The field equation may... be given a geometrical foundation, at least to a first approximation, by replacing it with the requirement that the mean cu… - Howard P. Robertson

Euclidean geometry is only one of several congruence geometries... Each of these geometries is characterized by a real number <math>K</math>, which for Euclidean geometry is 0, for the hyperbolic negative, and for the spherical and elliptic geometries, positive. In the case of 2-dimensional congruence spaces... <math>K</math> may be interpreted as the of the surface into the third dimension—whence it derives its name...

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...

What is the true geometry of the plate? ...Anyone examining the situation will prefer Poincaré's common-sense solution... to attribute it Euclidean geometry, and to consider the measured deviations... as due to the actions of a force (thermal stresses in the rule). ...On employing a brass rule in place of one of steel we would find that the local curvature is trebled—and an ideal rule (c = 0) would... lead to Euclidean geometry.