An Elementary Course In Synthetic Projective Geometry by Derrick Norman Lehmer

By Derrick Norman Lehmer

Meant to offer, As easily As attainable, The necessities of artificial Projective Geometry - Chapters: One-To-One Correspondence - kinfolk among basic types In One-To-One Correspondence With one another - mixture of 2 Projectively similar primary varieties - 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 homes Of Involutions - at the heritage of man-made Projective Geometry - Index

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Extra resources for An Elementary Course In Synthetic Projective Geometry

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Such a ray is clearly 58. Degenerate case 37 a line joining two corresponding points in the two point-rows. The result may be stated as follows: The system of rays joining corresponding points in two protective point-rows has at most two rays in common with any pencil in the plane. For that reason the system of rays is called a pencil of rays of the second order. 58. In the case of two perspective point-rows this system of rays degenerates into two pencils of rays of the first order, one of which has its center at the center of perspectivity of the two point-rows, and the other at the intersection of the two pointrows, any ray through which may be considered as joining two corresponding points of the two point-rows.

Construct now D, the fourth harmonic [31] 34 [32] An Elementary Course in Synthetic Projective Geometry of C with respect to A and B. D may coincide with P, in which case the sequence is closed; otherwise P lies in the stretch AD or in the stretch DB. If it lies in the stretch DB, construct the fourth harmonic of C with respect to D and B. This point D' may coincide with P, in which case, as before, the sequence is closed. If P lies in the stretch DD', we construct the fourth harmonic of C with respect to DD', etc.

It is clear that one point may be named arbitrarily and the other five named in 5! = 120 different ways, but since, as we have seen, two different assignments of names give the same line, it follows that there cannot be more than 60 different lines LMN obtained in this way from a given set of six points. As a matter of fact, the number obtained in this way is in general 60. The above theorem, which is of cardinal importance in the theory of the point-row of the second order, is due to Pascal and was discovered by him at the age of sixteen.

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