I've just read your July 9 posting on "25 = 27 - 2", which contains
preliminary translations of Articles 188 and 195 of Euler's "Algebra",
book II, chapter 12.
> Any comments? ...
Well, first a minor matter: On two occasions in your posting, you write
[Gauss's original follows:]
I assume you meant "Euler", not "Gauss".
But my overall comment is that it doesn't look like Euler is offering his
analysis as any kind of proof. It's more in the spirit of "Here's a good
way of generating solutions to problems like this", with a remark that
"Sometimes there are solutions that don't arise from this procedure, and
it'd be important to understand where they come from".
So, if this is all that Euler wrote on the subject, how did he ever get
falsely credited with having proved (or having claimed to prove) Fermat's
assertion? (Perhaps Ed Sandifer can help us out here, by pointing out
other places where Euler refers to this problem; maybe Euler later did
go so far as to claim he'd solved the problem.)
Jim Propp