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
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.
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 1⁰. Different Questions with respect to the same Figure 204 2⁰. 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