Howard P. Robertson: Geometry

“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."”

“Euclidean geometry is only one of several congruence geometries… Each of these geometries is characterized by a real number K, 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… K may be interpreted as the curvature of the surface into the third dimension—whence it derives its name…”

“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.”

“Now it is the practice of astronomers to assume that brightness falls off inversely with the square of the "distance" of an object—as it would do in Euclidean space, if there were no absorption… We must therefore examine the relation between this astronomer's "distance" d… and the distance r which appears as an element of the geometry.”

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

“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.”