An Elementary Course in Synthetic Projective Geometry

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

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 other. Magicians are apt to assign magic properties to many of the words and symbols which they are in the habit of using, and scientists are constantly confusing objective things with the subjective formulas for them. After the physicist has set up correspondences between physical facts and mathematical formulas, the "interpretation" of these formulas is his most important and difficult task.

*3.* In mathematics, effort is constantly being made to set up one-to-one correspondences between simple notions and more complicated ones, or between the well-explored fields of research and fields less known. Thus, by means of the mechanism employed in analytic geometry, algebraic theorems are made to yield geometric ones, and vice versa. In geometry we get at the properties of the conic sections by means of the properties of the straight line, and cubic surfaces are studied by means of the plane.

FIG. 1

FIG. 2

*4. One-to-one correspondence and enumeration.* If a one-to-one correspondence has been set up between the objects of one set and the objects of another set, then the inference may usually be drawn that they have the same number of elements. If, however, there is an infinite number of individuals in each of the two sets, the notion of counting is necessarily ruled out. It may be possible, nevertheless, to set up a one-to-one correspondence between the elements of two sets even when the number is infinite. Thus, it is easy to set up such a correspondence between the points of a line an inch long and the points of a line two inches long. For let the lines (Fig. 1) be AB and A’B’. Join AA’ and BB’, and let these joining lines meet in S. For every point C on AB a point C’ may be found on A’B’ by joining C to S and noting the point C’ where CS meets A’B’. Similarly, a point C may be found on AB for any point C’ on A’B’. The correspondence is clearly one-to-one, but it would be absurd to infer from this that there were just as many points on AB as on A’B’. In fact, it would be just as reasonable to infer that there were twice as many points on A’B’ as on AB. For if we bend A’B’ into a circle with center at S (Fig. 2), we see that for every point C on AB there are two points on A’B’. Thus it is seen that the notion of one-to-one correspondence is more extensive than the notion of counting, and includes the notion of counting only when applied to finite assemblages.

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