Pythagoras is for us at once the glorified and the actual founder of the Pythagoreans—a philosophical school or sect which exercised a profound and lasting influence on the course of ancient science, philosophy, and theology from pre-Socratic times in Lower Italy, and then, down to the days of the Roman empire, in Greece, in Alexandria, and elsewhere. It cannot be too carefully borne in mind that the earliest written information we have about Pythagoreanism is the fragments not of Pythagoras himself but only of Philolaus, a successor, that the school was at first mainly characterised by its ethico-religious and political tendencies, and that the founder became among later adherents the object of mystical reverence and contemplation; accordingly our knowledge about the doctrines of the Pythagoreans and about the personality of Pythagoras is extremely limited. Aristotle, for example, only speaks of the Pythagoreans in his résumé of Greek philosophy, and he with Plato probably only knew of Pythagoras through the oral utterances of Philolaus. Pythagoras was born in Samos about 582 B.C. As regards his education we know only that he was made acquainted with the teachings of the early Ionic philosophers, and, through his travels (which are said to have been not only among the Egyptians, but among the Phœnicians, the Chaldeans, the Persian Magi, the Indians, Jews, Druids, Thracians, &c.), with those of the Egyptian priests. About 530 he settled in Crotona, in Magna Græcia, where he founded the moral and religious school called by his name. Pythagoreanism was first a life and not a philosophy, a life of moral abstinence and purification, reactionary against the popular and the poetic religions, but yet sympathetic towards the old (Doric) aristocratic forms and institutions. All that can be certainly attributed to Pythagoras is the doctrine of transmigration of souls, the institution of certain religious and ethical regulations, and the beginning perhaps of those investigations into numbers and the relations of numbers which made the school famous. The Pythagoreans as an aristocratic party became unpopular after the defeat of the Sybarites by the Crotoniates in 510 (see CROTONA, SYBARIS), and at first were instrumental in putting down the democratic party in Lower Italy; but the tables were afterwards turned upon them, and they had to flee from persecution. How Pythagoras himself died is not exactly known; his death (according to tradition, at Metapontum) may be placed about 500 B.C.
The Pythagoreans adhered at first to certain mysteries—indeed, the Orphic mysteries; an examination as to fitness qualified for admission into their number; obedience and silence, abstinence and simplicity in dress and food and 'external goods,' and the habit of frequent self-examination were prescribed. The enjoined disposal of worldly goods may have helped to foster contemplation and scientific enthusiasm. This at least developed itself in the school. Pythagoras, for example, is said to have practised investigations into harmonies and the properties of numbers. Mathematical investigations were first begun by individuals, and then carried on prominently by the school. Their attention was early turned to the odd and even, to prime numbers, square numbers, &c.; and from this arithmetical standpoint they cultivated geometrical studies, number becoming for them the chief principle in space. The elementary relations of harmonies and the regular rotations of the spheres led the Pythagoreans to think of the cosmic order as a numerical one, and, like the early Greek Realists in philosophy, they took number to have a metaphysical significance—to be, as Aristotle tells us, not only the form, but the very substance of things: 'All is Number' came to be their thesis. As numerical proportions are repeated in different things, they regarded numbers also as archetypes, of which things were in a sense the ectypes. 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, the earth itself and the fixed stars all being in progress round the central fire. (It is interesting to notice this idea so early in science of the movement of the earth.) The scheme of revolution was given them first by the decade, each number of which had a peculiar significance, especially the unit, the duad, the square, &c. The table of contraries they also used in explaining the cosmos; this included such contrasts as the limited and the unlimited, the even and the odd, one and many, right and left, male and female, light and darkness, and so on. In all this room was naturally given to fanciful and arbitrary speculation, developed later among the Neo-Pythagoreans in such tables as 1, the point; 2, the line; 3, the surface; 4, body; 5, quality; 6, soul, and so on. To the virtues numbers were also given, justice being the square number; the soul, too, was in general a harmony chained to the body. As the Pythagoreans thought the heavenly bodies to be separated from each other by intervals corresponding to the harmonic lengths of strings, they held that the movement of the spheres gave rise to a pleasing sound called the 'harmony of the spheres.' Of the so-called 'elements' they had also numerical theories, fire being the tetrad, earth the cube, air the octahedron, water the equation.
The great mathematical discovery of Pythagoras is of course the hypothenuse theorem, where the square is equal to the sum of two squares. 'Pythagorean numbers' are such numbers as are related in the way the theorem indicates—e.g. 5, 4, and 3 (). Various other theorems are closely connected with this cardinal one; these concern chiefly the squares of the various perpendiculars which may be let fall from the different angles of the right-angled triangle upon the hypothenuse and sides. The speculations in general of the Pythagoreans may be regarded from various sides. Their formal principle of number is often said, and with truth, to mark a transition from the crude Hylozoism of Thales and the Ionic philosophers to a formal or rational or conceptual contemplation of the world, developed, say, by the Eleatics, and culminating in Plato. Their idea of a quantitative combination of elementary units became a commonplace of Greek speculative cosmology, constituting the ground for a deductive ontology. The conception—general of a measure or proportion in things is, of course, a most pronounced trait in the Greek mind. It is easy to trace in the Pythagorean doctrine of the elements and the contraries and of combination and of spheric completeness all the essential features of Greek cosmology. The influence of Pythagoras and geometrical conceptions over the mind of Plato can hardly be exaggerated. The chief interest of the Pythagoreans doubtless lay in the domain of physics, and their astronomical theories may be said to constitute their capital achievement. If we remember, too, that Pythagoras is perhaps the first Greek thinker who conceived of philosophy as first a life, a life in common, we shall see in this the beginning of the legislative and ethical view of the philosopher's function expressed in the fullest way in Plato's Republic. The ascetic and mystical aspects of Pythagoreanism linked it closely with Platonism in the mind of Christian idealists in later times. See NEOPLATONISM, NEO-PYTHAGOREANISM.
BIBLIOGRAPHY.—The fragments of Philolaus were published by Böckh in 1819. The brief notices Aristotle gives of the Pythagoreans in the first book of the Metaphysics contain almost all that is of philosophical importance in their theory. Zeller's account is quite exhaustive, and notices most that has been written on the subject (The Pre-Socratic Schools, Eng. trans. 1882). See also Grote's Greece, iv. 525–551; Chaignet, Pythag. et la Philos. Pythag. (1873); James Gow, Short History of Mathematics (1884).