# An Elementary Course in Synthetic Projective Geometry # An Elementary Course in Synthetic Projective Geometry

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CHAPTER I - ONE-TO-ONE CORRESPONDENCE

1. Definition of one-to-one correspondence

2. Consequences of one-to-one correspondence

3. Applications in mathematics

4. One-to-one correspondence and enumeration

5. Correspondence between a part and the whole

6. Infinitely distant point

7. Axial pencil; fundamental forms

8. Perspective position

9. Projective relation

10. Infinity-to-one correspondence

11. Infinitudes of different orders

12. Points in a plane

13. Lines through a point

14. Planes through a point

15. Lines in a plane

16. Plane system and point system

17. Planes in space

18. Points of space

19. Space system

20. Lines in space

21. Correspondence between points and numbers

22. Elements at infinity

PROBLEMS

CHAPTER II - RELATIONS BETWEEN FUNDAMENTAL FORMS IN ONE-TO-ONE CORRESPONDENCE WITH EACH OTHER

23. Seven fundamental forms

24. Projective properties

25. Desargues's theorem

26. Fundamental theorem concerning two complete quadrangles

27. Importance of the theorem

28. Restatement of the theorem

29. Four harmonic points

30. Harmonic conjugates

31. Importance of the notion of four harmonic points

32. Projective invariance of four harmonic points

33. Four harmonic lines

34. Four harmonic planes

35. Summary of results

36. Definition of projectivity

37. Correspondence between harmonic conjugates

38. Separation of harmonic conjugates

39. Harmonic conjugate of the point at infinity

40. Projective theorems and metrical theorems. Linear construction

41. Parallels and mid-points

42. Division of segment into equal parts

43. Numerical relations

44. Algebraic formula connecting four harmonic points

45. Further formulae

46. Anharmonic ratio

PROBLEMS

CHAPTER III - COMBINATION OF TWO PROJECTIVELY RELATED FUNDAMENTAL FORMS

47. Superposed fundamental forms. Self-corresponding elements

48. Special case

49. Fundamental theorem. Postulate of continuity

50. Extension of theorem to pencils of rays and planes

51. Projective point-rows having a self-corresponding point in common

52. Point-rows in perspective position

53. Pencils in perspective position

54. Axial pencils in perspective position

55. Point-row of the second order

56. Degeneration of locus

57. Pencils of rays of the second order

58. Degenerate case

59. Cone of the second order

PROBLEMS

CHAPTER IV - POINT-ROWS OF THE SECOND ORDER

60. Point-row of the second order defined

61. Tangent line

62. Determination of the locus

63. Restatement of the problem

64. Solution of the fundamental problem

65. Different constructions for the figure

66. Lines joining four points of the locus to a fifth

67. Restatement of the theorem

68. Further important theorem

69. Pascal's theorem

70. Permutation of points in Pascal's theorem

71. Harmonic points on a point-row of the second order

72. Determination of the locus

73. Circles and conics as point-rows of the second order

74. Conic through five points

75. Tangent to a conic

77. Inscribed triangle

78. Degenerate conic

PROBLEMS

CHAPTER V - PENCILS OF RAYS OF THE SECOND ORDER

79. Pencil of rays of the second order defined

80. Tangents to a circle

81. Tangents to a conic

82. Generating point-rows lines of the system

83. Determination of the pencil

84. Brianchon's theorem

85. Permutations of lines in Brianchon's theorem

86. Construction of the penvil by Brianchon's theorem

87. Point of contact of a tangent to a conic

89. Circumscribed triangle

90. Use of Brianchon's theorem

91. Harmonic tangents

92. Projectivity and perspectivity

93. Degenerate case

94. Law of duality

PROBLEMS

CHAPTER VI - POLES AND POLARS

96. Definition of the polar line of a point

97. Further defining properties

98. Definition of the pole of a line

99. Fundamental theorem of poles and polars

100. Conjugate 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