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 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. CHAPTER I. 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 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 Page