By Derrick Norman Lehmer

Meant to provide, As easily As attainable, The necessities of artificial Projective Geometry - Chapters: One-To-One Correspondence - family among primary varieties In One-To-One Correspondence With one another - mixture of 2 Projectively similar basic 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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Pascal's theorem furnishes an elegant solution of the problem of drawing a conic through five given points. To construct a sixth point on the conic, draw through the point numbered 1 an arbitrary line (Fig. 14), and [44] 50 An Elementary Course in Synthetic Projective Geometry let the desired point 6 be the second point of intersection of this line with the conic. The point L = 12-45 is obtainable at once; also the point N = 34-61. But L and N determine Pascal's line, and the intersection of 23 with 56 must be on this line.

Then, since the triangle DAS is similar to the triangle BAA', we may write the proportion AB : AD = BA' : SD. 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 [26] 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.

If any point is joined to four harmonic points, and the four lines thus obtained are cut by any fifth, the four points of intersection are again harmonic. 33. Four harmonic lines 21 33. Four harmonic lines. We are now able to extend the notion of harmonic elements to pencils of rays, and indeed to axial pencils. For if we define four harmonic rays as four rays which pass through a point and which pass one through each of four harmonic points, we have the theorem Four harmonic lines are cut by any transversal in four harmonic points.

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An Elementary Course In Synthetic Projective Geometry by Derrick Norman Lehmer


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