3>
+
+
+<p>However that may be, I must not finish this general
+appreciation of the calculus of finite differences without
+noticing a new conception to which it has given birth, and
+which has since acquired a great importance. It is the
+consideration of those periodic or discontinuous functions
+which preserve the same value for an infinite series of
+values of the corresponding variables, subjected to a certain
+law, and which must be necessarily added to the integrals
+of the equations of finite differences in order to
+render them sufficiently general, as simple arbitrary constants
+are added to all quadratures in order to complete
+their generality. This idea, primitively introduced by
+Euler, has since been the subject of extended investigation
+by M. Fourier, who has made new and important
+applications of it in his mathematical theory of heat.</p>
+
+
+
+
+<h3><a name="APPLICATIONS_OF_THIS_CALCULUS" id="APPLICATIONS_OF_THIS_CALCULUS">APPLICATIONS OF THIS CALCULUS.</a></h3>
+
+
+<p><i>Series.</i> Among the principal general applications
+which have been made of the calculus of finite differences,
+it would be proper to place in the first rank, as the
+most extended and the most important, the solution of
+questions relating to series; if, as has been shown, the
+general theory of series ought not to be considered as constituting,
+by its nature, the actual foundation of the calculus
+of Taylor.</p>
+
+
+<p><i>Interpolations.</i> This great class of problems being
+then set aside, the most essential of the veritable applications
+of the analysis of Taylor is, undoubtedly, thus
+far, the general method of <i>interpolations</i>, so frequently
+and so usefully employed in the investigation of the empirical<span class="pagenum"><a name="Page_174" id="Page_174">[Pg 174]</a></span>
+laws of natural phenomena. The question consists,
+as is well known, in intercalating between certain given
+numbers other intermediate numbers, subjected to the
+same law which we suppose to exist between the first.
+We can abundantly verify, in this principal application
+of the calculus of Taylor, how truly foreign and often inconvenient
+is the consideration of <i>differences</i> with respect
+to the questions which depend on that analysis. Indeed,
+Lagrange has replaced the formulas of interpolation, deduced
+from the ordinary algorithm of the calculus of
+finite differences, by much simpler general formulas,
+which are now almost always preferred, and which have
+been found directly, without making any use of the notion
+of <i>differences</i>, which only complicates the question.</p>
+
+
+<p><i>Approximate Rectification, &amp;c.</i> A last important
+class of applications of the calculus of finite differences,
+which deserves to be distinguished from the preceding,
+consists in the eminently useful employment made of it
+in geometry for determining by approximation the length
+and the area of any curve, and in the same way the cubature
+of a body of any form whatever. This procedure
+(which may besides be conceived abstractly as depending
+on the same analytical investigation as the question of
+interpolation) frequently offers a valuable supplement to
+the entirely logical geometrical methods which often lead
+to integrations, which we do not yet know how to effect,
+or to calculations of very complicated execution.</p>
+<p><span class="pagenum"><a name="Page_175" id="Page_175">[Pg 175]</a></span></p><p><span class="pagenum"><a name="Page_176" id="Page_176">[Pg 176]</a><br /><a name="Page_177" id="Page_177">[Pg 177]</a></span></p>
+<hr class="tb" />
+
+<p>Such are the various principal considerations to be
+noticed with respect to the calculus of finite differences.
+This examination completes the proposed philosophical
+outline of <span class="smcap">abstract Mathematics</span>.</p>
+
+
+<p><span class="smcap">Concrete Mathematics</span> will now be the subject of a
+similar labour. In it we shall particularly devote ourselves
+to examining how it has been possible (supposing
+the general science of the calculus to be perfect), by invariable
+procedures, to reduce to pure questions of analysis
+all the problems which can be presented by <i>Geometry</i> and
+<i>Mechanics</i>, and thus to impress on these two fundamental
+bases of natural philosophy a degree of precision and especially
+of unity; in a word, a character of high perfection,
+which could be communicated to them by such a
+course alone.</p>
+
+
+
+<p class="big center">BOOK II.</p>
+
+<p class="center">GEOMETRY.</p>
+
+<p><span class="pagenum"><a name="Page_178" id="Page_178">[Pg 178]</a><br /><a name="Page_179" id="Page_179">[Pg 179]</a></span></p>
+
+
+
+
+<p class="center big">BOOK II.</p>
+
+<h1>GEOMETRY.</h1>
+
+
+
+
+<h2><a name="CHAPTER_Ia" id="CHAPTER_Ia">CHAPTER I.</a></h2>
+
+<h3>GENERAL VIEW OF GEOMETRY.</h3>
+
+
+<p><i>Its true Nature.</i> After the general exposition of the
+philosophical character of concrete mathematics, compared
+with that of abstract mathematics, given in the introductory
+chapter, it need not here be shown in a special
+manner that geometry must be considered as a true natural
+science, only much more simple, and therefore much
+more perfect, than any other. This necessary perfection
+of geometry, obtained essentially by the application of
+mathematical analysis, which it so eminently admits, is
+apt to produce erroneous views of the real nature of this
+fundamental science, which most minds at present conceive
+to be a purely logical science quite independent of
+observation. It is nevertheless evident, to any one who
+examines with attention the character of geometrical reasonings,
+even in the present state of abstract geometry,
+that, although the facts which are considered in it are
+much more closely united than those relating to any other
+science, still there always exists, with respect to every
+body studied by geometers, a certain number of primitive
+phenomena, which, since they are not established by any<span class="pagenum"><a name="Page_180" id="Page_180">[Pg 180]</a></span>
+reasoning, must be founded on observation alone, and
+which form the necessary basis of all the deductions.</p>
+
+<p>The scientific superiority of geometry arises from the
+phenomena which it considers being necessarily the most
+universal and the most simple of all. Not only may all
+the bodies of nature give rise to geometrical inquiries, as
+well as mechanical ones, but still farther, geometrical
+phenomena would still exist, even though all the parts
+of the universe should be considered as immovable. Geometry
+is then, by its nature, more general than mechanics.
+At the same time, its phenomena are more simple,
+for they are evidently independent of mechanical phenomena,
+while these latter are always complicated with the
+former. The same relations hold good in comparing
+geometry with abstract thermology.</p>
+
+<p>For these reasons, in our classification we have made
+geometry the first part of concrete mathematics; that
+part the study of which, in addition to its own importance,
+serves as the indispensable basis of all the rest.</p>
+
+<p>Before considering directly the philosophical study of
+the different orders of inquiries which constitute our
+present geometry, we should obtain a clear and exact
+idea of the general destination of that science, viewed in
+all its bearings. Such is the object of this chapter.</p>
+
+
+<p><i>Definition.</i> Geometry is commonly defined in a very
+vague and entirely improper manner, as being <i>the science
+of extension</i>. An improvement on this would be to say
+that geometry has for its object the <i>measurement</i> of extension;
+but such an explanation would be very insufficient,
+although at bottom correct, and would be far from
+giving any idea of the true general character of geometrical
+science.</p><p><span class="pagenum"><a name="Page_181" id="Page_181">[Pg 181]</a></span></p>
+
+<p>To do this, I think that I should first explain <i>two fundamental
+ideas</i>, which, very simple in themselves, have
+been singularly obscured by the employment of metaphysical
+considerations.</p>
+
+
+<p><i>The Idea of Space.</i> The first is that of <i>Space</i>.
+This conception properly consists simply in this, that, instead
+of considering extension in the bodies themselves,
+we view it in an indefinite medium, which we regard as
+containing all the bodies of the universe. This notion is
+naturally suggested to us by observation, when we think
+of the <i>impression</i> which a body would leave in a fluid in
+which it had been placed. It is clear, in fact, that, as regards
+its geometrical relations, such an <i>impression</i> may
+be substituted for the body itself, without altering the
+reasonings respecting it. As to the physical nature of
+this indefinite <i>space</i>, we are spontaneously led to represent
+it to ourselves, as being entirely analogous to the
+actual medium in which we live; so that if this medium
+was liquid instead of gaseous, our geometrical <i>space</i>
+would undoubtedly be conceived as liquid also. This
+circumstance is, moreover, only very secondary, the essential
+object of such a conception being only to make
+us view extension separately from the bodies which manifest
+it to us. We can easily understand in advance the
+importance of this fundamental image, since it permits
+us to study geometrical phenomena in themselves, abstraction
+being made of all the other phenomena which
+constantly accompany them in real bodies, without, however,
+exerting any influence over them. The regular establishment
+of this general abstraction must be regarded
+as the first step which has been made in the rational
+study of geometry, which would have been impossible if<span class="pagenum"><a name="Page_182" id="Page_182">[Pg 182]</a></span>
+it had been necessary to consider, together with the form
+and the magnitude of bodies, all their other physical
+properties. The use of such an hypothesis, which is
+perhaps the most ancient philosophical conception created
+by the human mind, has now become so familiar to
+us, that we have difficulty in exactly estimating its importance,
+by trying to appreciate the consequences which
+would result from its suppression.</p>
+
+
+<p><i>Different Kinds of Extension.</i> The second preliminary
+geometrical conception which we have to examine
+is that of the different kinds of extension, designated by
+the words <i>volume</i>, <i>surface</i>, <i>line</i>, and even <i>point</i>, and of
+which the ordinary explanation is so unsatisfactory.<a name="FNanchor_13_13" id="FNanchor_13_13"></a><a href="#Footnote_13_13" class="fnanchor">[13]</a></p>
+
+<p>Although it is evidently impossible to conceive any extension
+absolutely deprived of any one of the three fundamental
+dimensions, it is no less incontestable that, in
+a great number of occasions, even of immediate utility,
+geometrical questions depend on only two dimensions,
+considered separately from the third, or on a single dimension,
+considered separately from the two others. Again,
+independently of this direct motive, the study of extension
+with a single dimension, and afterwards with two,
+clearly presents itself as an indispensable preliminary for
+facilitating the study of complete bodies of three dimensions,
+the immediate theory of which would be too complicated.<span class="pagenum"><a name="Page_183" id="Page_183">[Pg 183]</a></span>
+Such are the two general motives which oblige
+geometers to consider separately extension with regard to
+one or to two dimensions, as well as relatively to all three
+together.</p>
+
+<p>The general notions of <i>surface</i> and of <i>line</i> have been
+formed by the human mind, in order that it may be able
+to think, in a permanent manner, of extension in two
+directions, or in one only. The hyperbolical expressions
+habitually employed by geometers to define these notions
+tend to convey false ideas of them; but, examined in
+themselves, they have no other object than to permit us
+to reason with facility respecting these two kinds of extension,
+making complete abstraction of that which ought
+not to be taken into consideration. Now for this it is
+sufficient to conceive the dimension which we wish to
+eliminate as becoming gradually smaller and smaller,
+the two others remaining the same, until it arrives at
+such a degree of tenuity that it can no longer fix the attention.
+It is thus that we naturally acquire the real
+idea of a <i>surface</i>, and, by a second analogous operation,
+the idea of a <i>line</i>, by repeating for breadth what we had
+at first done for thickness. Finally, if we again repeat
+the same operation, we arrive at the idea of a <i>point</i>, or
+of an extension considered only with reference to its
+place, abstraction being made of all magnitude, and designed
+consequently to determine positions.</p>
+
+<p><i>Surfaces</i> evidently have, moreover, the general property
+of exactly circumscribing volumes; and in the same
+way, <i>lines</i>, in their turn, circumscribe <i>surfaces</i> and are
+limited by <i>points</i>. But this consideration, to which too
+much importance is often given, is only a secondary
+one.</p><p><span class="pagenum"><a name="Page_184" id="Page_184">[Pg 184]</a></span></p>
+
+<p>Surfaces and lines are, then, in reality, always conceived
+with three dimensions; it would be, in fact, impossible
+to represent to one's self a surface otherwise than
+as an extremely thin plate, and a line otherwise than as
+an infinitely fine thread. It is even plain that the degree
+of tenuity attributed by each individual to the dimensions
+of which he wishes to make abstraction is not
+constantly identical, for it must depend on the degree of
+subtilty of his habitual geometrical observations. This
+want of uniformity has, besides, no real inconvenience,
+since it is sufficient, in order that the ideas of surface
+and of line should satisfy the essential condition of their
+destination, for each one to represent to himself the dimensions
+which are to be neglected as being smaller than
+all those whose magnitude his daily experience gives him
+occasion to appreciate.</p>
+
+<p>We hence see how devoid of all meaning are the fantastic
+discussions of metaphysicians upon the foundations
+of geometry. It should also be remarked that these primordial
+ideas are habitually presented by geometers in
+an unphilosophical manner, since, for example, they explain
+the notions of the different sorts of extent in an
+order absolutely the inverse of their natural dependence,
+which often produces the most serious inconveniences in
+elementary instruction.</p>
+
+
+
+
+<h3><a name="THE_FINAL_OBJECT_OF_GEOMETRY" id="THE_FINAL_OBJECT_OF_GEOMETRY">THE FINAL OBJECT OF GEOMETRY.</a></h3>
+
+
+<p>These preliminaries being established, we can proceed
+directly to the general definition of geometry, continuing
+to conceive this science as having for its final object the
+<i>measurement</i> of extension.</p>
+
+<p>It is necessary in this matter to go into a thorough<span class="pagenum"><a name="Page_185" id="Page_185">[Pg 185]</a></span>
+explanation, founded on the distinction of the three kinds
+of extension, since the notion of <i>measurement</i> is not exactly
+the same with reference to surfaces and volumes
+as to lines.</p>
+
+
+<p><i>Nature of Geometrical Measurement.</i> If we take the
+word <i>measurement</i> in its direct and general mathematical
+acceptation, which signifies simply the determination
+of the value of the <i>ratios</i> between any homogeneous
+magnitudes, we must consider, in geometry, that the
+<i>measurement</i> of surfaces and of volumes, unlike that of
+lines, is never conceived, even in the most simple and the
+most favourable cases, as being effected directly. The
+comparison of two lines is regarded as direct; that of
+two surfaces or of two volumes is, on the contrary, always
+indirect. Thus we conceive that two lines may
+be superposed; but the superposition of two surfaces, or,
+still more so, of two volumes, is evidently impossible in
+most cases; and, even when it becomes rigorously practicable,
+such a comparison is never either convenient or
+exact. It is, then, very necessary to explain wherein
+properly consists the truly geometrical measurement of
+a surface or of a volume.</p>
+
+
+<p><i>Measurement of Surfaces and of Volumes.</i> For this
+we must consider that, whatever may be the form of a
+body, there always exists a certain number of lines, more
+or less easy to be assigned, the length of which is sufficient
+to define exactly the magnitude of its surface or of
+its volume. Geometry, regarding these lines as alone
+susceptible of being directly measured, proposes to deduce,
+from the simple determination of them, the ratio of the
+surface or of the volume sought, to the unity of surface,
+or to the unity of volume. Thus the general object of<span class="pagenum"><a name="Page_186" id="Page_186">[Pg 186]</a></span>
+geometry, with respect to surfaces and to volumes, is
+properly to reduce all comparisons of surfaces or of volumes
+to simple comparisons of lines.</p>
+
+<p>Besides the very great facility which such a transformation
+evidently offers for the measurement of volumes
+and of surfaces, there results from it, in considering it
+in a more extended and more scientific manner, the general
+possibility of reducing to questions of lines all questions
+relating to volumes and to surfaces, considered with
+reference to their magnitude. Such is often the most
+important use of the geometrical expressions which determine
+surfaces and volumes in functions of the corresponding
+lines.</p>
+
+<p>It is true that direct comparisons between surfaces or
+between volumes are sometimes employed; but such
+measurements are not regarded as geometrical, but only
+as a supplement sometimes necessary, although too rarely
+applicable, to the insufficiency or to the difficulty of
+truly rational methods. It is thus that we often determine
+the volume of a body, and in certain cases its surface,
+by means of its weight. In the same way, on other
+occasions, when we can substitute for the proposed volume
+an equivalent liquid volume, we establish directly
+the comparison of the two volumes, by profiting by the
+property possessed by liquid masses, of assuming any desired
+form. But all means of this nature are purely mechanical,
+and rational geometry necessarily rejects them.</p>
+
+<p>To render more sensible the difference between these
+modes of determination and true geometrical measurements,
+I will cite a single very remarkable example; the
+manner in which Galileo determined the ratio of the ordinary
+cycloid to that of the generating circle. The<span class="pagenum"><a name="Page_187" id="Page_187">[Pg 187]</a></span>
+geometry of his time was as yet insufficient for the rational
+solution of such a problem. Galileo conceived
+the idea of discovering that ratio by a direct experiment.
+Having weighed as exactly as possible two plates of the
+same material and of equal thickness, one of them having
+the form of a circle and the other that of the generated
+cycloid, he found the weight of the latter always
+triple that of the former; whence he inferred that the
+area of the cycloid is triple that of the generating circle,
+a result agreeing with the veritable solution subsequently
+obtained by Pascal and Wallis. Such a success evidently
+depends on the extreme simplicity of the ratio
+sought; and we can understand the necessary insufficiency
+of such expedients, even when they are actually practicable.</p>
+
+<p>We see clearly, from what precedes, the nature of that
+part of geometry relating to <i>volumes</i> and that relating to
+<i>surfaces</i>. But the character of the geometry of <i>lines</i> is
+not so apparent, since, in order to simplify the exposition,
+we have considered the measurement of lines as being
+made directly. There is, therefore, needed a complementary
+explanation with respect to them.</p>
+
+
+<p><i>Measurement of curved Lines.</i> For this purpose, it
+is sufficient to distinguish between the right line and
+curved lines, the measurement of the first being alone
+regarded as direct, and that of the other as always indirect.
+Although superposition is sometimes strictly practicable
+for curved lines, it is nevertheless evident that
+truly rational geometry must necessarily reject it, as
+not admitting of any precision, even when it is possible.
+The geometry of lines has, then, for its general object, to
+reduce in every case the measurement of curved lines to<span class="pagenum"><a name="Page_188" id="Page_188">[Pg 188]</a></span>
+that of right lines; and consequently, in the most extended
+point of view, to reduce to simple questions of
+right lines all questions relating to the magnitude of any
+curves whatever. To understand the possibility of such
+a transformation, we must remark, that in every curve
+there always exist certain right lines, the length of which
+must be sufficient to determine that of the curve. Thus,
+in a circle, it is evident that from the length of the radius
+we must be able to deduce that of the circumference;
+in the same way, the length of an ellipse depends
+on that of its two axes; the length of a cycloid upon the
+diameter of the generating circle, &amp;c.; and if, instead
+of considering the whole of each curve, we demand, more
+generally, the length of any arc, it will be sufficient to
+add to the different rectilinear parameters, which determine
+the whole curve, the chord of the proposed arc, or
+the co-ordinates of its extremities. To discover the relation
+which exists between the length of a curved line
+and that of similar right lines, is the general problem of
+the part of geometry which relates to the study of lines.</p>
+
+<p>Combining this consideration with those previously
+suggested with respect to volumes and to surfaces, we
+may form a very clear idea of the science of geometry,
+conceived in all its parts, by assigning to it, for its general
+object, the final reduction of the comparisons of all
+kinds of extent, volumes, surfaces, or lines, to simple comparisons
+of right lines, the only comparisons regarded as
+capable of being made directly, and which indeed could
+not be reduced to any others more easy to effect. Such
+a conception, at the same time, indicates clearly the veritable
+character of geometry, and seems suited to show
+at a single glance its utility and its perfection.</p><p><span class="pagenum"><a name="Page_189" id="Page_189">[Pg 189]</a></span></p>
+
+
+<p><i>Measurement of right Lines.</i> In order to complete
+this fundamental explanation, I have yet to show how
+there can be, in geometry, a special section relating to
+the right line, which seems at first incompatible with the
+principle that the measurement of this class of lines must
+always be regarded as direct.</p>
+
+<p>It is so, in fact, as compared with that of curved lines,
+and of all the other objects which geometry considers.
+But it is evident that the estimation of a right line cannot
+be viewed as direct except so far as the linear unit can
+be applied to it. Now this often presents insurmountable
+difficulties, as I had occasion to show, for another
+reason, in the introductory chapter. We must, then,
+make the measurement of the proposed right line depend
+on other analogous measurements capable of being effected
+directly. There is, then, necessarily a primary distinct
+branch of geometry, exclusively devoted to the right
+line; its object is to determine certain right lines from
+others by means of the relations belonging to the figures
+resulting from their assemblage. This preliminary part
+of geometry, which is almost imperceptible in viewing
+the whole of the science, is nevertheless susceptible of a
+great development. It is evidently of especial importance,
+since all other geometrical measurements are referred
+to those of right lines, and if they could not be determined,
+the solution of every question would remain
+unfinished.</p>
+
+<p>Such, then, are the various fundamental parts of rational
+geometry, arranged according to their natural dependence;
+the geometry of <i>lines</i> being first considered,
+beginning with the right line; then the geometry of <i>surfaces</i>,
+and, finally, that of <i>solids</i>.</p><hr class="chap" /><p><span class="pagenum"><a name="Page_190" id="Page_190">[Pg 190]</a></span></p>
+
+
+
+
+<h3><a name="INFINITE_EXTENT_OF_ITS_FIELD" id="INFINITE_EXTENT_OF_ITS_FIELD">INFINITE EXTENT OF ITS FIELD.</a></h3>
+
+
+<p>Having determined with precision the general and
+final object of geometrical inquiries, the science must
+now be considered with respect to the field embraced by
+each of its three fundamental sections.</p>
+
+<p>Thus considered, geometry is evidently susceptible,
+by its nature, of an extension which is rigorously infinite;
+for the measurement of lines, of surfaces, or
+of volumes presents necessarily as many distinct questions
+as we can conceive different figures subjected to
+exact definitions; and their number is evidently infinite.</p>
+
+<p>Geometers limited themselves at first to consider the
+most simple figures which were directly furnished them
+by nature, or which were deduced from these primitive
+elements by the least complicated combinations. But
+they have perceived, since Descartes, that, in order to constitute
+the science in the most philosophical manner, it
+was necessary to make it apply to all imaginable figures.
+This abstract geometry will then inevitably comprehend
+as particular cases all the different real figures which
+the exterior world could present. It is then a fundamental
+principle in truly rational geometry to consider, as
+far as possible, all figures which can be rigorously conceived.</p>
+
+<p>The most superficial examination is enough to convince
+us that these figures present a variety which is
+quite infinite.</p>
+
+
+<p><i>Infinity of Lines.</i> With respect to curved <i>lines</i>, regarding
+them as generated by the motion of a point governed
+by a certain law, it is plain that we shall have, in<span class="pagenum"><a name="Page_191" id="Page_191">[Pg 191]</a></span>
+general, as many different curves as we conceive different
+laws for this motion, which may evidently be determined
+by an infinity of distinct conditions; although it
+may sometimes accidentally happen that new generations
+produce curves which have been already obtained. Thus,
+among plane curves, if a point moves so as to remain constantly
+at the same distance from a fixed point, it will
+generate a <i>circle</i>; if it is the sum or the difference of
+its distances from two fixed points which remains constant,
+the curve described will be an <i>ellipse</i> or an <i>hyperbola</i>;
+if it is their product, we shall have an entirely different
+curve; if the point departs equally from a fixed
+point and from a fixed line, it will describe a <i>parabola</i>;
+if it revolves on a circle at the same time that this circle
+rolls along a straight line, we shall have a <i>cycloid</i>;
+if it advances along a straight line, while this line, fixed
+at one of its extremities, turns in any manner whatever,
+there will result what in general terms are called <i>spirals</i>,
+which of themselves evidently present as many
+perfectly distinct curves as we can suppose different relations
+between these two motions of translation and of
+rotation, &amp;c. Each of these different curves may then
+furnish new ones, by the different general constructions
+which geometers have imagined, and which give rise to
+evolutes, to epicycloids, to caustics, &amp;c. Finally, there
+exists a still greater variety among curves of double curvature.</p>
+
+
+<p><i>Infinity of Surfaces.</i> As to <i>surfaces</i>, the figures are
+necessarily more different still, considering them as generated
+by the motion of lines. Indeed, the figure may
+then vary, not only in considering, as in curves, the different
+infinitely numerous laws to which the motion of<span class="pagenum"><a name="Page_192" id="Page_192">[Pg 192]</a></span>
+the generating line may be subjected, but also in supposing
+that this line itself may change its nature; a circumstance
+which has nothing analogous in curves, since
+the points which describe them cannot have any distinct
+figure. Two classes of very different conditions may
+then cause the figures of surfaces to vary, while there
+exists only one for lines. It is useless to cite examples
+of this doubly infinite multiplicity of surfaces. It would
+be sufficient to consider the extreme variety of the single
+group of surfaces which may be generated by a right line,
+and which comprehends the whole family of cylindrical
+surfaces, that of conical surfaces, the most general class
+of developable surfaces, &amp;c.</p>
+
+
+<p><i>Infinity of Volumes.</i> With respect to <i>volumes</i>, there
+is no occasion for any special consideration, since they are
+distinguished from each other only by the surfaces which
+bound them.</p>
+
+<p>In order to complete this sketch, it should be added
+that surfaces themselves furnish a new general means of
+conceiving new curves, since every curve may be regarded
+as produced by the intersection of two surfaces. It
+is in this way, indeed, that the first lines which we may
+regard as having been truly invented by geometers were
+obtained, since nature gave directly the straight line and
+the circle. We know that the ellipse, the parabola, and
+the hyperbola, the only curves completely studied by the
+ancients, were in their origin conceived only as resulting
+from the intersection of a cone with circular base by
+a plane in different positions. It is evident that, by the
+combined employment of these different general means
+for the formation of lines and of surfaces, we could produce
+a rigorously infinitely series of distinct forms in<span class="pagenum"><a name="Page_193" id="Page_193">[Pg 193]</a></span>
+starting from only a very small number of figures directly
+furnished by observation.</p>
+
+
+<p><i>Analytical invention of Curves, &amp;c.</i> Finally, all
+the various direct means for the invention of figures
+have scarcely any farther importance, since rational geometry
+has assumed its final character in the hands of
+Descartes. Indeed, as we shall see more fully in chapter
+iii., the invention of figures is now reduced to the
+invention of equations, so that nothing is more easy than
+to conceive new lines and new surfaces, by changing at
+will the functions introduced into the equations. This
+simple abstract procedure is, in this respect, infinitely
+more fruitful than all the direct resources of geometry, developed
+by the most powerful imagination, which should
+devote itself exclusively to that order of conceptions. It
+also explains, in the most general and the most striking
+manner, the necessarily infinite variety of geometrical
+forms, which thus corresponds to the diversity of analytical
+functions. Lastly, it shows no less clearly that the
+different forms of surfaces must be still more numerous
+than those of lines, since lines are represented analytically
+by equations with two variables, while surfaces give
+rise to equations with three variables, which necessarily
+present a greater diversity.</p>
+
+<p>The preceding considerations are sufficient to show
+clearly the rigorously infinite extent of each of the three
+general sections of geometry.</p>
+
+
+
+
+<h3><a name="EXPANSION_OF_ORIGINAL_DEFINITION" id="EXPANSION_OF_ORIGINAL_DEFINITION">EXPANSION OF ORIGINAL DEFINITION.</a></h3>
+
+
+<p>To complete the formation of an exact and sufficiently
+extended idea of the nature of geometrical inquiries,
+it is now indispensable to return to the general definition<span class="pagenum"><a name="Page_194" id="Page_194">[Pg 194]</a></span>
+above given, in order to present it under a new point of
+view, without which the complete science would be only
+very imperfectly conceived.</p>
+
+<p>When we assign as the object of geometry the <i>measurement</i>
+of all sorts of lines, surfaces, and volumes, that
+is, as has been explained, the reduction of all geometrical
+comparisons to simple comparisons of right lines, we
+have evidently the advantage of indicating a general destination
+very precise and very easy to comprehend. But
+if we set aside every definition, and examine the actual
+composition of the science of geometry, we will at first
+be induced to regard the preceding definition as much
+too narrow; for it is certain that the greater part of the
+investigations which constitute our present geometry do
+not at all appear to have for their object the <i>measurement</i>
+of extension. In spite of this fundamental objection,
+I will persist in retaining this definition; for, in
+fact, if, instead of confining ourselves to considering the
+different questions of geometry isolatedly, we endeavour
+to grasp the leading questions, in comparison with which
+all others, however important they may be, must be regarded
+as only secondary, we will finally recognize that
+the measurement of lines, of surfaces, and of volumes, is
+the invariable object, sometimes <i>direct</i>, though most often
+<i>indirect</i>, of all geometrical labours.</p>
+
+<p>This general proposition being fundamental, since it
+can alone give our definition all its value, it is indispensable
+to enter into some developments upon this subject.</p><hr class="chap" /><p><span class="pagenum"><a name="Page_195" id="Page_195">[Pg 195]</a></span></p>
+
+
+
+
+<h3><a name="PROPERTIES_OF_LINES_AND_SURFACES" id="PROPERTIES_OF_LINES_AND_SURFACES">PROPERTIES OF LINES AND SURFACES.</a></h3>
+
+
+<p>When we examine with attention the geometrical investigations
+which do not seem to relate to the <i>measurement</i>
+of extent, we find that they consist essentially in
+the study of the different <i>properties</i> of each line or of each
+surface; that is, in the knowledge of the different modes
+of generation, or at least of definition, peculiar to each
+figure considered. Now we can easily establish in the
+most general manner the necessary relation of such a
+study to the question of <i>measurement</i>, for which the
+most complete knowledge of the properties of each form
+is an indispensable preliminary. This is concurrently
+proven by two considerations, equally fundamental, although
+quite distinct in their nature.</p>
+
+
+<p><span class="smcap">Necessity of their Study</span>: 1. <i>To find the most suitable
+Property.</i> The <i>first</i>, purely scientific, consists in
+remarking that, if we did not know any other characteristic
+property of each line or surface than that one according
+to which geometers had first conceived it, in
+most cases it would be impossible to succeed in the solution
+of questions relating to its <i>measurement</i>. In fact,
+it is easy to understand that the different definitions
+which each figure admits of are not all equally suitable
+for such an object, and that they even present the most
+complete oppositions in that respect. Besides, since the
+primitive definition of each figure was evidently not chosen
+with this condition in view, it is clear that we must
+not expect, in general, to find it the most suitable;
+whence results the necessity of discovering others, that
+is, of studying as far as is possible the <i>properties</i> of the
+proposed figure. Let us suppose, for example, that the<span class="pagenum"><a name="Page_196" id="Page_196">[Pg 196]</a></span>
+circle is defined to be "the curve which, with the same
+contour, contains the greatest area." This is certainly
+a very characteristic property, but we would evidently
+find insurmountable difficulties in trying to deduce from
+such a starting point the solution of the fundamental
+questions relating to the rectification or to the quadrature
+of this curve. It is clear, in advance, that the
+property of having all its points equally distant from a
+fixed point must evidently be much better adapted to
+inquiries of this nature, even though it be not precisely
+the most suitable. In like manner, would Archimedes
+ever have been able to discover the quadrature of the
+parabola if he had known no other property of that curve
+than that it was the section of a cone with a circular
+base, by a plane parallel to its generatrix? The purely
+speculative labours of preceding geometers, in transforming
+this first definition, were evidently indispensable
+preliminaries to the direct solution of such a question.
+The same is true, in a still greater degree, with respect
+to surfaces. To form a just idea of this, we need only
+compare, as to the question of cubature or quadrature,
+the common definition of the sphere with that one, no
+less characteristic certainly, which would consist in regarding
+a spherical body, as that one which, with the
+same area, contains the greatest volume.</p>
+
+<p>No more examples are needed to show the necessity
+of knowing, so far as is possible, all the properties of each
+line or of each surface, in order to facilitate the investigation
+of rectifications, of quadratures, and of cubatures,
+which constitutes the final object of geometry. We may
+even say that the principal difficulty of questions of this
+kind consists in employing in each case the property which<span class="pagenum"><a name="Page_197" id="Page_197">[Pg 197]</a></span>
+is best adapted to the nature of the proposed problem.
+Thus, while we continue to indicate, for more precision,
+the measurement of extension as the general destination
+of geometry, this first consideration, which goes to the
+very bottom of the subject, shows clearly the necessity
+of including in it the study, as thorough as possible, of
+the different generations or definitions belonging to the
+same form.</p>
+
+
+<p>2. <i>To pass from the Concrete to the Abstract.</i> A
+second consideration, of at least equal importance, consists
+in such a study being indispensable for organizing
+in a rational manner the relation of the abstract to the
+concrete in geometry.</p>
+
+<p>The science of geometry having to consider all imaginable
+figures which admit of an exact definition, it necessarily
+results from this, as we have remarked, that
+questions relating to any figures presented by nature
+are always implicitly comprised in this abstract geometry,
+supposed to have attained its perfection. But when
+it is necessary to actually pass to concrete geometry, we
+constantly meet with a fundamental difficulty, that of
+knowing to which of the different abstract types we are
+to refer, with sufficient approximation, the real lines or
+surfaces which we have to study. Now it is for the
+purpose of establishing such a relation that it is particularly
+indispensable to know the greatest possible number
+of properties of each figure considered in geometry.</p>
+
+<p>In fact, if we always confined ourselves to the single
+primitive definition of a line or of a surface, supposing
+even that we could then <i>measure</i> it (which, according to
+the first order of considerations, would generally be impossible),
+this knowledge would remain almost necessarily<span class="pagenum"><a name="Page_198" id="Page_198">[Pg 198]</a></span>
+barren in the application, since we should not ordinarily
+know how to recognize that figure in nature when
+it presented itself there; to ensure that, it would be necessary
+that the single characteristic, according to which
+geometers had conceived it, should be precisely that one
+whose verification external circumstances would admit:
+a coincidence which would be purely fortuitous, and on
+which we could not count, although it might sometimes
+take place. It is, then, only by multiplying as much as
+possible the characteristic properties of each abstract figure,
+that we can be assured, in advance, of recognizing
+it in the concrete state, and of thus turning to account
+all our rational labours, by verifying in each case the definition
+which is susceptible of being directly proven. This
+definition is almost always the only one in given circumstances,
+and varies, on the other hand, for the same
+figure, with different circumstances; a double reason for
+its previous determination.</p>
+
+
+<p><i>Illustration: Orbits of the Planets.</i> The geometry
+of the heavens furnishes us with a very memorable example
+in this matter, well suited to show the general necessity
+of such a study. We know that the ellipse was
+discovered by Kepler to be the curve which the planets
+describe about the sun, and the satellites about their
+planets. Now would this fundamental discovery, which
+re-created astronomy, ever have been possible, if geometers
+had been always confined to conceiving the ellipse
+only as the oblique section of a circular cone by a
+plane? No such definition, it is evident, would admit
+of such a verification. The most general property of the
+ellipse, that the sum of the distances from any of its points
+to two fixed points is a constant quantity, is undoubtedly<span class="pagenum"><a name="Page_199" id="Page_199">[Pg 199]</a></span>
+much more susceptible, by its nature, of causing the
+curve to be recognized in this case, but still is not directly
+suitable. The only characteristic which can here
+be immediately verified is that which is derived from the
+relation which exists in the ellipse between the length of
+the focal distances and their direction; the only relation
+which admits of an astronomical interpretation, as expressing
+the law which connects the distance from the
+planet to the sun, with the time elapsed since the beginning
+of its revolution. It was, then, necessary that the
+purely speculative labours of the Greek geometers on the
+properties of the conic sections should have previously
+presented their generation under a multitude of different
+points of view, before Kepler could thus pass from the
+abstract to the concrete, in choosing from among all these
+different characteristics that one which could be most
+easily proven for the planetary orbits.</p>
+
+
+<p><i>Illustration: Figure of the Earth.</i> Another example
+of the same order, but relating to surfaces, occurs in
+considering the important question of the figure of the
+earth. If we had never known any other property of the
+sphere than its primitive character of having all its points
+equally distant from an interior point, how would we ever
+have been able to discover that the surface of the earth
+was spherical? For this, it was necessary previously to
+deduce from this definition of the sphere some properties
+capable of being verified by observations made upon the
+surface alone, such as the constant ratio which exists between
+the length of the path traversed in the direction
+of any meridian of a sphere going towards a pole, and
+the angular height of this pole above the horizon at each
+point. Another example, but involving a much longer<span class="pagenum"><a name="Page_200" id="Page_200">[Pg 200]</a></span>
+series of preliminary speculations, is the subsequent proof
+that the earth is not rigorously spherical, but that its
+form is that of an ellipsoid of revolution.</p>
+
+<p>After such examples, it would be needless to give any
+others, which any one besides may easily multiply. All
+of them prove that, without a very extended knowledge
+of the different properties of each figure, the relation of
+the abstract to the concrete, in geometry, would be purely
+accidental, and that the science would consequently want
+one of its most essential foundations.</p>
+
+<p>Such, then, are two general considerations which fully
+demonstrate the necessity of introducing into geometry a
+great number of investigations which have not the <i>measurement</i>
+of extension for their direct object; while we
+continue, however, to conceive such a measurement as
+being the final destination of all geometrical science. In
+this way we can retain the philosophical advantages of
+the clearness and precision of this definition, and still include
+in it, in a very logical though indirect manner, all
+known geometrical researches, in considering those which
+do not seem to relate to the measurement of extension,
+as intended either to prepare for the solution of the final
+questions, or to render possible the application of the solutions
+obtained.</p>
+
+<p>Having thus recognized, as a general principle, the close
+and necessary connexion of the study of the properties of
+lines and surfaces with those researches which constitute
+the final object of geometry, it is evident that geometers,
+in the progress of their labours, must by no means constrain
+themselves to keep such a connexion always in
+view. Knowing, once for all, how important it is to
+vary as much as possible the manner of conceiving each<span class="pagenum"><a name="Page_201" id="Page_201">[Pg 201]</a></span>
+figure, they should pursue that study, without considering
+of what immediate use such or such a special property
+may be for rectifications, quadratures, and cubatures.
+They would uselessly fetter their inquiries by attaching
+a puerile importance to the continued establishment of
+that co-ordination.</p>
+
+<p>This general exposition of the general object of geometry
+is so much the more indispensable, since, by the very
+nature of the subject, this study of the different properties
+of each line and of each surface necessarily composes
+by far the greater part of the whole body of geometrical
+researches. Indeed, the questions directly relating to rectifications,
+to quadratures, and to cubatures, are evidently,
+by themselves, very few in number for each figure considered.
+On the other hand, the study of the properties
+of the same figure presents an unlimited field to the activity
+of the human mind, in which it may always hope
+to make new discoveries. Thus, although geometers have
+occupied themselves for twenty centuries, with more or
+less activity undoubtedly, but without any real interruption,
+in the study of the conic sections, they are far from
+regarding that so simple subject as being exhausted; and
+it is certain, indeed, that in continuing to devote themselves
+to it, they would not fail to find still unknown
+properties of those different curves. If labours of this
+kind have slackened considerably for a century past, it
+is not because they are completed, but only, as will be
+presently explained, because the philosophical revolution
+in geometry, brought about by Descartes, has singularly
+diminished the importance of such researches.</p>
+
+<p>It results from the preceding considerations that not
+only is the field of geometry necessarily infinite because<span class="pagenum"><a name="Page_202" id="Page_202">[Pg 202]</a></span>
+of the variety of figures to be considered, but also in virtue
+of the diversity of the points of view under the same
+figure may be regarded. This last conception is, indeed,
+that which gives the broadest and most complete idea of
+the whole body of geometrical researches. We see that
+studies of this kind consist essentially, for each line or for
+each surface, in connecting all the geometrical phenomena
+which it can present, with a single fundamental phenomenon,
+regarded as the primitive definition.</p>
+
+
+
+
+<h3><a name="THE_TWO_GENERAL_METHODS_OF_GEOMETRY" id="THE_TWO_GENERAL_METHODS_OF_GEOMETRY">THE TWO GENERAL METHODS OF GEOMETRY.</a></h3>
+
+
+<p>Having now explained in a general and yet precise
+manner the final object of geometry, and shown how the
+science, thus defined, comprehends a very extensive class
+of researches which did not at first appear necessarily to
+belong to it, there remains to be considered the <i>method</i>
+to be followed for the formation of this science. This
+discussion is indispensable to complete this first sketch
+of the philosophical character of geometry. I shall here
+confine myself to indicating the most general consideration
+in this matter, developing and summing up this important
+fundamental idea in the following chapters.</p>
+
+<p>Geometrical questions may be treated according to
+<i>two methods</i> so different, that there result from them two
+sorts of geometry, so to say, the philosophical character
+of which does not seem to me to have yet been properly
+apprehended. The expressions of <i>Synthetical Geometry</i>
+and <i>Analytical Geometry</i>, habitually employed to designate
+them, give a very false idea of them. I would much
+prefer the purely historical denominations of <i>Geometry of
+the Ancients</i> and <i>Geometry of the Moderns</i>, which have
+at least the advantage of not causing their true character<span class="pagenum"><a name="Page_203" id="Page_203">[Pg 203]</a></span>
+to be misunderstood. But I propose to employ henceforth
+the regular expressions of <i>Special Geometry</i> and
+<i>General Geometry</i>, which seem to me suited to characterize
+with precision the veritable nature of the two
+methods.</p>
+
+
+<p><i>Their fundamental Difference.</i> The fundamental
+difference between the manner in which we conceive
+Geometry since Descartes, and the manner in which the
+geometers of antiquity treated geometrical questions, is
+not the use of the Calculus (or Algebra), as is commonly
+thought to be the case. On the one hand, it is certain
+that the use of the calculus was not entirely unknown
+to the ancient geometers, since they used to make continual
+and very extensive applications of the theory of
+proportions, which was for them, as a means of deduction,
+a sort of real, though very imperfect and especially
+extremely limited equivalent for our present algebra.
+The calculus may even be employed in a much more
+complete manner than they have used it, in order to obtain
+certain geometrical solutions, which will still retain
+all the essential character of the ancient geometry; this
+occurs very frequently with respect to those problems of
+geometry of two or of three dimensions, which are commonly
+designated under the name of <i>determinate</i>. On
+the other hand, important as is the influence of the calculus
+in our modern geometry, various solutions obtained
+without algebra may sometimes manifest the peculiar
+character which distinguishes it from the ancient geometry,
+although analysis is generally indispensable. I will
+cite, as an example, the method of Roberval for tangents,
+the nature of which is essentially modern, and which,
+however, leads in certain cases to complete solutions,<span class="pagenum"><a name="Page_204" id="Page_204">[Pg 204]</a></span>
+without any aid from the calculus. It is not, then, the
+instrument of deduction employed which is the principal
+distinction between the two courses which the human
+mind can take in geometry.</p>
+
+<p>The real fundamental difference, as yet imperfectly
+apprehended, seems to me to consist in the very nature
+of the questions considered. In truth, geometry, viewed
+as a whole, and supposed to have attained entire perfection,
+must, as we have seen on the one hand, embrace
+all imaginable figures, and, on the other, discover
+all the properties of each figure. It admits, from this
+double consideration, of being treated according to two
+essentially distinct plans; either, 1°, by grouping together
+all the questions, however different they may be,
+which relate to the same figure, and isolating those relating
+to different bodies, whatever analogy there may
+exist between them; or, 2°, on the contrary, by uniting
+under one point of view all similar inquiries, to whatever
+different figures they may relate, and separating the
+questions relating to the really different properties of the
+same body. In a word, the whole body of geometry
+may be essentially arranged either with reference to the
+<i>bodies</i> studied or to the <i>phenomena</i> to be considered.
+The first plan, which is the most natural, was that of
+the ancients; the second, infinitely more rational, is that
+of the moderns since Descartes.</p>
+
+
+<p><i>Geometry of the Ancients.</i> Indeed, the principal characteristics
+of the ancient geometry is that they studied,
+one by one, the different lines and the different surfaces,
+not passing to the examination of a new figure till they
+thought they had exhausted all that there was interesting
+in the figures already known. In this way of proceeding,<span class="pagenum"><a name="Page_205" id="Page_205">[Pg 205]</a></span>
+when they undertook the study of a new curve,
+the whole of the labour bestowed on the preceding ones
+could not offer directly any essential assistance, otherwise
+than by the geometrical practice to which it had
+trained the mind. Whatever might be the real similarity
+of the questions proposed as to two different figures,
+the complete knowledge acquired for the one could not
+at all dispense with taking up again the whole of the investigation
+for the other. Thus the progress of the mind
+was never assured; so that they could not be certain, in
+advance, of obtaining any solution whatever, however
+analogous the proposed problem might be to questions
+which had been already resolved. Thus, for example,
+the determination of the tangents to the three conic sections
+did not furnish any rational assistance for drawing
+the tangent to any other new curve, such as the conchoid,
+the cissoid, &amp;c. In a word, the geometry of the
+ancients was, according to the expression proposed above,
+essentially special.</p>
+
+
+<p><i>Geometry of the Moderns.</i> In the system of the
+moderns, geometry is, on the contrary, eminently <i>general</i>,
+that is to say, relating to any figures whatever. It
+is easy to understand, in the first place, that all geometrical
+expressions of any interest may be proposed with
+reference to all imaginable figures. This is seen directly
+in the fundamental problems&mdash;of rectifications, quadratures,
+and cubatures&mdash;which constitute, as has been
+shown, the final object of geometry. But this remark
+is no less incontestable, even for investigations which relate
+to the different <i>properties</i> of lines and of surfaces,
+and of which the most essential, such as the question of
+tangents or of tangent planes, the theory of curvatures,<span class="pagenum"><a name="Page_206" id="Page_206">[Pg 206]</a></span>
+&amp;c., are evidently common to all figures whatever. The
+very few investigations which are truly peculiar to particular
+figures have only an extremely secondary importance.
+This being understood, modern geometry consists
+essentially in abstracting, in order to treat it by itself,
+in an entirely general manner, every question relating
+to the same geometrical phenomenon, in whatever
+bodies it may be considered. The application of the
+universal theories thus constructed to the special determination
+of the phenomenon which is treated of in each
+particular body, is now regarded as only a subaltern labour,
+to be executed according to invariable rules, and
+the success of which is certain in advance. This labour
+is, in a word, of the same character as the numerical calculation
+of an analytical formula. There can be no other
+merit in it than that of presenting in each case the solution
+which is necessarily furnished by the general
+method, with all the simplicity and elegance which the
+line or the surface considered can admit of. But no real
+importance is attached to any thing but the conception
+and the complete solution of a new question belonging
+to any figure whatever. Labours of this kind are alone
+regarded as producing any real advance in science. The
+attention of geometers, thus relieved from the examination
+of the peculiarities of different figures, and wholly
+directed towards general questions, has been thereby able
+to elevate itself to the consideration of new geometrical
+conceptions, which, applied to the curves studied by the
+ancients, have led to the discovery of important properties
+which they had not before even suspected. Such is
+geometry, since the radical revolution produced by Descartes
+in the general system of the science.</p><p><span class="pagenum"><a name="Page_207" id="Page_207">[Pg 207]</a></span></p>
+
+
+<p><i>The Superiority of the modern Geometry.</i> The mere
+indication of the fundamental character of each of the
+two geometries is undoubtedly sufficient to make apparent
+the immense necessary superiority of modern geometry.
+We may even say that, before the great conception
+of Descartes, rational geometry was not truly constituted
+upon definitive bases, whether in its abstract or
+concrete relations. In fact, as regards science, considered
+speculatively, it is clear that, in continuing indefinitely
+to follow the course of the ancients, as did the
+moderns before Descartes, and even for a little while afterwards,
+by adding some new curves to the small number
+of those which they had studied, the progress thus
+made, however rapid it might have been, would still be
+found, after a long series of ages, to be very inconsiderable
+in comparison with the general system of geometry,
+seeing the infinite variety of the forms which would still
+have remained to be studied. On the contrary, at each
+question resolved according to the method of the moderns,
+the number of geometrical problems to be resolved
+is then, once for all, diminished by so much with respect
+to all possible bodies. Another consideration is, that it
+resulted, from their complete want of general methods,
+that the ancient geometers, in all their investigations,
+were entirely abandoned to their own strength, without
+ever having the certainty of obtaining, sooner or later,
+any solution whatever. Though this imperfection of the
+science was eminently suited to call forth all their admirable
+sagacity, it necessarily rendered their progress
+extremely slow; we can form some idea of this by the
+considerable time which they employed in the study of
+the conic sections. Modern geometry, making the progress<span class="pagenum"><a name="Page_208" id="Page_208">[Pg 208]</a></span>
+of our mind certain, permits us, on the contrary, to
+make the greatest possible use of the forces of our intelligence,
+which the ancients were often obliged to waste
+on very unimportant questions.</p>
+
+<p>A no less important difference between the two systems
+appears when we come to consider geometry in the
+concrete point of view. Indeed, we have already remarked
+that the relation of the abstract to the concrete
+in geometry can be founded upon rational bases only so
+far as the investigations are made to bear directly upon
+all imaginable figures. In studying lines, only one by
+one, whatever may be the number, always necessarily
+very small, of those which we shall have considered, the
+application of such theories to figures really existing in
+nature will never have any other than an essentially
+accidental character, since there is nothing to assure us
+that these figures can really be brought under the abstract
+types considered by geometers.</p>
+
+<p>Thus, for example, there is certainly something fortuitous
+in the happy relation established between the
+speculations of the Greek geometers upon the conic sections
+and the determination of the true planetary orbits.
+In continuing geometrical researches upon the same plan,
+there was no good reason for hoping for similar coincidences;
+and it would have been possible, in these special
+studies, that the researches of geometers should have
+been directed to abstract figures entirely incapable of any
+application, while they neglected others, susceptible perhaps
+of an important and immediate application. It is
+clear, at least, that nothing positively guaranteed the
+necessary applicability of geometrical speculations. It
+is quite another thing in the modern geometry. From<span class="pagenum"><a name="Page_209" id="Page_209">[Pg 209]</a></span>
+the single circumstance that in it we proceed by general
+questions relating to any figures whatever, we have in
+advance the evident certainty that the figures really existing
+in the external world could in no case escape the
+appropriate theory if the geometrical phenomenon which
+it considers presents itself in them.</p>
+
+<p>From these different considerations, we see that the
+ancient system of geometry wears essentially the character
+of the infancy of the science, which did not begin
+to become completely rational till after the philosophical
+resolution produced by Descartes. But it is evident, on
+the other hand, that geometry could not be at first conceived
+except in this <i>special</i> manner. <i>General</i> geometry
+would not have been possible, and its necessity could
+not even have been felt, if a long series of special labours
+on the most simple figures had not previously furnished
+bases for the conception of Descartes, and rendered apparent
+the impossibility of persisting indefinitely in the
+primitive geometrical philosophy.</p>
+
+
+<p><i>The Ancient the Base of the Modern.</i> From this last
+consideration we must infer that, although the geometry
+which I have called <i>general</i> must be now regarded as
+the only true dogmatical geometry, and that to which
+we shall chiefly confine ourselves, the other having no
+longer much more than an historical interest, nevertheless
+it is not possible to entirely dispense with <i>special</i> geometry
+in a rational exposition of the science. We undoubtedly
+need not borrow directly from ancient geometry
+all the results which it has furnished; but, from the
+very nature of the subject, it is necessarily impossible entirely
+to dispense with the ancient method, which will
+always serve as the preliminary basis of the science, dogmatically<span class="pagenum"><a name="Page_210" id="Page_210">[Pg 210]</a></span>
+as well as historically. The reason of this is
+easy to understand. In fact, <i>general</i> geometry being
+essentially founded, as we shall soon establish, upon the
+employment of the calculus in the transformation of geometrical
+into analytical considerations, such a manner of
+proceeding could not take possession of the subject immediately
+at its origin. We know that the application
+of mathematical analysis, from its nature, can never commence
+any science whatever, since evidently it cannot
+be employed until the science has already been sufficiently
+cultivated to establish, with respect to the phenomena
+considered, some <i>equations</i> which can serve as starting
+points for the analytical operations. These fundamental
+equations being once discovered, analysis will enable us
+to deduce from them a multitude of consequences which
+it would have been previously impossible even to suspect;
+it will perfect the science to an immense degree,
+both with respect to the generality of its conceptions and
+to the complete co-ordination established between them.
+But mere mathematical analysis could never be sufficient
+to form the bases of any natural science, not even to demonstrate
+them anew when they have once been established.
+Nothing can dispense with the direct study of
+the subject, pursued up to the point of the discovery of
+precise relations.</p>
+
+<p>We thus see that the geometry of the ancients will
+always have, by its nature, a primary part, absolutely necessary
+and more or less extensive, in the complete system
+of geometrical knowledge. It forms a rigorously
+indispensable introduction to <i>general</i> geometry. But it
+is to this that it must be limited in a completely dogmatic
+exposition. I will consider, then, directly, in the<span class="pagenum"><a name="Page_211" id="Page_211">[Pg 211]</a></span>
+following chapter, this <i>special</i> or <i>preliminary</i> geometry
+restricted to exactly its necessary limits, in order to occupy
+myself thenceforth only with the philosophical examination
+of <i>general</i> or <i>definitive</i> geometry, the only one
+which is truly rational, and which at present essentially
+composes the science.</p><hr class="chap" /><p><span class="pagenum"><a name="Page_212" id="Page_212">[Pg 212]</a></span></p>
+
+
+
+
+<h2><a name="CHAPTER_IIa" id="CHAPTER_IIa">CHAPTER II.</a></h2>
+
+<h3>ANCIENT OR SYNTHETIC GEOMETRY.</h3>
+
+
+<p>The geometrical method of the ancients necessarily
+constituting a preliminary department in the dogmatical
+system of geometry, designed to furnish <i>general</i> geometry
+with indispensable foundations, it is now proper to
+begin with determining wherein strictly consists this preliminary
+function of <i>special</i> geometry, thus reduced to
+the narrowest possible limits.</p>
+
+
+
+
+<h3><a name="ITS_PROPER_EXTENT" id="ITS_PROPER_EXTENT">ITS PROPER EXTENT.</a></h3>
+
+
+<p><i>Lines; Polygons; Polyhedrons.</i> In considering it
+under this point of view, it is easy to recognize that we
+might restrict it to the study of the right line alone for
+what concerns the geometry of <i>lines</i>; to the <i>quadrature</i>
+of rectilinear plane areas; and, lastly, to the <i>cubature</i> of
+bodies terminated by plane faces. The elementary propositions
+relating to these three fundamental questions
+form, in fact, the necessary starting point of all geometrical
+inquiries; they alone cannot be obtained except by
+a direct study of the subject; while, on the contrary,
+the complete theory of all other figures, even that of the
+circle, and of the surfaces and volumes which are connected
+with it, may at the present day be completely
+comprehended in the domain of <i>general</i> or <i>analytical</i>
+geometry; these primitive elements at once furnishing
+<i>equations</i> which are sufficient to allow of the application<span class="pagenum"><a name="Page_213" id="Page_213">[Pg 213]</a></span>
+of the calculus to geometrical questions, which would not
+have been possible without this previous condition.</p>
+
+<p>It results from this consideration that, in common practice,
+we give to <i>elementary</i> geometry more extent than
+would be rigorously necessary to it; since, besides the
+right line, polygons, and polyhedrons, we also include in
+it the circle and the "round" bodies; the study of which
+might, however, be as purely analytical as that, for example,
+of the conic sections. An unreflecting veneration
+for antiquity contributes to maintain this defect in method;
+but the best reason which can be given for it is the
+serious inconvenience for ordinary instruction which there
+would be in postponing, to so distant an epoch of mathematical
+education, the solution of several essential questions,
+which are susceptible of a direct and continual application
+to a great number of important uses. In fact,
+to proceed in the most rational manner, we should employ
+the integral calculus in obtaining the interesting
+results relating to the length or the area of the circle, or
+to the quadrature of the sphere, &amp;c., which have been
+determined by the ancients from extremely simple considerations.
+This inconvenience would be of little importance
+with regard to the persons destined to study
+the whole of mathematical science, and the advantage
+of proceeding in a perfectly logical order would have a
+much greater comparative value. But the contrary case
+being the more frequent, theories so essential have necessarily
+been retained in elementary geometry. Perhaps
+the conic sections, the cycloid, &amp;c., might be advantageously
+added in such cases.</p>
+
+
+<p><i>Not to be farther restricted.</i> While this preliminary
+portion of geometry, which cannot be founded on the application<span class="pagenum"><a name="Page_214" id="Page_214">[Pg 214]</a></span>
+of the calculus, is reduced by its nature to a
+very limited series of fundamental researches, relating to
+the right line, polygonal areas, and polyhedrons, it is certain,
+on the other hand, that we cannot restrict it any
+more; although, by a veritable abuse of the spirit of
+analysis, it has been recently attempted to present the
+establishment of the principal theorems of elementary geometry
+under an algebraical point of view. Thus some
+have pretended to demonstrate, by simple abstract considerations
+of mathematical analysis, the constant relation
+which exists between the three angles of a rectilinear
+triangle, the fundamental proposition of the theory
+of similar triangles, that of parallelopipedons, &amp;c.; in a
+word, precisely the only geometrical propositions which
+cannot be obtained except by a direct study of the subject,
+without the calculus being susceptible of having
+any part in it. Such aberrations are the unreflecting
+exaggerations of that natural and philosophical tendency
+which leads us to extend farther and farther the influence
+of analysis in mathematical studies. In mechanics,
+the pretended analytical demonstrations of the parallelogram
+of forces are of similar character.</p>
+
+<p>The viciousness of such a manner of proceeding follows
+from the principles previously presented. We have already,
+in fact, recognized that, since the calculus is not,
+and cannot be, any thing but a means of deduction, it
+would indicate a radically false idea of it to wish to
+employ it in establishing the elementary foundations of
+any science whatever; for on what would the analytical
+reasonings in such an operation repose? A labour of this
+nature, very far from really perfecting the philosophical
+character of a science, would constitute a return towards<span class="pagenum"><a name="Page_215" id="Page_215">[Pg 215]</a></span>
+the metaphysical age, in presenting real facts as mere
+logical abstractions.</p>
+
+<p>When we examine in themselves these pretended analytical
+demonstrations of the fundamental propositions
+of elementary geometry, we easily verify their necessary
+want of meaning. They are all founded on a vicious
+manner of conceiving the principle of <i>homogeneity</i>, the
+true general idea of which was explained in the second
+chapter of the preceding book. These demonstrations
+suppose that this principle does not allow us to admit the
+coexistence in the same equation of numbers obtained by
+different concrete comparisons, which is evidently false,
+and contrary to the constant practice of geometers. Thus
+it is easy to recognize that, by employing the law of homogeneity
+in this arbitrary and illegitimate acceptation,
+we could succeed in "demonstrating," with quite as much
+apparent rigour, propositions whose absurdity is manifest
+at the first glance. In examining attentively, for example,
+the procedure by the aid of which it has been attempted
+to prove analytically that the sum of the three
+angles of any rectilinear triangle is constantly equal to
+two right angles, we see that it is founded on this preliminary
+principle that, if two triangles have two of their
+angles respectively equal, the third angle of the one will
+necessarily be equal to the third angle of the other. This
+first point being granted, the proposed relation is immediately
+deduced from it in a very exact and simple manner.
+Now the analytical consideration by which this
+previous proposition has been attempted to be established,
+is of such a nature that, if it could be correct, we
+could rigorously deduce from it, in reproducing it conversely,
+this palpable absurdity, that two sides of a triangle<span class="pagenum"><a name="Page_216" id="Page_216">[Pg 216]</a></span>
+are sufficient, without any angle, for the entire determination
+of the third side. We may make analogous
+remarks on all the demonstrations of this sort, the sophisms
+of which will be thus verified in a perfectly apparent
+manner.</p>
+
+<p>The more reason that we have here to consider geometry
+as being at the present day essentially analytical, the
+more necessary was it to guard against this abusive exaggeration
+of mathematical analysis, according to which
+all geometrical observation would be dispensed with, in
+establishing upon pure algebraical abstractions the very
+foundations of this natural science.</p>
+
+
+<p><i>Attempted Demonstrations of Axioms, &amp;c.</i> Another
+indication that geometers have too much overlooked the
+character of a natural science which is necessarily inherent
+in geometry, appears from their vain attempts, so
+long made, to <i>demonstrate</i> rigorously, not by the aid of
+the calculus, but by means of certain constructions, several
+fundamental propositions of elementary geometry.
+Whatever may be effected, it will evidently be impossible
+to avoid sometimes recurring to simple and direct observation
+in geometry as a means of establishing various
+results. While, in this science, the phenomena
+which are considered are, by virtue of their extreme simplicity,
+much more closely connected with one another
+than those relating to any other physical science, some
+must still be found which cannot be deduced, and which,
+on the contrary, serve as starting points. It may be
+admitted that the greatest logical perfection of the science
+is to reduce these to the smallest number possible,
+but it would be absurd to pretend to make them completely
+disappear. I avow, moreover, that I find fewer<span class="pagenum"><a name="Page_217" id="Page_217">[Pg 217]</a></span>
+real inconveniences in extending, a little beyond what
+would be strictly necessary, the number of these geometrical
+notions thus established by direct observation,
+provided they are sufficiently simple, than in making
+them the subjects of complicated and indirect demonstrations,
+even when these demonstrations may be logically
+irreproachable.</p>
+
+<p>The true dogmatic destination of the geometry of the
+ancients, reduced to its least possible indispensable developments,
+having thus been characterized as exactly as
+possible, it is proper to consider summarily each of the
+principal parts of which it must be composed. I think
+that I may here limit myself to considering the first and
+the most extensive of these parts, that which has for its
+object the study of <i>the right line</i>; the two other sections,
+namely, the <i>quadrature of polygons</i> and the <i>cubature
+of polyhedrons</i>, from their limited extent, not being capable
+of giving rise to any philosophical consideration of
+any importance, distinct from those indicated in the preceding
+chapter with respect to the measure of areas and
+of volumes in general.</p>
+
+
+
+
+<h3><a name="GEOMETRY_OF_THE_RIGHT_LINE" id="GEOMETRY_OF_THE_RIGHT_LINE">GEOMETRY OF THE RIGHT LINE.</a></h3>
+
+
+<p>The final question which we always have in view in
+the study of the right line, properly consists in determining,
+by means of one another, the different elements
+of any right-lined figure whatever; which enables us
+always to know indirectly the length and position of a
+right line, in whatever circumstances it may be placed.
+This fundamental problem is susceptible of two general
+solutions, the nature of which is quite distinct, the one
+<i>graphical</i>, the other <i>algebraic</i>. The first, though very<span class="pagenum"><a name="Page_218" id="Page_218">[Pg 218]</a></span>
+imperfect, is that which must be first considered, because
+it is spontaneously derived from the direct study
+of the subject; the second, much more perfect in the
+most important respects, cannot be studied till afterwards,
+because it is founded upon the previous knowledge
+of the other.</p>
+
+
+
+
+<h3><a name="GRAPHICAL_SOLUTIONS" id="GRAPHICAL_SOLUTIONS">GRAPHICAL SOLUTIONS.</a></h3>
+
+
+<p>The graphical solution consists in constructing at will
+the proposed figure, either with the same dimensions, or,
+more usually, with dimensions changed in any ratio whatever.
+The first mode need merely be mentioned as being
+the most simple and the one which would first occur
+to the mind, for it is evidently, by its nature, almost entirely
+incapable of application. The second is, on the
+contrary, susceptible of being most extensively and most
+usefully applied. We still make an important and continual
+use of it at the present day, not only to represent
+with exactness the forms of bodies and their relative positions,
+but even for the actual determination of geometrical
+magnitudes, when we do not need great precision.
+The ancients, in consequence of the imperfection of their
+geometrical knowledge, employed this procedure in a
+much more extensive manner, since it was for a long time
+the only one which they could apply, even in the most
+important precise determinations. It was thus, for example,
+that Aristarchus of Samos estimated the relative distance
+from the sun and from the moon to the earth, by
+making measurements on a triangle constructed as exactly
+as possible, so as to be similar to the right-angled
+triangle formed by the three bodies at the instant when
+the moon is in quadrature, and when an observation of<span class="pagenum"><a name="Page_219" id="Page_219">[Pg 219]</a></span>
+the angle at the earth would consequently be sufficient to
+define the triangle. Archimedes himself, although he was
+the first to introduce calculated determinations into geometry,
+several times employed similar means. The
+formation of trigonometry did not cause this method to
+be entirely abandoned, although it greatly diminished its
+use; the Greeks and the Arabians continued to employ
+it for a great number of researches, in which we now regard
+the use of the calculus as indispensable.</p>
+
+<p>This exact reproduction of any figure whatever on a
+different scale cannot present any great theoretical difficulty
+when all the parts of the proposed figure lie in the
+same plane. But if we suppose, as most frequently happens,
+that they are situated in different planes, we see,
+then, a new order of geometrical considerations arise.
+The artificial figure, which is constantly plane, not being
+capable, in that case, of being a perfectly faithful image
+of the real figure, it is necessary previously to fix with
+precision the mode of representation, which gives rise to
+different systems of <i>Projection</i>.</p>
+
+<p>It then remains to be determined according to what
+laws the geometrical phenomena correspond in the two
+figures. This consideration generates a new series of
+geometrical investigations, the final object of which is
+properly to discover how we can replace constructions in
+relief by plane constructions. The ancients had to resolve
+several elementary questions of this kind for various
+cases in which we now employ spherical trigonometry,
+principally for different problems relating to the celestial
+sphere. Such was the object of their <i>analemmas</i>,
+and of the other plane figures which for a long time supplied
+the place of the calculus. We see by this that the<span class="pagenum"><a name="Page_220" id="Page_220">[Pg 220]</a></span>
+ancients really knew the elements of what we now name
+<i>Descriptive Geometry</i>, although they did not conceive it
+in a distinct and general manner.</p>
+
+<p>I think it proper briefly to indicate in this place the
+true philosophical character of this "Descriptive Geometry;"
+although, being essentially a science of application,
+it ought not to be included within the proper domain of
+this work.</p>
+
+
+
+
+<h3><a name="DESCRIPTIVE_GEOMETRY" id="DESCRIPTIVE_GEOMETRY">DESCRIPTIVE GEOMETRY.</a></h3>
+
+
+<p>All questions of geometry of three dimensions necessarily
+give rise, when we consider their graphical solution,
+to a common difficulty which is peculiar to them;
+that of substituting for the different constructions in relief,
+which are necessary to resolve them directly, and
+which it is almost always impossible to execute, simple
+equivalent plane constructions, by means of which we
+finally obtain the same results. Without this indispensable
+transformation, every solution of this kind would be
+evidently incomplete and really inapplicable in practice,
+although theoretically the constructions in space are usually
+preferable as being more direct. It was in order to
+furnish general means for always effecting such a transformation
+that <i>Descriptive Geometry</i> was created, and
+formed into a distinct and homogeneous system, by the
+illustrious <span class="smcap">Monge</span>. He invented, in the first place, a uniform
+method of representing bodies by figures traced on a
+single plane, by the aid of <i>projections</i> on two different
+planes, usually perpendicular to each other, and one of
+which is supposed to turn about their common intersection
+so as to coincide with the other produced; in this
+system, or in any other equivalent to it, it is sufficient<span class="pagenum"><a name="Page_221" id="Page_221">[Pg 221]</a></span>
+to regard points and lines as being determined by their
+projections, and surfaces by the projections of their generating
+lines. This being established, Monge&mdash;analyzing
+with profound sagacity the various partial labours of
+this kind which had before been executed by a number
+of incongruous procedures, and considering also, in a general
+and direct manner, in what any questions of that
+nature must consist&mdash;found that they could always be
+reduced to a very small number of invariable abstract
+problems, capable of being resolved separately, once for
+all, by uniform operations, relating essentially some to
+the contacts and others to the intersections of surfaces.
+Simple and entirely general methods for the graphical
+solution of these two orders of problems having been
+formed, all the geometrical questions which may arise in
+any of the various arts of construction&mdash;stone-cutting,
+carpentry, perspective, dialling, fortification, &amp;c.&mdash;can
+henceforth be treated as simple particular cases of a single
+theory, the invariable application of which will always
+necessarily lead to an exact solution, which may
+be facilitated in practice by profiting by the peculiar
+circumstances of each case.</p>
+
+<p><span class="pagenum"><a name="Page_222" id="Page_222">[Pg 222]</a></span><span class="pagenum"><a name="Page_223" id="Page_223">[Pg 223]</a></span></p>
+
+<hr class="tb" />
+
+<p>This important creation deserves in a remarkable degree
+to fix the attention of those philosophers who consider
+all that the human species has yet effected as a
+first step, and thus far the only really complete one, towards
+that general renovation of human labours, which
+must imprint upon all our arts a character of precision
+and of rationality, so necessary to their future progress.
+Such a revolution must, in fact, inevitably commence
+with that class of industrial labours, which is essentially
+connected with that science which is the most simple,
+the most perfect, and the most ancient. It cannot fail
+to extend hereafter, though with less facility, to all other
+practical operations. Indeed Monge himself, who conceived
+the true philosophy of the arts better than any one
+else, endeavoured to sketch out a corresponding system
+for the mechanical arts.</p>
+
+<p>Essential as the conception of descriptive geometry
+really is, it is very important not to deceive ourselves
+with respect to its true destination, as did those who,
+in the excitement of its first discovery, saw in it a means
+of enlarging the general and abstract domain of rational
+geometry. The result has in no way answered to these
+mistaken hopes. And, indeed, is it not evident that descriptive
+geometry has no special value except as a science
+of application, and as forming the true special theory of
+the geometrical arts? Considered in its abstract relations,
+it could not introduce any truly distinct order of
+geometrical speculations. We must not forget that, in
+order that a geometrical question should fall within the
+peculiar domain of descriptive geometry, it must necessarily
+have been previously resolved by speculative geometry,
+the solutions of which then, as we have seen,
+always need to be prepared for practice in such a way as
+to supply the place of constructions in relief by plane
+constructions; a substitution which really constitutes the
+only characteristic function of descriptive geometry.</p>
+
+<p>It is proper, however, to remark here, that, with regard
+to intellectual education, the study of descriptive geometry
+possesses an important philosophical peculiarity, quite
+independent of its high industrial utility. This is the
+advantage which it so pre-eminently offers&mdash;in habituating
+the mind to consider very complicated geometrical
+combinations in space, and to follow with precision their
+continual correspondence with the figures which are actually
+traced&mdash;of thus exercising to the utmost, in the
+most certain and precise manner, that important faculty
+of the human mind which is properly called "imagination,"
+and which consists, in its elementary and positive
+acceptation, in representing to ourselves, clearly and easily,
+a vast and variable collection of ideal objects, as if
+they were really before us.</p>
+
+<p>Finally, to complete the indication of the general nature
+of descriptive geometry by determining its logical
+character, we have to observe that, while it belongs to
+the geometry of the ancients by the character of its solutions,
+on the other hand it approaches the geometry of
+the moderns by the nature of the questions which compose
+it. These questions are in fact eminently remarkable
+for that generality which, as we saw in the preceding
+chapter, constitutes the true fundamental character
+of modern geometry; for the methods used are always
+conceived as applicable to any figures whatever, the peculiarity
+of each having only a purely secondary influence.
+The solutions of descriptive geometry are then graphical,
+like most of those of the ancients, and at the same time
+general, like those of the moderns.</p>
+
+<p><span class="pagenum"><a name="Page_224" id="Page_224">[Pg 224]</a></span></p>
+
+<hr class="tb" />
+
+<p>After this important digression, we will pursue the
+philosophical examination of <i>special</i> geometry, always
+considered as reduced to its least possible development,
+as an indispensable introduction to <i>general</i> geometry.
+We have now sufficiently considered the <i>graphical</i> solution
+of the fundamental problem relating to the right line&mdash;that
+is, the determination of the different elements of any
+right-lined figure by means of one another&mdash;and have
+now to examine in a special manner the <i>algebraic</i> solution.</p>
+
+
+
+
+<h3><a name="ALGEBRAIC_SOLUTIONS" id="ALGEBRAIC_SOLUTIONS">ALGEBRAIC SOLUTIONS.</a></h3>
+
+
+<p>This kind of solution, the evident superiority of which
+need not here be dwelt upon, belongs necessarily, by the
+very nature of the question, to the system of the ancient
+geometry, although the logical method which is employed
+causes it to be generally, but very improperly, separated
+from it. We have thus an opportunity of verifying, in
+a very important respect, what was established generally
+in the preceding chapter, that it is not by the employment
+of the calculus that the modern geometry is essentially
+to be distinguished from the ancient. The ancients
+are in fact the true inventors of the present trigonometry,
+spherical as well as rectilinear; it being only much
+less perfect in their hands, on account of the extreme inferiority
+of their algebraical knowledge. It is, then, really
+in this chapter, and not, as it might at first be thought,
+in those which we shall afterwards devote to the philosophical
+examination of <i>general</i> geometry, that it is proper
+to consider the character of this important preliminary
+theory, which is usually, though improperly, included in
+what is called <i>analytical geometry</i>, but which is really
+only a complement of <i>elementary geometry</i> properly so
+called.</p>
+
+<p>Since all right-lined figures can be decomposed into
+triangles, it is evidently sufficient to know how to determine
+the different elements of a triangle by means of one
+another, which reduces <i>polygonometry</i> to simple <i>trigonometry</i>.</p><hr class="chap" /><p><span class="pagenum"><a name="Page_225" id="Page_225">[Pg 225]</a></span></p>
+
+
+
+
+<h3><a name="TRIGONOMETRY" id="TRIGONOMETRY">TRIGONOMETRY.</a></h3>
+
+
+<p>The difficulty in resolving algebraically such a question
+as the above, consists essentially in forming, between
+the angles and the sides of a triangle, three distinct equations;
+which, when once obtained, will evidently reduce all
+trigonometrical problems to mere questions of analysis.</p>
+
+
+<p><i>How to introduce Angles.</i> In considering the establishment
+of these equations in the most general manner,
+we immediately meet with a fundamental distinction
+with respect to the manner of introducing the angles
+into the calculation, according as they are made to enter
+<i>directly</i>, by themselves or by the circular arcs which are
+proportional to them; or <i>indirectly</i>, by the chords of
+these arcs, which are hence called their <i>trigonometrical
+lines</i>. Of these two systems of trigonometry the second
+was of necessity the only one originally adopted, as being
+the only practicable one, since the condition of geometry
+made it easy enough to find exact relations between the
+sides of the triangles and the trigonometrical lines which
+represent the angles, while it would have been absolutely
+impossible at that epoch to establish equations between
+the sides and the angles themselves.</p>
+
+
+<p><i>Advantages of introducing Trigonometrical Lines.</i>
+At the present day, since the solution can be obtained by
+either system indifferently, that motive for preference no
+longer exists; but geometers have none the less persisted
+in following from choice the system primitively admitted
+from necessity; for, the same reason which enabled these
+trigonometrical equations to be obtained with much more
+facility, must, in like manner, as it is still more easy to
+conceive <i>à priori</i>, render these equations much more simple,<span class="pagenum"><a name="Page_226" id="Page_226">[Pg 226]</a></span>
+since they then exist only between right lines, instead
+of being established between right lines and arcs
+of circles. Such a consideration has so much the more
+importance, as the question relates to formulas which are
+eminently elementary, and destined to be continually
+employed in all parts of mathematical science, as well
+as in all its various applications.</p>
+
+<p>It may be objected, however, that when an angle is
+given, it is, in reality, always given by itself, and not by
+its trigonometrical lines; and that when it is unknown, it
+is its angular value which is properly to be determined,
+and not that of any of its trigonometrical lines. It seems,
+according to this, that such lines are only useless intermediaries
+between the sides and the angles, which have
+to be finally eliminated, and the introduction of which
+does not appear capable of simplifying the proposed research.
+It is indeed important to explain, with more
+generality and precision than is customary, the great real
+utility of this manner of proceeding.</p>
+
+
+<p><i>Division of Trigonometry into two Parts.</i> It consists
+in the fact that the introduction of these auxiliary
+magnitudes divides the entire question of trigonometry
+into two others essentially distinct, one of which has
+for its object to pass from the angles to their trigonometrical
+lines, or the converse, and the other of which
+proposes to determine the sides of the triangles by the trigonometrical
+lines of their angles, or the converse. Now
+the first of these two fundamental questions is evidently
+susceptible, by its nature, of being entirely treated and
+reduced to numerical tables once for all, in considering
+all possible angles, since it depends only upon those angles,
+and not at all upon the particular triangles in which<span class="pagenum"><a name="Page_227" id="Page_227">[Pg 227]</a></span>
+they may enter in each case; while the solution of the
+second question must necessarily be renewed, at least in
+its arithmetical relations, for each new triangle which it
+is necessary to resolve. This is the reason why the first
+portion of the complete work, which would be precisely
+the most laborious, is no longer taken into the account,
+being always done in advance; while, if such a decomposition
+had not been performed, we would evidently have
+found ourselves under the obligation of recommencing
+the entire calculation in each particular case. Such is
+the essential property of the present trigonometrical system,
+which in fact would really present no actual advantage,
+if it was necessary to calculate continually the
+trigonometrical line of each angle to be considered, or the
+converse; the intermediate agency introduced would then
+be more troublesome than convenient.</p>
+
+<p>In order to clearly comprehend the true nature of this
+conception, it will be useful to compare it with a still
+more important one, designed to produce an analogous
+effect either in its algebraic, or, still more, in its arithmetical
+relations&mdash;the admirable theory of <i>logarithms</i>.
+In examining in a philosophical manner the influence
+of this theory, we see in fact that its general result is
+to decompose all imaginable arithmetical operations into
+two distinct parts. The first and most complicated of
+these is capable of being executed in advance once for
+all (since it depends only upon the numbers to be considered,
+and not at all upon the infinitely different combinations
+into which they can enter), and consists in considering
+all numbers as assignable powers of a constant
+number. The second part of the calculation, which must
+of necessity be recommenced for each new formula which<span class="pagenum"><a name="Page_228" id="Page_228">[Pg 228]</a></span>
+is to have its value determined, is thenceforth reduced
+to executing upon these exponents correlative operations
+which are infinitely more simple. I confine myself here
+to merely indicating this resemblance, which any one can
+carry out for himself.</p>
+
+<p>We must besides observe, as a property (secondary
+at the present day, but all-important at its origin) of the
+trigonometrical system adopted, the very remarkable circumstance
+that the determination of angles by their trigonometrical
+lines, or the converse, admits of an arithmetical
+solution (the only one which is directly indispensable for
+the special destination of trigonometry) without the previous
+resolution of the corresponding algebraic question.
+It is doubtless to such a peculiarity that the ancients
+owed the possibility of knowing trigonometry. The investigation
+conceived in this way was so much the more
+easy, inasmuch as tables of chords (which the ancients
+naturally took as the trigonometrical lines) had been previously
+constructed for quite a different object, in the
+course of the labours of Archimedes on the rectification
+of the circle, from which resulted the actual determination
+of a certain series of chords; so that when Hipparchus
+subsequently invented trigonometry, he could
+confine himself to completing that operation by suitable
+intercalations; which shows clearly the connexion of ideas
+in that matter.</p>
+
+
+<p><i>The Increase of such Trigonometrical Lines.</i> To
+complete this philosophical sketch of trigonometry, it is
+proper now to observe that the extension of the same considerations
+which lead us to replace angles or arcs of circles
+by straight lines, with the view of simplifying our
+equations, must also lead us to employ concurrently several<span class="pagenum"><a name="Page_229" id="Page_229">[Pg 229]</a></span>
+trigonometrical lines, instead of confining ourselves
+to one only (as did the ancients), so as to perfect this
+system by choosing that one which will be algebraically
+the most convenient on each occasion. In this point of
+view, it is clear that the number of these lines is in itself
+no ways limited; provided that they are determined by
+the arc, and that they determine it, whatever may be the
+law according to which they are derived from it, they are
+suitable to be substituted for it in the equations. The
+Arabians, and subsequently the moderns, in confining
+themselves to the most simple constructions, have carried
+to four or five the number of <i>direct</i> trigonometrical
+lines, which might be extended much farther.</p>
+
+<p>But instead of recurring to geometrical formations,
+which would finally become very complicated, we conceive
+with the utmost facility as many new trigonometrical
+lines as the analytical transformations may require,
+by means of a remarkable artifice, which is not
+usually apprehended in a sufficiently general manner.
+It consists in not directly multiplying the trigonometrical
+lines appropriate to each arc considered, but in introducing
+new ones, by considering this arc as indirectly
+determined by all lines relating to an arc which is a very
+simple function of the first. It is thus, for example, that,
+in order to calculate an angle with more facility, we will
+determine, instead of its sine, the sine of its half, or of
+its double, &amp;c. Such a creation of <i>indirect</i> trigonometrical
+lines is evidently much more fruitful than all
+the direct geometrical methods for obtaining new ones.
+We may accordingly say that the number of trigonometrical
+lines actually employed at the present day by
+geometers is in reality unlimited, since at every instant,<span class="pagenum"><a name="Page_230" id="Page_230">[Pg 230]</a></span>
+so to say, the transformations of analysis may lead us to
+augment it by the method which I have just indicated.
+Special names, however, have been given to those only
+of these <i>indirect</i> lines which refer to the complement of
+the primitive arc, the others not occurring sufficiently
+often to render such denominations necessary; a circumstance
+which has caused a common misconception
+of the true extent of the system of trigonometry.</p>
+
+
+<p><i>Study of their Mutual Relations.</i> This multiplicity
+of trigonometrical lines evidently gives rise to a third
+fundamental question in trigonometry, the study of the
+relations which exist between these different lines; since,
+without such a knowledge, we could not make use, for
+our analytical necessities, of this variety of auxiliary
+magnitudes, which, however, have no other destination.
+It is clear, besides, from the consideration just indicated,
+that this essential part of trigonometry, although simply
+preparatory, is, by its nature, susceptible of an indefinite
+extension when we view it in its entire generality, while
+the two others are circumscribed within rigorously defined
+limits.</p>
+
+<p>It is needless to add that these three principal parts
+of trigonometry have to be studied in precisely the inverse
+order from that in which we have seen them necessarily
+derived from the general nature of the subject;
+for the third is evidently independent of the two others,
+and the second, of that which was first presented&mdash;the
+resolution of triangles, properly so called&mdash;which must
+for that reason be treated in the last place; which rendered
+so much the more important the consideration of
+their natural succession and logical relations to one another.</p><p><span class="pagenum"><a name="Page_231" id="Page_231">[Pg 231]</a></span></p>
+
+<p>It is useless to consider here separately <i>spherical trigonometry</i>,
+which cannot give rise to any special philosophical
+consideration; since, essential as it is by the importance
+and the multiplicity of its uses, it can be treated
+at the present day only as a simple application of rectilinear
+trigonometry, which furnishes directly its fundamental
+equations, by substituting for the spherical triangle
+the corresponding trihedral angle.</p>
+
+<p>This summary exposition of the philosophy of trigonometry
+has been here given in order to render apparent,
+by an important example, that rigorous dependence and
+those successive ramifications which are presented by
+what are apparently the most simple questions of elementary
+geometry.</p>
+<p><span class="pagenum"><a name="Page_232" id="Page_232">[Pg 232]</a></span></p>
+<hr class="tb" />
+
+<p>Having thus examined the peculiar character of <i>special</i>
+geometry reduced to its only dogmatic destination,
+that of furnishing to general geometry an indispensable
+preliminary basis, we have now to give all our attention
+to the true science of geometry, considered as a whole,
+in the most rational manner. For that purpose, it is
+necessary to carefully examine the great original idea of
+Descartes, upon which it is entirely founded. This will
+be the object of the following chapter.</p>
+
+
+
+
+<h2><a name="CHAPTER_IIIa" id="CHAPTER_IIIa">CHAPTER III.</a></h2>
+
+<h3>MODERN OR ANALYTICAL GEOMETRY.</h3>
+
+
+<p><i>General</i> (or <i>Analytical</i>) geometry being entirely
+founded upon the transformation of geometrical considerations
+into equivalent analytical considerations, we
+must begin with examining directly and in a thorough
+manner the beautiful conception by which Descartes has
+established in a uniform manner the constant possibility
+of such a co-relation. Besides its own extreme importance
+as a means of highly perfecting geometrical science,
+or, rather, of establishing the whole of it on rational
+bases, the philosophical study of this admirable conception
+must have so much the greater interest in our eyes
+from its characterizing with perfect clearness the general
+method to be employed in organizing the relations of the
+abstract to the concrete in mathematics, by the analytical
+representation of natural phenomena. There is no
+conception, in the whole philosophy of mathematics
+which better deserves to fix all our attention.</p>
+
+
+
+
+<h3><a name="ANALYTICAL_REPRESENTATION_OF_FIGURES" id="ANALYTICAL_REPRESENTATION_OF_FIGURES">ANALYTICAL REPRESENTATION OF FIGURES.</a></h3>
+
+
+<p>In order to succeed in expressing all imaginable geometrical
+phenomena by simple analytical relations, we
+must evidently, in the first place, establish a general
+method for representing analytically the subjects themselves
+in which these phenomena are found, that is, the
+lines or the surfaces to be considered. The <i>subject</i> being<span class="pagenum"><a name="Page_233" id="Page_233">[Pg 233]</a></span>
+thus habitually considered in a purely analytical
+point of view, we see how it is thenceforth possible to
+conceive in the same manner the various <i>accidents</i> of
+which it is susceptible.</p>
+
+<p>In order to organize the representation of geometrical
+figures by analytical equations, we must previously surmount
+a fundamental difficulty; that of reducing the
+general elements of the various conceptions of geometry
+to simply numerical ideas; in a word, that of substituting
+in geometry pure considerations of <i>quantity</i> for all
+considerations of <i>quality</i>.</p>
+
+
+<p><i>Reduction of Figure to Position.</i> For this purpose
+let us observe, in the first place, that all geometrical
+ideas relate necessarily to these three universal categories:
+the <i>magnitude</i>, the <i>figure</i>, and the <i>position</i> of the
+extensions to be considered. As to the first, there is
+evidently no difficulty; it enters at once into the ideas
+of numbers. With relation to the second, it must be
+remarked that it will always admit of being reduced to
+the third. For the figure of a body evidently results
+from the mutual position of the different points of which
+it is composed, so that the idea of position necessarily
+comprehends that of figure, and every circumstance of
+figure can be translated by a circumstance of position.
+It is in this way, in fact, that the human mind has proceeded
+in order to arrive at the analytical representation
+of geometrical figures, their conception relating directly
+only to positions. All the elementary difficulty is then
+properly reduced to that of referring ideas of situation
+to ideas of magnitude. Such is the direct destination
+of the preliminary conception upon which Descartes has
+established the general system of analytical geometry.</p><p><span class="pagenum"><a name="Page_234" id="Page_234">[Pg 234]</a></span></p>
+
+<p>His philosophical labour, in this relation, has consisted
+simply in the entire generalization of an elementary operation,
+which we may regard as natural to the human mind,
+since it is performed spontaneously, so to say, in all
+minds, even the most uncultivated. Thus, when we
+have to indicate the situation of an object without directly
+pointing it out, the method which we always adopt,
+and evidently the only one which can be employed, consists
+in referring that object to others which are known,
+by assigning the magnitude of the various geometrical
+elements, by which we conceive it connected with the
+known objects. These elements constitute what Descartes,
+and after him all geometers, have called the <i>co-ordinates</i>
+of each point considered. They are necessarily
+two in number, if it is known in advance in what plane
+the point is situated; and three, if it may be found indifferently
+in any region of space. As many different
+constructions as can be imagined for determining the
+position of a point, whether on a plane or in space, so
+many distinct systems of co-ordinates may be conceived;
+they are consequently susceptible of being multiplied to
+infinity. But, whatever may be the system adopted, we
+shall always have reduced the ideas of situation to simple
+ideas of magnitude, so that we will consider the change
+in the position of a point as produced by mere numerical
+variations in the values of its co-ordinates.</p>
+
+
+<p><i>Determination of the Position of a Point.</i> Considering
+at first only the least complicated case, that of <i>plane
+geometry</i>, it is in this way that we usually determine
+the position of a point on a plane, by its distances from
+two fixed right lines considered as known, which are
+called <i>axes</i>, and which are commonly supposed to be<span class="pagenum"><a name="Page_235" id="Page_235">[Pg 235]</a></span>
+perpendicular to each other. This system is that most
+frequently adopted, because of its simplicity; but geometers
+employ occasionally an infinity of others. Thus
+the position of a point on a plane may be determined, 1°,
+by its distances from two fixed points; or, 2°, by its distance
+from a single fixed point, and the direction of that
+distance, estimated by the greater or less angle which it
+makes with a fixed right line, which constitutes the system
+of what are called <i>polar</i> co-ordinates, the most frequently
+used after the system first mentioned; or, 3°, by
+the angles which the right lines drawn from the variable
+point to two fixed points make with the right line which
+joins these last; or, 4°, by the distances from that point
+to a fixed right line and a fixed point, &amp;c. In a word,
+there is no geometrical figure whatever from which it is
+not possible to deduce a certain system of co-ordinates
+more or less susceptible of being employed.</p>
+
+<p>A general observation, which it is important to make
+in this connexion, is, that every system of co-ordinates is
+equivalent to determining a point, in plane geometry, by
+the intersection of two lines, each of which is subjected
+to certain fixed conditions of determination; a single
+one of these conditions remaining variable, sometimes
+the one, sometimes the other, according to the system
+considered. We could not, indeed, conceive any other
+means of constructing a point than to mark it by the
+meeting of two lines. Thus, in the most common system,
+that of <i>rectilinear co-ordinates</i>, properly so called,
+the point is determined by the intersection of two right
+lines, each of which remains constantly parallel to a
+fixed axis, at a greater or less distance from it; in the
+<i>polar</i> system, the position of the point is marked by the<span class="pagenum"><a name="Page_236" id="Page_236">[Pg 236]</a></span>
+meeting of a circle, of variable radius and fixed centre,
+with a movable right line compelled to turn about this
+centre: in other systems, the required point might be
+designated by the intersection of two circles, or of any
+other two lines, &amp;c. In a word, to assign the value of
+one of the co-ordinates of a point in any system whatever,
+is always necessarily equivalent to determining a
+certain line on which that point must be situated. The
+geometers of antiquity had already made this essential
+remark, which served as the base of their method of
+geometrical <i>loci</i>, of which they made so happy a use to
+direct their researches in the resolution of <i>determinate</i>
+problems, in considering separately the influence of each
+of the two conditions by which was defined each point
+constituting the object, direct or indirect, of the proposed
+question. It was the general systematization of this
+method which was the immediate motive of the labours
+of Descartes, which led him to create analytical geometry.</p>
+
+<p>After having clearly established this preliminary conception&mdash;by
+means of which ideas of position, and thence,
+implicitly, all elementary geometrical conceptions are capable
+of being reduced to simple numerical considerations&mdash;it
+is easy to form a direct conception, in its entire
+generality, of the great original idea of Descartes, relative
+to the analytical representation of geometrical figures:
+it is this which forms the special object of this
+chapter. I will continue to consider at first, for more
+facility, only geometry of two dimensions, which alone
+was treated by Descartes; and will afterwards examine
+separately, under the same point of view, the theory of
+surfaces and curves of double curvature.</p><hr class="chap" /><p><span class="pagenum"><a name="Page_237" id="Page_237">[Pg 237]</a></span></p>
+
+
+
+
+<h3><a name="PLANE_CURVES" id="PLANE_CURVES">PLANE CURVES.</a></h3>
+
+
+<p><i>Expression of Lines by Equations.</i> In accordance
+with the manner of expressing analytically the position
+of a point on a plane, it can be easily established that,
+by whatever property any line may be defined, that definition
+always admits of being replaced by a corresponding
+equation between the two variable co-ordinates of the
+point which describes this line; an equation which will
+be thenceforth the analytical representation of the proposed
+line, every phenomenon of which will be translated
+by a certain algebraic modification of its equation. Thus,
+if we suppose that a point moves on a plane without its
+course being in any manner determined, we shall evidently
+have to regard its co-ordinates, to whatever system
+they may belong, as two variables entirely independent
+of one another. But if, on the contrary, this point is
+compelled to describe a certain line, we shall necessarily
+be compelled to conceive that its co-ordinates, in all the
+positions which it can take, retain a certain permanent
+and precise relation to each other, which is consequently
+susceptible of being expressed by a suitable equation;
+which will become the very clear and very rigorous analytical
+definition of the line under consideration, since
+it will express an algebraical property belonging exclusively
+to the co-ordinates of all the points of this line.
+It is clear, indeed, that when a point is not subjected to
+any condition, its situation is not determined except in
+giving at once its two co-ordinates, independently of each
+other; while, when the point must continue upon a defined
+line, a single co-ordinate is sufficient for completely
+fixing its position. The second co-ordinate is then a<span class="pagenum"><a name="Page_238" id="Page_238">[Pg 238]</a></span>
+determinate <i>function</i> of the first; or, in other words,
+there must exist between them a certain <i>equation</i>, of a
+nature corresponding to that of the line on which the
+point is compelled to remain. In a word, each of the
+co-ordinates of a point requiring it to be situated on a
+certain line, we conceive reciprocally that the condition,
+on the part of a point, of having to belong to a line defined
+in any manner whatever, is equivalent to assigning
+the value of one of the two co-ordinates; which is found
+in that case to be entirely dependent on the other. The
+analytical relation which expresses this dependence may
+be more or less difficult to discover, but it must evidently
+be always conceived to exist, even in the cases in
+which our present means may be insufficient to make it
+known. It is by this simple consideration that we may
+demonstrate, in an entirely general manner&mdash;independently
+of the particular verifications on which this fundamental
+conception is ordinarily established for each special
+definition of a line&mdash;the necessity of the analytical
+representation of lines by equations.</p>
+
+
+<p><i>Expression of Equations by Lines.</i> Taking up again
+the same reflections in the inverse direction, we could
+show as easily the geometrical necessity of the representation
+of every equation of two variables, in a determinate
+system of co-ordinates, by a certain line; of which such
+a relation would be, in the absence of any other known
+property, a very characteristic definition, the scientific
+destination of which will be to fix the attention directly
+upon the general course of the solutions of the equation,
+which will thus be noted in the most striking and the
+most simple manner. This picturing of equations is one
+of the most important fundamental advantages of analytical<span class="pagenum"><a name="Page_239" id="Page_239">[Pg 239]</a></span>
+geometry, which has thereby reacted in the highest
+degree upon the general perfecting of analysis itself;
+not only by assigning to purely abstract researches a
+clearly determined object and an inexhaustible career,
+but, in a still more direct relation, by furnishing a new
+philosophical medium for analytical meditation which
+could not be replaced by any other. In fact, the purely
+algebraic discussion of an equation undoubtedly makes
+known its solutions in the most precise manner, but in
+considering them only one by one, so that in this way
+no general view of them could be obtained, except as the
+final result of a long and laborious series of numerical
+comparisons. On the other hand, the geometrical <i>locus</i>
+of the equation, being only designed to represent distinctly
+and with perfect clearness the summing up of all these
+comparisons, permits it to be directly considered, without
+paying any attention to the details which have furnished
+it. It can thereby suggest to our mind general analytical
+views, which we should have arrived at with much
+difficulty in any other manner, for want of a means of
+clearly characterizing their object. It is evident, for example,
+that the simple inspection of the logarithmic
+curve, or of the curve <i>y</i> = sin. <i>x</i>, makes us perceive
+much more distinctly the general manner of the variations
+of logarithms with respect to their numbers, or of
+sines with respect to their arcs, than could the most attentive
+study of a table of logarithms or of natural sines.
+It is well known that this method has become entirely
+elementary at the present day, and that it is employed
+whenever it is desired to get a clear idea of the general
+character of the law which reigns in a series of precise
+observations of any kind whatever.</p><p><span class="pagenum"><a name="Page_240" id="Page_240">[Pg 240]</a></span></p>
+
+
+<p><i>Any Change in the Line causes a Change in the
+Equation.</i> Returning to the representation of lines by
+equations, which is our principal object, we see that this
+representation is, by its nature, so faithful, that the line
+could not experience any modification, however slight it
+might be, without causing a corresponding change in the
+equation. This perfect exactitude even gives rise oftentimes
+to special difficulties; for since, in our system of
+analytical geometry, the mere displacements of lines affect
+the equations, as well as their real variations in magnitude
+or form, we should be liable to confound them
+with one another in our analytical expressions, if geometers
+had not discovered an ingenious method designed
+expressly to always distinguish them. This method is
+founded on this principle, that although it is impossible
+to change analytically at will the position of a line with
+respect to the axes of the co-ordinates, we can change in
+any manner whatever the situation of the axes themselves,
+which evidently amounts to the same; then, by
+the aid of the very simple general formula by which this
+transformation of the axes is produced, it becomes easy
+to discover whether two different equations are the analytical
+expressions of only the same line differently situated,
+or refer to truly distinct geometrical loci; since, in
+the former case, one of them will pass into the other by
+suitably changing the axes or the other constants of the
+system of co-ordinates employed. It must, moreover, be
+remarked on this subject, that general inconveniences of
+this nature seem to be absolutely inevitable in analytical
+geometry; for, since the ideas of position are, as we have
+seen, the only geometrical ideas immediately reducible to
+numerical considerations, and the conceptions of figure<span class="pagenum"><a name="Page_241" id="Page_241">[Pg 241]</a></span>
+cannot be thus reduced, except by seeing in them relations
+of situation, it is impossible for analysis to escape
+confounding, at first, the phenomena of figure with simple
+phenomena of position, which alone are directly expressed
+by the equations.</p>
+
+
+<p><i>Every Definition of a Line is an Equation.</i> In order
+to complete the philosophical explanation of the fundamental
+conception which serves as the base of analytical
+geometry, I think that I should here indicate a new
+general consideration, which seems to me particularly
+well adapted for putting in the clearest point of view this
+necessary representation of lines by equations with two
+variables. It consists in this, that not only, as we have
+shown, must every defined line necessarily give rise to a
+certain equation between the two co-ordinates of any one
+of its points, but, still farther, every definition of a line
+may be regarded as being already of itself an equation of
+that line in a suitable system of co-ordinates.</p>
+
+<p>It is easy to establish this principle, first making a
+preliminary logical distinction with respect to different
+kinds of definitions. The rigorously indispensable condition
+of every definition is that of distinguishing the object
+defined from all others, by assigning to it a property
+which belongs to it exclusively. But this end may be
+generally attained in two very different ways; either by
+a definition which is simply <i>characteristic</i>, that is, indicative
+of a property which, although truly exclusive,
+does not make known the mode of generation of the object;
+or by a definition which is really <i>explanatory</i>, that
+is, which characterizes the object by a property which expresses
+one of its modes of generation. For example, in
+considering the circle as the line, which, under the same<span class="pagenum"><a name="Page_242" id="Page_242">[Pg 242]</a></span>
+contour, contains the greatest area, we have evidently a
+definition of the first kind; while in choosing the property
+of its having all its points equally distant from a fixed
+point, we have a definition of the second kind. It is, besides,
+evident, as a general principle, that even when any
+object whatever is known at first only by a <i>characteristic</i>
+definition, we ought, nevertheless, to regard it as susceptible
+of <i>explanatory</i> definitions, which the farther study
+of the object would necessarily lead us to discover.</p>
+
+<p>This being premised, it is clear that the general observation
+above made, which represents every definition
+of a line as being necessarily an equation of that line in
+a certain system of co-ordinates, cannot apply to definitions
+which are simply <i>characteristic</i>; it is to be understood
+only of definitions which are truly <i>explanatory</i>.
+But, in considering only this class, the principle is easy
+to prove. In fact, it is evidently impossible to define the
+generation of a line without specifying a certain relation
+between the two simple motions of translation or of rotation,
+into which the motion of the point which describes it
+will be decomposed at each instant. Now if we form the
+most general conception of what constitutes <i>a system of
+co-ordinates</i>, and admit all possible systems, it is clear
+that such a relation will be nothing else but the <i>equation</i>
+of the proposed line, in a system of co-ordinates of a nature
+corresponding to that of the mode of generation considered.
+Thus, for example, the common definition of
+the <i>circle</i> may evidently be regarded as being immediately
+the <i>polar equation</i> of this curve, taking the centre
+of the circle for the pole. In the same way, the elementary
+definition of the <i>ellipse</i> or of the <i>hyperbola</i>&mdash;as
+being the curve generated by a point which moves in<span class="pagenum"><a name="Page_243" id="Page_243">[Pg 243]</a></span>
+such a manner that the sum or the difference of its distances
+from two fixed points remains constant&mdash;gives at
+once, for either the one or the other curve, the equation
+<i>y</i> + <i>x</i> = <i>c</i>, taking for the system of co-ordinates that in
+which the position of a point would be determined by its
+distances from two fixed points, and choosing for these
+poles the two given foci. In like manner, the common
+definition of any <i>cycloid</i> would furnish directly, for that
+curve, the equation <i>y</i> = <i>mx</i>; adopting as the co-ordinates
+of each point the arc which it marks upon a circle of invariable
+radius, measuring from the point of contact of that
+circle with a fixed line, and the rectilinear distance from
+that point of contact to a certain origin taken on that
+right line. We can make analogous and equally easy verifications
+with respect to the customary definitions of spirals,
+of epicycloids, &amp;c. We shall constantly find that
+there exists a certain system of co-ordinates, in which we
+immediately obtain a very simple equation of the proposed
+line, by merely writing algebraically the condition
+imposed by the mode of generation considered.</p>
+
+<p>Besides its direct importance as a means of rendering
+perfectly apparent the necessary representation of every
+line by an equation, the preceding consideration seems to
+me to possess a true scientific utility, in characterizing
+with precision the principal general difficulty which occurs
+in the actual establishment of these equations, and in
+consequently furnishing an interesting indication with respect
+to the course to be pursued in inquiries of this kind,
+which, by their nature, could not admit of complete and
+invariable rules. In fact, since any definition whatever
+of a line, at least among those which indicate a mode of
+generation, furnishes directly the equation of that line in<span class="pagenum"><a name="Page_244" id="Page_244">[Pg 244]</a></span>
+a certain system of co-ordinates, or, rather, of itself constitutes
+that equation, it follows that the difficulty which
+we often experience in discovering the equation of a
+curve, by means of certain of its characteristic properties,
+a difficulty which is sometimes very great, must proceed
+essentially only from the commonly imposed condition of
+expressing this curve analytically by the aid of a designated
+system of co-ordinates, instead of admitting indifferently
+all possible systems. These different systems
+cannot be regarded in analytical geometry as being all
+equally suitable; for various reasons, the most important
+of which will be hereafter discussed, geometers think
+that curves should almost always be referred, as far as is
+possible, to <i>rectilinear co-ordinates</i>, properly so called.
+Now we see, from what precedes, that in many cases these
+particular co-ordinates will not be those with reference to
+which the equation of the curve will be found to be directly
+established by the proposed definition. The principal
+difficulty presented by the formation of the equation
+of a line really consists, then, in general, in a certain
+transformation of co-ordinates. It is undoubtedly true
+that this consideration does not subject the establishment
+of these equations to a truly complete general method, the
+success of which is always certain; which, from the very
+nature of the subject, is evidently chimerical: but such a
+view may throw much useful light upon the course which
+it is proper to adopt, in order to arrive at the end proposed.
+Thus, after having in the first place formed the
+preparatory equation, which is spontaneously derived
+from the definition which we are considering, it will be
+necessary, in order to obtain the equation belonging to
+the system of co-ordinates which must be finally admitted,<span class="pagenum"><a name="Page_245" id="Page_245">[Pg 245]</a></span>
+to endeavour to express in a function of these last co-ordinates
+those which naturally correspond to the given
+mode of generation. It is upon this last labour that it
+is evidently impossible to give invariable and precise precepts.
+We can only say that we shall have so many
+more resources in this matter as we shall know more of
+true analytical geometry, that is, as we shall know the
+algebraical expression of a greater number of different algebraical
+phenomena.</p>
+
+
+
+
+<h3><a name="CHOICE_OF_CO-ORDINATES" id="CHOICE_OF_CO-ORDINATES">CHOICE OF CO-ORDINATES.</a></h3>
+
+
+<p>In order to complete the philosophical exposition of the
+conception which serves as the base of analytical geometry,
+I have yet to notice the considerations relating to
+the choice of the system of co-ordinates which is in general
+the most suitable. They will give the rational explanation
+of the preference unanimously accorded to the
+ordinary rectilinear system; a preference which has hitherto
+been rather the effect of an empirical sentiment of
+the superiority of this system, than the exact result of a
+direct and thorough analysis.</p>
+
+
+<p><i>Two different Points of View.</i> In order to decide
+clearly between all the different systems of co-ordinates,
+it is indispensable to distinguish with care the two general
+points of view, the converse of one another, which
+belong to analytical geometry; namely, the relation of
+algebra to geometry, founded upon the representation of
+lines by equations; and, reciprocally, the relation of geometry
+to algebra, founded on the representation of equations
+by lines.</p>
+
+<p>It is evident that in every investigation of general geometry
+these two fundamental points of view are of necessity<span class="pagenum"><a name="Page_246" id="Page_246">[Pg 246]</a></span>
+always found combined, since we have always to
+pass alternately, and at insensible intervals, so to say,
+from geometrical to analytical considerations, and from
+analytical to geometrical considerations. But the necessity
+of here temporarily separating them is none the
+less real; for the answer to the question of method which
+we are examining is, in fact, as we shall see presently,
+very far from being the same in both these relations, so
+that without this distinction we could not form any clear
+idea of it.</p>
+
+
+<p>1. <i>Representation of Lines by Equations.</i> Under <i>the
+first point of view</i>&mdash;the representation of lines by equations&mdash;the
+only reason which could lead us to prefer one
+system of co-ordinates to another would be the greater
+simplicity of the equation of each line, and greater facility
+in arriving at it. Now it is easy to see that there does
+not exist, and could not be expected to exist, any system
+of co-ordinates deserving in that respect a constant preference
+over all others. In fact, we have above remarked
+that for each geometrical definition proposed we can conceive
+a system of co-ordinates in which the equation of
+the line is obtained at once, and is necessarily found to
+be also very simple; and this system, moreover, inevitably
+varies with the nature of the characteristic property
+under consideration. The rectilinear system could not,
+therefore, be constantly the most advantageous for this object,
+although it may often be very favourable; there is
+probably no system which, in certain particular cases,
+should not be preferred to it, as well as to every other.</p>
+
+
+<p>2. <i>Representation of Equations by Lines.</i> It is by no
+means so, however, under the <i>second point of view</i>. We
+can, indeed, easily establish, as a general principle, that<span class="pagenum"><a name="Page_247" id="Page_247">[Pg 247]</a></span>
+the ordinary rectilinear system must necessarily be better
+adapted than any other to the representation of equations
+by the corresponding geometrical <i>loci</i>; that is to
+say, that this representation is constantly more simple
+and more faithful in it than in any other.</p>
+
+<p>Let us consider, for this object, that, since every system
+of co-ordinates consists in determining a point by the
+intersection of two lines, the system adapted to furnish
+the most suitable geometrical <i>loci</i> must be that in which
+these two lines are the simplest possible; a consideration
+which confines our choice to the <i>rectilinear</i> system. In
+truth, there is evidently an infinite number of systems
+which deserve that name, that is to say, which employ
+only right lines to determine points, besides the ordinary
+system which assigns the distances from two fixed lines
+as co-ordinates; such, for example, would be that in
+which the co-ordinates of each point should be the two
+angles which the right lines, which go from that point to
+two fixed points, make with the right line, which joins
+these last points: so that this first consideration is not
+rigorously sufficient to explain the preference unanimously
+given to the common system. But in examining in a
+more thorough manner the nature of every system of co-ordinates,
+we also perceive that each of the two lines,
+whose meeting determines the point considered, must
+necessarily offer at every instant, among its different conditions
+of determination, a single variable condition, which
+gives rise to the corresponding co-ordinate, all the rest
+being fixed, and constituting the <i>axes</i> of the system,
+taking this term in its most extended mathematical acceptation.
+The variation is indispensable, in order that
+we may be able to consider all possible positions; and<span class="pagenum"><a name="Page_248" id="Page_248">[Pg 248]</a></span>
+the fixity is no less so, in order that there may exist
+means of comparison. Thus, in all <i>rectilinear</i> systems,
+each of the two right lines will be subjected to a fixed
+condition, and the ordinate will result from the variable
+condition.</p>
+
+
+<p><i>Superiority of rectilinear Co-ordinates.</i> From these
+considerations it is evident, as a general principle, that
+the most favourable system for the construction of geometrical
+<i>loci</i> will necessarily be that in which the variable
+condition of each right line shall be the simplest
+possible; the fixed condition being left free to be made
+complex, if necessary to attain that object. Now, of
+all possible manners of determining two movable right
+lines, the easiest to follow geometrically is certainly that
+in which, the direction of each right line remaining invariable,
+it only approaches or recedes, more or less, to
+or from a constant axis. It would be, for example, evidently
+more difficult to figure to one's self clearly the
+changes of place of a point which is determined by the
+intersection of two right lines, which each turn around
+a fixed point, making a greater or smaller angle with a
+certain axis, as in the system of co-ordinates previously
+noticed. Such is the true general explanation of the
+fundamental property possessed by the common rectilinear
+system, of being better adapted than any other to the
+geometrical representation of equations, inasmuch as it
+is that one in which it is the easiest to conceive the
+change of place of a point resulting from the change in
+the value of its co-ordinates. In order to feel clearly all
+the force of this consideration, it would be sufficient to
+carefully compare this system with the polar system, in
+which this geometrical image, so simple and so easy to<span class="pagenum"><a name="Page_249" id="Page_249">[Pg 249]</a></span>
+follow, of two right lines moving parallel, each one of
+them, to its corresponding axis, is replaced by the complicated
+picture of an infinite series of concentric circles,
+cut by a right line compelled to turn about a fixed
+point. It is, moreover, easy to conceive in advance what
+must be the extreme importance to analytical geometry
+of a property so profoundly elementary, which, for that
+reason, must be recurring at every instant, and take a
+progressively increasing value in all labours of this kind.</p>
+
+
+<p><i>Perpendicularity of the Axes.</i> In pursuing farther
+the consideration which demonstrates the superiority of
+the ordinary system of co-ordinates over any other as to
+the representation of equations, we may also take notice
+of the utility for this object of the common usage of taking
+the two axes perpendicular to each other, whenever
+possible, rather than with any other inclination. As regards
+the representation of lines by equations, this secondary
+circumstance is no more universally proper than
+we have seen the general nature of the system to be;
+since, according to the particular occasion, any other inclination
+of the axes may deserve our preference in that
+respect. But, in the inverse point of view, it is easy to
+see that rectangular axes constantly permit us to represent
+equations in a more simple and even more faithful
+manner; for, with oblique axes, space being divided by
+them into regions which no longer have a perfect identity,
+it follows that, if the geometrical <i>locus</i> of the equation
+extends into all these regions at once, there will be presented,
+by reason merely of this inequality of the angles,
+differences of figure which do not correspond to any
+analytical diversity, and will necessarily alter the rigorous
+exactness of the representation, by being confounded<span class="pagenum"><a name="Page_250" id="Page_250">[Pg 250]</a></span>
+with the proper results of the algebraic comparisons.
+For example, an equation like: <i>x<sup>m</sup></i> + <i>y<sup>m</sup></i> = <i>c</i>, which, by its
+perfect symmetry, should evidently give a curve composed
+of four identical quarters, will be represented, on
+the contrary, if we take axes not rectangular, by a geometric
+<i>locus</i>, the four parts of which will be unequal.
+It is plain that the only means of avoiding all inconveniences
+of this kind is to suppose the angle of the two
+axes to be a right angle.</p>
+
+<p>The preceding discussion clearly shows that, although
+the ordinary system of rectilinear co-ordinates has no constant
+superiority over all others in one of the two fundamental
+points of view which are continually combined in
+analytical geometry, yet as, on the other hand, it is not
+constantly inferior, its necessary and absolute greater
+aptitude for the representation of equations must cause
+it to generally receive the preference; although it may
+evidently happen, in some particular cases, that the necessity
+of simplifying equations and of obtaining them
+more easily may determine geometers to adopt a less
+perfect system. The rectilinear system is, therefore, the
+one by means of which are ordinarily constructed the
+most essential theories of general geometry, intended to
+express analytically the most important geometrical phenomena.
+When it is thought necessary to choose some
+other, the polar system is almost always the one which
+is fixed upon, this system being of a nature sufficiently
+opposite to that of the rectilinear system to cause the
+equations, which are too complicated with respect to the
+latter, to become, in general, sufficiently simple with respect
+to the other. Polar co-ordinates, moreover, have
+often the advantage of admitting of a more direct and<span class="pagenum"><a name="Page_251" id="Page_251">[Pg 251]</a></span>
+natural concrete signification; as is the case in mechanics,
+for the geometrical questions to which the theory of
+circular movement gives rise, and in almost all the cases
+of celestial geometry.</p>
+
+<hr class="tb" />
+
+<p>In order to simplify the exposition, we have thus far
+considered the fundamental conception of analytical geometry
+only with respect to <i>plane curves</i>, the general
+study of which was the only object of the great philosophical
+renovation produced by Descartes. To complete
+this important explanation, we have now to show
+summarily how this elementary idea was extended by
+Clairaut, about a century afterwards, to the general
+study of <i>surfaces</i> and <i>curves of double curvature</i>. The
+considerations which have been already given will permit
+me to limit myself on this subject to the rapid examination
+of what is strictly peculiar to this new case.</p>
+
+
+
+
+<h3><a name="SURFACES" id="SURFACES">SURFACES.</a></h3>
+
+
+<p><i>Determination of a Point in Space.</i> The complete
+analytical determination of a point in space evidently requires
+the values of three co-ordinates to be assigned; as,
+for example, in the system which is generally adopted,
+and which corresponds to the <i>rectilinear</i> system of plane
+geometry, distances from the point to three fixed planes,
+usually perpendicular to one another; which presents the
+point as the intersection of three planes whose direction
+is invariable. We might also employ the distances from
+the movable point to three fixed points, which would
+determine it by the intersection of three spheres with a
+common centre. In like manner, the position of a point
+would be defined by giving its distance from a fixed point,<span class="pagenum"><a name="Page_252" id="Page_252">[Pg 252]</a></span>
+and the direction of that distance, by means of the two
+angles which this right line makes with two invariable
+axes; this is the <i>polar</i> system of geometry of three dimensions;
+the point is then constructed by the intersection
+of a sphere having a fixed centre, with two right
+cones with circular bases, whose axes and common summit
+do not change. In a word, there is evidently, in this
+case at least, the same infinite variety among the various
+possible systems of co-ordinates which we have already
+observed in geometry of two dimensions. In general,
+we have to conceive a point as being always determined
+by the intersection of any three surfaces whatever,
+as it was in the former case by that of two lines: each
+of these three surfaces has, in like manner, all its conditions
+of determination constant, excepting one, which
+gives rise to the corresponding co-ordinates, whose peculiar
+geometrical influence is thus to constrain the point
+to be situated upon that surface.</p>
+
+<p>This being premised, it is clear that if the three co-ordinates
+of a point are entirely independent of one another,
+that point can take successively all possible positions
+in space. But if the point is compelled to remain
+upon a certain surface defined in any manner whatever,
+then two co-ordinates are evidently sufficient for determining
+its situation at each instant, since the proposed
+surface will take the place of the condition imposed by
+the third co-ordinate. We must then, in this case, under
+the analytical point of view, necessarily conceive this
+last co-ordinate as a determinate function of the two
+others, these latter remaining perfectly independent of
+each other. Thus there will be a certain equation between
+the three variable co-ordinates, which will be permanent,<span class="pagenum"><a name="Page_253" id="Page_253">[Pg 253]</a></span>
+and which will be the only one, in order to correspond
+to the precise degree of indetermination in the
+position of the point.</p>
+
+
+<p><i>Expression of Surfaces by Equations.</i> This equation,
+more or less easy to be discovered, but always possible,
+will be the analytical definition of the proposed surface,
+since it must be verified for all the points of that surface,
+and for them alone. If the surface undergoes any change
+whatever, even a simple change of place, the equation
+must undergo a more or less serious corresponding modification.
+In a word, all geometrical phenomena relating
+to surfaces will admit of being translated by certain equivalent
+analytical conditions appropriate to equations of
+three variables; and in the establishment and interpretation
+of this general and necessary harmony will essentially
+consist the science of analytical geometry of three
+dimensions.</p>
+
+
+<p><i>Expression of Equations by Surfaces.</i> Considering
+next this fundamental conception in the inverse point of
+view, we see in the same manner that every equation of
+three variables may, in general, be represented geometrically
+by a determinate surface, primitively defined by
+the very characteristic property, that the co-ordinates of
+all its points always retain the mutual relation enunciated
+in this equation. This geometrical locus will evidently
+change, for the same equation, according to the
+system of co-ordinates which may serve for the construction
+of this representation. In adopting, for example,
+the rectilinear system, it is clear that in the equation between
+the three variables, <i>x</i>, <i>y</i>, <i>z</i>, every particular value
+attributed to <i>z</i> will give an equation between at <i>x</i> and <i>y</i>, the
+geometrical locus of which will be a certain line situated<span class="pagenum"><a name="Page_254" id="Page_254">[Pg 254]</a></span>
+in a plane parallel to the plane of <i>x</i> and <i>y</i>, and at a distance
+from this last equal to the value of <i>z</i>; so that the
+complete geometrical locus will present itself as composed
+of an infinite series of lines superimposed in a series
+of parallel planes (excepting the interruptions which
+may exist), and will consequently form a veritable surface.
+It would be the same in considering any other system
+of co-ordinates, although the geometrical construction
+of the equation becomes more difficult to follow.</p>
+
+<p>Such is the elementary conception, the complement of
+the original idea of Descartes, on which is founded general
+geometry relative to surfaces. It would be useless
+to take up here directly the other considerations which
+have been above indicated, with respect to lines, and
+which any one can easily extend to surfaces; whether
+to show that every definition of a surface by any method
+of generation whatever is really a direct equation of that
+surface in a certain system of co-ordinates, or to determine
+among all the different systems of possible co-ordinates
+that one which is generally the most convenient.
+I will only add, on this last point, that the necessary superiority
+of the ordinary rectilinear system, as to the representation
+of equations, is evidently still more marked in
+analytical geometry of three dimensions than in that of
+two, because of the incomparably greater geometrical
+complication which would result from the choice of any
+other system. This can be verified in the most striking
+manner by considering the polar system in particular,
+which is the most employed after the ordinary rectilinear
+system, for surfaces as well as for plane curves, and for
+the same reasons.</p>
+
+<p>In order to complete the general exposition of the fundamental<span class="pagenum"><a name="Page_255" id="Page_255">[Pg 255]</a></span>
+conception relative to the analytical study of
+surfaces, a philosophical examination should be made of
+a final improvement of the highest importance, which
+Monge has introduced into the very elements of this theory,
+for the classification of surfaces in natural families,
+established according to the mode of generation, and expressed
+algebraically by common differential equations, or
+by finite equations containing arbitrary functions.</p>
+
+
+
+
+<h3><a name="CURVES_OF_DOUBLE_CURVATURE" id="CURVES_OF_DOUBLE_CURVATURE">CURVES OF DOUBLE CURVATURE.</a></h3>
+
+
+<p>Let us now consider the last elementary point of view
+of analytical geometry of three dimensions; that relating
+to the algebraic representation of curves considered in
+space, in the most general manner. In continuing to
+follow the principle which has been constantly employed,
+that of the degree of indetermination of the geometrical
+locus, corresponding to the degree of independence of the
+variables, it is evident, as a general principle, that when
+a point is required to be situated upon some certain curve,
+a single co-ordinate is enough for completely determining
+its position, by the intersection of this curve with the surface
+which results from this co-ordinate. Thus, in this
+case, the two other co-ordinates of the point must be conceived
+as functions necessarily determinate and distinct
+from the first. It follows that every line, considered in
+space, is then represented analytically, no longer by a
+single equation, but by the system of two equations between
+the three co-ordinates of any one of its points. It
+is clear, indeed, from another point of view, that since
+each of these equations, considered separately, expresses
+a certain surface, their combination presents the proposed
+line as the intersection of two determinate surfaces.<span class="pagenum"><a name="Page_256" id="Page_256">[Pg 256]</a></span>
+Such is the most general manner of conceiving the algebraic
+representation of a line in analytical geometry of
+three dimensions. This conception is commonly considered
+in too restricted a manner, when we confine ourselves
+to considering a line as determined by the system
+of its two <i>projections</i> upon two of the co-ordinate planes;
+a system characterized, analytically, by this peculiarity,
+that each of the two equations of the line then contains
+only two of the three co-ordinates, instead of simultaneously
+including the three variables. This consideration,
+which consists in regarding the line as the intersection
+of two cylindrical surfaces parallel to two of the
+three axes of the co-ordinates, besides the inconvenience
+of being confined to the ordinary rectilinear system, has
+the fault, if we strictly confine ourselves to it, of introducing
+useless difficulties into the analytical representation
+of lines, since the combination of these two cylinders
+would evidently not be always the most suitable for
+forming the equations of a line. Thus, considering this
+fundamental notion in its entire generality, it will be
+necessary in each case to choose, from among the infinite
+number of couples of surfaces, the intersection of which
+might produce the proposed curve, that one which will
+lend itself the best to the establishment of equations, as
+being composed of the best known surfaces. Thus, if
+the problem is to express analytically a circle in space,
+it will evidently be preferable to consider it as the intersection
+of a sphere and a plane, rather than as proceeding
+from any other combination of surfaces which could
+equally produce it.</p>
+
+<p>In truth, this manner of conceiving the representation
+of lines by equations, in analytical geometry of three dimensions,<span class="pagenum"><a name="Page_257" id="Page_257">[Pg 257]</a></span>
+produces, by its nature, a necessary inconvenience,
+that of a certain analytical confusion, consisting
+in this: that the same line may thus be expressed, with
+the same system of co-ordinates, by an infinite number
+of different couples of equations, on account of the infinite
+number of couples of surfaces which can form it;
+a circumstance which may cause some difficulties in recognizing
+this line under all the algebraical disguises of
+which it admits. But there exists a very simple method
+for causing this inconvenience to disappear; it consists
+in giving up the facilities which result from this variety
+of geometrical constructions. It suffices, in fact, whatever
+may be the analytical system primitively established
+for a certain line, to be able to deduce from it the
+system corresponding to a single couple of surfaces uniformly
+generated; as, for example, to that of the two
+cylindrical surfaces which <i>project</i> the proposed line upon
+two of the co-ordinate planes; surfaces which will evidently
+be always identical, in whatever manner the line
+may have been obtained, and which will not vary except
+when that line itself shall change. Now, in choosing
+this fixed system, which is actually the most simple, we
+shall generally be able to deduce from the primitive equations
+those which correspond to them in this special construction,
+by transforming them, by two successive eliminations,
+into two equations, each containing only two of
+the variable co-ordinates, and thereby corresponding to
+the two surfaces of projection. Such is really the principal
+destination of this sort of geometrical combination,
+which thus offers to us an invariable and certain means
+of recognizing the identity of lines in spite of the diversity
+of their equations, which is sometimes very great.</p><hr class="chap" /><p><span class="pagenum"><a name="Page_258" id="Page_258">[Pg 258]</a></span></p>
+
+
+
+
+<h3><a name="IMPERFECTIONS_OF_ANALYTICAL_GEOMETRY" id="IMPERFECTIONS_OF_ANALYTICAL_GEOMETRY">IMPERFECTIONS OF ANALYTICAL GEOMETRY.</a></h3>
+
+
+<p>Having now considered the fundamental conception of
+analytical geometry under its principal elementary aspects,
+it is proper, in order to make the sketch complete,
+to notice here the general imperfections yet presented by
+this conception with respect to both geometry and to
+analysis.</p>
+
+<p><i>Relatively to geometry</i>, we must remark that the
+equations are as yet adapted to represent only entire
+geometrical loci, and not at all determinate portions of
+those loci. It would, however, be necessary, in some circumstances,
+to be able to express analytically a part of
+a line or of a surface, or even a <i>discontinuous</i> line or
+surface, composed of a series of sections belonging to distinct
+geometrical figures, such as the contour of a polygon,
+or the surface of a polyhedron. Thermology, especially,
+often gives rise to such considerations, to which
+our present analytical geometry is necessarily inapplicable.
+The labours of M. Fourier on discontinuous functions
+have, however, begun to fill up this great gap, and
+have thereby introduced a new and essential improvement
+into the fundamental conception of Descartes. But
+this manner of representing heterogeneous or partial figures,
+being founded on the employment of trigonometrical
+series proceeding according to the sines of an infinite
+series of multiple arcs, or on the use of certain definite
+integrals equivalent to those series, and the general integral
+of which is unknown, presents as yet too much
+complication to admit of being immediately introduced
+into the system of analytical geometry.</p>
+
+<p><i>Relatively to analysis</i>, we must begin by observing<span class="pagenum"><a name="Page_259" id="Page_259">[Pg 259]</a></span>
+that our inability to conceive a geometrical representation
+of equations containing four, five, or more variables, analogous
+to those representations which all equations of two
+or of three variables admit, must not be viewed as an imperfection
+of our system of analytical geometry, for it
+evidently belongs to the very nature of the subject.
+Analysis being necessarily more general than geometry,
+since it relates to all possible phenomena, it would be
+very unphilosophical to desire always to find among geometrical
+phenomena alone a concrete representation of
+all the laws which analysis can express.</p>
+
+<p>There exists, however, another imperfection of less
+importance, which must really be viewed as proceeding
+from the manner in which we conceive analytical geometry.
+It consists in the evident incompleteness of our
+present representation of equations of two or of three variables
+by lines or surfaces, inasmuch as in the construction
+of the geometric locus we pay regard only to the
+<i>real</i> solutions of equations, without at all noticing any
+<i>imaginary</i> solutions. The general course of these last
+should, however, by its nature, be quite as susceptible as
+that of the others of a geometrical representation. It
+follows from this omission that the graphic picture of the
+equation is constantly imperfect, and sometimes even so
+much so that there is no geometric representation at all
+when the equation admits of only imaginary solutions.
+But, even in this last case, we evidently ought to be
+able to distinguish between equations as different in
+themselves as these, for example,</p>
+
+<p>
+<i>x<sup>2</sup></i> + <i>y<sup>2</sup></i> + 1 = 0, <i>x<sup>6</sup></i> + <i>y<sup>4</sup></i> + 1 = 0, <i>y<sup>2</sup></i> + <i>e<sup>x</sup></i> = 0.<br />
+</p>
+
+<p>We know, moreover, that this principal imperfection often
+brings with it, in analytical geometry of two or of<span class="pagenum"><a name="Page_260" id="Page_260">[Pg 260]</a></span>
+three dimensions, a number of secondary inconveniences,
+arising from several analytical modifications not corresponding
+to any geometrical phenomena.</p>
+
+<hr class="tb" />
+
+<p>Our philosophical exposition of the fundamental conception
+of analytical geometry shows us clearly that this
+science consists essentially in determining what is the
+general analytical expression of such or such a geometrical
+phenomenon belonging to lines or to surfaces; and,
+reciprocally, in discovering the geometrical interpretation
+of such or such an analytical consideration. A detailed
+examination of the most important general questions
+would show us how geometers have succeeded in actually
+establishing this beautiful harmony, and in thus imprinting
+on geometrical science, regarded as a whole, its present
+eminently perfect character of rationality and of
+simplicity.</p>
+
+<div class="blockquot"><p><i>Note.</i>&mdash;The author devotes the two following chapters of his course to
+the more detailed examination of Analytical Geometry of two and of three
+dimensions; but his subsequent publication of a separate work upon this
+branch of mathematics has been thought to render unnecessary the reproduction
+of these two chapters in the present volume.</p></div>
+
+
+<p>THE END.</p>
+
+<div class="footnote">
+<p>FOOTNOTES:</p>
+
+<div class="footnote"><p><a name="Footnote_1_1" id="Footnote_1_1"></a><a href="#FNanchor_1_1"><span class="label">[1]</span></a> The investigation of the mathematical phenomena of the laws of heat
+by Baron Fourier has led to the establishment, in an entirely direct manner,
+of Thermological equations. This great discovery tends to elevate our philosophical
+hopes as to the future extensions of the legitimate applications of
+mathematical analysis, and renders it proper, in the opinion of author,
+to regard <i>Thermology</i> as a third principal branch of concrete mathematics.</p></div>
+
+<div class="footnote"><p><a name="Footnote_2_2" id="Footnote_2_2"></a><a href="#FNanchor_2_2"><span class="label">[2]</span></a> The translator has felt justified in employing this very convenient word
+(for which our language has no precise equivalent) as an English one, in its
+most extended sense, in spite of its being often popularly confounded with
+its Differential and Integral department.</p></div>
+
+<div class="footnote"><p><a name="Footnote_3_3" id="Footnote_3_3"></a><a href="#FNanchor_3_3"><span class="label">[3]</span></a> With the view of increasing as much as possible the resources and the
+extent (now so insufficient) of mathematical analysis, geometers count this
+last couple of functions among the analytical elements. Although this inscription
+is strictly legitimate, it is important to remark that circular functions
+are not exactly in the same situation as the other abstract elementary
+functions. There is this very essential difference, that the functions of the
+four first couples are at the same time simple and abstract, while the circular
+functions, which may manifest each character in succession, according
+to the point of view under which they are considered and the manner in
+which they are employed, never present these two properties simultaneously.
+</p>
+<p>
+Some other concrete functions may be usefully introduced into the number
+of analytical elements, certain conditions being fulfilled. It is thus, for
+example, that the labours of M. Legendre and of M. Jacobi on <i>elliptical</i>
+functions have truly enlarged the field of analysis; and the same is true of
+some definite integrals obtained by M. Fourier in the theory of heat.</p></div>
+
+<div class="footnote"><p><a name="Footnote_4_4" id="Footnote_4_4"></a><a href="#FNanchor_4_4"><span class="label">[4]</span></a> Suppose, for example, that a question gives the following equation between
+an unknown magnitude x, and two known magnitudes, <i>a</i> and <i>b</i>,
+</p>
+
+<p><br />
+<i>x<sup>3</sup></i> + 3<i>ax</i> = 2<i>b</i>,<br />
+</p>
+
+<p>
+as is the case in the problem of the trisection of an angle. We see at once
+that the dependence between <i>x</i> on the one side, and <i>ab</i> on the other, is
+completely determined; but, so long as the equation preserves its primitive
+form, we do not at all perceive in what manner the unknown quantity is
+derived from the data. This must be discovered, however, before we can
+think of determining its value. Such is the object of the algebraic part of
+the solution. When, by a series of transformations which have successively
+rendered that derivation more and more apparent, we have arrived at presenting
+the proposed equation under the form
+</p>
+
+<p><br />
+<i>x</i> = &#8731;(<i>b</i> + &#8730;(<i>b<sup>2</sup></i> + <i>a<sup>3</sup></i>)) + &#8731;(<i>b</i> - &#8730;(<i>b<sup>2</sup></i> + <i>a<sup>3</sup></i>)),<br />
+</p>
+
+<p>
+the work of <i>algebra</i> is finished; and even if we could not perform the arithmetical
+operations indicated by that formula, we would nevertheless have
+obtained a knowledge very real, and often very important. The work of
+<i>arithmetic</i> will now consist in taking that formula for its starting point, and
+finding the number <i>x</i> when the values of the numbers <i>a</i> and <i>b</i> are given.</p></div>
+
+<div class="footnote"><p><a name="Footnote_5_5" id="Footnote_5_5"></a><a href="#FNanchor_5_5"><span class="label">[5]</span></a> I have thought that I ought to specially notice this definition, because
+it serves as the basis of the opinion which many intelligent persons, unacquainted
+with mathematical science, form of its abstract part, without considering
+that at the time of this definition mathematical analysis was not
+sufficiently developed to enable the general character of each of its principal
+parts to be properly apprehended, which explains why Newton could
+at that time propose a definition which at the present day he would certainly
+reject.</p></div>
+
+<div class="footnote"><p><a name="Footnote_6_6" id="Footnote_6_6"></a><a href="#FNanchor_6_6"><span class="label">[6]</span></a> This is less strictly true in the English system of numeration than in
+the French, since "twenty-one" is our more usual mode of expressing this
+number.</p></div>
+
+<div class="footnote"><p><a name="Footnote_7_7" id="Footnote_7_7"></a><a href="#FNanchor_7_7"><span class="label">[7]</span></a> Simple as may seem, for example, the equation
+</p>
+
+<p><br />
+<i>a<sup>x</sup></i> + <i>b<sup>x</sup></i> = <i>c<sup>x</sup></i>,<br />
+
+</p>
+<p>
+we do not yet know how to resolve it, which may give some idea of the
+extreme imperfection of this part of algebra.</p></div>
+
+<div class="footnote"><p><a name="Footnote_8_8" id="Footnote_8_8"></a><a href="#FNanchor_8_8"><span class="label">[8]</span></a> The same error was afterward committed, in the infancy of the infinitesimal
+calculus, in relation to the integration of differential equations.</p></div>
+
+<div class="footnote"><p><a name="Footnote_9_9" id="Footnote_9_9"></a><a href="#FNanchor_9_9"><span class="label">[9]</span></a> The fundamental principle on which reposes the theory of equations,
+and which is so frequently applied in all mathematical analysis&mdash;the decomposition
+of algebraic, rational, and entire functions, of any degree whatever,
+into factors of the first degree&mdash;is never employed except for functions
+of a single variable, without any one having examined if it ought to be extended
+to functions of several variables. The general impossibility of such
+a decomposition is demonstrated by the author in detail, but more properly
+belongs to a special treatise.</p></div>
+
+<div class="footnote"><p><a name="Footnote_10_10" id="Footnote_10_10"></a><a href="#FNanchor_10_10"><span class="label">[10]</span></a> The only important case of this class which has thus far been completely
+treated is the general integration of <i>linear</i> equations of any order
+whatever, with constant coefficients. Even this case finally depends on
+the algebraic resolution of equations of a degree equal to the order of differentiation.</p></div>
+
+<div class="footnote"><p><a name="Footnote_11_11" id="Footnote_11_11"></a><a href="#FNanchor_11_11"><span class="label">[11]</span></a> Leibnitz had already considered the comparison of one curve with an
+other infinitely near to it, calling it "<i>Differentiatio de curva in curvam</i>."
+But this comparison had no analogy with the conception of Lagrange, the
+curves of Leibnitz being embraced in the same general equation, from which
+they were deduced by the simple change of an arbitrary constant.</p></div>
+
+<div class="footnote"><p><a name="Footnote_12_12" id="Footnote_12_12"></a><a href="#FNanchor_12_12"><span class="label">[12]</span></a> I propose hereafter to develop this new consideration, in a special work
+upon the <i>Calculus of Variations</i>, intended to present this hyper-transcendental
+analysis in a new point of view, which I think adapted to extend its
+general range.</p></div>
+
+<div class="footnote"><p><a name="Footnote_13_13" id="Footnote_13_13"></a><a href="#FNanchor_13_13"><span class="label">[13]</span></a> Lacroix has justly criticised the expression of <i>solid</i>, commonly used by
+geometers to designate a <i>volume</i>. It is certain, in fact, that when we wish
+to consider separately a certain portion of indefinite space, conceived as gaseous,
+we mentally solidify its exterior envelope, so that a <i>line</i> and a <i>surface</i>
+are habitually, to our minds, just as <i>solid</i> as a <i>volume</i>. It may also be remarked
+that most generally, in order that bodies may penetrate one another
+with more facility, we are obliged to imagine the interior of the <i>volumes</i> to
+be hollow, which renders still more sensible the impropriety of the word
+<i>solid</i>.</p></div>
+</div>
+
+
+
+
+
+
+
+
+
+<pre>
+
+
+
+
+
+End of Project Gutenberg's The philosophy of mathematics, by Auguste Comte
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+</pre>
+
+</body>
+</html>
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