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 18
  2. 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 70
• 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
• 1. Derived Functions 108
  2. An extension of ordinary Analysis 108
  3. Example: Tangents 109
  4. Fundamental Identity of the three Methods 110
  5. Their comparative Value 113
  6. That of Leibnitz 113
  7. That of Newton 115
  8. 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 144
  9. Definite Integrals 146
  10. 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 182
• 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. 1. Different Questions with respect to the same Figure 204
     2. 2. 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
• 1. 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 258
  2. Relatively to Analysis 258

THE
PHILOSOPHY OF MATHEMATICS.