Ideas held over the next two centuries by followers of Pythagoras of Samos (6th century BC).
Pythagoras is said to have founded a semi-religious brotherhood which developed doctrines about reincarnation and purification. He is also credited with noticing that simple harmonies (octave, fifth and so on) are associated with simple arithmetical ratios.
He or his followers (who reverently attributed their own ideas to him) developed a metaphysics of limit (good) and unlimited (bad), and in some way saw number as lying at the basis of reality; but the mathematical ideas traditionally attributed to them (including 'Pythagoras' theorem') are now thought unlikely to belong to Pythagoreans earlier than Archytas of Tarentum (fl. 400 BC).
Scientific ideas, such as that the Earth is not at the centre of the universe, may have a Pythagorean origin, but were probably developed for religious rather than scientific reasons.
Also see: Orphism
J A Philip, Pythagoras and Early Pythagoreanism (1966)