Discussion of a set of points in terms of their mutual distances

Necessary and sufficient conditions are given for a set of numbers to be the mutual distances of a set of real points in Euclidean space, and matrices are found whose ranks determine the dimension of the smallest Euclidean space containing such points. Methods are indicated for determining the configuration of these points, and for approximating to them by points in a space of lower dimensionality.This paper was written in response to suggestions by Harold Gulliksen and by M. W. Richardson. The latter is working on a psychophysical problem in which the dimensionality of a set of points whose mutual distances are available is a central idea.