Contributions to Geometry.—Other contributions to etry by the comprise the : we are told, himself formulated definitions in (2) The proved that the sum of the three of any is equal to two angles; we have their proof, which, like Euclid's, uses the property of hence they knew the theory of parallels. (3) They discovered the powerful method in geometry of the "application of areas" (cf. Eucl. i. 44, 45), includ ing application with "excess" and "defect" (cf. Eucl. vi. 28, 29) which amounts to the geometrical solution of any quadratic equa tion in algebra having real roots. (4) Pythagoras himself is said to have discovered the theory of proportion (for when Proclus says that he discovered the theory of irrationals, TCJV alryceP, the reading should almost certainly be avo.Xcryov, of propor tionals) and of the three means, arithmetic, geometric and har monic. The arithmetic and harmonic means are shown in the middle terms of the proportion a: a + 2ab :b, a particular case 2 ad- b being 12:9=8:6, from the terms of which the three musical in tervals can be obtained. The Pythagorean theory of proportion was arithmetical (after the manner of Euclid, Book vii.) and did not apply to incommensurable magnitudes; it must not therefore be confused with the general theory due to Eudoxus which is ex pounded in Euclid v. (5) Proclus also says that Pythagoras discovered the construction of the five regular solids. It was more probably Theaetetus who (as another scholium says) discovered the octahedron and the icosahedron ; but the Pythagoreans were clearly acquainted with the pyramid or tetrahedron and the dodec ahedron. The construction of the dodecahedron requires that of a regular pentagon, which again depends (as in Eucl. iv., io, I I) on the problem of Eucl. ii. i 1. about the division of a line "in extreme and mean ratio," a particular case of the "application of areas." The assumption that the Pythagoreans could construct a regular pentagon is confirmed by the fact that the "pentagram," the "triple interwoven triangle," or the star-pentagon, was used by the Pythagoreans as a symbol of recognition between the members of the school and was called by them Health. (6) The Pythagoreans discovered how to construct a rectilineal figure equal to one and similar to another rectilineal figure.
Summing up the Pythagorean geometry, we may say that it covered the bulk of the subject matter of Euclid's Books i., ii., iv., vi. (and probably iii.), with the qualification that the Pythagorean theory of proportion was inadequate in that it did not apply to incommensurable magnitudes.
Pythagorean Astronomy.—It remains to speak of the Pyth agorean astronomy. Pythagoras was one of the first to hold that the earth and the universe are spherical in shape. He realized that the sun, moon and planets have a motion of their own independent of the daily rotation and in the opposite sense. It is improbable that Pythagoras himself was responsible for the astronomical system known as Pythagorean, which deposed the earth from its place in the centre and made it a "planet" like the sun, the moon and the other planets ; for Pythagoras apparently the earth was still at the centre. The later Pythagorean system is attributed al
ternatively to Philolaus and to Hicetas, a native of Syracuse. The system may be thus described. The universe is spherical in shape and finite in size. Outside it is infinite void which enables the universe to breathe, as it were. At the centre is the central fire, called the Hearth of the Universe (among other names), wherein is situated the governing principle, the force which directs the movement and activity of the universe. In the universe there re volve round the central fire the following bodies : nearest to the central fire is the "counter-earth" which always accompanies the earth ; next in order (reckoning from the centre outwards) is the earth, then the moon, then the sun, then the five planets and then, last of all, the sphere of the fixed stars. The counter-earth, re volving in a smaller orbit than the earth, is not seen by us because the hemisphere in which we live is always turned away from the counter-earth (the analogy of the moon which always turns the same side to us may have suggested this). This part of the theory involves the assumption that the earth rotates about its own axis in the same time as it takes to complete its orbit round the central fire; and, as the latter revolution was held to produce day and night, it is a fair inference that the earth was thought to revolve round the central fire in a day and a night, or in 24 hours.
The system amounts to a first step towards an anticipation of the Copernican hypothesis, and Copernicus himself referred to it as such. The curious thing in the system is the introduction of the counter-earth. Aristotle in one place says that its object was to bring the number of the revolving "bodies" up to ten, the perfect number according to the Pythagoreans; but elsewhere he hints at the truer explanation, when he says that eclipses of the moon were considered to be due sometimes to the interposition of the earth, sometimes to the interposition of the counter-earth, whence it would appear that the counter-earth was invented in order to explain the frequency of lunar eclipses as compared with solar.
For accounts of the sources and for further details reference may be made to the following works in addition to Zeller, Die Philosophie der Griechen (4th ed. 1892, etc.) ; Diels, Doxographi Graeci (1879) and Die Fragmente der Vorzokratiker (4th ed., 1922) ; A. Delatte, Etudes sur la litterature Pythagoricienne (1915) and La vie de Pythagore de Diogine Lairce (1922); Gomperz, Griechische Denker, vol. i. (Eng. trans., 1901), and especially Burnet, Early Greek Philosophy (3rd ed., 192o). For the mathematics see, besides other histories of mathematics, James Gow, A Short History of Greek Mathematics (1884) ; Sir T. L. Heath, A History of Greek Mathematics, vol. i. (1921) ; Eva Sachs, Die fiinf Platonischen Korper (1917). (T. L. H.)