[FOM] Understanding Euclid

[Antonino Drago]

Saturday, December 13, 2008 6:31 AM Vaughan Pratt asked:
"What is a model of Euclidean geometry?
More precisely, what surfaces does Euclidean geometry describe?  (We can
take two-dimensionality for granted.)  And of those, which are
reasonable in some sense?"

I think that it will be useful to refer to Poincaré's paper  H. Poincaré, 
Les hypothèses sur les fondements de la
Géométrie, in Oeuvres, XI, Gauthier-Villars, Paris, 1956, 79-91, where he
obtains *four* goemetries from the study of::
1)  the quadratic forms (the Euclidean model is the elliptic paraboloid),
2) the Lie groups and
3) a set of axioms from which he derives by arguing in a very interesting
way, i.e. ad excludendum, the same geometries.

I seem that it is little known. Do someone knows why?

Best greetings
Antonino Drago