  The assumption that \(ds^{2} = F^{2}_{p}\) is a  quadratic differential form is not only the simplest one, but also  the preferred one for other important reasons. Riemann himself was  well aware of other possibilities; for example, the possibility that  \(ds\) could be the 4th root of a homogeneous polynomial of 4th  order in the differentials. But Riemann restricted himself to the  special case \(m = 1\) because he was pressed for time and because he  wanted to give specific geometric interpretations of his results. As  Weyl points out Riemann’s own answer to the space problem is  inadequate since Riemann’s mathematical justification for the  restriction to the Pythagorean case are not very compelling. The  first satisfactory justification of the Pythagorean form of the  Riemannian metric, although limited in scope because it presupposed  the full homogeneity of Euclidean space, was provided by the  investigations of Hermann von Helmholtz. Helmholtz diverged from  Riemann’s analytic approach and made use merely of the fundamental  concept of geometry, namely, the concept of congruent  mapping, and characterized the geometric structure of space by  requiring of space the full homogeneity of Euclidean space. His  analysis was thereby restricted to the cases of constant positive,  zero, or negative curvature. Abstracting from our experience of the  movement of rigid bodies, Helmholtz was able to mathematically derive Riemann’s distance formula from a number of axioms about rigid body  motion in space. Helmholtz (1868) argued that Riemann’s hypothesis  that the metric structure of space is determined locally by a  quadratic differential form, is really a consequence of the facts  (Tatsachen) of rigid-body motion. 