roduction of infinite point; center of involution
142. Fundamental metrical theorem
143. Existence of double points
144. Existence of double rays
145. Construction of an involution by means of circles
146. Circular points
147. Pairs in an involution of rays which are at right angles. Circular
involution
148. Axes of conics
149. Points at which the involution determined by a conic is circular
150. Properties of such a point
151. Position of such a point
152. Discovery of the foci of the conic
153. The circle and the parabola
154. Focal properties of conics
155. Case of the parabola
156. Parabolic reflector
157. Directrix. Principal axis. Vertex
158. Another definition of a conic
159. Eccentricity
160. Sum or difference of focal distances
PROBLEMS
CHAPTER X - ON THE HISTORY OF SYNTHETIC PROJECTIVE GEOMETRY
161. Ancient results
162. Unifying principles
163. Desargues
164. Poles and polars
165. Desargues's theorem concerning conics through four points
166. Extension of the theory of poles and polars to space
167. Desargues's method of describing a conic
168. Reception of Desargues's work
169. Conservatism in Desargues's time
170. Desargues's style of writing
171. Lack of appreciation of Desargues
172. Pascal and his theorem
173. Pascal's essay
174. Pascal's originality
175. De la Hire and his work
176. Descartes and his influence
177. Newton and Maclaurin
178. Maclaurin's construction
179. Descriptive geometry and the second revival
180. Duality, homology, continuity, contingent relations
181. Poncelet and Cauchy
182. The work of Poncelet
183. The debt which analytic geometry owes to synthetic geometry
184. Steiner and his work
185. Von Staudt and his work
186. Recent developments
INDEX

CHAPTER I - ONE-TO-ONE CORRESPONDENCE

*1. Definition of one-to-one correspondence.* Given any two sets of
individuals, if it is possible to set up such a correspondence between the
two sets that to any individual in one set corresponds one and only one
individual in the other, then the two sets are said to be in _one-to-one
correspondence_ with each other. This notion, simple as it is, is of
fundamental importance in all branches of science. The process of counting
is nothing but a setting up of a one-to-one correspondence between the
objects to be counted and c