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Written by the nineteenth-century French philosophical founder of positivism, this comprehensive map of mathematical science assigns to each part of the complex whole its true position and value. The two-part treatment begins with a general view of mathematical analysis and advances to algebra, continuing with an exploration of geometry's ancient and modern methods.
| Introduction.: General Considerations on Mathematical Science | 17 | |
| The Object of Mathematics | 18 | |
| Measuring Magnitudes | 18 | |
| 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 | General View of Mathematical Analysis | 45 |
| The True Idea of an Equation | 46 | |
| Divisions of the Calculus | 53 | |
| Chapter II | Ordinary Analysis; or, Algebra | 69 |
| Algebraic Equations | 71 | |
| Algebraic Resolution of Equations | 72 | |
| Numerical Resolution of Equations | 75 | |
| 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 |
| Method of Leibnitz | 91 | |
| Method of Newton | 103 | |
| Method of Lagrange | 108 | |
| Chapter IV | The Differential and Integral Calculus | 120 |
| Its two fundamental Divisions | 120 | |
| Their Relations to each Other | 121 | |
| The Differential Calculus | 127 | |
| The Integral Calculus | 135 | |
| Chapter V | The Calculus of Variations | 151 |
| Problems giving rise to it | 151 | |
| Analytical Nature of these Questions | 154 | |
| Methods of the older Geometers | 155 | |
| Method of Lagrange | 156 | |
| Its Relations to the ordinary Calculus | 163 | |
| Chapter VI | The Calculus of Finite Differences | 167 |
| General Theory of Series | 170 | |
| Periodic or Discontinuous Functions | 173 | |
| Applications of this Calculus | 173 | |
| Book II | Geometry | |
| Chapter I | A General View of Geometry | 179 |
| The final Object of Geometry | 184 | |
| The infinite extent of its Field | 190 | |
| Expansion of Original Definition | 193 | |
| The two general Methods of Geometry | 202 | |
| Chapter II | Ancient or Synthetic Geometry | |
| Its proper Extent | 212 | |
| Geometry of the right Line | 217 | |
| Graphical Solutions | 218 | |
| Algebraical Solutions | 224 | |
| Chapter III | Modern or Analytical Geometry | |
| The analytical Representation of Figures | 232 | |
| Plane Curves | 237 | |
| Surfaces | 251 | |
| Curves in Space | 255 | |
| Imperfections of Analytical Geometry | 258 |
Overview
Written by the nineteenth-century French philosophical founder of positivism, this comprehensive map of mathematical science assigns to each part of the complex whole its true position and value. The two-part treatment begins with a general view of mathematical analysis and advances to algebra, continuing with an exploration of geometry's ancient and modern methods.