[FOM] Gauss and non-Euclidean geometry

[Antonino Drago]

 Kutateladze wrote:

> Gauss wrote to Schumacher on Novermber 28, 1846  (I apologize for
> a rough translation):
>
>    Recently I have a chance to look again through the Lobachevsky
>    book ... , Berlin 1840.... You know  that I have the same
>    beliefs (since 1792) during 54 years (with some later enrichment
>    that I am disinclined to dwell upon here); so, as far as the
>    contents is concerned, I found nothing new for me in the
>    composition of Lobachevsky; however, the author, developing this
>    approach, travels in an another way completely different
>    from mine. I feel myself obliged to draw your attention to thus
>    book that will surely bring  to you  a quite exceptional
>    pleasure.
>
> If you recall that Gauss was in full possession of surface theory
> as its inventor (the ``Gauss curvature'' in particular) in contrast
> to Lobachevsky and Bolyau who stood farther from differential geometry,
> you will see that Gauss's technique and understanding of surface theory 
> were at a higher level
> than those by Lobachevsky and Bolyai. He was sincere and polite in his
> views of the contributions by Lobachevsky and Bolyai as
> appropriate for a great master appreciating the gems of genius of younger
> persons but knowing much more about the area.
> Gauss does not deserve a shade of reproach in my opinion.

To my knowledge, the history is slightly different from the above version.
Gauss' letter of 28/11/46 to Schumacher, quoted in the above, says (in 
Houel's French that I have at hand):
"the exposition is at all different from that I planned, and the author 
deals with the subject with a master hand and with a very geometrical 
spirit.",
not the shortened version "... travels in an another way completely 
different from mine."
Moreover, Gauss promoted Lobachevsky in the Goettengen Academy.
But he never manifested his knowledge about non-Eclidean geometries, not 
even when he maliciously chose the third subject out of the three subjects 
Riemann submitted for his Habilitation thesis. Riemann was obliged to employ 
a long time for having an idea of non-Euclidean geometries and then he chose 
a different viewpoint with respect  to both Lobachevsky and Bolyai. Gauss 
exited form the room of the discussion of this thesis by rubbing his hands.
At present Gauss' covert attitude about this subject is known by his 
correspondence he had with several mathematicians of his time. When the 
father of Bolyai sent his geometrical work including the appendix wrote by 
his son about the so-alled absolute geometry, Gauss answered: I cannot 
praise your son because I would prise myself, since I had the same ideas 
since so long time..." A very starnge semtence, that led Bolyai son to think 
a Gauss' plagiarism; so that he abandoned his researches.
A last but not least point of theoretical nature. As shwoed Katz in Historia 
Mathematica (1970?), the invention of non-Euclidean geometry required the 
development of trigonometry as an an idependent theory. The differential 
geometry attitude was a posterior attitude and it was considered relevant 
more than half a century after Rieman's thesis, when general relativity 
applied it.
As a matter of fact, the curvature radius was invented by Lazare Carnot, as 
showed C. Boyer by a specific paper (this historian recalls this fact in his 
book History of Mathematics, Princeton 1985). Lazare Carnot is the same 
geometer that first gave an independent foundation to trigonometry in his 
Géométrie de Position, 1803.
Sorry for the long message; the last point seems to me relevant for 
focussing the attention abotu a uncommon fact; the birth of the more 
abstract geometrical theories from a theory belonging to a practical 
mathematical tradition.
Best regards
Antonino Drago