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
Annulus (mathematics) within the annulus, which is the chord tangent to the inner circle, in the accompanying diagram. That can be shown using the Pythagorean theorem since this line is tangent to the smaller circle and perpendicular to its radius at that point, so and are sides of a right-angled triangle with hypotenuse , and the area of the annulus is given by The area can also be obtained via calculus by dividing the annulus up into an infinite number of annuli of infinitesimal width and area and then integrating from to : The area of an annulus sector of angle ,