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 ukousmata, o• 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 beside a light. Two very eorious 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 unmistakably 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 he 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 religions 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 he 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 mological; 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 determined." (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 rather 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 'rope-stretchers' concern ing the right-angled triangle (see HYPOTENUSE), 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 he 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 thago•eans, a treatment of whose discoveries, as far as known, is given in the article PYTBA COREANTSM.
Consult: Zeller, Philosophic der Gricchcn (5th ed., Leipzig, 1892; Eng. trans. of 4th ed., London, 1881) ; Hitter and Preller, Historia Philosophice Grcrecr (7th ed., Gotha, 1888) ; Feberweg, 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 and die hider (Leipzig, 1884) ; Cantor, Vorlesungcn iiber Gcschichte der Mathcmatik (ib., 1900) ; Gov, Greek Mathe matics (Cambridge, 1884) ; Fink, History of Mathematics (Chicago, 1900) ; and for the later developments of Pythago•eanism, see that article.