The Pythagoras myth
- January 4, 2024
- Mark Ronan
- Themes: Maths
Contrary to popular belief, Pythagoras was not in fact the first to develop his eponymous theorem – it had already been known for over a thousand years.
Is Pythagoras the father of modern mathematics? In popular perception he was the source of the famous theorem about right-angled triangles: The Square on the Hypotenuse is equal to the Sum of the Squares on the Other Two Sides. It’s a great result – serious mathematics indeed – but the theorem had been known for 1,000 years or so. Euclid’s Elements, the indispensable text for Greek geometry, never mentions Pythagoras in connection with the theorem named after him. That theorem (and its converse), the final results at the end of Book I, illustrate how the new Greek methods using angles could be used to prove an ancient result. Mathematicians love such things – using new ideas to prove results already known.
The Elements is a marvellous treatise by an author who compiles the geometry due to himself and previous Greek mathematicians, notably Hippocrates of Chios in the fifth century BC, and presents it rigorously. There are 13 books in all. In Book I he lays out definitions (points, lines, angles and so on), postulates such as the one saying that all right-angles are equal, and axioms such as the one that says that if two things equal the same thing, then they equal one another. It is all very explicit, leaving nothing to doubt, and of a quite different character to the earlier geometry of the Ancient Near East or Egypt.
Angles were a new idea, thanks to the Greeks. Right-angles were familiar, but the idea of measuring the size of an angle was not so obvious. Imagine you have two walls, or two lines in a plane, that meet at a corner. The word gon is the Greek word for corner, as in pentagon (five corners), hexagon, and so on, and they used the same word for angle. How, or even why, would you measure a gon? For the new Greek geometry they were fundamental, however, and Euclid refers to one angle being twice another angle, and uses the idea of adding two angles, though he never measures angles in degrees. That came from astronomy, where the Babylonians divided the sun’s path around the earth into 360 degrees. Theirs was the mathematics that long predated the Greeks, but Euclid’s work is independent, a truly Greek invention. So, back to Pythagoras.
As Walter Burkert writes in his excellent Lore and Science in Ancient Pythagoreanism: ‘From the very beginning, his influence was mainly felt in an atmosphere of miracle, secrecy, and revelation… he represents not the origin of the new, but the survival or revival of ancient pre-scientific lore, based on superhuman authority and expressed in ritual obligation.’ He goes on to say that: ‘As the old and the new interpenetrated and influenced each other, the picture of Pythagoras became distorted until, with the victory of rational science, he came to seem its true founder.’
Like any sensible civilisation, the Greeks borrowed ideas from others and built on what had gone before. They took their alphabet from the Phoenicians – a Semitic-speaking people – and adapted it to their own (Indo-European) language. It was soon standardised to 24 letters, and used to write literature. The earliest Greek epics: Homer’s Iliad and Odyssey, borrowed from traditional tales from the Ancient Near East, particularly the Epic of Gilgamesh. Homer himself, like the geometer Hippocrates, supposedly came from Chios, an island off the west coast of Anatolia, and the Greek debt to Ancient Near Eastern myth is very well assessed by Martin West in his book The East Face of Helicon.
Why was the theorem on right-angled triangles named after Pythagoras? Answering this question moves the story to around 700 years on from Euclid to the fifth century AD when Proclus became head of the Neoplatonist Academy in Athens. He wrote commentaries on Book I of the Elements that make an allusion to this legendary figure, but is doubtful about some of the claims, for example that Pythagoras sacrificed an ox to celebrate his discovery. This hardly fits the image of Pythagoras as leader of a vegetarian brotherhood strongly opposed to animal sacrifice.
Proclus’s commentaries refer to a second proof (in Elements Book VI) that does not use angles, and its key ingredient – that the areas of two triangles having the same shape are in the same proportion as the squares of corresponding sides – was known to the Babylonians. Proclus was a capable expositor and critic, but more philosopher than mathematician, though he did have access to books and manuscripts now lost. One was a history of geometry by Eudemus of Rhodes. This student of Plato, writing in the fourth century BC, before Euclid, may be the source for the story that the Egyptians were the first to discover geometry, as the rising of the Nile wiped out everyone’s proper boundaries and they needed to compute new areas every year. Certainly the Egyptians were ahead of the Greeks in this field, but the mathematician Jöran Friberg shows how they had learned in turn from the Babylonians during the early second millennium BC. Unfortunately, Proclus was wedded to the idea that geometry came from Egypt and even claimed that Thales, whom the Greeks proudly claimed as the first philosopher, had travelled there and introduced geometry into Greece, completely excluding the Babylonians from the picture.
It all seemed so simple: first, the Egyptians then the Greeks, and Proclus gives a summary of developments in the Greek world, mentioning numerous people who contributed, including Plato. The trouble is that although Proclus appears to quote Eudemus, he seems to have used a later source by a Neoplatonist named Iamblicus, writing in about 300 AD. In fact one ‘quotation’ credited to Eudemus turns out to originate word for word from Iamblichus, whom Proclus copies extensively in his commentary on Euclid.
Since Iamblichus was writing more than 700 years after the Pythagoreans flourished, his claims about the ‘purity, subtlety, and exactitude’ of their methods, how they purify the soul and lead on to the highest principles and a realm of pure, immaterial Being, cannot be taken seriously. There are no texts, so he makes great efforts to reconstruct ‘what they would probably have said if one of them could have taught his doctrine publicly’. This is folly, but Iamblichus persists: ‘If we are to pursue mathematics in the Pythagorean manner, we must follow its upward path, full of divinity, which brings purification and perfection.’
This is nonsense. You cannot understand the past by imagining that the present emerged from it in a simple upward trajectory.
It seems Proclus had found a gap that needed filling in the history of geometry and since Eudemus gave little information about Pythagoras – even none at all – he conveniently attributed to Pythagoras the major early achievements in geometry.