In considerations involving the nature of the world as a whole the irregularities caused by the aggregation of matter into stars and stellar systems … - 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...

All the light which is radiated... will, after it has traveled a distance <math>r</math>, lie on the surface of a sphere whose area <math>S</math> is given by the first of the formulae (3). And since the practical procedure... in determining <math>d</math> is equivalent to assuming that all this light lies on the surface of a Euclidean sphere of radius <math>d</math>, it follows...<math>4 \pi d^2 = S = 4 \pi r^2 (1 - \frac{K r^2}{3} + ...);</math>whence, to our approximation 4)<math>d = r (1- \frac{K r^2}{6} + ...),</math> or
<math>r = d (1 + \frac{K d^2}{6} + ...).</math>

[T]he space constant <math>K</math>... "" may in principle at least be determined by measurement on the surface, without recourse to its embodiment in a higher dimensional space.