TITLE: An axiomatic treatment of hyperbolic trigonometry?
QUESTION [7 upvotes]: I would like to see results derived in hyperbolic trigonometry synthetically, i.e. just by working from axioms, for example the ones given by Hilbert (or even Tarski). Most authors seem to discuss hyperbolic trigonometry by working in a specific model, for example Hartshorne in Geometry: Euclid and Beyond gives a lot of results that model Hilbert's axioms but he derives hyperbolic trigonometry by considering the conformal disk model.  I have read that Lobachevsky (and Minding) derived trigonometric identities by working from Euclid. Unfortunately I don't think their relevant papers have been translated to English so I wonder if some modern authors have tried this approach. Does somebody know of relevant paper or a textbook?

REPLY [3 votes]: The 1968 edition of Hilbert's Grundlagen der Geometrie by Paul Bernays appendix 3 contains an article from the Mathematische Annalen, vol. 57, where Hilbert indeed founds the geometry of the Bolyai-Lobačevskij plane upon axioms.
I find the text quite concise and therefore difficult, but -of course- tremendously interesting. I think that an English translation exists.