Chapter III: 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.