There is a system in the Pythagorean theory not to be found in the teaching of the other reconcilers. Although all the numbers, and with them all the world, are divided into two opposing classes, these are, nevertheless, united in a harmony. The harmony of a dualism reminds us of Heracleitus’ harmony of antitheses. All series of numbers have their unity and harmony in the odd-even number, One. To the Pythagorean the opposites of life――the good and the bad, the limited and unlimited, the perfect and imperfect, the odd and even――exist in an harmonious whole. As the Pythagorean school grew in years, the realms to which it applied its theory increased. While we have stated its metaphysical theory first in order to give it prominence, the school came to the formulation of its theory through its investigations in mathematics, music, and astronomy. Then it applied the theory to geometrical structures and to other fields with a procedure that was arbitrary and unmethodical. Yet so universal was the application of the theory that it lived to have superstitious authority for the human mind in the Middle Ages. =3. Pythagorean Astronomy.= The formation of the world-all began from the One, or central fire, which attracted and limited the nearest portions of the unlimited. This fire became the centre of the world-all, which had the shape of a hollow globe. Around the central fire the celestial bodies move in globular transparent shells. Their movements are concentric to the fire. This is the beginning of the astronomical theory of the crystalline spheres. The world-all is divided into three concentric portions. The periphery or outer rim is Olympus, where all is perfection and where the gods dwell. Between Olympus and the moon is Cosmos, where all is orderly and all movements are in circles. Between the moon and the central fire is the region called Uranus, where all is disorderly and the movements are up and down. The earth is in this lower section of disorder, and moves in a transparent globular shell like the celestial bodies around the central fire. The number of the heavenly bodies is the perfect number, ten. The world-all is conceived as a heavenly heptachord, with the orbits of the seven planets as the sounding strings. Upon this notion was founded the harmony of the spheres, which harmony is not heard by man because it is constant. In modifying this astronomical theory and then accepting it, the most important change that Aristotle made was to conceive the earth as at the centre of the world-all with the sun revolving about it. This was the form in which the Ptolemaic astronomers received it.