[FOM] Beltrami and non-Euclidean geometry

[Alasdair Urquhart]

A brief comment on Andre Rodin's posting about non-Euclidean
geometry:

> ... Lobachevsky didn't model his
> space in the Euclidean one but made it the other way round. Namely he found
> on Lobachevsky's plane (what we now call a model of) Euclidean line (which
> he called "oricircle" and found in Lobachevsky's 3D space (what we now call
> a model of) Euclidean plane (which he called "orisphere"). These models are
> much nicer than Beltrami's since they don't have singularities just like
> models of spheric (sometimes referred to as Riemanean) geometry in Euclidean
> space.

It's commonly believed that Beltrami only discussed the surface of a 
pseudosphere as a partial model for hyperbolic geometry, which suffers
from the singularity problem mentioned here.  But in fact, in two
papers of 1868, Beltrami discussed all three of the models that we
now call the "Poincare disc model", the "Poincare half-plane model"
and the "Klein disc model."

Beltrami's papers are now conveniently available in English translation
by John Stillwell (with commentary) in "Sources of Hyperbolic Geometry",
AMS History of Mathematics Volume 10.

Alasdair Urquhart