PYTHAGOREANISM. 583 PYTHAGOKEANISM. ism. According to Zellei's exjiositiou "the Pytha- gorean system .started from the proposition that all is, in its essence, number. From this results the doctrine of the primitive opposites; and consequently, the opposition of the odd and the even, the limited and the unlimited precede all others. The unity likewise of these opposites was sought in number alone, which was therefore defined more particularly as harmonj-. Many of our authorities, however, represent the matter dilferently. They assert that the entire system was founded on the opposition of unity and dual- ity, which is then reduced to the opposition of spiritual and corporeal, of form andj substance, of the Deity and matter, and is itself derived from the Deity as the original Unity. According to another theory, the starting point of the sys- tem was not the arithmetical conce])tion of num- ber and its constituents, but the geometrical con- ception of the limits of space and of unlimited space." A fourth opinion "hascs the system not on the consideration of number, but on the dis- tinction of the limited and unlimited." This is not the place to canvass the arguments for and against each of these interpretations. The probability is that there was no one single consistent theory accepted by all Pythagoreans, but that each of these theories was held by some one or more of their number. The real question is not what Pythagoreans taught, but what was the earliest statement of philosophical problems given by accredited Pythagoreans. Even this question cannot be answered with assurance, as far as the fundamental principle of the system is concerned. Pythagoras himself probably gave no clear expression of philosophical opinion, be- cause he was not so much interested in philo- sophical theory as in religious and moral reform. See Ptthagosas. Philolaus, a contemporary of Socrates and De- niocritus, was probably the first distinctively philoso])hical Pythagorean, and although he com- mitted his views to writing, unfortunately we have onlj" fragments of his works, and even they are of doubtful authenticity. Xor is the witness of Plato and Aristotle to his teachings wholly unambiguous. Although the weight of Zeller's great name is given to the arithmetical inter- pretation of his views, it seems more satisfactory to regard Philolaus as having started from geo- metrical facts and the phenomena of sounds pre- sented by the strings of the heptachord. If this hypothesis be correct Philolaus held an atomistic view of the constitution of the world. The ulti- mate units of reality were considered to be per- ceptil)le spatial points of material character. Two such points made a line, three made a surface, and four made a solid. By number he did not mean an abstraction, but the concrete quantum of such points. "The Prthagorean Un- limited is, in fact, the res extensa; it is an early attempt to conceive Space in a realistic way and not merely as the place of body. Being an early attempt, it was not very successful; and. if the Pythagoreans did not make the I'nliniited a mere predicate of Air like Anaximenes. they fell into the opposite extreme of simply identifying it with Air and the Void. Tlie Limit must, of course, be strictly correlative with the Unlimited. It will then be a spatial limit, and not an iileal one. The theory that things are numbers, then, comes simply to this, that things are built up of geometrical figures, that they are portions of space limited in a variety of ways." (Burnet.) The smallest constituent parts of "the earth were considered cubical, those of lire tctrahedral, those of water icosahedral. while those of "the fifth element which embraces all the others" were dodecahedral. But the Pythagoreans went fur- ther and gave quantitative values to things im- material, which were thus construed in a ma- terial waj'. The soul was correlated in some way with the number six; reason, health, and light with seven; love, friendship, and prudence with eight. Such phantasy is the result of an attempt to reduce all reality to terms found sat- isfactory in explaining sensible reality. Along with this curious fiction went a mystical signifi- cance of numbers. The Pythagoreans were fond also of arranging things by opposites and finding ten such pairs. Thus one favorite classification gives us the following ten antitheses: (1) Lim- ited and unlimited; (2) odd and even; (3) one and many; (4) right and left: (5) male and female; (0) rest and motion; (7) straight and crooked; (8) light and darkness; (9) good and evil; (10) square and rectangle. The Pythagorean cosmology is interesting as it was a guess that came so near the truth con- cerning the solar system. Much of it was fanci- ful, but in spite of these vagaries we must recognize the fact that Pythagoreanism taught that the earth is a sphere revolving around a central fire, the centre of gravity of the universe, around which the stars likewise revolve, carried around by transparent shells. The central fire, however, was not the sun, but an invisible object, because toward it the farther side of the earth is always turned. The sun and stars shine by light rellected from this self-luminous centre. The lieavcn of the fixed stars, the sun. the moon, the five then known planets, and the earth made only nine objects ; hence to fill out the ])erfect number of ten, a counter-earth was invented. Solar eclipses were due to the intervention of the earth between the central fire and the sun : lunar eclipses to the intervention of some heavenly body, sometimes the counter-sun, between the central fire and the moon. "When the earth is on the same side of the central fire as the sun, we have day ; when it is on the other side, night." (Zeller. ) "The distance of the spheres from the central fire was determined according to simple numerical relationships. Corresponding to this, they assumed that from the revolution of the spheres there resulted a melodious musical sound, the so-called harmony of the spheres." ( Windelband. ) Following Pythagoras, the Pythagoreans accepted the iloctriiu; of metempsy- chosis, but the doctrine of the world soul, some- times ascribed to the Pythagoreans, was probably not a i)art of their system. The Pj-thagoreans laid much emphasis on music, as can be seen from their doctrine of the music of the spheres and from their insistence on the all-importance of harmony. But besides the discovery of the relation between the length of the strings of the lyre and the tones emitted, they did not contribute much to the theory of music. ^^■hile the Ionic school founded geometry, the main progress was due to the Pvthagorean school in Italy. The Prthagoreans were the first to give the rigorous proofs now demanded and to use mathematics in a specialized meaning. To P's-thagoras himself is due the first rigorous
