Two Schools
Arguably Pythagoras' most important contribution to society was the Pythagorean Theorem. This states that in any right triangle with side lengths, and, where the latter is the length of the hypotenuse, . According to tradition, Pythagoreanism developed at some point into two separate schools of thought, the mathēmatikoi Μαθηματικοι ("learners") and the akousmatikoi Ακουσματικοι, ("listeners"). The mathēmatikoi were supposed to have extended and developed the more mathematical and scientific work begun by Pythagoras, while the akousmatikoi focused on the more religious and ritualistic aspects of his teachings. The akousmatikoi claimed that the mathēmatikoi were not genuinely Pythagorean, but followers of the "renegade" Pythagorean Hippasus. The mathēmatikoi, on the other hand, allowed that the akousmatikoi were Pythagorean, but felt that their own group was more representative of Pythagoras.
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