self-supported in empty space, revolving with the other planets round
a central luminary. They thus anticipated the heliocentric theory,
and Copernicus has left it on record that the Pythagorean doctrine
of the planetary movement of the earth gave him the first hint of
its true hypothesis. The Pythagoreans did not, however, put the
sun in the centre of the system. That place was filled by the central
fire to which they gave the names of Hestia, the hearth of the
universe, the watch-tower of Zeus, and other mythological expressions.
It had then been recently discovered that the moon shone
by reflected light, and the Pythagoreans (adapting a theory of
Empedocles), explained the li ht of the sun also as due to reflection
from the central fire. Rounci this fire revolve ten bodies, first the
Antichthon or counter-earth, then the earth, followed in order
by the moon, the sun, the five then known planets and the heaven
of the fixed stars. The central fire and the counter-earth are
invisible to us because the side of the' earth on which we live is
always turned away from them, and our light and heat come to us,
as already stated, by reflection from the sun. When the earth is
on the same side of the central fire as the sun, the side of the earth
on which we live is turned towards the sun and we have day;
when the earth and the sun are on opposite sides of the central fire
we are turned away from the sun and it is night. The distance of
the revolving orbs from the central fire was determined according
to simple numerical relations, and the Pythagoreans combined their
astronomical and their musical discoveries in the famous doctrine
of “ the harmony of the spheres.” The velocities of the bodies
depend upon their distances from the centre, the slower and nearer
bodies giving out a deep note and the swifter a high note, the
concert of the whole yielding the cosmic octave. The reason why
we do not hear this music is that we are like men in a smith's forge,
who cease to be aware of a sound which they constantly hear and
are never in a position to contrast with silence.
Authorities.-Zeller's account of Pythagoreanism in his Philosophie
der Griechen gives a full account of the sources, with critical
references in the notes to the numerous monographs on the subject,
but the labour and ingenuity of more recent scholars has succeeded
in clearing up a number of points since he wrote. Diels, Doxographi
graeci (1879), and Die Fragrnente der Vorsokratiker, vol. i.
(2nd ed., 1906). Gomperz, Greek Thinkers, vol. i., and especially
Burnet's Early Greek Philosophy (2nd ed., 1908), give the results
of the latest investigations. Tannery's Science hellene; Milhaud's
La Science grecque and Philosophes géornetres; Cantor's History of
Mathematics; and Gow's Short History of Greek Mathematics, refer
to the mathematical and physical doctrines of the school.
(A. S. P.-P.)

PYTHAGOREAN Gnounrnv

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 mathematics 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 (Meta. 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 oint of view (dbkws Kal voep&s).”1 (c) Diogenes Laértius (viii. Ilg relates that “it was Pythagoras who carried geometry to perfection, after Moeris2 had first found out the principles 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 Aristoxenus, 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.” (e) Diogenes Laértius (viii. 13) reports on the same authority that Pythagoras was the first person who introduced measures and weights among the Greeks. (f) He discovered the numerical relations of the musical scale (Diog. 1 Proclus Diadochus, In prirnurn Euclidis elementorum librurn commentarii, ed. Friedlein. p. 65.

Moeris was a king of Egypt who, Herodotus tells us, lived 900 years before his visit to that country.

Aristox. Fragm. ap. Stob. Eclog. Phys. i. 2, 6. Laért. viii. xl). (g) Proclus* says that “the word 'mathematics We learn also from the

made a fourfold division

originated with the Pythagoreans." (h)

sa1ne authority' that the Pythagoreans

of mathematical science, attributing one of its parts to the “how many ” (16 irécov) and the other to the “ how much ” (-ni 1r1;)ir<ov); 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 continued quantity or the “how much" is either stable or in motion. Hence they laid down that arithmetic contemplates 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, but that astronomy (15 aoatptmi) contemplates continued quantity so far as it is of a self-motive nature. (i) Diogenes Laertius (viii. 25) states, on the authority of Favorinus, that Pythagoras “employed definitions in the mathematical subjects to which he applied himself.” The following notices of the geometrical work of Pythagoras and the early Pythagoreans are also preserved. (1) The Pythagoreans define a point as “ unity having position” (Procl. op. cit. p. 95). (2) They considered a point as analogous to the monad, a line to the dyad, a superficies to the triad, and a body to the tetrad (ibid. p. 97). (3) They showed that the plane around a point is completely filled by six equilateral triangles, four squares, or three regular hexagons (ibid. p. 305). (4) Eudemus ascribes to them the discovery of the theorem that the interior angles of a triangle are equal to two right angles, and gives their proof, which was substantially the sane as that in Euclid I. 32 5 (ibid. p. 379). (5) Proclus informs us in his commentary on Euclid l. 44 that Eudemus says that the problems concerning the application of areas-where the term “ application " is not to be taken in its restricted sense (1rapaBoM;), in which it is used in this proposition, but also in its wider signification, embracing D1repBoM, and é>>fu, tLs, in which it is used in Book VI. Props. 28, 29-are old, and inventions of the Pythagoreans" (ibid. p. 419). (6) This is confirmed by Plutarchf who says, after Apollodorus, that Pythagoras sacrificed an ox on finding the geometrical diagram, either the one relating to the hypotenuse, viz. that the square on it is equal to the sum of the squares on the sides, or that relating to the problem concerning the application of an area.9 (7) Plutarch 1° also ascribes to Pythagoras the solution of the problem, To construct a figure equal to one and similar to another given figure. (8) Eudemus states that Pythagoras discovered the construction of the regular solids (Procl. op. cit. p. 65). (9) Hippasus, the Pythagorean, is said to have perished in the sea on account of his impiety, inasmuch as he boasted that he first divulged the knowledge of the sphere with the twelve pentagons (the inscribed ordinate dodecahedron): Hippasus assumed the glory of the discovery to himself, whereas everything belonged to Him-“ for thus they designate Pythagoras, and do not call him by name.”“ (10) The triple interwoven triangle or pentagram-star-shaped regular pentagon-was used as a symbol or sign of recognition by the Pythagoreans and was called by them “health ” (0yte£a.).'2 (11) The discovery of the law of the three 4 Procl. op. cit. p. 45.

5 Op. cit. p. 35.

6 We learn from a fragment of Geminus, which has been handed down by Eutocius in his commentary on the Conics of Apollonius (Apoll. Conica, ed. Halleius, p- 9), that the ancient geometers observed two right angles in each species of triangle, in the equilateral first, then in the isosceles, and lastly in the scalene, whereas later writers proved the theorem generally thus-“ The three internal angles of every triangle are equal to two right angles.” 7 The words of Proclus are interesting. “ According to Eudemus the inventions respecting the application, excess and defect of areas are ancient, and are due to the Pythagoreans. Moderns, borrowing these names, transferred them to the so-called conic lines, the parabola, the hyperbola, the ellipse, as the older school, in their nomenclature concerning the description of areas in plano on a finite right line, regarded the terms thus: An area is said to be applied (impa-B<i}}e/.v) to a given right line when an area equal in content to some given one is described thereon; but when the base of the area is greater than the given line, then the area is said to be in excess (ir/repB¢i))etv); but when the base is less, so that some part of the given line lies without the described area, then the area is said to be in defect (é>}ei1reu/). Euclid uses in this way in his sixth book the terms excess and defect. . The term application (1rapaBé.)>ew), which we owe to the Pythagoreans, has this signification." 3 Non posse suaviter vivi sec. Epicur-urn, c. xi. 9 Eire 1rpéB)'q;/.a '/repi 'roii Xwpiou riis 1rapa{3o}s. Some authors, rendering the last five words “ concerning the area of the parabola, " have ascribed to Pythagoras the quadrature of the parabola, which was one of the great discoveries of Archimedes; 1° Syrnp. viii., Quaest. 2, c. 4.

11 Iamblichus, De vit. Pyth. c. 18, § 88.

2 Lucian, Pro lapsu in salut. § 5; also schol. on Aristoph. Nub. 611. That the Pythagoreans used such symbols we learn from lamblichus (De vit. Pyth. c. 33, §§ 237 and 238). This figure is referred to Pythagoras himself, and in the middle ages was called Pythagorae jigura; even so late as Paracelsus it was regarded by