In the second memoir published during the same year (1868), "Fundamental theory of spaces of constant curvature", Beltrami continued this logic and gave an abstract proof of equiconsistency of hyperbolic and Euclidean geometry for any dimension. He accomplished this by introducing several models of non-Euclidean geometry that are now known as the Beltrami–Klein model, the Poincaré disk model, and the Poincaré half-plane model, together with transformations that relate them. For the half-plane model, Beltrami cited a note by Joseph Liouville in the treatise of Gaspard Monge on differential geometry. Beltrami also showed that "n"-dimensional Euclidean geometry is realized on a horosphere of the ("n" + 1)-dimensional hyperbolic space, so the logical relation between consistency of the Euclidean and the non-Euclidean geometries is symmetric. Beltrami acknowledged the influence of Bernhard Riemann's groundbreaking "Habilitation" lecture "On the hypotheses on which geometry is based" (1854; published posthumously in 1868).