A Dictionary of Greek and Roman biography and mythology. By various writers. Ed. by William Smith. Illustrated by numerous engravings on wood.

PYTHAGORAS PYTHAGORAS. 623
tnity. And: in this aspect of the matter Aristotle crated to four deities, Kronos, Hades, Pan, and
speaks of unity as the principium and essence and Dionysus; the angle of a square to Rhea, Demeter,
element of all things (Met. xii. 6, i. 6, p. 987, b. and Hestia; the angle of a dodecagon to Zeus;
202); the divine unity being the first principle and apparently to shadow forth the sphere of their
cause, and one, as the first of the limiting numbers operations (Procl. in.Euclid. Elem. i. p. 36;
and the element of all, being the basis of positive Bdckh, 1. c. p. 152, &c.). As we learn that he
existence, and when itself become possessed of connected solid extension with the number four
extension (.Met. xii. 3, p. 1091, a. 15) the element (Theol. Arithnz. p. 56), it is not unlikely that, as
of all that possesses extension (comp. Brandis,. c. others did (Nicom. Arithin. ii. 6), he connected the
p. 511, &c.). In its development, however, the number one with a point, two with a line, three
Pythagorean system seemsto have taken a twofold with a surface (Xpoia). To the number five he
direction, one school of Pythagoreans regarding appropriated quality and colour; to six life; to
numbers as the inherent, fundamental elements of seven intelligence, health, and light; to eight love,
things (Arist. de Caelo, iii. 1); another section, friendship, understanding, insight (Theol. Ariftm.
of which Hippasus seems to have been the head, 1. c.). Others connected marriage, justice, &c. with,
regarding numbers as the patterns merely, but not different numbers (Alex. in Arist. Met. i. 5, 13).
as entering into the essence of things (Arist. Met. Guided by similar fanciful analogies they assumed
i. 6. Though Aristotle speaks of the Pythago- the existence of five elements, connected with
reans generally here, there can be no doubt that geometrical figures, the cube being earth; the
the assertion, in which the Greek commentators pyramid, fire; the octaedron, air; the eikosaedron,
found a difficulty, should be restricted to a section water; the dodecaedron, the fifth element, to
of the Pythagoreans. Comp. Iambl. in Nicom. which Philolaus gives the curious appellation of a
Arithm2. p. 11; Syrian. in Arist. Met. xii. p. 1080, Tas rpapapao dAKaCs (Stob. I.c. i. p. 10; Bickh,
b. 18; Simplic. in lhys. f. 104, b.; IambI. 1. c. p. 161; comp. Plut. de Plac. Phil. ii. 6).
Pyth. 81; Stob. Eel. Plhys. p. 302; Brandis, I. c. In the Pythagorean system the element fire was
p. 444). the most dignified and important. It accordingly
As in the octave and its different harmonical occupied the most honourable position in the unirelations, the Pythagoreans found the ground of verse —the extreme (re'pas), rather than intermeconnection between the opposed primary elements, diate positions; and by extreme they understood
and the mutual relations of existing things, so in both the centre and the remotest region ('n 8'
the properties of particular numbers, and their Eo'xa'ov Kal Tr3 poCov o ripaF, Arist. de Caelo, ii.
relation to the principia, did they attempt to find 13). The central fire Philolaus terms the hearth
the explanation of the particular properties of dif- of the universe, the house or watch-tower of Zeus,
ferent things, and therefore addressed themselves the mother of the gods, the altar and bond and
to the investigation of the properties of numbers, measure of nature (Stob. 1. c. p. 488; Bickh, 1. c.
dividing them into various species. Thus they p. 94, &c.). It was the enlivening principle of the
had three kinds of even, according as the number universe. By this fire they probably understood
was a power of two (dpTraKeLcs aprIov), or a multi- something purer and more ethereal than the comple of two, or of some power of two, not itself a mon element fire (Brandis, 1. c. p. 491). Round
power of two (7repLaodpTrov), or the sum of an odd this central fire the heavenly bodies performed
and an even number (adpTLorrp'TrTov-a word their circling dance (Xopeveiv is the expression of
which seems to have been used in more than one Philolaus); —farthest off, the sphere of the fixed
sense. Nicom. Arithml. i. 7, 8). In like manner stars; then, in order, the five planets, the sun, the
they had three kinds of odd. It was probably the moon, the earth and the counter-earth (dvsfxOOwv)
use of the decimal system of notation which led - a sort of other half of the earth, a distinct body
co the number ten being supposed to be possessed from it, but always moving parallel to it, which
of extraordinary powers. "One must contemplate they seem to have introduced merely to make up
the works and essential nature of number accord- the number ten. The most distant region, which
ing to the power which is in the number ten; for was at the same time the purest, was termed
it is great, and perfect, and all-working, and the Olympus (Brandis, 1. c. p. 476). The space befirst principle (apXd) and guide of divine and tween the heaven of the fixed stars and the moon
heavenly and human life." (Philolaus ap. Stob. was termed Kcdoos; the space between the moon
Ecl. Phys. p. 8; Bdckh, p. 139.) This, doubtless, and the earth ospavo's (Stob. l. c.). Philolaus ashad to do with the formation of the list of ten pairs sumed a daily revolution of the earth round the
of opposite principles, which was drawn out by some central fire, but not round its own axis. The revoPythagoreans (Arist. Ilet. i. 5). In like manner lution of the earth round its axis was taught
the tetractys (possibly the sum of the first four (after Hicetas of Syracuse; see Cic. Acad. iv. 39)
numbers, or 10) was described as containing the by the Pythagorean Ecphantus and Heracleides
source and root of ever-flowing nature (Carm. Aur. Ponticus (Plut. Plac. iii. 13; Procl. in Tim. p. 281 ):
1. 48). The number three was spoken of as de- a combined motion round the central fire and round
fining or limiting the universe and all things, having its own axis, by Aristarchus of Samos (Plut. de
end, middle, and beginning, and so being the Fac. Lun. p. 933). The infinite (aIretpov) beyond
number of the wehole (Arist. de Caelo, i. 1). This the mundane sphere was, at least according to
part of their system they seem to have helped out Archytas (Simpl. in Phys. f. 108), not void space,
by considerations as to the connection of numbers but corporeal. The physical existence of the uniwith lines, surfaces, and solids, especially the regu- verse, which in the view of the Pythagoreans was a
lar geometrical figures (TIleolog. Arithm. 10, p. 61, huge sphere (Stob. 1. c. p. 452, 468), was represented
&c.), and to have connected the relations of things as a sort of' vital process, time, space, and breath
with various geometrical relations, among which (7rvo0) being, as it were, inhaled out of the &arelpov
angles played an important part. Thus, according (e&rsTrodyeO9al P' ieK'o0 adreLpo Xpvo'v Te Keal
to Philolaus, the angle of a triangle was conse- 7roiv cKal To Keyv, Stob. I. c. p. 380; see espe