On Mar 1, 5:21=A0pm, Gene Ward Smith <g...@[EMAIL PROTECTED]
> wrote:
> Butch Malahide <fred.gal...@[EMAIL PROTECTED]
> wrote in news:7ce37c6d-9f9b-485e-
> bec7-93f3440a2...@[EMAIL PROTECTED]
>
> >> Euler scores with the generalized Fermat little theorem, the
V-E+F=3D2
> >> theorem, the summation of the reciprocals of the square numbers,
solvin=
g
> >> Pell's equation, the pentagonal number theorem for partitions, the
> >> Konigsberg bridges problem, and the divergence of the sum of the
> >> reciprocals of the primes, for a total of seven.
>
> > One of those seven is not like the others. I'm not sure if Euler
> > actually proved (or even stated) the characterization of "Eulerian"
> > graphs, and I'm too lazy to look it up, but even if he did, it seems
> > to lack the depth and im****tance of the other six. I'm not sure it
> > even belongs on a list of *Euler's* top 100 theorems.
>
> Euler proved the theorem for convex polyhedra. Like many things, it's
> im****tant in good measure because of what it leads to.
>
>
>
> > =A0Gauss scores with the
> >> fundamental theorem of algebra, quadratic reciprocity, and the
> independenc
> > e
> >> of the parallel postulate, for a total of three.
>
> > Did Gauss prove the independence of the parallel postulate? I thought
> > I read somewhere that that was done (model for non-Euclidean geometry)
> > later and by some less notorious mathematician.
That would have been Janos Bolyai -
" '... denote by the system of geometry based on the hypothesis that
Euclid's Fifth Postulate is true, and by S the system based on the
opposite hypothesis. All theorems we state without explicitly
specifying the system or S in which the theorem is valid are meant to
be absolute, that is, valid independently of whether or S is true. '
Farkas Bolyai, Janos' father, sent a reprint to Gauss who, on reading
the Appendix, wrote to a friend saying:-
I regard this young geometer Bolyai as a genius of the first order .
To Farkas Bolyai, however, Gauss wrote:-
To praise it would amount to praising myself. For the entire content
of the work ... coincides almost exactly with my own meditations which
have occupied my mind for the past thirty or thirty-five years .
There is no doubt that Gauss was simply stating facts here. The
clearest reference in Gauss's letters to his work on non-euclidean
geometry, which shows the depth of his understanding, occurs in a
letter he wrote to Taurinus on 8 November 1824 when he wrote:-
The assumption that the sum of the three angles of a triangle is less
than 180 leads to a curious geometry, quite different from ours [i.e.
Euclidean geometry] but thoroughly consistent, which I have developed
to my entire satisfaction, so that I can solve every problem in it
excepting the determination of a constant, which cannot be fixed a
priori. .... the three angles of a triangle become as small as one
wishes, if only the sides are taken large enough, yet the area of the
triangle can never exceed, or even attain a certain limit, regardless
of how great the sides are. "
Even though Gauss had discovered non-Euclidean Geometry earlier,
I think his reply to Farks showed Gauss to be a jerk.
Euler is the most prolific mathematician ever, so it's not
surprising that he would top the list- A. McIntire
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