Pythagoras achieved influence at Croton for a period, but this eventually waned, and he retired, perhaps fleeing elsewhere. It appears that Pythagoras wrote nothing himself—or at least nothing that has survived—and his disciples and subsequent followers, the Pythagoreans, ascribed their own ideas to their master out of respect.
Assessing his contribution accurately is subject to speculation; however, the enormous and unusual levels of respect shown to him by his followers suggest that his personal contribution was significant.
Re-creation of his life and times is assisted by fragmentary mentions in the works of Plato and Aristotle and hints that he may have visited or studied with Anaximander and Thales.
The contribution of Pythagoras and the Pythagoreans is divided between the religious and the scientific, the latter is better known today, especially the theorem and its impact on musical theory.
Pythagorean geometry featured the speciﬁ cation of the hypotenuse to identify the lengths of the sides of right-angled triangles, although this was probably not the first statement of the proposition.
The use of ratios to explain the mathematical relationships underlying the structure of music have also been very inﬂ uential. The underlying concept was that it was relationships between the ratios of the first four integers (known as the Decad), which gave rise to the melodic intervals.
A series of 10 opposite pairs, or dualities, was established, both of which were necessary for existence. One half of these pairs were "limited" and the other half "unlimited", and together they connected all the many points that constituted the universe.
Eventually, this gave rise to the concept of the music of the spheres, in which complex numerical relationships existed, which regulated the movements and natures of all physical phenomena according to moral principles.
In religious or spiritual terms, Pythagoreans led a rather ascetic life that stressed moral principles underlying the nature of the universe and the transmigration of the soul—that is, reincarnation—that all people would undergo on numerous occasions, not always taking human form.
His disciples followed strict dietary guidelines and may have avoided making animal sacrifices, an important feature of Greek religious practices at the time, as part of their commitment to a vegetarian way of life. In any case the Pythagoreans excited compassion and sympathy rather than dislike or contempt.