An Elementary Course in Synthetic Projective Geometry | Page 3

points and lines 101. Construction of the polar line of a given point 102. Self-polar triangle 103. Pole and polar projectively related 104. Duality 105. Self-dual theorems 106. Other correspondences PROBLEMS

CHAPTER VII
- METRICAL PROPERTIES OF THE CONIC SECTIONS 107. Diameters. Center 108. Various theorems 109. Conjugate diameters 110. Classification of conics 111. Asymptotes 112. Various theorems 113. Theorems concerning asymptotes 114. Asymptotes and conjugate diameters 115. Segments cut off on a chord by hyperbola and its asymptotes 116. Application of the theorem 117. Triangle formed by the two asymptotes and a tangent 118. Equation of hyperbola referred to the asymptotes 119. Equation of parabola 120. Equation of central conics referred to conjugate diameters PROBLEMS

CHAPTER VIII
- INVOLUTION 121. Fundamental theorem 122. Linear construction 123. Definition of involution of points on a line 124. Double-points in an involution 125. Desargues's theorem concerning conics through four points 126. Degenerate conics of the system 127. Conics through four points touching a given line 128. Double correspondence 129. Steiner's construction 130. Application of Steiner's construction to double correspondence 131. Involution of points on a point-row of the second order. 132. Involution of rays 133. Double rays 134. Conic through a fixed point touching four lines 135. Double correspondence 136. Pencils of rays of the second order in involution 137. Theorem concerning pencils of the second order in involution 138. Involution of rays determined by a conic 139. Statement of theorem 140. Dual of the theorem PROBLEMS

CHAPTER IX
- METRICAL PROPERTIES OF INVOLUTIONS 141. Introduction 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 Credits A Word from Project Gutenberg The Full Project Gutenberg License

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 certain words, 'one,' 'two,' 'three,' etc., in the mind. Many savage peoples have discovered no better method of counting than by setting up a one-to-one correspondence between the objects to be counted and their fingers. The scientist who busies himself with naming and classifying the objects of nature is only setting up a one-to-one correspondence between the objects and certain words which serve, not as a means of counting the objects, but of listing them in a convenient way. Thus he may be able to marshal and array his material in such a way as to bring to light relations that may exist between the objects themselves. Indeed, the whole notion of language springs from this idea of one-to-one correspondence.

*2. Consequences of one-to-one correspondence.* The most useful and interesting problem that may arise in connection with any one-to-one correspondence is to determine just what relations existing between the individuals of one assemblage may be carried over to another assemblage in one-to-one correspondence with it. It is a favorite error to assume that whatever holds for one set must also hold for the