By Derrick Norman Lehmer
Meant to provide, As easily As attainable, The necessities of man-made Projective Geometry - Chapters: One-To-One Correspondence - relatives among primary varieties In One-To-One Correspondence With one another - mix of 2 Projectively similar primary kinds - Point-Rows Of the second one Order - Pencils Of Rays Of the second one Order - Poles And Polars - Metrical houses Of The Conic Sections - Involution - Metrical houses Of Involutions - at the historical past of man-made Projective Geometry - Index
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Extra info for An Elementary Course In Synthetic Projective Geometry
Cut across, now, by any plane, and we get a conic section which is thus exhibited as the locus of intersection of two projective pencils. It thus appears that a conic section is a point-row of the second order. It will later 74. Conic through five points 49 appear that a point-row of the second order is a conic section. In the future, therefore, we shall refer to a point-row of the second order as a conic. FIG. 14 74. Conic through five points. Pascal's theorem furnishes an elegant solution of the problem of drawing a conic through five given points.
If two points of Pascal's hexagon approach coincidence, then the line joining them approaches as a limiting position the tangent line at that point. Pascal's theorem thus affords a ready method of drawing the tangent line to a conic at a given point. If the conic is determined by the points 1, 2, 3, 4, 5 (Fig. 15), and it is desired to draw the tangent at the point 1, we may call that point 1, 6. The points L and M are obtained as usual, and the intersection of 34 with LM gives N. Join N to the point 1 for the desired tangent at that point.
Also, from the similar triangles DSC and BCC', we have CD : CB = SD : B'C. From these two proportions we have, remembering that BA' = BC', AB · CD = −1, AD · CB the minus sign being given to the ratio on account of the fact  28 An Elementary Course in Synthetic Projective Geometry that A and C are always separated from B and D, so that one or three of the segments AB, CD, AD, CB must be negative. 45. Writing the last equation in the form CB : AB = -CD : AD, and using the fundamental relation connecting three points on a line, PR + RQ = PQ, which holds for all positions of the three points if account be taken of the sign of the segments, the last proportion may be written (CB - BA) : AB = -(CA - DA) : AD, or (AB - AC) : AB = (AC - AD) : AD; so that AB, AC, and AD are three quantities in hamonic progression, since the difference between the first and second is to the first as the difference between the second and third is to the third.
An Elementary Course In Synthetic Projective Geometry by Derrick Norman Lehmer