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Pythagoras & The Numbers Game

Richard Lewis on Pythagoras of Samos.

Pythagoras discovered a rule about right-angled triangles, as every schoolchild knows. What fewer people these days know was that Pythagoras was one of the founders of modern science; his ideas about harmony and numbers inflamed the imaginations of generations to come. He was also a mystic and founded a religious order whose members worshipped mathematics.


Pythagoras grew up on the rich and cultured island of Samos, in the Aegean, during the reign there of the tyrant Polycrates (6th Century BC). His father was a silversmith called Mnesarchos. He was a pupil of both Anaximander the philosopher, and Pherecydes the mystic. After establishing a reputation as a thinker, Pythagoras left Samos to see the world, visiting Egypt, among other places, before settling in the Greek colony of Croton in southern Italy. Soon after arriving in Croton, he established a secretive religious cult (the Brotherhood) whose recruits gave up everything to practice communal living. The sect’s tenets included a veneration of harmonious numbers, and a refusal to eat beans. They developed Pythagoras’ mathematical theories, and held all their discoveries in common, so that it isn’t possible to say which ideas came from Pythagoras, and which from his disciples. Within the Brotherhood, Pythagoras’ word was law. The sect grew quickly, for a while held power in Croton and was briefly influential throughout the Greek world. It then disintegrated just as rapidly; the Crotonians kicked out Pythagoras and his followers. Pythagoras moved to another city, Metapontion, where he died in 496 BC.

Pythagoreanism for Beginners

Pythagoras’ first great discovery was that the strings on a lyre are in tune with one another if their lengths are related by ratios of whole numbers (2/1, 3/2, 4/3 etc.). For example, if you halve the length of the string, the note produced goes up by exactly an octave. Pythagoras was deeply impressed by this evidence that the world obeyed mathematical rules. He searched for other relationships in the world which appeared to involve ratios of whole numbers. He found them everywhere; in geometry, in astronomy, in medicine. It was the beginning of science. After a while he and his followers concluded that the whole universe depended on these ‘harmonious numbers’, and they began to worship them as well as to study them.

The Pythagoreans believed that the Earth was a sphere surrounded by concentric hollow spheres containing the Sun, the Moon and the various planets. The diameters of the spheres were related to one another as ratios of whole numbers, naturally, and it was thought that their rotation therefore created a harmonious music, which a sufficiently well-trained ear could hear. This Pythagorean notion of the ‘music of the spheres’ was still widely believed 2,000 years later.

The Pythagoreans believed that certain numbers had particular significance. The number one represented the pure, undivided unity from which all of existence takes its form. The number two (the ‘dyad’) represented pairs of opposites, and the idea of distinction. They recognised 10 such sets of oppositions, which categorised all things that exist:

  • finite : infinite
  • quiescent : moving
  • odd : even
  • straight : crooked
  • one : many
  • light : darkness
  • right : left
  • good : evil
  • male : female
  • square : parallelogram

The number three (the ‘divine triad’) represented completed existence, combining the creative unity with the dyad, which organises matter into categories. The triad also encapsulates the beginning, the middle, and the end.

Pythagoras also believed in the transmigration of souls – in other words, that after death a person’s soul is reborn in a new body, either animal or human.

Enduring achievements

• Pythagoras’s Theorem for right-angled triangles: the square on the hypotenuse is equal to the sum of the squares on the other two sides. The Egyptians had known this through empirical measurements, but Pythagoras was the first to find a geometrical proof of the rule.

• The discovery of the existence of irrational numbers (see next section.)

• Pythagoras was the grandfather of musical theory and musical notation

• In astronomy, Pythagoras may have been the first to realise that the Morning Star and the Evening Star were in fact the same object (the planet Venus), and that the Moon shines by the reflected light of the Sun.

• The Pythagoreans also left their mark on medicine: when we talk about ‘muscle tone’ or a ‘tonic’ we are using a Pythagorean analogy between the human body and a musical instrument, which much be kept in tune.

The collapse of Pythagoreanism

The whole Pythagorean world-view rested on the idea that all relationships involved ratios of whole numbers, harmonious numbers. Then they discovered ‘irrational’ numbers, such as the square root of 2. Such numbers are common; for instance the diagonal of a square with 1 inch sides is v2 inches. Irrational numbers are both even and odd at the same time. They can’t be expressed as ratios of whole numbers. The discovery of irrational numbers was the greatest mathematical advance of the Pythagoreans, but it fatally undermined their most fundamental beliefs. The very label ‘irrational’ expresses the Pythagoreans’ horror of these unharmonious numbers. They tried to hush up the discovery, but a disciple called Hippasos let the secret leak out. The Pythagoreans assassinated him. People really cared about maths in those days.

The other cause of the Pythagoreans’ demise was their desire to impose their views on the rest of the Greek world by seizing secular power wherever they could. This caused a violent reaction and the sect was severely persecuted. Pythagorean meeting houses were burned down and the organisation was effectively stamped out. Individual Pythagoreans were generally tolerated, but their obsession with numbers may not have inspired universal respect. The Brotherhood’s insistence that women be admitted to membership on equal terms also seemed eccentric to their contemporaries. So that makes two out of three to the Pythagoreans. Better cut back on the beans in your diet, in case they turn out to be right about that as well!

The Fifteen WACKY Rules of the Pythagorean Brotherhood!!!!

It has been traditional to pay homage to the scientific genius of the Pythagoreans while at the same time pouring scorn on the various little oddities of their sect. This is a highly unfair, as we don’t fully understand their belief-system and therefore can’t properly appreciate the symbolic significance of their religious observances. However, far be it from me to break with tradition…;

  1. To abstain from beans
  2. Not to pick up what has fallen
  3. Not to touch a white cock
  4. Not to break bread
  5. Not to step over a crossbar
  6. Not to stir the fire with iron
  7. Not to eat from a whole loaf
  8. Not to pluck a garland
  9. Not to sit on a quart measure
  10. Not to eat the heart
  11. Not to walk on highways
  12. Not to let swallows share one’s roof
  13. When the pot is taken off the fire, not to leave the mark of it in the ashes, but to stir them together
  14. Do not look in a mirror beside a light
  15. When you rise from the bedclothes, roll them together and smooth out the impress of the body

(Rules quoted from Bertrand Russell’s History of Western Philosophy )

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