One of the simplest proofs comes from ancient China, and probably dates from well before Pythagoras’ birth. Pythagoras’ Theorem and the properties of right-angled triangles seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, and it was touched on in some of the most ancient mathematical texts from Babylon and Egypt, dating from over a thousand years earlier. It should be noted, however that (6, 8, 10) is not what is known as a “primitive” Pythagorean triple, because it is just a multiple of (3, 4, 5). The simplest and most commonly quoted example of a Pythagorean triangle is one with sides of 3, 4 and 5 units (3 2 + 4 2 = 5 2, as can be seen by drawing a grid of unit squares on each side as in the diagram at right), but there are a potentially infinite number of other integer “ Pythagorean triples”, starting with (5, 12 13), (6, 8, 10), (7, 24, 25), (8, 15, 17), (9, 40, 41), etc. What Pythagoras and his followers did not realize is that this also works for any shape: thus, the area of a pentagon on the hypotenuse is equal to the sum of the pentagons on the other two sides, as it does for a semi-circle or any other regular (or even irregular( shape.
He is mainly remembered for what has become known as Pythagoras’ Theorem (or the Pythagorean Theorem): that, for any right-angled triangle, the square of the length of the hypotenuse (the longest side, opposite the right angle) is equal to the sum of the square of the other two sides (or “legs”). Pythagoras discovered that a complete system of mathematics could be constructed, where geometric elements corresponded with numbers, and where integers and their ratios were all that was necessary to establish an entire system of logic and truth.
Before Pythagoras, for example, geometry had been merely a collection of rules derived by empirical measurement. However, Pythagoras and his school – as well as a handful of other mathematicians of ancient Greece – was largely responsible for introducing a more rigorous mathematics than what had gone before, building from first principles using axioms and logic. It is a great tribute to the Pythagoreans’ intellectual achievements that they deduced the special place of the number 10 from an abstract mathematical argument rather than from something as mundane as counting the fingers on two hands.
The holiest number of all was “ Tetractys” or ten, a triangular number composed of the sum of one, two, three and four. Odd numbers were thought of as female and even numbers as male.
For example, the number one was the generator of all numbers two represented opinion three, harmony four, justice five, marriage six, creation seven, the seven planets or “ wandering stars” etc. The over-riding dictum of Pythagoras’s school was “ All is number” or “ God is number”, and the Pythagoreans effectively practised a kind of numerology or number-worship, and considered each number to have its own character and meaning. Resentment built up against the secrecy and exclusiveness of the Pythagoreans and, in 460 BCE, all their meeting places were burned and destroyed, with at least 50 members killed in Croton alone. There was always a certain amount of friction between the two groups and eventually the sect became caught up in some fierce local fighting and ultimately dispersed. The members were divided into the “ mathematikoi” (or “ learners“), who extended and developed the more mathematical and scientific work that Pythagoras himself began, and the “ akousmatikoi” (or “ listeners“), who focused on the more religious and ritualistic aspects of his teachings. Although Pythagorean thought was largely dominated by mathematics, it was also profoundly mystical, and Pythagoras imposed his quasi-religious philosophies, strict vegetarianism, communal living, secret rites and odd rules on all the members of his school (including bizarre and apparently random edicts about never urinating towards the sun, never marrying a woman who wears gold jewellery, never passing an ass lying in the street, never eating or even touching black fava beans, etc). The school he established at Croton in southern Italy around 530 BCE was the nucleus of a rather bizarre Pythagorean sect.
Indeed, it is by no means clear whether many (or indeed any) of the theorems ascribed to him were in fact solved by Pythagoras personally or by his followers. He left no mathematical writings himself, and much of what we know about Pythagorean thought comes to us from the writings of Philolaus and other later Pythagorean scholars. But, although his contribution was clearly important, he nevertheless remains a controversial figure. It is sometimes claimed that we owe pure mathematics to Pythagoras, and he is often called the first “true” mathematician.