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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
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