THE
PHILOSOPHY
OF
MATHEMATICS.

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
• 1. Measuring Magnitudes   

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1. Difficulties    19
2. General Method    20
3. Illustrations    21
4. 1. 1. Falling Bodies    21
   2. 2. Inaccessible Distances    23
   3. 3. Astronomical Facts    24

True Definition of Mathematics    25

1. A Science, not an Art    25

Its Two Fundamental Divisions    26

1. Their different Objects    27
2. Their different Natures    29
3. Concrete Mathematics    31
4. Geometry and Mechanics    32
5. Abstract Mathematics    33
6. The Calculus, or Analysis    33

Extent of Its Field    35

1. Its Universality    36
2. Its Limitations    37

BOOK I.
ANALYSIS.

CHAPTER I.

•     Page
• GENERAL VIEW OF MATHEMATICAL ANALYSIS    45
• The True Idea of an Equation    46
• 1. Division of Functions into Abstract and Concrete    47
  2. Enumeration of Abstract Functions    50
• Divisions of the Calculus    53
• 1. The Calculus of Values, or Arithmetic    57
  2. Its Extent    57
  3. Its true Nature    59
  4. The Calculus of Functions    61
  5. Two Modes of obtaining Equations    61
  6. 1. 1. By the Relations between the given Quantities    61
     2. 2. By the Relations between auxiliary Quantities    64
  7. Corresponding Divisions of the Calculus of Functions    67

CHAPTER II.

• ORDINARY ANALYSIS; OR, ALGEBRA.    69
• 1. Its Object    69
  2. Classification of Equations   

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Algebraic Equations    71

1. Their Classification    71

Algebraic Resolution of Equations    72

1. Its Limits    72
2. General Solution    72
3. What we know in Algebra    74

Numerical Resolution of Equations    75

1. 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: its different conceptions    88
• 1. Preliminary Remarks    88
  2. Its early History    89
• Method of Leibnitz    91
• 1. Infinitely small Elements    91
  2. Examples:
  3. 1. 1. Tangents    93
     2. 2. Rectification of an Arc    94
     3. 3. Quadrature of a Curve    95
     4. 4. Velocity in variable Motion    95
     5. 5. Distribution of Heat    96
  4. Generality of the Formulas    97
  5. Demonstration of the Method    98
  6. 1. Illustration by Tangents    102
• Method of Newton    103
• 1. Method of Limits    103
  2. Examples:
  3. 1. 1. Tangents    104
     2. 2. Rectifications    105
  4. Fluxions and Fluents    106
• Method of Lagrange    108

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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.

• THE DIFFERENTIAL AND INTEGRAL CALCULUS    120
• Its two fundamental Divisions    120
• Their Relations to each Other    121
• 1. 1. Use of the Differential Calculus as preparatory to that of the Integral    123
  2. 2. Employment of the Differential Calculus alone    125
  3. 3. Employment of the Integral Calculus alone    125
  4. 1. Three Classes of Questions hence resulting    126
• The Differential Calculus    127
• 1. Two Cases: Explicit and Implicit Functions    127
  2. 1. Two sub-Cases: a single Variable or several    129
     2. Two other Cases: Functions separate or combined    130
  3. Reduction of all to the Differentiation of the ten elementary Functions    131
  4. Transformation of derived Functions for new Variables    132
  5. Different Orders of Differentiation    133
  6. Analytical Applications    133
• The Integral Calculus    135
• 1. Its fundamental Division: Explicit and Implicit Functions    135
  2. Subdivisions: a single Variable or several    136
  3. Calculus of partial Differences    137
  4. Another Subdivision: different Orders of Differentiation    138
  5. Another equivalent Distinction    140
  6. Quadratures    142
  7. 1. Integration of Transcendental Functions    143
     2. Integration by Parts    143
     3. Integration of Algebraic Functions    143
  8. Singular Solutions   

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Definite Integrals    146

Prospects of the Integral Calculus    148

CHAPTER V.

• THE CALCULUS OF VARIATIONS    151
• Problems giving rise to it    151
• 1. Ordinary Questions of Maxima and Minima    151
  2. A new Class of Questions    152
  3. 1. Solid of least Resistance; Brachystochrone; Isoperimeters    153
• Analytical Nature of these Questions    154
• Methods of the older Geometers    155
• Method of Lagrange    156
• 1. Two Classes of Questions    157
  2. 1. 1. Absolute Maxima and Minima    157
     2. Equations of Limits    159
     3. 1. A more general Consideration    159
     4. 2. Relative Maxima and Minima    160
  3. Other Applications of the Method of Variations    162
• Its Relations to the ordinary Calculus    163

CHAPTER VI.

• THE CALCULUS OF FINITE DIFFERENCES    167
• 1. Its general Character    167
  2. Its true Nature    168
• General Theory of Series    170
• 1. Its Identity with this Calculus    172
• Periodic or discontinuous Functions    173
• Applications of this Calculus    173
• 1. Series    173
  2. Interpolation    173
  3. Approximate Rectification, &c.    174

BOOK II.
GEOMETRY.

CHAPTER I.

• A GENERAL VIEW OF GEOMETRY    179
• 1. The true Nature of Geometry    179
  2. Two fundamental Ideas    181
  3. 1. 1. The Idea of Space    181
     2. 2. Different kinds of Extension   

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The final object of Geometry    184

1. Nature of Geometrical Measurement    185
2. 1. Of Surfaces and Volumes    185
   2. Of curve Lines    187
   3. Of right Lines    189

The infinite extent of its Field    190

1. Infinity of Lines    190
2. Infinity of Surfaces    191
3. Infinity of Volumes    192
4. Analytical Invention of Curves, &c.    193

Expansion of Original Definition    193

1. Properties of Lines and Surfaces    195
2. Necessity of their Study    195
3. 1. 1. To find the most suitable Property    195
   2. 2. To pass from the Concrete to the Abstract    197
4. Illustrations:
5. 1. Orbits of the Planets    198
   2. Figure of the Earth    199

The two general Methods of Geometry    202

1. Their fundamental Difference    203
2. 1. 1â°. Different Questions with respect to the same Figure    204
   2. 2â°. Similar Questions with respect to different Figures    204
3. Geometry of the Ancients    204
4. Geometry of the Moderns    206
5. Superiority of the Modern    207
6. The Ancient the base of the Modern    209

CHAPTER II.

• ANCIENT OR SYNTHETIC GEOMETRY    212
• Its proper Extent    212
• 1. Lines; Polygons; Polyhedrons    212
  2. Not to be farther restricted    213
  3. Improper Application of Analysis    214
  4. Attempted Demonstrations of Axioms    216
• Geometry of the right Line    217
• Graphical Solutions    218

Descriptive Geometry    220

Algebraical Solutions    224

1. Trigonometry    225
2. Two Methods of introducing Angles    226
3. 1. 1. By Arcs    226
   2. 2. By trigonometrical Lines    226
4. Advantages of the latter    226
5. Its Division of trigonometrical Questions    227
6. 1. 1. Relations between Angles and trigonometrical Lines    228
   2. 2. Relations between trigonometrical Lines and Sides    228
7. Increase of trigonometrical Lines    228
8. Study of the Relations between them    230

CHAPTER III.

• MODERN OR ANALYTICAL GEOMETRY    232
• The analytical Representation of Figures    232
• 1. Reduction of Figure to Position    233
  2. Determination of the position of a Point    234
• Plane Curves    237
• 1. Expression of Lines by Equations    237
  2. Expression of Equations by Lines    238
  3. Any change in the Line changes the Equation    240
  4. Every "Definition" of a Line is an Equation    241
  5. Choice of Co-ordinates    245
  6. Two different points of View    245
  7. 1. 1. Representation of Lines by Equations    246
     2. 2. Representation of Equations by Lines    246
  8. Superiority of the rectilinear System    248
  9. 1. Advantages of perpendicular Axes    249
• Surfaces    251
• 1. Determination of a Point in Space    251
  2. Expression of Surfaces by Equations    253
  3. Expression of Equations by Surfaces    253
• Curves in Space    255
• Imperfections of Analytical Geometry    258
• 1. Relatively to Geometry   

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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.