The usual "purely geometrical proof" of the incommensurability of the
diagonal and the side of a square comes from a
TEXTBOOK OF ALGEBRA
By G. Chrystal, first edition, Edingurgh, 1886; fifth, 1904. I own the Chelsea
reprint, 1952: Page 270 of Vol I.
I too have a geometrical version that seems to me simpler than the Chrystal
proof. It is printed in
Essays in Honor of Robert Bart, St. John's College, 1993.
Best wishes,
Sam Kutler
>There exist, of course, purely geometric proofs of the incomensurability of
>the diagonal with the side, but I do not dispose of exact references at the
>moment, nor dates thereof.
>AJFO.
>At 10:09 21-12-1998 -0500, you wrote:
>>On Mon, 21 Dec 1998, Fernando Q. Gouvea wrote:
>>>
>>> It certainly seems natural to reconstruct the proof(s) as you have... but
>>> aren't there other possibilities? I think it is Grattan-Guinness who
>>> suggested that the "odd numbers" that are equal to "even" in Aristotle
>>> could also be the *number of prime divisors* of a^2=2b^2. I have no idea
>>> how probable it is that the Greeks could have reasoned in this way, but it
>>> does give a proof too.
>>
>> But one that depends on the theory of unique factorization into
>>primes, which it's far from clear was available at the time. (Euclid's
>>statement is a weaker one.)
>>
>> John Conway
>>
>>