Pythagoreanism to have survived as part of, or deeply influenced, the Bacchic cults and Orphism. Pythagoras was already in ancient times well known for the mathematical achievement of the Pythagorean theorem. Although the Babylonians had used this formula for centuries, Pythagoras had discovered that "in a right-angled triangle the square of the hypotenuse is equal to the squares of the other two sides". In ancient times Pythagoras was also noted for his discovery that music had mathematical foundations. Antique sources that credit Pythagoras as the philosopher who first discovered music intervals also credit him as the inventor of the monochord, a
Group of rational points on the unit circle Group of rational points on the unit circle In mathematics, the rational points on the unit circle are those points ("x", "y") such that both "x" and "y" are rational numbers ("fractions") and satisfy "x" + "y" = 1. The set of such points turns out to be closely related to primitive Pythagorean triples. Consider a primitive right triangle, that is, with integral side lengths "a", "b", "c", with "c" the hypotenuse, such that the sides have no common factor larger than 1. Then on the unit circle there exists the rational point ("a"/"c", "b"/"c"), which, in the complex plane,