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PYTHAGORAS (6th century B.C.), Greek philosopher, was, in all probability, a native of Samos or one of the neighbouring islands (others say a Tyrrhenian, a Syrian or a Tyrian), and the first part of his life may therefore be said to belong to that Ionian seaboard which had already witnessed the first development of philosophic thought in Greece (see IONIAN SCHOOL). The exact year of his birth has been variously placed between 586 and 569 B.C., but 582 may be taken as the most probable date. He was a pupil of Pherecydes (q.v.), and later of Hermodamas (Diog. Laert. viii. 2). He left in Ionia the reputation of a learned and universally informed man. " Of all men Pythagoras, the son of Mnesarchus, was the most assiduous inquirer," says Heracleitus, and then proceeds in his contemptuous fashion to brand his predecessor's wisdom as only eclectically compiled information or polymathy (iroKvuadia.) . This accumulated wisdom, as well as most of the tenets of the Pythagorean school, was attributed in antiquity to the extensive travels of Pythagoras, which brought him in contact (so it was said) not only with the Egyptians, the Phoenicians, the Chaldaeans, the Jews and the Arabians, but also with the Druids of Gaul, the Persian Magi and the Brahmans. But these tales represent only the tendency of a later age to connect the beginnings of Greek speculation with the hoary religions and priesthoods of the East. There is no intrinsic improbability, however, in the statement of Isocrates (Laud. Busir. 28, p. 227 Steph.) that Pythagoras visited Egypt and other countries of the Mediterranean, for travel was one of the few ways of gathering knowledge. Some of the accounts (e.g. Callimachus) represent Pythagoras as deriving much of his mathematical knowledge from Egyptian sources, but, however it may have been with the practical beginnings of geometrical knowledge, the scientific development of mathematical principles can be shown to be an independent product of Greek genius. Some of the rules of the Pythagorean ritual have their Egyptian parallels, as Herodotus points out, but it does not necessarily follow that they were borrowed from that quarter, and he is certainly wrong in tracing the doctrine of metempsychosis (q.v.) to Egypt.

The historically important part of his career begins with his migration to Crotona, one of the Dorian colonies in the south of Italy, about the year 529. According to tradition, he was driven from Samos by the tyranny of Polycrates. At Crotona Pythagoras speedily became the centre of a widespread and influential organization, which seems to have resembled a religious brotherhood or an association for the moral reformation of society much more than a philosophic school. Pythagoras appears, indeed, in all the accounts more as a moral reformer than as a speculative thinker or scientific teacher; and the doctrine of the school which is most clearly traceable to Pythagoras himself in the ethico-mystical doctrine of transmigration. The Pythagorean brotherhood had its rise in the wave of religious revival which swept over Hellas in the 6th century B.C., and it had much in common with the Orphic communities which sought by rites and abstinences to purify the believer's soul and enable it to escape from " the wheel of birth." Its aims were undoubtedly those of a religious order rather than a political league. But a private religious organization of this description had no place in the traditions of Greek life, and could only maintain itself by establishing " the rule of the saints " on a political basis. The Pythagoreans appear to have established their supremacy for a time over a considerable part of Magna Graecia, but this entanglement with politics led in the end to the dismemberment and suppression of the society. The authorities differ hopelessly in chronology, but according to the balance of evidence the first reaction against the Pythagoreans took place in the lifetime of Pythagoras after the victory gained by Crotona over Sybaris in 510. Dissensions seem to have arisen about the allotment of the conquered territory, and an adverse party was formed in Crotona under the leadership of Cylon. This was probably the cause of Pythagoras's withdrawal to Metapontum, which an almost unanimous tradition assigns as the place of his death in the end of the 6th or the beginning of the 5th century. The order appears to have continued powerful in Magna Graecia till the middle of the 5th century, when it was violently trampled out. The meeting-houses of the Pythagoreans were everywhere sacked and burned; mention is made in particular of "the house of Milo" in Crotona, where fifty or sixty leading Pythagoreans were surprised and slain.

The persecution to which the brotherhood was subjected throughout Magna Graecia was the immediate cause of the spread of the Pythagorean philosophy in Greece proper. Philolaus, who resided at Thebes in the end of the 5th century (cf. Plato, Phaedo, 61 D), was the author of the first written exposition of the system. Lysis, the instructor of Epaminondas, was another of these refugees. This Theban Pythagoreanism had an important influence upon Plato's thought, and Philolaus had also disciples in the stricter sense. But as a philosophic school Pythagoreanism became extinct in Greece about the middle of the 4th century. In Italy where, after a temporary suppression, it attained a new importance in the person of Archytas of Tarentum the school finally disappeared about the same time.

Aristotle in his accounts of Pythagorean doctrines never refers to Pythagoras but always with a studied vagueness to " the Pythagoreans " (oJ na.\obiu>voi UvOa.y6ptu>t). Nevertheless, certain doctrines may be traced to the founder's teaching. Foremost among these is the theory of the immortality and transmigration of the soul (see METEMPSYCHOSIS). Pythagoras's teaching on this point is connected by one of the most trustworthy authorities with the doctrine of the kinship of all living beings; and in the light of anthropological research it is easy to recognize the close relationship of the two beliefs. The Pythagorean rule of abstinence from flesh is thus, in its origin, a taboo resting upon the blood-brotherhood of men and beasts; and the same line cf thought shows a number of the Pythagorean rules of life which we find embedded in the different traditions to be genuine taboos belonging to a similar level of primitive thought. The moral and religious application which Pythagoras gave to the doctrine of transmigration continued to be the teaching of the school. The view of the body (o&pa) as the tomb (OTJIM) of the soul, and the account of philosophy in the Phaedo as a meditation of death, are expressly connected by Plato with the teaching of Philolaus; and the strain of asceticism and other worldliness which meets us here and elsewhere in Plato is usually traced to Pythagorean influence. Plato's mythical descriptions of a future life of retribution and purificatory wandering can also be shown to reproduce Pythagorean teaching, though the substance of them may have been drawn from a common source in the Mysteries.

The scientific doctrines of the Pythagorean school have no apparent connexion with the religious mysticism of the society or their rules of living. They have their origin in the same disinterested desire of Knowledge which gave rise to the other philosophical schools of Greece, and the idea of " philosophy " or the " theoretic life " as a method of emancipation from the evils of man's present state of existence, though a genuine Pythagorean conception, is clearly an afterthought. The discourses and speculations of the Pythagoreans all connect themselves with the idea of number, and the school holds an important place in the history of mathematical and astronomical science. An unimpeached tradition carries back the Pythagorean theory of numbers to the teaching of the founder himself. Working on hints contained in the oldest traditions, recent investigators have shown that the _ discpveries attributed to Pythagoras connect themselves with a primitive numerical symbolism, according to which numbers were represented by dots arranged in symmetrical patterns, such as are still to be seen in the marking of dice or dominoes. Each pattern of units becomes on this plan a fresh unit. The " holy tetractys," by which the later Pythagoreans used to swear, was a figure of this kind representing the number 10 as the triangle of 4, and showing at a glance that 1+2+3+4 = 10. The sums of the series of any successive numbers may be graphically represented in a similar way, and are hence spoken of as " triangular numbers," while the sums of the series of successive odd numbers are called " square numbers," and those of successive even numbers "oblong numbers"; thus 3 and 5 added to the unit give a figure of this description -I- 1 while 4 and 6, added to 2, are thus represented -I ! Such a method of representing number in areas leads naturally to problems of a geometrical nature, and as the practical use of the right-angled triangle was already familiar in the arts and crafts, there is no reason to dispute the well-established tradition which assigns to Pythagoras the discovery of the proposition that in such a triangle the square on the hypotenuse is equal to the sum of the squares on the other two sides. And it is probably also correct to attribute to him the discovery of the harmonic intervals which underlie the production of musical sounds. Impressed by this reduction of musical sounds to numbers and by the presence of numerical relations in every department of phenomena, Pythagoras and his early followers enunciated the doctrine that " all things are numbers." Numbers seemed to them, as Aristotle put it, to be the first things in the whole of nature, and they supposed the elements of numbers to be the elements of all things, and the whole heaven to be a musical scale and a number (Mela. A. 9863). Numbers, in other words, were conceived at that early stage of thought not as relations or qualities predicable of things, but as themselves constituting the substance or essence of the phenomena the rational reality to which the appearances of sense are reducible.

But the development of these ideas into a comprehensive metaphysical system was no doubt the work of Philolaus in the latter part of the 5th century. His formulation of the theory implies a knowledge of the teaching of Parmenides and Empedocles, and had itself in turn a great influence upon Plato. The " elements of numbers," of which Aristotle speaks in the passage quoted above, were, according to the Pythagoreans, the Odd and the Even, which they identified with the Limit and the Unlimited; and Aristotle distinctly says that they did not treat these as " priorities of certain other substances" such as fire, water or anything else of that sort, but that the unlimited itself and the one were the reality of the things of which they were predicated, and that is why they said that number was " the reality of everything " (Meta. A. 587). Numbers, therefore, are spatially conceived, " one " being identified with the point in the sense of a unit having position and magnitude. From combinations of such units the higher numbers and geometrical figures arise " two " being identified with the line, " three " with the surface, and " four with the solid and the Pythagoreans proceeded to explain the elements of Empedocles as built up out of geometrical figures in the manner followed by Plato in the Timaeus. The identification of the numerical opposites, the Odd and the Even, with. the Limit and the Unlimited otherwise difficult to explain may perhaps be understood, as Burnet suggests, by reference to the arrangement of the units or "terms" (6pm.) in patterns. " When the odd is divided into_ two equal parts," he quotes from Stobaeus, "a unit is left over in the middle; but when the even is so divided, an empty field is left over, without a master and without a number, showing that it is defective and incomplete." The idea of opposites, derived, perhaps, originally from Heracleitus, was developed by the Pythagoreans in a list of ten fundamental oppositions, bearing a certain resemblance to the tables of categories framed by later philosophers, but in its arbitrary mingling of mathematical, physical and ethical contrasts characteristic of the uncritical beginnings of speculative thought: (i) limited and unlimited, (2) odd and even, (3)one and many,(4)right and left, (5) male and female, (6) rest and motion, (7) straight and curved, (8) light and darkness, (9) good and evil, (lo) square and oblong. To the Pythagoreans, as to Heracleitus, the universe was in a sense the realized union of these opposites, but interpretations of Pythagoreanism which represent the whole system as founded on the opposition of unity and duality, and proceed to identify this with the opposition of form and matter, of _ divine activity and passive material, betray on the surface their post-Platonic origin. Still more is this the case when in Neoplatonic fashion they go on to derive this original opposition from the supreme unity or God. The further speculations of the Pythagoreans on the subject of number rest mainly on analogies, which often become capricious and tend to lose themselves at last in a barren symbolism. "Seven" is called Topftvot and 'A0i>n), because within the decade it has neither factors nor product. " Five," on the other hand, signifies marriage, because it is the union of the first masculine with the first feminine number (3+2, unity being considered as a number apart). The thought already becomes more fanciful when " one is identified with reason, ^because it is unchangeable ; " two " with opinion, because it is unlimited and indeterminate ; " four " with justice, because it is the first square number, the product of equals.

The astronomy of the Pythagoreans was their most notable contribution to scientific thought, and its importance- lies in the fact that they were the first to conceive the earth as a globe, self-supported in empty space, revolving with the other planets round a central luminary. They thus anticipated the heliocentric theory, and Copernicus has left it on record that the Pythagorean doctrine of the planetary movement of the earth gave him the first hint of its true hypothesis. The Pythagoreans did not, however, put the Sun in the centre of the system. That place was filled by the central fire to which they gave the names of Hestia, the hearth of the universe, the watch-tower of Zeus, and other mythological expressions. It had then been recently discovered that the Moon shone by reflected light, and the Pythagoreans (adapting a theory of Empedocles), explained the light of the Sun also as due to reflection from the central fire. Round this fire revolve ten bodies, first the Antichthon or counter-earth, then the earth, followed in order by the Moon, the Sun, the five then known planets and the heaven of the fixed stars. The central fire and the counter-earth are invisible to us because the side of the earth on which we live is always turned away from them, and our light and heat come to us, as already stated, by reflection from the Sun. When the earth is on the same side of the central fire as the Sun, the side of the earth on which we live is turned towards the Sun and we have day; when the earth and the Sun are on opposite sides of the central fire we are turned away from the Sun and it is night. The distance of the revolving orbs from the central fire was determined according to simple numerical relations, and the Pythagoreans combined their astronomical and their musical discoveries in the famous doctrine of " the harmony of the spheres." The velocities of the bodies depend upon their distances from the centre, the slower and nearer bodies giving out a deep note and the swifter a high note, the concert of the whole yielding the cosmic octave. The reason why we do not hear this music is that we are like men in a smith's forge, who cease to be aware of a sound which they constantly hear and are never in a position to contrast with silence.

AUTHORITIES. Zeller's account of Pythagoreanism in his Philosophic der Griechen gives a full account of the sources, with critical references in the notes to the numerous monographs on the subject, but the labour and ingenuity of more recent scholars has succeeded in clearing up a number of points since he wrote. Diels, Doxographi graeci (1879), and Die Fragmente der Vorsokraliker, vol. i. (2nd ed., 1906). Gomperz, Greek Thinkers, vol. i., and especially Burnet's Early Greek Philosophy (2nd ed., 1908), give the results of the latest investigations. Tannery's Science hellbne; Milhaud's La Science grecque and Philosophes geontetres; Cantor's History of Mathematics; and Gow's Short History of Greek Mathematics, refer to the mathematical and physical doctrines of the school.

(A. S. P.-P.)

Note - this article incorporates content from Encyclopaedia Britannica, Eleventh Edition, (1910-1911)

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