Re: Pythagorean triples

Siem: The Egyptians certainly knew of the 3:4:5 right triangle and
undoubtedly so did Thales. However, I suspect he dealt only with
rational triangles. Most historians of mathematics consider the
Pythagorean (not necessarily by Pythagoras) discovery of the
irrationality of the square root of 2 as the defining moment in number
theory.

However, the history of logistic or arithmentic/algebra is not as clear
as that of geometry. The Babylonians could approximate square and cube
roots quite accurately and undoubtedly, the so-called "Euclidean"
algorithm and something like Continued Fractions was known very early.

I think this is pretty good evidence that Aristoxenos did not invent
equal-temperament or intend to. It's quite clear that if he had,
Eratosthenes and Ptolemy would not have presented his tunings in as
grotesquely distorted form as they appear in the Harmonics.

--John