 While Riemann tries to describe the general properties of space, Helmholtz asks what the most general geometrical axioms are that can account for our empirical measurement of objects. These will be the axioms that preserve observed rigid motions, given an underlying system of geometry. Helmholtz investigates the question of which systems of geometry (Euclidean, Lobachevskian, and Riemannian) are possible under which assumptions. Here Helmholtz makes an initial mistake, and argues that only Euclidean geometry can account for our actual physical measurements. In April 1869, Eugenio Beltrami pointed out in a letter to Helmholtz that Lobachevskian geometry could be adequate to the task under the assumption that we live in a “pseudosphere” (see Other Internet Resources). Helmholtz concedes the point immediately (Königsberger 1906, 263). In 1870, in “On the Origin and Significance of the Axioms of Geometry,” Helmholtz investigates in much closer detail the question of whether non-Euclidean geometries can be visualized. This essay, perhaps even more than “On the Facts Underlying Geometry,” was widely influential. Moritz Schlick and Hans Reichenbach engaged with Helmholtz’s views in the 1870 essay (see, e.g., Reichenbach 1920, Schlick’s notes to Helmholtz 1921). A related article, “The Origin and Meaning of Geometrical Axioms,” was published in Mind in 1878, and Russell responds to it in his doctoral dissertation.