When rigorously proving theorems, Archimedes often used what are now called Riemann sums. In "On the Sphere and Cylinder," he gives upper and lower bounds for the surface area of a sphere by cutting the sphere into sections of equal width. He then bounds the area of each section by the area of an inscribed and circumscribed cone, which he proves have a larger and smaller area correspondingly. He adds the areas of the cones, which is a type of Riemann sum for the area of the sphere considered as a surface of revolution.