is a congruence geometry, or equivalently the space comprising its elements is homogeneous and isotropic; the intrinsic relations between... elements… - Howard P. Robertson

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...

Now it is the practice of astronomers to assume that brightness falls off inversely with the square of the "distance" of an object—as it would do in Euclidean space, if there were no absorption... We must therefore examine the relation between this astronomer's "distance" <math>d</math>... and the distance <math>r</math> which appears as an element of the geometry.

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.