By Derrick Norman Lehmer
Meant to provide, As easily As attainable, The necessities of artificial Projective Geometry - Chapters: One-To-One Correspondence - family among basic varieties In One-To-One Correspondence With one another - mixture of 2 Projectively similar primary types - Point-Rows Of the second one Order - Pencils Of Rays Of the second one Order - Poles And Polars - Metrical homes Of The Conic Sections - Involution - Metrical houses Of Involutions - at the historical past of artificial Projective Geometry - Index
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Extra resources for An Elementary Course In Synthetic Projective Geometry
This theorem throws into our hands the following fundamental theorem concerning two complete quadrangles, a complete quadrangle being defined as the figure obtained by joining any four given points by straight lines in the six possible ways. Given two complete quadrangles, K, L, M, N and K', L', M', N', so related that KL, K'L', MN, M'N' all meet in a point A; LM, L'M', NK, N'K' all meet in a point Q; and LN, L'N' meet in a point B on the line AC; then the lines KM and K'M' also meet in a point D on the line AC.
18] 27. Importance of the theorem. The importance of this theorem lies in the fact that, A, B, and C being given, an indefinite number of quadrangles K', L', M', N' my be found such that K'L' and M'N' meet in A, K'N' and L'M' in C, with L'N' passing through B. Indeed, the lines AK' and AM' may be drawn arbitrarily through A, and any line through B may be used to determine L' and N'. By joining these two points to C the points K' and M' are determined. Then the line joining K' and M', found in this way, must pass through the point D already determined by the quadrangle K, L, M, N.
It is natural to speak of four such points as four harmonic points on the locus, and to use this notion to define projective correspondence between point-rows of the second order, or between a point-row of the second order and any fundamental form of the first order. Thus, in particular, the point-row of the second order, Ã, is said to be perspectively related to the  48 An Elementary Course in Synthetic Projective Geometry pencil S when every ray on S goes through the point on Ã which corresponds to it.