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
Isosceles triangle a special isosceles triangle with the property that the other two inscribed squares, with sides collinear with the sides of the triangle, are of the same size as the base square. A much older theorem, preserved in the works of Hero of Alexandria, states that, for an isosceles triangle with base formula_3 and height formula_1, the side length of the inscribed square on the base of the triangle is For any integer formula_35, any triangle can be partitioned into formula_36 isosceles triangles. In a right triangle, the median from the hypotenuse (that is, the line segment from the midpoint of