That the existence of these motions (the "axiom of free mobility") is a desideratum, if not... a necessity, for a geometry applicable to physical spa… - Howard P. Robertson

[O]nly in a homogeneous and isotropic space can the traditional concept of a rigid body be maintained.

The search for the curvature <math>K</math> indicates that, after making all known corrections, the number N seems to increase faster with <math>d</math> than the third power, which would be expected in a Euclidean space, hence <math>K</math> is positive. The space implied thereby is therefore bounded, of finite total volume, and of a present "radius of curvature" <math>R = \frac{1}{K^\frac{1}{2}}</math> which is found to be of the order of 500 million light years. Other observations, on the "red shift" of light from these distant objects, enable us to conclude with perhaps more assurance that this radius is increasing...

The field equation may... be given a geometrical foundation, at least to a first approximation, by replacing it with the requirement that the mean curvature of the space vanish at any point at which no heat is being applied to the medium—in complete analogy with... the general theory of relativity by which classical field equations are replaced by the requirement that the Ricci contracted curvature tensor vanish.

In all these congruence geometries, except the Euclidean, there is at hand a natural unit of length <math>R = \frac{1}{K^\frac{1}{2}}</math>; this length we shall, without prejudice, call the "radius of curvature" of the space.