PYTHAGORAS unity. And in this aspect of the matter Aristotle speaks of unity as the principium and essence and element of all things {Met. xii. 6, i. 6, p. 987, b. 2'2) ; the diviTie imity being the first principle and cause, and owe, as the first of the limiting numbers and the element of all, being the basis of positive existence, and when itself become possessed of extension (Met. xii. 3, p. 1091, a. 15) the element of all that possesses extension (comp. Brandis, l. e. p. 511, &c.). In its development, however, the Pythagorean system seems to have taken a twofold direction, one school of Pythagoreans regarding numbers as the inherent, fundamental elements of things ( Arist. de Caelo, iii. 1 ) ; another section, of which Hippasus seems to have been the head, regarding numbers as the patterns merely, but not as entering into the essence of things (Arist. Met. i. 6. Though Aristotle speaks of the Pythago- reans generally here, there can be no doubt that the assertion, in which the Greek commentators found a difficulty, should be restricted to a section of the Pythagoreans. Comp. Iambi, in Nicom. Arithm. p. 11 ; Syrian, in Arist. Met. xii. p. 1080, b. 18 ; Simplic. in Phys. f. 104, b. ; Iambi. Pyth. 81 ; Stob. Ed. Phys. p. 302 ; Brandis, I. c. p. 444). As in the octave and its different harmonical relations, the Pythagoreans found the ground of connection between the opposed primary elements, and the mutual relations of existing things, so in the properties of particular numbers, and their relation to the principia, did they attempt to find the explanation of the particular properties of dif- ferent things, and therefore addressed themselves to the investigation of the properties of numbers, dividing them into various species. Thus they had three kinds of even, according as the number Avas a power of two (dpTiaKis apriov), or a multi- ple of two, or of some power of two, not itself a power of two {Trepiaadpriov), or the sura of an odd and an even number (dprioirepiTTov — a word which seems to have been used in more than one sense. Kicom. Arithm. i. 7, 8). In like manner they had three kinds of odd. It was probably the use of the decimal system of notation which led to the number ten being supposed to be possessed of extraordinary powers. '* One must contemplate the works and essential nature of inimber accord- ing to the power which is in the number ten ; for it is great, and perfect, and all-working, and the first principle (opx<*) ^"^^ guide of divine and heavenly and human life." (Philolaus ap. Stob. Eel. Phys. p. 8 ; Biickh, p. 139.) This, doubtless, had to do with the formation of the list of ieji pairs of opposite principles, which was drawn out by some Pythagoreans (Arist. Met. i. 5). In like manner the tetractys (possibly the sura of the first four numbers, or 10) was described as containing the source and root of ever-flowing nature (Carm. Aur. 1. 48). The number three was spoken of as de- fining or limiting the universe and all things, having end, middle, and beginning, and so being the number of the whole (Arist. de Caelo, i. 1). This part of their system they seem to have helped out by considerations as to the connection of numbers with lines, surfaces, and solids, especially the regu- l.ar geometrical figures (T/icolog. Arithm. 10, p. 61, &c.), and to have connected the relations of things with various geometrical relations, among which angles played an important part. Thus, according "to Philolaus, the angle of a triangle was conse- PYTHAGORAS. e2$ crated to four deities, Kronos, Hades, I'an, and Dionysus ; the angle of a square to Rhea, Demeter, and Hestia ; the angle of a dodecagon to Zeus ; apparently to shadow forth the sphere of their operations ( Procl. in Euclid. Elem. i. p. 36 ; Bcickh, I. c. p. 152, &c.). As we learn that he connected solid extension with the number fou?' (T/ieol. Arithm. p. 56), it is not unhkely that, as others did CNicom. Arithm. ii. 6), he connected the number one with a point, two with a line, three with a surface (xpoid). To the number five he appropriated quality and colour ; to six life ; to seven intelligence, health, and light ; to eight love, friendship, understanding, insight (T/ieol. Arithm.. I. c). Others connected marriage, justice, &c. with different numbers (Alex, in Arist. Met. i. 5, 13). Guided by similar fanciful analogies they assumed the existence of five elements, connected with geometrical figures, the cube being earth ; the pyramid, fire ; the octaedron, air ; the eikosaedron, water ; the dodecaedron, the fifth element, to which Philolaus gives the curious appellation of a rds (Tcpalpas 6Kd.s (Stob. I.e. i. p- 10; Bockh, I.e. p. 161 ; comp. Plut. de Plac. Phil. ii. 6). In the Pythagorean system the element Jire was the most dignified and important. It accordingly occupied the most honourable position in tlie uni- verse — the extreme (ircpas), rather than interme- diate positions ; and by extreme they understood both the centre and the remotest region (to 5' eaxo-Tov KoX TO ulaov irepay, Arist. de Caelo^ ii. 13). The central fire Philolaus terms the hearth of the universe, the house or watch-tower of Zeus, the mother of the gods, the altar and bond and measure of nature (Stob. l. c. p. 488 ; Bockh, /. c. p. 94, &c.). It was the enlivening principle of the universe. By this fire they probably understood something purer and more ethereal than the com- mon element fire (Brandis, ^. c. p. 491). Round this central fire the heavenly bodies performed their circling dance (xopeveiv is the expression of Philolaus) ; — farthest off, the sphere of the fixed stars ; then, in order, the five planets, the sun, the moon, the earth and the counter-earth (dvTix^wu) — a sort of other half of the earth, a distinct body from it, but always moving parallel to it, which they seem to have introduced merely to make up the number ten. The most distant region, which was at the same time the purest, was termed Olympus (Brandis, /.c. p. 476). The space be- tween the heaven of the fixed stars and the moon was termed k6oixos ; the space between the moon and the earth ovpav6s (Stob. 1. c). Philolaus as- sumed a daily revolution of the earth round the central fire, but not round its own axis. The revo- lution of the earth round its axis was taught (after Hicetas of Syracuse ; see Cic. Acad. iv. 39) by the Pythagorean Ecphantus and Heracleides Ponticus (Plut. Plac. iii. 1 3 ; Procl. in Tim. p. 281 ) : a combined motion round the central fire Jind round its own axis, by Aristarchus of Samos (Plut. de Fac. Lun. p. 933). The infinite {dneipou) beyond the mundane sphere was, at least according to Archytas (Simpl. in Phys. f. 108), not void space, but corporeal. The physical existence of the uni- verse, which in the view of the Pythagoreans was a huge sphere (Stob. I.e. p. 452, 468), was represented as a sort of vital process, time, space, and breath (ttj/ot^) being, as it were, inhaled out of the direipou leTTStadyeadai S' e/c tou direipov XP^^*^^ "^^ f<^ iryoiiu koI rd kzvov^ Stob. I, c. p. 380 ; see esfe-
