Pythagoras, a Greek philosopher, founder of a philosophical, religious, and political association in southern Italy, born in Samos about 580 B. C, died probably in Metapontum about 500. He was the son of Mnesarchus, an opulent merchant, and according to some accounts was a disciple of Pherecydes of Syros, and of Thales and Anaximander. He is said to have spent 30 years in travel for the purpose of collecting all attainable knowledge, especially the esoteric doctrines of priests concerning the worship of the gods. Egypt, Arabia, Phoenicia, Judea, Babylonia, and even Gaul and India, are among the countries in which he is said to have travelled. Herodotus traces the doctrine of metempsychosis and certain religious regulations of the Pythagoreans and Orphists back to the Egyptians, thus apparently implying that Pythagoras visited Egypt. Aristotle testifies that the mathematical sciences originated in Egypt, and were there cultivated by the priests; and according to Callimachus Pythagoras brought his mathematical knowledge from that country. On the authority of Apol-lodorus, Diogenes Laërtius ascribes to Pythagoras, among other things, the discovery of the relation between the hypothenuse and the sides of the right-angled triangle.

It is easy to see that many of the statements made by later writers are mere embellishments and fables. Following Aristoxenus, Diogenes Laërtius again says that Pythagoras emigrated to Crotona in lower Italy, in order to escape the tyranny of Polycrates, and according to Cicero he came to Italy about 529 B. C. In Crotona he succeeded in winning the aristocratic party to his project of an ethical and religious reform, and in uniting them into a powerful political faction. Then it is said that about 20 years later the democratic party of Crotonians, under Cylon, obtained his banishment, and he withdrew to Metapontum, where he soon died. - It is generally held that Philolaus, a contemporary of Socrates, was the first to publish the Pythagorean system of philosophy; but though a considerable number of fragments that pass under his name are extant, their authenticity is very questionable. Böckh's collection of the fragments has been in part, if not wholly, rejected by Zeller, Rose, and others. The writings reported to come from Pythagoras himself are undoubtedly spurious.

The most important indications of his doctrines are obtained from the writings of Aristotle. The fundamental doctrines are, that the essences of all things rest upon numerical relations; that numbers are the principle of all that exists; and that the world subsists by the rhythmical order of its elements. Everywhere in nature appear the two elements of the finite and the infinite, which give rise to the elementary opposites of the universe, the odd and even, one and many, right and left, male and female, fixed and moved, straight and curved, light and darkness, square and oblong, good and bad. The essence of number is unity, which is at once odd and even, and contains in itself in germ all the universe. It is both the form and the substance of all things, and identical with the Deity. Proceeding from itself it begets duality, and returning upon itself it begets trinity. Added to itself it produces the line; a third point placed on the other two gives the surface; and a fourth point placed on the other three gives the pyramid or solid. The quadrate or tetractys and the decade are, like unity, sacred numbers and first principles.

The universe was produced by the breathing of the first principle into the infinite void of the world, which thus became both finite and infinite, and therefore capable of development into a multiplicity of numbers or things. In the actual world every single whole is a unit, capable of further development by the vital process of breathing. Every abstract idea is a number, and material objects are symbols of numbers. Thus the Pythagoreans called justice a square number, intending by this to express the correspondence between action and suffering or retribution. There are five elements, earth, fire, air, water, and ether, represented respectively by the cube, pyramid, octahedron, icosahedron, and dodecahedron. The universe is a harmonious whole, consisting of ten great bodies revolving around a common centre. The doctrine of the harmony of the spheres was based on the idea that the celestial spheres were separated from each other by intervals corresponding with the relative lengths of strings arranged to produce harmonious tones. The centre is the sun, the seat of Jupiter, the principle of life, and the most perfect object in nature.

That his hypothesis of the sun's immobility, and of the revolution of the earth around it, agrees with the facts of nature, was shown much later (about 280 B. C.) by the astronomer Aristarchus of Samos. The stars also are divinities, and men and even inferior animals are akin to the Supreme Being. The souls of men are moving numbers, light particles from the universal soul, capable of combining with any body, and destined to pass successively through several. They are chained to the body as a punishment, and dwell in it as in a prison. With the theory of metempsychosis he combined the doctrine of moral retribution. The reason and understanding have their seats in the brain; the passions are placed in the heart. Moral good is identified with unity, evil with multiplicity; virtue is the harmony of the soul and its similitude to God. The aim of life is to make it represent the beautiful order of the universe. The whole practical tendency of Pythagoreanism was ascetic (according to some accounts including abstinence from animal food), and inculcated a strict self-control, promoted, as is said, by a novitiate of silence, and an earnest culture, in which music was considered important.

Though it seems to have been founded on the mysticism of numbers, yet Aristotle called the Pythagoreans a school of mathematicians. - See Schaarschmidt, Die angebliche Schriftstellerei des Philolaus (Bonn, 1864); Zeller, Die Pythagorassage (Leipsic, 1865); and Ueberweg's "History of Philosophy" (translated into English, New York, 1872).