An Elementary Course in Synthetic Projective Geometry by Derrick Norman Lehmer

By Derrick Norman Lehmer

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Extra info for An Elementary Course in Synthetic Projective Geometry

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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.

Very frequent use will be made of these fundamental theorems. [34] 55. Point-row of the second order. The question naturally arises, What is the locus of points of intersection of corresponding rays of two projective pencils which are not in perspective position? This locus, which will be discussed in detail in subsequent chapters, is easily seen to have at most two points in common with any line in the plane, and on account of this fundamental property will be called a point-row of the second order.

B = 6, the points L and N are interchanged, but the line is left unchanged. 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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