In the sum <math>\sigma</math> of the three angles of a triangle (whose sides are arcs of s) is greater than two right angles [180°]; it can... be sh… - Howard P. Robertson

That the existence of these motions (the "axiom of free mobility") is a desideratum, if not... a necessity, for a geometry applicable to physical space, has been forcefully argued on a priori grounds by von Helmholtz, Whitehead, Russell and others; for only in a homogeneous and isotropic space can the traditional concept of a rigid body be maintained.

The value of the intrinsic approach is especially apparent in considering 3-dimensional congruence spaces... The intrinsic geometry of such a space of curvature <math>K</math> provides formulae for the surface area <math>S</math> and the volume <math>V</math> of a "small sphere" of radius <math>r</math>, whose leading terms are 3)<math>S = 4 \pi r^2 (1 - \frac{Kr^2}{3} + ...)</math>,
<math>V = \frac{4}{3} \pi r^3 (1 - \frac{Kr^2}{5} + ...)</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.