Helmholtz. "Ueber den Ursprung und die Bedeutung der geometrischen Axiome," Vorträge und Reden, Bd. 2, S. 1.
[Note 7] (p. 26). The postulate that finite rigid bodies are to be capable of free motions, can be most strikingly illustrated in the realm of two-dimensions. Let us imagine a triangle to be drawn upon a sphere, and also upon a plane: the former being bounded by arcs of great circles and the latter by straight lines; one can then slide these triangles over their respective surfaces at will, and can make them coincide with other triangles, without thereby altering the lengths of the sides or the angles. Gauss has shown that this is possible because the curvature at every point of the sphere (or the plane, respectively) has exactly the same value. And yet the geometry of curves traced upon a sphere is different from that of curves traced upon a plane, for the reason that these two configurations cannot be deformed into one another without tearing (vide [Note 27]). But upon both of them planimetrical figures can be freely shifted about, and, therefore, theorems of congruence hold upon them. If, however, we were to define a curvilinear triangle upon an egg-shaped surface by the three shortest lines connecting three given points upon it, we should find that triangles could be constructed at different places on this surface, having the same lengths for the sides; but these sides would enclose angles different from those included by the corresponding sides of the initial triangle, and, consequently, such triangles would not be congruent, in spite of the fact that corresponding sides are equal. Figures upon an egg-shaped surface cannot, therefore, be made to slide over the surface without altering their dimensions: and in studying the geometrical conditions upon such a surface, we do not arrive at the usual theorems of congruence. Quite analogous arguments can be applied to three- and four-dimensional realms: but the latter cases offer no corresponding pictures to the mind. If we demand that bodies are to be freely movable in space without suffering a change of dimensions, the "curvature" of the space must be the same at every point. The conception of curvature, as applied to any manifold of more than two dimensions, allows of strict mathematical formulation; the term itself only hints at its analogous meaning, as compared with the conception of curvature of a surface. In three-dimensional space, too, various cases can be distinguished, similarly to plane- and spherical-geometry in two-dimensional space. Corresponding to the sphere, we have a non-Euclidean space with constant positive curvature; corresponding to the plane we have Euclidean space with curvature zero. In both these spaces bodies can be moved about without their dimensions altering; but Euclidean space is furthermore infinitely extended: whereas "spherical" space, though unbounded, like the surface of a sphere, is not infinitely extended. These questions are to be found extensively treated in a very attractive fashion in Helmholtz's familiar essay: "Ueber den Ursprung und die Bedeutung der geometrischen Axiome" (Vorträge und Reden, Bd. 2, S. 1).
[Note 8] (p. 26). The properties, which the analytical expression for the length of the line-element must have, may be understood from the following:
Let the numbers
,
denote any point of any continuous two-dimensional manifold, e.g. a surface. Then, together with this point, a certain "domain" around the point is given, which includes points all of which lie in the plane.—D. Hilbert has strictly defined the conception of a multiply-extended magnitude (i.e. a manifold) upon the basis of the theory of aggregates in his "Grundlagen der Geometrie" (p. 177). In this definition the conception of the "domain" encircling a point is made to give Riemann's postulate of the continuous connection existing between the elements of a manifold and a strict form.
Setting out from the point