Much of what we know about Pythagoras is conjecture. We do know that he was born on the Greek island of Samos in about 570 BC and died in about 495 BC. Around about 530 BC he founded what can only be called a rather bizarre sect in Croton, in Italy: it was semi-mystical, and Pythagoras imposed strict vegetarianism, communal living, secret rites and strange rules – such as instructions to his followers never to urinate towards the sun! Part of the problem for later historians is that members of the sect were sworn to strict secrecy: the punishment for revealing any of the sect’s discoveries could be death. As a result, Pythagoras left no writings himself, so that we do not even know whether he personally made any discoveries, or simply provided the environment in which discoveries were made.
We all know (I hope) Pythagoras’ Theorem. However, it is unlikely that Pythagoras, or his sect, discovered the theorem: it is fairly certain, for example, that Chinese and Egyptian mathematicians were aware of it long before Pythagoras. However, it was Pythagoras who gave the theorem its definitive form, and probably constructed a proof (there are now over 400 distinct proofs). So what contributions did the Pythagorean sect actually make to mathematics?
Their work was centred on number, to the extent that they indulged in a form of “number worship.” Many numbers were linked to particular ideas: for example, one was the generator of all other numbers; two represented opinion; three represented harmony; and so on. Ten was seen as the holiest of numbers, partly because it is the sum 1 + 2 + 3 + 4. Perhaps the greatest of all their achievements was building a rigorous mathematical system from axioms and logic, rather than just cataloguing random discoveries. Through logic they encountered seemingly impossible problems. For example, when using Pythagoras theorem on a right-angled triangle with two short sides of 1 and 1, they tried to deal with a hypotenuse of length √2, and hence opened up the whole new world of irrational numbers – numbers which cannot be written exactly as fractions. This discovery by Pythagoras’ pupil Hippasus resulted in his death by drowning, because he revealed the conundrum to the outside world. But the discovery of the existence of a number which could not be written as the ratio of two of “God’s creation” (integers) marked the real birth of Greek geometry, and its dominance for the next 2000 years.
So, what else? Pythagoras and his followers discovered much about triangles, including that the sum of the angles is 180°; and possibly also the general theorem about the sum of the interior angles of a polygon. They laid the foundations of number theory, with their investigations of square, triangular and perfect numbers. Pythagoras is also credited with the links between numbers and the intervals between musical notes; that is, that harmonious notes always have whole number ratios. For example, a note played on two strings, one double the length of the other, will be exactly an octave apart. In fact, Pythagoras was so excited by such discoveries, that he decided that the whole universe was based on numbers, that the planets and stars moved according to mathematical equations (he was right!), and we still talk about “the music of the spheres.”