Thales and his theorem
Thales was an ancient Greek mathematician. He was one of the earliest mathematicians who used proofs to demonstrate things, as opposed to the Egyptian and Babylonian method of using what worked and not proving it. Thales proved this theorem for the first time and got his name put on it, Thales' Theorem states,
An angle inscribed in a semi-circle is a right angle.
Thales wasn't just a mathematician, he was a philosopher as well. He had a theory about the universe, He said "all is water". His idea was that water, when arranged in different ways produces different things and everything is made of water in some form.
The Pythagoreans
The Pythagoreans came after Thales. They were a Greek cult which was very interested in numbers. They had a theory of the universe based on numbers. They believed that the number 10 was a magic number. The first four integers were thought to be very important because they added up to 10. These numbers were used to explain the creation of the universe and it's underlying mathematical perfection, symbolized by the sum of these numbers, 10. As the name would suggest, the Pythagoreans Discovered the Pythagorean theorem. They also did work in the mathematics of music. They created a scale, with ratios defining the length of a string relative to that of the Tonic.
C D E F Gg A B C
tonic supertonic mediant subdominant dominant submediant subtonic octave
1 8/9 64/81 3/4 2/3 16/27 128/243 1/2

The problem with this system was that some of the ratios are rather complicated for actual use. Ptolemy later simplified the scale and created a new version
C D E F Gg A B C
tonic supertonic mediant subdominant dominant submediant subtonic octave
1 8/9 4/5 3/4 2/3 3/5 8/15 1/2

This table was simpler to use but wasn't as accurate and it didn't sound right. Later the equal tempered scale was introduced
C C# D D# E F F# G G# A A# B C
1 1/(12*square root of2) 1/(6*square root of2) 1/(4*square root of2) 1/(3*square root of2) 1/(2+5/15) 1/(square root of2) 1/(2+1/17) 1/(2+2/3) 1/(2+3/4) 1/(2+5/6) 1/(2+11/12) 1/2
This scale is the modern scale.
If you take a triangle were both legs are 1 and apply the Pythagorean theorem to it, you get the square root of 2. This may not seem like a problem but the for the Pythagoreans, who believed that the universe was ordered and harmonious because it was based on mathematical order. The idea of an irrational number was contrary to this idea of order and harmony. According to legend Hippasus, the Pythagorean who discovered this came to a sticky end.
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