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Pythagoras was a Greek philosopher and mathematician, whose doctrines were strongly influenced Plato. Pythagoras was born on the island of Samos around 582 B.C. and died around 500 B.C. Pythagoras was instructed in the teachings of the early Ionian philosophers Thales, Anaximander, and Anaximenes. Pythagoras is said to have been driven from Samos by his disgust for the tyranny of Polycrates. About 530 B.C. Pythagoras settled in Crotona, a Greek colony in southern Italy, where he founded a movement with religious, political, and philosophical aims, known as Pythagoreanism. The philosophy of Pythagoras is known only through the work of his disciples.

The Pythagoreans adhered to certain mysteries, similar in many respects to the Orphic mysteries . Obedience and silence, abstinence from food, simplicity in dress and possessions, and the habit of frequent self-examination were prescribed. The Pythagoreans believed in immortality and in the transmigration of souls. Pythagoras himself was said to have claimed that he had been Euphorbus, a warrior in the Trojan War, and that he had been permitted to bring into his earthly life the memory of all his previous existences.

Among the extensive mathematical investigations carried on by the Pythagoreans were their studies of odd and even numbers and of prime and square numbers From this arithmetical standpoint they cultivated the concept of number, which became for them the ultimate principle of all proportion, order, and harmony in the universe. Through such studies they established a scientific foundation for mathematics. In geometry the great discovery of the school was the hypotenuse theorem, or Pythagorean theorem, which states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides.

The astronomy of the Pythagoreans marked an important advance in ancient scientific thought, for they were the first to consider the earth as a globe revolving with the other planets around a central fire. They explained the harmonious arrangement of things as that of bodies in a single, all-inclusive sphere of reality, moving according to a numerical scheme. Because the Pythagoreans thought that the heavenly bodies are separated from one another by intervals corresponding to the harmonic lengths of strings, they held that the movement of the spheres gives rise to a musical sound—the "harmony of the spheres."