PYTHAGORAS 139 number, the product of equals. More legitimate is their application of number to geometry, according to which "one" was identified with the point, "two" with the line, "three" with the surface, and "four" with the cube. In the history of music the Pytha- gorean school is also of considerable importance from the develop- ment which the theory of the octave owes to its members ; according to some accounts the discovery of the harmonic system is due to Pythagoras himself. As already mentioned, the movements of the heavenly bodies formed for the Pythagoreans an illustration on a grand scale of the truth of their theory. Their cosmological system is also interest- ing on account of peculiarities which mark it out from the current conceptions of antiquity and bring it curiously near to the modern theory. Conceiving the universe, like many early thinkers, as a sphere, they placed in the heart of it the central fire, to which they gave the name of Hestia, the hearth or altar of the universe, the citadel or throne of Zeus. Around this move the ten heavenly bodies farthest off the heaven of the fixed stars, then the five planets known to antiquity, then the sun, the moon, the earth, and lastly the counter-earth (avrixOui 1 ), which revolves between the earth and the central fire and thus completes the sacred decade. Revolving along with the earth, the last-mentioned body is always interposed as a shield between us and the direct rays of the central fire. Our light and heat come to us indirectly by way of reflexion from the sun. 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. This attribution of the changes of day and night to the earth's own motion led up directly to the true theory, as soon as the machinery of the central fire and the counter-earth was dispensed with. The counter-earth became the western hemisphere, and the earth re- volved on its own axis instead of round an imaginary centre. But, as appears from the above, the Pythagorean astronomy is also remarkable as having attributed a planetary motion to the earth instead of making our globe the centre of the universe. Long after- wards, when the church condemned the theory of Copernicus, the indictment that lay against it was its heathen and " Pythagorean " character. The doctrine which the memory of mankind associates most closely with Pythagoras's name is that of the transmigration of souls METEMPSYCHOSIS (q.v.). Though evidently of great import- ance for Pythagoras himself, it does not stand in any very obvious connexion with his philosophy proper. He seems to have adopted the idea from the Orphic Mysteries. The bodily life of the soul, according to this doctrine, is an imprisonment suffered for sins committed in a former state of existence. At death the soul reaps what it has sown in the present life. The reward of the best is to enter the cosmos, or the higher and purer regions of the universe, while the direst crimes receive their punishment in Tartarus. But the general lot is to live afresh in a series of human or animal forms, the nature of the bodily prison being determined in each case by the deeds done in the life just ended. This is the same doctrine of retribution and purificatory wandering which meets us in Plato's mythical descriptions of a future life. They are borrowed by him in their substance from the Pythagoreans or from a common source in the Mysteries. In accordance with this religious view of life as a stage of probation were the ethical precepts of the school, inculcat- ing reverence towards the gods and to parents, justice, gentleness, temperance, purity of life, prayer, regular self-examination, and the observance of various ritual requirements. Connecting its ethics in this way with religion and the idea of a future life, the Pythagorean societies had in them from the be- ginning a germ of asceticism and contemplative mysticism which it was left for a later age fully to develop. The Pythagorean life was destined to survive the peculiar doctrines of the Pythagorean philosophy and to be grafted on later philosophic ideas. The asceticism which characterized it appears in the 4th century B.C. in close connexion with the Orphic Mysteries ; and the " Pytha- goreans " of that time are frequently the butts of the New Athenian Comedy. In the Alexandrian period the Pythagorean tradition struck deeper roots ; in Alexandria and elsewhere schools of men arose calling themselves Pythagoreans, but more accurately dis- tinguished by modern criticism as Neopythagoreans, seeing that their philosophical doctrines are evidently derived in varying pro- portions from Plato, Aristotle, and the Stoics. In general it may be said that they develop the mystic side of the Platonic doctrine ; and only so far as this is connected with the similar speculations of Pythagoras can they claim to be followers of the latter. Hence men like Plutarch, who personally prefer to call themselves Platonists, may also be considered as within the scope of this Pythagorean revival. The link that really connects these Neo- pythogoreans with the Samian philosopher and distinguishes them from the other schools of their time is their ascetic ideal of life and their preoccupation with religion. In religious speculation they paved the way for the Neoplatonic conception of God as immeasur- ably transcending the world ; and in their thirst for prophecies, oracles, and signs they gave expression to the prevalent longing for a supernatural revelation of the divine nature and will. The asceticism of the Jewish sect of the Essenes seems, as Zeller con- tends, to be due to a strong infusion of Neopythagorean elements. At a still later period Neopythagoreanism set up Pythagoras and Apollonius of Tyana not only as ideals of the philosophic life but also as prophets and wonder-workers in immediate communication with another world, and in the details of their "lives" it is easy to read the desire to emulate the narrative of the Gospels. The Life of Apollonius by Philostratus, which is for the most part an historical romance, belongs to the 3d Christian century. Zeller's discussion of Pythagoreanism, in his Philosophic d. Griechen, book i. , is very full ; he also deals at considerable length in the last volume of the work with the Neopythagoreans, considered as the precursors of Neoplatonism ami the probable origin of the Essenes. The numerous monographs dealing with special parts of the subject are there examined and sifted. (A. SE.) Pythagorean Geometry. As the introduction of geometry into Greece is by common consent attributed to Thales, so all are agreed that to Pythagoras is due the honour of having raised mathe- matics to the rank of a science. We know that the early Pythagoreans published nothing, and that, moreover, they referred all their discoveries back to their master. (See PHILOLAUS.) Hence it is not possible to separate his work from that of his early disciples, and we must therefore treat the geometry of the early Pythagorean school as a whole. We know that Pythagoras made numbers the basis of his philosophical system, as well physical as metaphysical, and that he united the study of geometry with that of arithmetic. The following statements have been handed down to us. (a) Aristotle (Met., i. 5, 985) says "the Pythagoreans first applied themselves to mathematics, a science which they improved ; and, penetrated with it, they fancied that the principles of mathematics were the principles of all things." (b) Eudemus informs us that "Pythagoras changed geometry into the form of a liberal science, regarding its principles in a purely abstract manner, and investigated its theorems from the immaterial and intellectual point of view (avAws KCU vocpws)." l (c) Diogenes Laertius (viii. 11) relates that "it was Pythagoras who carried geometry to perfection, after Mreris' 2 had first found out the prin- ciples of the elements of that science, as Anticlides tells us in the second book of his History of Alexander and the part of the science to which Pythagoras applied himself above all others was arithmetic." (d) According to Aris- toxenus, the musician, Pythagoras seems to have esteemed arithmetic above everything, and to have advanced it by diverting it from the service of commerce and by likening all things to numbers. 3 (e) Diogenes Laertius (viii. 13) reports on the same authority that Pythagoras was the first person who introduced measures and weights among the Greeks. (/) He discovered the numerical relations of the musical scale (Diog. Laert., viii. 11). (g) Proclus 4 says that "the word 'mathematics' originated with the Pythagoreans." (h) We learn also from the same author- ity 5 that the Pythagoreans made a fourfold division of mathematical science, attributing one of its parts to the " how many" (TO TTOO-OV) and the other to the "how much" (TO Tn/Aucov) ; and they assigned to each of these parts a twofold division. They said that discrete quantity or the "how many " is either absolute or relative, and that con- tinued quantity or the "how much" is either stable or in motion. Hence they laid down that arithmetic contem- plates that discrete quantity which subsists by itself, but music that which is related to another; and that geometry considers continued quantity so far as it is immovable, 1 Proclus Diadochus, In primum Euclidis Elementorum librum Commentarii, ed. Friedlein, p. 65. 2 Mosris was a king of Egypt who, Herodotus tells us, lived 900 years before his visit to that country. 3 Aristox., Fragm., ap. Stob., Eclog. Phys., i. 2, 6. 4 Prod., op. cit., p. 45. 5 Op. cit., p. 35.
