On extensions of Pascal's theorem

H. W. Richmond
1936 Proceedings of the Edinburgh Mathematical Society  
1. The object of this paper is firstly to extend the theorem of Pascal concerning six points of a conic to sets of 2 (n + 1) points of the rational normal cur v^e of order n in space of n dimensions; secondly to explain why a wider extension to other sets of 2 (n + 1) points in \n] must be sought; and lastly to give briefly an extension to [3] and [4] which will be further generalised in a later paper. The striking feature of Pascal's theorem-that each of the sixty ways of arranging the points
more » ... ranging the points in a cycle, or as vertices of a closed polygon, leads to a different version of the theorem-is retained in the following extension to [n]. If 2 (n + 1) points of a rational normal C n in [n] are taken as the vertices of a polygon, the [n -2]'s which are determined each as the intersection of one of the n + 1 pairs of opposite primes of the polygon have the property that every line which intersects all but one of these [n -2]'s must also intersect the last. To establish this the procedure is the same for all values of n; proofs for the values 3 and 4 of n will explain it. 2. In space of three dimensions, if eight points P lt P 2 , P3, Pi, -P 5 , P 6 , P 7 , P a of a twisted cubic curve are taken as vertices of a skew polygon of eight sides, the four lines of intersection of opposite planes of the polygon, [P-^P^P^ and P 6 P G P 7 ; P 2 P 3 P 4 and P e P 7 P s ; P^PiPs and P 7 P a P 1 ; P± P 5 P 6 and P s P x P 2 ) will be generators of a hyperboloid. It is clear that, if a u a 2 , a 3 , a 4 , a 5 , a 6 , a 7 , a 8 denote any eight numbers, then (cii -a b ) (a 2 a 3 a 4 -a 6 a 7 a s ) + (a 2 -%) («3 «4 a 5 -a 7 a 8 a x ) + (% -^7) ( a 4 «s «e -a s a 1 a 2 ) + (a 4 -<*$) (% «e «7 -a i a i ^3) = °-Replace a n by (6 -e n )/( -e n ), and clear away the fractions by multiplying by the product of all the quantities ( -e n ). The first factor of the first term, a 1 -a 5 , gives us (6e i ) ( -e 5 ) -(6 -e b ) ( -e x ) = {e t -e b ) (6 -).
doi:10.1017/s0013091500008282 fatcat:iikq3ilrjreptdpjzpjhnippne