The three-dimensional analogue of Ptolemy’s theorem (Simson’s Euclid, Prop. D) has been given by Cayley (Collected Works, I, 1). The following note contains the theory of such generalizations to space of any number of dimensions.

Taking Euclidean space of n dimensions we denote the Cartesian coordinates of a point As by (xs,a) where the Greek suffix is a dummy and takes the values 1, 2, 3, ..., n, n + 1. We make the convention that xs, n+1 = 1 for every point As.