Contents - An Elementary Course in Synthetic Projective Geometry - Derrick Norman Lehmer

[Preface]
[Contents]
[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]
[76. Inscribed quadrangle]
[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]
[88. Circumscribed quadrilateral]
[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]
[95. Inscribed and circumscribed quadrilaterals]
[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]

[pg 1]