TITLE: What sort of models did Bolyai and Lobachevsky use to demonstrate the consistency of their models of non-Euclidean Geometry?
QUESTION [17 upvotes]: As is well-known, in the 1820s both Bolyai and Lobachevsky showed, at long last, the independence of the Parallel Postulate from the rest of the axioms of Euclidean geometry by developing what we now call hyperbolic geometry. Of course, from a modern point of view there are very nice models—the Poincaré Disk and the hyperbolic metric on the upper half-plane, but these came later, and even the models provided by surfaces of constant negative Gauss curvature was a theory only developed by Bianchi and Bäcklund towards the end of the Nineteenth century.  I have been trying to discover what sort of models Bolyai and Lobachevsky had in mind or that were appealed to by them or others just after their work, to demonstrate the consistency of their non-Euclidean geometry, but I have been unsuccessful and would much appreciate any pointers.

REPLY [11 votes]: I believe that Riemann realized that the sphere was a model for elliptic geometry (1854), while Beltrami (in 1868) introduced his model for hyperbolic space, but neither Bolyai nor Lobachevsky proved that their geometry was internally consistent - their primary argument was aesthetic (you got a nice theory with beautiful properties).