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 Ninteenth 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.

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.