This paper is principally based on the following general theorem, which is a remarkable extension of Pascal's **hexagram**: "If a polygon move so that each of its sides passes through a fixed point, and if all its summits except one describe curves of the degrees m, n, p, &c., respectively, then the free summit moves on a curve of the degree 2mnp. ..

This proposition, which he called the mystic **hexagram**, he made the keystone of his theory; from it alone he deduced more than 400 corollaries, embracing, according to his own account, the conics of Apollonius, and other results innumerable.