[FOM] Gauss and non-Euclidean geometry

[Alasdair Urquhart]

Robert Black writes (about the myth of Gauss testing non-Euclidean
geometry in his surveying practice):

> I think you'll find it's a myth - see for example Buehler's biography
> of Gauss. Actually, I'm a bit sceptical about Gauss and non-euclidean
> geometry in general.

There is strong evidence that Gauss had the basic ideas of hyperbolic
geometry before Lobachevsky's publication of 1829.  In a letter to
Taurinus of 1824, he summarizes his work in such a way as to make
clear that he was in command of the basic ideas by that date.
There's an excerpt in the very fine survey article by John Milnor
"Hyperbolic Geometry: the first 150 years" in The Mathematical
Heritage of Henri Poincar'e (also Milnor's Collected Papers Volume 1) --
this paper also makes it clear that the so-called Klein and
Poincar'e models were already known to Beltrami in 1868.

As for Gauss, there is no doubt that he anticipated Lobachevsky
and Bolyai, but his dog-in-the-manger attitude to the discoveries
of Bolyai and Lobachevsky did him little credit.

Alasdair Urquhart