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 le… - Howard P. Robertson

Measurements which may be made on the surface of the earth... is an example of a 2-dimensional congruence space of positive curvature <math>K = \frac{1}{R^2}</math>... [C]onsider... a "small circle" of radius <math>r</math> (measured on the surface!)... its perimeter <math>L</math> and area <math>A</math>... are clearly less than the corresponding measures <math>2\pi r</math> and <math>\pi r^2</math>... in the Euclidean plane. ...for sufficiently small <math>r</math> (i.e., small compared with <math>R</math>) these quantities on the sphere are given by 1):<math>L = 2 \pi r (1 - \frac{Kr^2}{6} + ...)</math>,
<math>A = \pi r^2 (1 - \frac{Kr^2}{12} + ...)</math>

In considerations involving the nature of the world as a whole the irregularities caused by the aggregation of matter into stars and stellar systems may be ignored; and if we further assume that the total matter in the world has but little effect on its macroscopic properties, we may consider them as being determined by the solution of an empty world.

[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.