Do any of you happen to know the history of the standard prime factorization proof of $\sqrt 2$ is irrational? I know this theorem was known to Aristotle, and that the Fundamental Theorem of Arithmetic, on which the proof rests, is found already in Euclid, but I've not been able to track down the origin of this particular proof. 


These sites I know about: <http://www.cut-the-knot.org/proofs/sq_root.shtml>, <http://www.math.ufl.edu/~rcrew/texts/pythagoras.html>, and of course <http://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic>


But any other references, online or in paper form, would be greatly appreciated!



First known proof of $\sqrt 2$ is irrational with prime factorization?
"The Discovery of Incommensurability" by Kurt von Fritz [ <http://www.jstor.org/stable/1969021> ] indicates that the early Greek mathematicians did not explicitly use the Fundamental Theorem to prove the irrationality of √2. The proof known to Aristotle ("the diagonal of the square is incommensurate with the side, because odd numbers are equal to evens if it is supposed to be commensurate") uses a restricted version of the Fundamental Theorem, as explained in <http://en.wikipedia.org/wiki/Quadratic_irrational>


Apparently, the explicit use of the Fundamental Theorem to prove the irrationality of √2 is post-Gauss. This is argued convincingly by [Barry Mazur](http://www.math.harvard.edu/~mazur/preprints/Eva.Nov.20.pdf):


*This fundamental theorem of arithmetic has a peculiar history. It is not trivial, and any of its proofs take work, and, indeed, are interesting in themselves. But it is nowhere stated in the ancient literature. It was used, implicitly, by the early modern mathematicians, Euler included, without anyone noticing that it actually required some verification, until Gauss finally realized the need for stating it explicitly, and proving it.*


