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 triang… - Howard P. Robertson

The general theory of relativity considers physical space-time as a four-dimensional manifold whose line element coefficients <math>g_{\mu \nu}</math> satisfy the differential equations<math>G_{\mu \nu} = \lambda g_{\mu \nu} \qquad .\;.\;.\;.\;.\;.\; (1)</math>in all regions free from matter and electromagnetic field, where <math>G_{\mu \nu}</math> is the contracted Riemann-Christoffel tensor associated with the fundamental tensor <math>g_{\mu \nu}</math>, and <math>\lambda</math> is the .

Let a thin, flat metal plate be heated... so that the temperature T is not uniform... clamp or otherwise constrain the plate to keep it from buckling... [and] remain [reasonably] flat... Make simple geometric measurements... with a short metal rule, which has a certain coefficient of expansion c... What is the geometry of the plate as revealed by the results of those measurements? ...[T]he geometry will not turn out to be Euclidean, for the rule will expand more in the hotter regions... [T]he plate will seem to have a negative curvature <math>K</math>... the kind of structure exhibited... in the neighborhood of a "."

These formulae [in (1) and (2) above] may be shown to be valid for a circle or a triangle in the hyperbolic plane... for which <math>K < 0</math>. Accordingly here the perimeter and area of a circle are greater, and the sum of the three angles of a triangle are less, than the corresponding quantities in the Euclidean plane. It can also be shown that each full line is of infinite length, that through a given point outside a given line an infinity of full lines may be drawn which do not meet the given line (the two lines bounding the family are said to be "parallel" to the given line), and that two full lines which meet do so in but one point.