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Read Ebook: The philosophy of mathematics by Comte Auguste Gillespie W M William Mitchell Translator
Font size: Background color: Text color: Add to tbrJar First Page Next PageEbook has 243 lines and 66035 words, and 5 pagesINTRODUCTION. Page GENERAL CONSIDERATIONS ON MATHEMATICAL SCIENCE 17 TRUE DEFINITION OF MATHEMATICS 25 A Science, not an Art 25 EXTENT OF ITS FIELD 35 Its Universality 36 Its Limitations 37 ANALYSIS. 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 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 TRANSCENDENTAL ANALYSIS: Page ITS DIFFERENT CONCEPTIONS 88 Preliminary Remarks 88 Its early History 89 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 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 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 GEOMETRY. 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 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 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 Add to tbrJar First Page Next Page |
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