PYTHAGOKAS. 582 PYTHAGOREANISM. discourse on abstract mathematics and recondite philosophy. It was perhaps rather a religious cult in which primitive ideas of totemism and taboo were revived. We know that throughout Greece in his day there was a strong reaction against the newer national theology. The old clan and tribal deities had been long displaced by the Homeric gods, and yet these greater gods were not showing themselves able to save their devoted worshipers from the impending Persian peril. In the deep depression and gloom that settled over the Hellenic world in prospect of an Oriental invasion, it was small wonder that the older cults should be revived and eagerly cultivated in the hope that the local deities might be of service in preserving Greek civilization. This motive no doubt accounted for the rapid spread of the many Greek mysteries in the sixth century B.C. It seems reasonable, in the light of all we know, to suppose that the early Pythagorean brotherhood was one of these mystic circles, founded with a view of purifying its members from some imaginary guilt, and accomplishing this end by the observance of taboo. Among the akoiismata, or exoteric teachings of the later Py- thagoreans, we find such prohibitions as these : not to sit on a quart measure : not to step across the beam of a balance; not to eat beans or the heart of animals; not to stir fire with iron; not to look in a mirror be-side a light. Two very curious precepts enjoined the stirring of the ashes when a pot has been lifted from resting on them, so as to obliterate the marks it has made, and the smoothing of the bedclothes when one has risen from one's couch, so as to smooth out the impress of the body. All these punctilios point almost immistakably to primi- tive magic. As Burnet remarks, we find in such practices, so senseless to the outsider, an ex- planation of the popular outburst against the so- ciety. The domination of such a religious order ruling the State must have been galling enough. "The 'rule of the saints' would be nothing to it ; and we can still imagine and sympathize with the irritation felt by the plain man of those days at having all his legislation done for him by a set of incomprehensible pedants, who made a point of abstaining from beans, and would not let him beat his own dog, because they recognized in its howls the voice of a departed friend. This feeling would be ag- gravated by the private religious worship of the society. Greek democracies could never pardon the introduction of new gods. . . This intro- duced, as it were, an unknown and incalculable element into the arrangements of the State, which might very likely be hostile to the democracy, and was in any case a standing menace to the mass of citizens, who had no means of propitiat- ing the intruding divinity." But although the main motive of the brother- hood was thus superstitious, there is no doubt that a certain philosophic doctrine was taught to the brethren by its learned founder. Like all the early Greek philosophies, it was probably cos- niological; and it was likewise dualistic. "The two primary opposites, the Limited and the Un- limited, were brought together in a 'harmony' which could he numerically detei-rained." (Bur- net.) The Unlimited was space, the Limited were the definite forms in which space manifested itself. Space was not regarded as an abstract entity; it was ratlier a material sensible thing, probably identified with air. Hence the universe was said to breathe. The unlimited air is in its essence dark ; the principle of limitation is fire. the bright element which reveals definite spatial outlines. Such is the most plausible reconstruc- tion of early Pythagoreanism as taught by its founder. How much mathematics Pythagoras knew is likewise uncertain. To him without question is to be ascribed the first proof of the theorem known to the Egyptian Tope-stretchers' concern- ing the right-angled triangle (see HyPoTtxusE), which they knew in the case of the triangle with sides 3, 4, 5, without giving a rigorous proof. Of other matters, what is to be ascribed to Pythag- oras himself, and what to his pupils, it is dif- ficult to decide. Therefore we generally speak of a mathematical truth as being due to the Py- thagoreans, a treatment of whose discoveries, as far as known, is given in the article Pytha- GOBE.XISM. Consult: Zeller, Philosophic der Gricchen (5th ed., Leipzig, 1892 ; Eng. trans, of 4th ed.. London, 1881) ; Ritter and Preller. Ili.storiu Philosojjhia; Oracm (7th ed., Gotha, 1888) : Ueberweg. His- tory of Philosophy ( Eng. trans.. New York, 1887) ; Windelband, History of Ancient Philoso- phy (Eng. trans, by Cushman, New York, 1899) ; Burnet, Early Greek Philosophy (London. 1892) ; Schroeder, Pythagoras nnd die Inder (Leipzig, 1884) ; Cantor, Vorlesungen uber Geschichte der ilathematik (ib., 1900); Gow, Greek llathe- maties (Cambridge, 1884); Fink, History of Mathematics (Chicago, 1900) ; and for the later developments of PyJ;hagoreanism. see that article. PYTHAGORAS OF RHE'GITJM. A famous Greek sculptor of the first half of the fifth cen- tury -B.C. He is commonly called a Rhegian. but on "a pedestal at Olympia bearing his signature he calls himself a Samian, and it is probable that he was one of the emigrants from Asia Minor to Magna Gnecia about B.C. 496. He be- longs to the period of transition from the archaic art to the great masters of the time of Phidias. He was especially celebrated for his statues of athletes and, we are told, first introduced "sym- metry and rhythm' into his works. This seems to refer to the careful adjustment of the harmonv between the parts by a study of proportions and to the endeavor to secure graceful and fiowing lines, removed from the stiffness and schematic treatment of the archaic school. He also was said to have been the first to render the hair, veins, and muscles in a natural manner. His importance was evidently great in the develop- ment of Greek art. but none of his works can be ' identified with certainty, though some gems may preserve reminiscences of his statue of the lame man, probably Philoctetes. who made the spec- tators feel the pain of his wound. The Choiseul- Gouffier Apollo has been thought by Waldstein to be a copy of his Olympian statue of the pugil- ' ist Euthymus of Locris. but the attribution has | not been generally accepted. Consult the his- tories of Greek sculpture by Collignon and Gard- ner, and Waldstein. Essays on the Art of Phidias [ (London and Xew York. 188.5). , PYTHAGO'REANISM. The philosophical system ndvocateil hy the fdllowers of Pythagoras. Xo point in Greek philosophy is more disputed than the proper interpretation of Pythagorean-
