Be that as it may, Thales’ geometry, on the other hand, is very real, and whether he applied it to the Great Pyramid or to an olive tree, the shadow method is no less appealing. This method represents a particular case of a property now known as Thales’ theorem. Several other mathematical results are also attributed to Thales: the circle is divided in two equal parts by any diameter (fig. 1); the angles at the base of an isosceles triangle are equal (fig. 2); the opposite angles produced at the point of intersection of two straight lines are equal (fig. 3); if the three vertices of a triangle lie on a circle and one side passes through the centre of this circle, then the triangle is a right-angled triangle (fig. 4). In fact, this last statement is sometimes also referred to as Thales’ theorem.
Fig. 1
Fig. 2
Fig. 3
Fig. 4
I want to look at this strange word that is as fascinating as it is awesome: what is a theorem? Etymologically, the word comes from the Greek thea (contemplation) and horáō (to see), so a theorem would be a sort of observation on the world of mathematics, a fact that would have been noted, examined and then recorded by mathematicians. Theorems can be transmitted orally or in writing, and are similar to one’s grandmother’s recipes or to weather lore that has been tested over the generations and is confidently believed to be true. One swallow does not make a summer, bay leaves soothe rheumatism, and the 3-4-5 triangle has a right angle. These are things we believe to be true and that we keep in mind in order to use them when they are relevant.
According to this definition, the Mesopotamians, the Egyptians and the Chinese also stated theorems. However, beginning with Thales, the Greeks gave them a new dimension. For them, a theorem not only had to state a mathematical truth, but that truth must be formulated in the most general way possible, and accompanied by a proof of its validity.
Back to one of the properties attributed to Thales: that the diameter of a circle divides the circle into two equal parts. Such an assertion may seem disappointing coming from a scholar of Thales’ calibre. It seems self-evident; why did we have to wait until the sixth century BC for such a trivial assertion to be stated? There is no doubt that Egyptian and Babylonian scholars must have known that a long time beforehand.
Make no mistake. The audacious thing about that property attributed to Thales is not so much its content as its formulation. Thales dared to speak about a circle without saying precisely which one. In order to state the same rule, the Babylonians, Egyptians and Chinese would have used an example. Draw a circle of radius 3 and draw one of its diameters, they would have said, and this circle is divided into two equal parts by this diameter. And when one example did not suffice for an understanding of the rule, they would have given a second one, a third, and a fourth if necessary. They would have given as many examples as needed to enable readers to understand that they could repeat the same procedure on every circle they might meet. But the general assertion was never formulated.
Thales reached a milestone.
‘Take a circle, any one you like, I don’t need to know which. It may be enormous or tiny. Draw it in the horizontal, in the vertical or on an inclined plane, it doesn’t matter to me. I don’t care at all about your particular circle and how you have drawn it. However, I assert that its diameter divides it into two equal parts.’
With this operation, Thales definitively assigned the status of abstract mathematical objects to geometric figures. This step in thinking was similar to that taken 2,000 years earlier by the Mesopotamians when they considered numbers independently from the objects counted. A circle was no longer a figure drawn on the ground, on a tablet or on a papyrus. The circle became a fiction, an idea, an abstract ideal all of whose real representations are merely imperfect instances.
From that point on, mathematical truths were stated in a concise and general manner, independently of the various particular cases they covered. It is these statements that the Greeks then called theorems.
Thales had several disciples in Miletus. The two most famous were Anaximenes and Anaximander. Anaximander in turn had his own disciples, who included a certain Pythagoras, who would give his name to the most famous theorem of all time.
Pythagoras was born at the start of the sixth century BC on the island of Samos, situated off the coast of what is now Turkey, and less than 40 kilometres away from the town of Miletus. As a young man he gained experience through his travels in the ancient world, and chose to settle in the town of Croton, in the south-east of present-day Italy. It was there that he founded his school in 532 BC.
Pythagoras and his disciples were not just mathematicians and scholars, but also philosophers, monks and politicians. Yet it must be said that if we were to transpose it to our time, the community begun by Pythagoras would undoubtedly be perceived to be among the most obscure and most dangerous of sects. The life of the Pythagoreans was governed by a set of precise rules. For example, anyone wishing to join the school had to go through a period of five years of silence. The Pythagoreans had no individual possessions: all their belongings were shared. They used various symbols such as the tetractys or the pentagram in the shape of a star with five branches to recognize one another. Moreover, the Pythagoreans thought of themselves as enlightened people, and thought it normal that political power should come to them. They firmly repressed the rebellions of towns that refused to accept their authority. In fact, it was in one such riot that Pythagoras died at the age of eighty-five.
The number of myths of all kinds that were invented around Pythagoras is also impressive. His disciples were scarcely lacking in imagination, as we can now see. According to them, Pythagoras was the son of the god Apollo. The name Pythagoras also means literally ‘he who was announced by the Oracle’: the Oracle of Delphi was in fact the oracle of the temple of Apollo, and it is she who is said to have told Pythagoras’ parents about the forthcoming birth of their child. According to the Oracle, Pythagoras would be the most handsome and wisest of men. After such a birth, the Greek scholar was predestined for great things. Pythagoras remembered all his previous lives. According to this, he had in particular been one of the heroes of the Trojan War called Euphorbus. In his youth, Pythagoras took part in the Olympic Games and took the laurels in all the pygmachia events (pugilism, the ancestor of our boxing). Pythagoras invented the very first musical scales. Pythagoras was able to walk in the air. Pythagoras died and was resuscitated. Pythagoras was a talented soothsayer and healer. Pythagoras had control over animals. Pythagoras had a golden thigh.
While most of these legends are so far-fetched that no one believes them, in other cases the jury is still out. Is it true, for example, that Pythagoras was the first to use the word ‘mathematics’? The facts are so sketchy that some historians have even come to speculate that Pythagoras was a purely fictional person, dreamed up by the Pythagoreans to serve as their tutelary figure.
Therefore, since it is not possible to learn more about the man, let us return to the subject for which he is still known to schoolchildren the whole world over more than 2,500 years after his death: Pythagoras’ theorem. What does this famous theorem tell us? Its statement may seem astonishing, because it establishes a link between two apparently unconnected mathematical concepts: right-angled triangles and the square numbers.
Let us return to our favourite triangle, the 3-4-5. From the lengths of the three sides we can construct three square numbers: 9, 16 and 25.
We now spot a curious coincidence: 9 + 16 = 25. The sum