Between science and wisdom: Pythagoras and the beginning of Greek mathematics

Talete di Mileto

Pitagora di Samo

Ippocrate di ChioTeeteto di

AteneEudosso di Cnido

Archimede di Siracusa

Apollonio di Perga

Euclide di Alessandria

Conone di SamoAristarco di Samo

Autolico di Pitane

Eudemo di Rodi

Eratostene di Cirene

Ipsicle di AlessandriaDidimo di AlessandriaFilone di Alessandria Menelao di AlessandriaErone di AlessandriaTolomeo di AlessandriaDiofanto di AlessandriaPappo di AlessandriaTeone di Alessandria

Between science and wisdom

(VI century b. C.)Thales of Miletus

Pythagoras of Samos

Between science and wisdom

They say that Thales first proved that the circle is divided into two [equal] parts by its diameter

(VI century b. C.)Thales of Miletus

Between science and wisdom

(VI century b. C.)Thales of Miletus

It is said that first he has established that the angles at the base of each isosceles triangle are equal.

Between science and wisdom

Thales of Miletus was able to measure the height of the pyramids

(VI century b. C.)Thales of Miletus

[Thales] led Pythagoras to sail to Egypt and meet with the priests of Memphis and Diospolis, because they were the ones who had instructed him in those disciplines for which he was considered wise by the people.

Porphyry, Vita Pythagorae

Between science and wisdom

It is said that when Cambyses took possession of Egypt, took prisoner Pythagoras who lived there with the priests, and that Pythagoras, once in Babylon, was initiated into the mysteries. Cambyses lived precisely at the time of Polycrates, to escape whose tyranny Pythagoras had gone to Egypt.

Theologumena Arithmetica

Between science and wisdom

They say that the one who first divulged the nature of commensurability and incommensurability to men who did not deserve to be made part of this knowledge, was so hated by the other Pythagoreans, that not only drove him from the community, but also built him a tomb as if he had died, he that once had been their friend

Iamblichus, De vita pythagorica

Between science and wisdom

A proof of this type, for example, is that of the incommensurability of the diagonal [and the side of the square], which is based on the fact that if we assume that they are commensurable, odd and even numbers are equal.

Aristotle, Prior analytics

Between science and wisdom

Let ABCD be a square and suppose that the diagonal BC is commensurable with the side AB. Let E and Z be the smaller numbers that are to one another in the ratio of BC to AB : they are relatively prime. But also their squares, respectively, I and K are relatively prime. On the other hand, the square of the diagonal is twice the square of the side [by Pythagora’s theorem]. So I = 2K, and I is even. In addition, half of the square of an even number is also even, and therefore I / 2, i.e. K, will be even. But I and K are prime to each other, while two even numbers can not be prime. Therefore either I or K, or both, must be odd. On the other hand it has been shown that both must be even. This is contradictory, and thus the incommensurability is proved.

D

B

C

A

Alexander of Aphrodisias, Analytica

Between science and wisdom

In any right triangle, the square which is described upon the side subtending the right angle is equal to the squares described upon the side which contain the right angle.

Between science and wisdom

Between science and wisdom

30

1;24,51,10

42;25,35

1,414213

42,42639

Between science and wisdom

Between science and wisdom

A B

C

D

ABC = ACD + CBD

AB2 = AC2 + CB2

Between science and wisdom

Between science and wisdom