An Elementary Course in Synthetic Projective Geometry

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. Correspond...

CORRESPONDENCE WITH EACH OTHER 23. Seven fundamental forms 24. Projective properties 25. Desargues’s theorem 26. Fundamental theorem concerning two complete quadrangles 27. Impo...

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 p...

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 co...

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....

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 t...

107. Diameters. Center 108. Various theorems 109. Conjugate diameters 110. Classification of conics 111. Asymptotes 112. Various theorems 113. Theorems concerning asymptotes 114...

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 conic...

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...

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...

*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...

*23. Seven fundamental forms.* In the preceding chapter we have called attention to seven fundamental forms: the point-row, the pencil of rays, the axial pencil, the plane syste...

*47. Superposed fundamental forms. Self-corresponding elements.* We have seen (§ 37) that two projective point-rows may be superposed upon the same straight line. This happens,...

*60. Point-row of the second order defined.* We have seen that two fundamental forms in one-to-one correspondence may sometimes generate a form of higher order. Thus, two point-...

*79. Pencil of rays of the second order defined.* If the corresponding points of two projective point-rows be joined by straight lines, a system of lines is obtained which is ca...

_If a quadrilateral be inscribed in a conic, two pairs of opposite sides and the tangents at opposite vertices intersect in four points, all of which lie on a straight line._

*107. Diameters. Center.* After what has been said in the last chapter one would naturally expect to get at the metrical properties of the conic sections by the introduction of...

*121. Fundamental theorem.* The important theorem concerning two complete quadrangles (§ 26), upon which the theory of four harmonic points was based, can easily be extended to...

*141. Introduction of infinite point; center of involution.* We connect the projective theory of involution with the metrical, as usual, by the introduction of the elements at i...

*161. Ancient results.* The theory of synthetic projective geometry as we have built it up in this course is less than a century old. This is not to say that many of the theorem...