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but only of the Pythagoreans. However that may be, one thing is certain : the first impetus towards arithmetical speculation known under the name of Pythagorean philosophy was given by the great mathematician of Samos, and even though direct and positive proofs are wanting, nothing can hinder us from proclaiming him as the originator of the doctrines set forth in this section.

PYTHAGORAS, like Thales, of Ionian origin, was born at Samos during the first half of the sixth century. He was at first the pupil of the theologian Pherecydes and perhaps also of Anaximander, the physicist. According to a tradition which, it must be confessed, has nothing to warrant it among the ancients, he visited Phænicia, Egypt, and Babylon, where he was initiated into the Eastern theologica] speculations, and introduced to the study of geometry, which had already attained a high degree of perfection on its native soil. Returning to Greece about 520, he realized his ideals of religious, social, and philosophical reform at Crotona in Magna Græcia, by founding a kind of brotherhood, the members of which entertained the same opinions concerning morality, politics, and religion.

1 When we compare the doctrines, aims, and organization of this brotherhood, as portrayed by the Neo-Platonic historians (especially Jamblichus), with Buddhistic monachism, we are almost tempted (with. Alexander Polyhistor and Clement of Alexandria) to regard Pythagoras as the pupil of the Brahmans, nay, to identify him with Buddha himself. Indeed, not only do the names (Túdwv, Ilvdayópas = an inspired one, a soothsayer, and Buddha enlightened) bear such close resemblance to each other that even the most fastidious philologist can find no objection in translating Ilvdayópelos by “preacher of Buddhism,” but the Pythagorean and Buddhistic teachings are very much alike. Dualism, pessimism, metempsychosis, celibacy, a common life according to rigorous rules, frequent self-examinations, medita tions, devotions, prohibitions against bloody sacrifices and animal nourishment, kindliness towards all men, truthfulness, fidelity, justice, - all these elements are common to both. The fact that most ancient authors and above all Aristotle himself have comparatively little to say

Nothing certain is known of the end of the philosopher. His work prospered. The Pythagoreans were the possessors of all the sciences known in their time, - geometry, astronomy, music, and medicine,1 -- and consequently acquired an overpowering influence among the Doric people, who were less advanced than the Ionians. They preponderated at Crotona, at Tarentum, and in the Sicilian republics, until the middle of the fifth century, when the victorious democracy partly expelled them. The exiles repaired to Thebes or to Athens. Here their influence counteracted that of the Sophists, and brought about the spiritualistic reaction of Socrates and Plato against the materialism and scepticism which had, in the same epoch, been imported from Sicily, Thrace, and Ionia.

Ionian metaphysics springs from physics ; Pythagorean metaphysics is grafted on mathematics, and is consequently totally different from the former at the very outset. What interests the philosophers of Miletus is matter and its

concerning the person and life of Pythagoras, would tend to confirm the hypothesis of the identity of Pythagoreanism and Buddhism. However, the existence of Pythagoras, the mathematician, five centuries before the Christian era, is placed beyond doubt by the testimony of Heraclitus, Herodotus, etc. Furthermore, Buddhism in the form of Manichæism (that is to say, monachism) did not begin to spread westward before the third century of our era. We may perhaps explain everything satisfactorily by distinguishing between the Pythagoreanism of the Neo-Platonic historians and primitive and genuine Pythagoreanism. The biographers of Pythagoras were without exact and sufficient data regarding the life and work of the sage of Samos, and somewhat unscrupulous, besides, in the choice of their sources. They likewise allowed themselves to be misled by certain analogies; the essential features of their imaginary portrait are derived from Persian dualism and Hindoo pessimism.

1 These sciences, which constituted the subject-matter of Pythagorean instruction, were called padhuara, — the term from which the word mathematics is derived. The original meaning of the word embraces the totality of human knowledge.

perpetual movement; what impresses Pythagoras and the Pythagoreans is the. immaterial in matter, the order which prevails in the world, the unity, proportion, and harmony in its contrasts, the mathematical relations underlying all things. In geometry, in astronomy, and in music, everything is ultimately reduced to number. Hence number is the principle and innermost essence of the world; and things are sensible numbers. Every being represents a number, and the final goal of science is to find for each being the number for which it stands. The infinite series of numbers, and consequently of things, is derived from unity. As number is the essence of things, unity is the essence of number. Pythagoreanism distinguishes, two kinds of unities : (1) the Unity from which the series of numbers (beings) is derived, and which therefore contains and comprehens them all; the absolute and unopposed unity, the Monad of monails (ń uovás), the God of gods : and (2) the One, the first in the series of derived numbers which is opposed to the numbers tvu, three, and every plu rality (Tandos), and consequently limited by the two, the three, and the plurality; it is a relative unity, a created monad (év). The opposition between the one and the many is the source of all the rest. All the contrasts of nature, the dry and the moist, the warm and the cold, the clear and the obscure, the male and the female, the good and the evil, the finite (TTETepao uévov) and the infinite (άπειρον), are but varieties of the έν and tlie πλήθος, Or of the odd (Teputtóv) and the even (áptiov). Plurality as such is without consistency and may be divided into unities; the even number is reducible to the odd unit. The absolute unity is neither even nor odd; or rather, it is as yet both even and odd, singular and plural, God and the world. It is to Pythagoreanism what the ämrelpov is in the system of Anaximander: the neuter being that is superior and anterior to sexual contrasts, the absoluta indifference which precedes and creates the dualism of forces and elements. But the Pythagoreans guard against calling it ömrelpov, since the ameipov is, according to them, opposed to the prépalvov, as passivity to activity, or matter to the workman, or form, or plastic principle. Inasmuch as everything is, according to them, reduced to number, numerical relations, and ultimately to Idea, the matter and motion of the Ionians are, in their opinion, merely negative, the absence of ideal unity. Concerning the question of movement and origin, the conclusions of the Pythagoreans do not differ from the Eleatic doctrines. Movement and origin seem to be incompatible with their idealism. Although they have their own cosmogony, like the other schools of the period, they do not assume that the universe had a beginning in time, and consequently that there was a time when the universe did not exist. The world has existed εξ αιώνος και εις αιώνα, and the cosmogony simply aims to explain the order, law, or series, according to which things eternally emanate from their principle.

Pythagorean physics therefore accommodates itself to human sensualism, just like the physics of Parmenides. It makes what is in itself immutable, variable. It places itself on the sensualistic standpoint held by the novices among its followers (åkovo patikoi), and represents the eternal unity as a sphere (o toŮ Tavtòs opalpa), as a compact sphere, in which the parts are not distinguished (πλήρες, συνεχές), and which floats in the infinite (ärelpov). The ideal opposition between the even and the odd, the one and the many, becomes the real opposition of the full and the void. At the origin of things, the full was without the void, or, at least, the void was external to it. The formation of the cosmos begins by the void breaking in upon the full. This process is like a perpetual breath which agitates the world (πνοή, πνεύμα). The void penetrates the σφαίρα and establishes itself in it, thereby breaking it up into an infinite number of infinitesimal particles, reduced images of the opaîpa (the aroma of the atomists). Since, from the geometrical point of view, quality is reduced to quantity and form, these particles differ only in quantity and in figure. They form either cubes or pyramids (tetrahedrons) or octahedrons or icosahedrons or dodecahedrons. The unity reacts against this endless separation, and the particles are joined together again according to their geometric affinities and form elementary bodies: earth, fire, air, water, and ether. Fire is the element par excellence, being formed of tetrahedric particles. It is the symbol of the divine principle in nature and is concentrated into a central sun, the hearth of the universe and the abode of the Supreme God (fotla toll Tavtós), around which revolve (1) the Quranos, embracing the counter-earth (avtíxowv) and the earth; (2) the Cosmos proper, consisting of the moon, the sun (?) and the planets ; (3) the Olympus with the fixed stars. Pythagoras substitutes for the earth a central fire (which is invisible because the earth keeps facing it with the part that is opposite to the one we inhabit), and makes the earth revolve around this centre. But this does not mean, of course, that he advanced the heliocentric theory; he merely foreshadowed the system which his school formulated during the following centuries without succeeding in having it accepted by the majority of scientists. The distances separating the spheres are proportional to the numbers which express the relations that exist between tones and the respective lengths of vibrating strings; and the result of their revolutions around the axis of the world is a divine, harmony which the musical genius alone can perceive. This harmony is the soul of the universe. The different

. beings form an ascending scale according to the degree of perfection with which they reflect the universal harmony. The motion of the elementary being, the physical point, produces the line; the line moves and produces the plane

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