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.
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asked May 11, 2015 at 23:41
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5$\begingroup$ My take, from having published on this, is that Bolyai did not make any model in our sense, he predicted everything that had to happen and said he had created a new world. Gauss was less than gracious, having come to many of the same conclusions; I give Bolyai greater credit, he actually said this is it. $\endgroup$– Will JagyCommented May 11, 2015 at 23:52
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5$\begingroup$ The popular history is that they didn't prove consistency; Beltrami did, a 38 years later, using, most cleanly, the projective or Beltrami-Klein disk model. $\endgroup$– Dylan ThurstonCommented May 11, 2015 at 23:52
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1$\begingroup$ Also, why do you say 1820s? Lobachevsky and Bolyai published in 1830 and 1832. If you're going to credit unpublished work, then Gauss deserves credit as well. $\endgroup$– Dylan ThurstonCommented May 11, 2015 at 23:54
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4$\begingroup$ Relevant math.stackexchange question: math.stackexchange.com/questions/665981/… $\endgroup$– Dylan ThurstonCommented May 11, 2015 at 23:56
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1$\begingroup$ I think anyone interested in this topic will want to read this review by R. Osserman. $\endgroup$– Marcus JohnsonCommented May 12, 2015 at 22:45
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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).
answered May 11, 2015 at 23:51
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1$\begingroup$ >... but neither Bolyai nor Lobachevsky proved that their geometry was internally consistent... That's what I was beginning to suspect, Igor, but is that something that is "As far as I know...", or do you have some strong reason to believe this---say perhaps a well-researched historical reference. $\endgroup$ Commented May 11, 2015 at 23:53
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2$\begingroup$ @DickPalais Well, here is what the wikipedia says about Beltrami (which seems to strongly imply that at the time, the question of consistency was open, and Beltrami closed it): en.wikipedia.org/wiki/Eugenio_Beltrami $\endgroup$ Commented May 12, 2015 at 0:06