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Taken from the Parrot’s Theorem a novel by Denis Guedj
ISBN 0 297 645781
Chapter 5
THE SCHOOL OF PYTHAGORAS; a cult of secrecy and the music of numbers.
Sixth century BC
Greece
No written work of Pythagoras survives. Even the dates of his birth and death are unknown. What is known, however, is that he was born on the island of Samos in the Aegean sea in the 6th century BC, and died in the town of Crotona in southern Italy. He was only 18 when he participated in the Olympic Games, where he won every single boxing tournament. After that, he decided to travel. He went first to nearby Ionia, where he spent some years with Thales and his pupil Anaximander. Then he travelled to Syria, where he stayed with the Phoenician sages. From there he went to Mount Carmel in what we now call the Lebanon, and then to Egypt, where he stayed for twenty years. In the temples on the banks of the Nile he learned the wisdom of the Egyptian high priests.
It is said that philosophy has two origins. One was Thales and the other Pythagoras. Both travelled to Egypt and down the banks of the Nile. Thales came back with a tale of shadows and Pythagoras with a story of numbers. Pythagoras talked to animals. Did you know that he once convinced a bear to stop terrorizing a local village?
When the Persians invaded Egypt, he was taken prisoner and brought to Babylon. He spent twelve years there in the capital of Mesopotamia, where he learned much from the scribes and from the wise men. He returned to Samos forty years after he had left, older and wiser. But Samos was ruled by the tyrant Polycrates and unable to bear his tyranny, Pythagoras left for Greece and then on to Sybarius in southern Italy famous as being a city of pleasure- but it was in the nearby town of Crotona that he made his home and founded his school which lasted 150 years. Pythagoras was leader of the mathematicians.
A rich and powerful man named Cylon who lived in the town wanted to be admitted to the Pythagorean school, but was rejected. Cylon was an autocratic man: he was not in the habit of being refused something he wanted. Furious at his refusal, Cylon decided to take his revenge. The members of the Pythagorean school met regularly to discuss city politics. One night, Cylon and his supporters set fire to the meeting house. All but one of those inside died in the blaze. One man survived the fire Philolaus, an astronomer and cosmographer, who had proposed an astounding theory about the world two thousand years before Copernicus and Galileo. Not only did the earth turn, according to Philolaus, but it was not the centre of the universe. Philolaus further suggested that there was a fire at the centre of the universe around which the earth, the planets and even the sun turned
Just across a small strait from Crotona in southern Italy was the town of Tarentum, where Archytas invented the number 1. Surely number one has always existed! Well apparently not because the Greek thinkers believed numbers began with 2. To them, there was ‘one’ and ‘more than one’. They believed that ‘one’ was not a quantity but a statement of existence, whereas numbers were about multiplicity. – but Achytas insisted on the singularity of one and made a number of it. Archytas was not only the father of ‘one’ but the ‘first engineer’. Marshalling his knowledge of mathematics and geometry, he developed a theory of mechanics and is reputed to have made a mechanical bird – a wooden dove which could fly, powered by a small engine. He was also the first Graffiti artist in history. He wrote graffiti because he could not bring himself to swear. When he felt he absolutely had to, he simply wrote the offending word on a nearby wall.
The world of mathematics was greatly expanded by the Pythagoreans. They brought disciplines into mechanics. Their often mystical views of numbers did not stop them from establishing arithmetic as a science. The first proofs in the history of maths were developed by them. They proved that the square root of 2 was an irrational number. They demonstrated that the sum of the angles in any triangle is 180 degrees.
Take four identical vases on a table. The first is empty, the second is half filled with water, the third one quarter filled and the fourth filled to one third. Tapping the empty vase and then the vase half full lets two distinctive tones ring out. Known as an octave.
Then the vase empty and the one that is a third full. Rings out what is known as a fifth.
Then the empty and the one that is a quarter full. Which rings out a fourth.
Pythagoras saw numbers everywhere but he first found numbers in music.
Using a simple device, Pythagoras had made an extraordinary discovery: that the difference in musical notes is a difference between two numbers. The octave we hear by hitting the two vases, one empty and one half-filled; is defined as a ratio of one half; a musical fifth is two thirds and a musical fourth is three quarters. In fact harmony is simply a series of sounds based on numerical ratios.
Pythagoras and his school were determined to discover the mathematical laws inherent in the way nature regulated itself.
Pythagoras began by creating a catalogue of numbers, beginning with 1 – which seems so natural to us today that we assume it always existed. He divided whole numbers into odd and even numbers: those which are divisible by 2 and those which aren’t. He then went on to establish the rules of calculation….
Even plus even equals even; odd plus odd equals even; odd plus even equals odd.
In multiplication: Even times even equals even; odd times odd equals even; odd times even equals even.
Before Pythagoras was born the Egyptians and the Babylonians had discovered a link between triple numbers. A collector named Plimpton acquired a Babylonian tablet on which a scribe had engraved a dozen triplets, indicating definitely that the sum of the squares of the first two was equal to the square of the third.
Example: 3 squared is 9; 4 squared is 16; added together they make 25 which is 5 squared.
A triangle with sides of these lengths is aright angled triangle. So the theorem tells us that there is a relationship between the lengths of the sides of a right angled triangle which can be expressed as a squared plus b squared equals c squared.
In summary:
‘The square on the hypotenuse is equal to the sum of the squares on the other two sides.'
The Egyptians and the Babylonians gave examples but they were not able to prove the theorem however Pythagoras did.
He founded a sort of sect. One of its rules was never to divulge the things its members had learned. To stop their secrets from falling into the wrong hands, Pythagoreans rarely wrote anything down and their knowledge was passed on by word of mouth. What is written down is permanent, while the spoken word is ephemeral. To make sure their wisdom did not disappear, they developed exercises to train the memory. What did it feel like to be an apprentice applying to join Pythagoras’ school. Pythagoras was rigorous in his selection. He began by seeing whether the disciple could hold his tongue. Could he remain silent during lessons and more importantly, keep secret what he had learned? The first test depended not on what the disciple said, but what he did not say. The school room was divided into two by a curtain. Pythagoras would sit on one side and the candidates on the other. They could listen but not see. This period lasted for 5 years.
The writings of the Pythagoras school were also secret and were deliberately written to have a double meaning: one which anyone could understand, but the other comprehensible only to the initiated. Most of the knowledge was passed on by word of mouth. Every pupil of Pythagoras every morning, before getting up had to remember the precise events of the day before- every thing he had seen or said or done and everyone he had met.
The Parrots owner faced with the threat of death if he did not divulge his life’s work on mathematical proofs to extortionists writes to a dear friend---
‘You should know that I would not be the first person in the history of maths to be secretive, though secrecy is no longer much in fashion. Nowadays people rush to print before they’ve even finished the proof.’
‘Do you know what first interested me about Pythagoras? He invented the word ‘friendship’. When someone asked him what a friend was, he replied ‘Someone who is another me, like the numbers 220 and 284’. Two numbers are friends if each is the sum of everything that measures the other. The most famous of these so called amicable numbers are 220 and 284. They’re a fine couple. You should try checking them sometime.