# Geometry, philosophy, triangles and music

Along with other mathematicians of the time, Pythagoras was largely responsible for introducing a more rigorous mathematics than what had gone before, building from first principles using axioms and logic. Prior to this, geometry for example, had been merely a collection of rules derived by empirical measurement.

Pythagoras was interested in the principles of mathematics – the concept of number, the concept of a triangle or other mathematical figures, the abstract idea of a proof. He discovered that a complete system of mathematics could be constructed, where geometric elements correspond with numbers and those integers and their ratios were all that was necessary to establish an entire system of logic and truth. Today we have become so mathematically sophisticated that we fail even to recognise 2 as an abstract quantity – thank you Pythagoras!

He is mainly remembered for what has become known as Pythagoras’ Theorem: that, for any right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the square of the other two sides, written as an equation: a2 + b2 = c2.

Pythagoras also made remarkable contributions to the mathematical theory of music. He noticed that vibrating strings produce harmonious tones when the ratios of the lengths of the strings are whole numbers, and that these ratios could be extended to other instruments. It is said to have discovered the chromatic, diatonic and enharmonic scales, which he thought reflected the divine mathematical order inherent in the universe.

Plato and Aristotle were influenced by Pythagoras’s way of thinking. In ‘The Academy’ it was stressed that mathematics as a way of understanding more about reality. Plato, in particular, was convinced that geometry was the key to unlocking the secrets of the universe and apparently there was a sign above the Academy entrance reading something akin to ‘Let no-one unversed in geometry enter here.’

Bertrand Russell called Pythagoras the most influential philosopher in the history of Western philosophy.