The Philosophy of Mathematics

CHAPTER I.. Page

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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

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:

TRANSCENDENTAL ANALYSIS:

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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

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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

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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

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.

GEOMETRY.

CHAPTER I.

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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

ANCIENT OR SYNTHETIC GEOMETRY

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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

MODERN OR ANALYTICAL GEOMETRY

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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.

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