[T]he astronomical data give the number N of nebulae counted out to a given inferred "distance" <math>d</math>, and in order to determine the curvatu… - Howard P. Robertson

The solution of (1), which represents a homogeneous manifold, may be written in the form:<math>ds^2 = \frac{d\rho^2}{1 - \kappa^2\rho^2} - \rho^2 (d\theta^2 + sin^2 \theta \; d\phi^2) + (1 - \kappa^2 \rho^2)\; c^2 d\tau^2, \qquad (2)</math>where <math>\kappa = \sqrt \frac{\lambda}{3}</math>. If we consider <math>\rho</math> as determining distance from the origin... and <math>\tau</math> as measuring the proper-time of a clock at the origin, we are led to the de Sitter spherical world...

We have merely (!) to measure the volume <math>V</math> of a sphere of radius <math>r</math> or the sum <math>\sigma</math> of the angles of a triangle of measured are <math>\delta</math>, and from the results to compute the value of <math>K</math>.

is a congruence geometry, or equivalently the space comprising its elements is homogeneous and isotropic; the intrinsic relations between... elements of a configuration are unaffected by the position or orientation of the configuration. ...[M]otions of are the familiar translations and rotations... made in proving the theorems of Euclid.