The Pythagorean cosmology is interesting as it was a guess that came so near the truth con cerning the solar system. Much of it was fanci ful. but in spite of these vagaries we must recognize the fact that Pythagoreanism taught that the earth is a sphere revolving around a central tire, the centre of gravity of the universe. around which the stars revolve, carried by transparent shells. The central fire, however, was not the sun, but an invisible object. because toward it the farther side of the earth is always turned. The sun and stars shine by light reflected from this self-luminous centre. The heaven of the fixed stars, the sun, the moon, the five then known planets, and the earth made only nine objects: hence to fill out the perfect number of ten. a counter-earth was invented. Solar eclipses were due to the intervention of the earth between the central tire and the sun; lunar eclipses to the intervention of some heavenly body, sometimes the counter-sun. between the central tire and the moon. "When the earth is on the same side of the central lire as the sun. we have day; when it is On the other side. night." (Zeller.) "The distance of the spheres from the central fire was determined according to simple numerical relationships. Corresnontli»g to this, they assumed that from the revolution of the spheres there resulted a melodious musical sound. the so-called harmony of the spheres." 1Windelband.) Following Pythagoras. the Pythagoreans accepted the doctrine of metempsy chosis:but the doctrine of the world soul, some times ascribed to the Pythagoreans, was probably not a part of their system.
The Pythaoreans laid much emphasis on music. as can ho seen from their doctrine of the mush- of the spheres and from their insistence on the all-importance of harmony. But besides the discovery of the relation between the length of the strings of the lyre and the tones emitted.
they did not contribute much to the theory of music.
While the Ionic school founded geometry. the main progress was due to the Pythagorean school in Italy. The Pythagoreans were the first to give the rigorous proofs now demanded and to use mathematics in a specialized meaning. To Pythagoras himself is due the first rigorous proof of the proposition known by his name (Euclid i. 47: see HyroTEsusE). The school was concerned chiefly with the questions 'how many' and 'how great.' i.e. with number and magnitude, and the Pythagorean geometry is mostly con cerned with those relations of areas. volumes, and lines which admit of arithmetical expres sion. Geometry was to them a means for inves tigation in the theory of numbers. This is seen in the remarks concerning gnomon-numbers. Among the Pythagoreans a square out of which a corner was cut in the shape of a square was called a gnomon. The gnomon-number of the Pythagoreans is 2s + 1, since the square on a. can be equal to the square on a + 1 by adding the square / '1 and the two rectangles I -n, we then have it + 2a ± 1 = (n 1)=. pressions like plane and solid numbers used for the contents of spatial magnitudes of two and three dimensions also serve to indicate the con stant tendency to objectify mathematical thought by means of geometry. The knowledge of the Pythagoreans in the field of elementary series was quite comprehensive (see SERIES ) , and the three proportions, arithmetical, geometrical, and harmonienl, were known to them. The so-called most perfect or musical proportions, e.g. 6 : S = 0 : 12, was invented by the Babylonians and is said to have been first brought to Greece by Pythagoras. By improvement in definition, by systematization, and by the use of deduction. the study of geometry at the hands of the Pytha goreans was made a factor of liberal education.