Howard P. Robertson: Measure

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

“[T]he space constant K… "curvature" may in principle at least be determined by measurement on the surface, without recourse to its embodiment in a higher dimensional space.”

“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 K… the kind of structure exhibited… in the neighborhood of a "saddle point."”

“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.”

“[W]e propose… to deal exclusively with properties intrinsic to the space… measured within the space itself… in terms of… inner properties.”

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