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 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 light of the sun also as due to reflection from the central fire. Round 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.

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 ). 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 point of view ( ).” (c) Diogenes Laërtius (viii. 11) relates that “it was Pythagoras who carried geometry to perfection, after Moeris 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 Laërtius (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.

Laërt. viii. xl). (g) Proclus says that “the word ‘mathematics’ originated with the Pythagoreans.” (h) We learn also from the same authority that the Pythagoreans made a fourfold division of mathematical science, attributing one of its parts to the “how many” ( ) and the other to the “how much” ( ); 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 ( ) contemplates continued quantity so far as it is of a self-motive nature. (i) Diogenes Laërtius (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 same as that in Euclid I. 32 (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 , in which it is used in this proposition, but also in its wider signification, embracing  and, 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 Plutarch 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. (7) Plutarch 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”. (11) The discovery of the law of the three