2 1 8 D O C . 2 2 0 N O N - E U C L I D E A N G E O M E T R Y

Riemann’s contribution to the development of our ideas about the relationship

between geometry and physics is twofold. Firstly, he developed spherical-elliptical

geometry, which is the counterpart to Lobachevsky’s hyperbolic

geometry.[10]

He

thus presented for the first time the possibility that geometric space could be of fi-

nite extent in a metrical sense. This idea was soon understood and has led to the

often considered question of whether physical space might not be finite.

But secondly, Riemann had the daring thought of constructing a geometry that

is incomparably more general than that of Euclid or than the non-Euclidean geom-

etries in the narrower sense. He thus created “Riemannian geometry,” which (like

the non-Euclidean geometries in the narrower sense) is Euclidean only in the in-

finitely small; it represents the application of Gaussian surface theory to a contin-

uum of arbitrarily many dimensions. According to this more general geometry, the

metric properties of space, in particular the possibility of stacking infinitely many

infinitely small rigid bodies over finite regions, are not determined by the axioms

of geometry alone. Instead of letting himself be discouraged by this realization, and

concluding from it that his system was physically meaningless, Riemann had the

daring thought that the geometric behavior of bodies might be determined by phys-

ical realities, in particular, forces. He thus arrived through purely mathematical

speculation at the idea of the inseparability of geometry and physics, an idea that

actually prevailed seventy years later in the general theory of relativity through

which geometry and the theory of gravitation were fused into one.

After

Levi-Civita[11]

brought Riemannian geometry to its simplest form by in-

troducing the concept of infinitesimal parallel-shift, Weyl and Eddington further

generalized Riemannian geometry in the hope that the electromagnetic laws would

also find a place in the thus expanded conceptual

system.[12]

Whatever may be the

result of these endeavors, one may in any case say with good reason: The ideas that

developed out of non-Euclidean geometry have proven themselves eminently fruit-

ful in modern theoretical physics.

Translator’s note: This translation is based in part on an unpublished English typescript, Muehsam

Family Papers, AR 25021, Box 7, Folder 43, Leo Baeck Institute, New York.

[p. 20]