 Helmholtz’s work on geometry was made public from 1868, the year he gave a lecture, “On the Actual Foundations of Geometry,” in Heidelberg, and published his “On the Facts Underlying Geometry.” The latter “astonished” the academic public, who had considered Helmholtz primarily an experimental scientist and physiologist (Königsberger 1906, 254). However, Helmholtz had initially wanted to be a physicist, and had always been alive to the mathematical consequences of his work in physiology. Riemann’s work revolutionized mathematics and physics, and philosophy by extension, and Helmholtz was perhaps the first to recognize the extent of the revolution. Indeed, Helmholtz claimed that he had come to results similar to Riemann’s, though perhaps slightly later. Although the publication of Riemann’s Habilitationsrede, “On the Hypotheses Underlying Geometry,” meant that Helmholtz’s own results would not be seen as original, Helmholtz says in his own essay that he is content that “so distinguished a mathematician [as Riemann] should have considered these questions as worthy of his attention” (Helmholtz 1883 [1868], cited in Königsberger 261). With his 1868 essay, Helmholtz saw himself as contributing to the Riemannian approach to the foundations of geometry (Königsberger 1906, 254). Helmholtz’s title, “On the Facts Underlying Geometry,” is a deliberate echo of Riemann’s “On the Hypotheses Underlying Geometry.” The difference in the titles, facts versus hypotheses, underscores the fact that while Helmholtz’s approach is very similar to Riemann’s, there is at least one significant difference.