PYTHAGOREANISM, the philosophical doctrine of the Pythagoreans, or followers of Pythagoras (q.v.). The system of the Pytha goreans was comprehensive and included a theory of being; that is, a religious cult; •met aphysic.; a cosmological theory, and a mathe-. maticai theory.

Pythagorean Cult.— The lieved in tmmortality and the transmigration of souls. As they•consequently considered all ani-. mals to partake of human. nature, they forbade, the eating of nerd, and even that of beans, which they somehow associated with flesh. They formed a close corporation, and it was considered sinful to. reveal any imperfections in the mathematical work of the school to those. outside.

. Pythagorean. Metaphysics.— The Pytha. goreans taught that the 'essence' of ail Things was number; that everything in its Anal analysis could be resolved into number. This statement, Which is recorded in Aristotle's Metapbyines,,where he is enumerating the Greek. schools of philosophy,. has 'occasioned much dispute. We may perhaps see in this doctrine the basis of the 10 antitheses of Pythagorean teaching, especially 'that of the opposites, odd and even, the definite and indefinite, which are placed first in the list. Number was also an idea in which these opposites were • each in-. eluded, and was, therefore,' sometimes spoken• of as harmony. But other interpretations of the Pythagorean number make unity and duality as the root notion, and pronounce that these terms may be reduced to the opposition of the spiritual and corporeal, of form and of substance, of the Supreme Being and the material world. The Deity is the one, the Original Unity, the Infi nite, out of which all finite things have come. The opposition between the limited or finite and the unlimited or infinite is by some phi losophers regarded as the fundamental idea in the Pythagorean number. It is possible that the doctrine was from the first propounded as a vague generalization which might he and v.'.1-3 interpreted in different ways by different mem-, bets of the school. It is evident that nothing can exist without number, as is stated in the apocryphal Book of Wisdom, in some respects a product of Alexandrian Neo-Pythagoreanism. The numbers themselves are divisible into odd and even, thus • suggesting the contrast be tween the limited-and uniirmted, the conditioned and the unconditioned, the relative and the abso lute, matter and spirit, man and God. On the other hand, it is possible that the Pythagorean number was not arithmetical but geometrical. The great disciple of Pythagoras in the time of Socrates was Philolaus, but of . the writings in which he expounded his vidwr only :fragments survive, and these are of doubtful authenticity. Philolaus may have been under the influence of Democritus, and his theory of number have been based on geometrical axioms and the in tervals in the sounds struck from the .seven stringed lyre. Probably he:. wag an? atornist. The individual atom would in that. case repre sent to' him a material 'spatial -point, two • of which made a line; three a surface, four a solid of these solids,. represented always. by even numbers, the constitutents of earth Were cubical; those of fire tetrahedral, those of water .icesaa hedrai, etc. From the use,of. -munbert as. the

ontological basis of things the passage was easy enough to the. .wild and fanciful application of them as mere symbols. Than the later Pytha goreians made the soul correspond with number six, while seven was the counterpart of reason and health. The imagination, here stepped in • and with curious _ingenuity labored• to give a 'rational bags . to -these atkionis. Hence the famous oppositions of this :philosophical sect; namely (1) limited 'and unlimited; • (2) even and odd; (3) one and Many; (4) right and left; (5) male and female; (6) rest and motion; (7) straight and crooked; (8) light and darkness; (9) good and evil; (10) square and rectangle. It will be perceived that in these oppositions the idea of completeness, as repre sented by an even number, and incompleteness by an odd number is the ruling element, and in the idea of incompleteness is implied the poten tiality of indefinite extension, multiplication or variation. From the very beginning of Pytha goreanism many semi-mystical, semi-scientific speculations were made concerning square numbers; triangular numbers, of the form Xi +3X 2 , representing triangular regularly spaced aggregates of points ; and rectangular numbers of the form X' -I- X, representing rec tangular aggregates of points.

Pythagorean Pythagoras, or at least the Pythagoreans, had some vague idea of a heliocentric solar system. They taught, that the universe had as its centre a fire round which the earth and stars revolved. This cen tral fire was not identified with the sun; the stars were luminous from reflecting its light. They taught that this fire was not visible from the earth; that there was a counter earth which made up with the five known planets, the fixed stars, sun and moon, 10 celestial phenomena. The distance of the spheres from the central fire was determined according to simple nu merical relationships., The harmony of the. spheres was a melodious sound resulting from the revolution of the heavenly bodies in ac cordance with the intervals of their distance from the central fire. The Pythagoreans dis covered the connection between the length of the string in a lyre and the character of the note which was sounded on percussion, and developed a theory of the musical scale. In deed this seems to have been the source of their medical and mathematical researches.

Pythagogpan Geometry.— The Ionic school of Greek philosophy imported geometry from Egypt into the Greek world of intellectual ac tivity. Its early development in Europe was mainly due to the • followers of Pythagoras, who himself enunciated the theorem now known as the 47th proposition of the first book of Euclid, which says that the square of the hypothenuse of a right angled triangle is the sum of the squares on its legs. The three proposi tions, arithmetical, geometrical and harmoni cal, were known to them, having been intro ducat into Grdece by Pytbagorast whoi them from the inventors, the •abylonians. (See Nilo-Pull ',Rona PYTHAGORAS). Consult Burnet, (Greelc phileisophyi (Pant I, London 1914) ; Gomperz, Thinkers> (tr. New York 1901) ; Zeller, 'Die Philosophic der Griechen in ihrer geschichtlichen Entwicklun gen' (5th ed., Leipzig 1892; tr. London 1881).