Chapter 11, The Sphere and the Cylinder Summary and Analysis

The most important work Archimedes did, in his opinion, was his work on spheres and cylinders, along with cones and pyramids. He also related flat objects like circles and triangles to their three-dimensional counterparts, like spheres and trapezoids. He found that taking a sphere and creating a cylinder as wide as the greatest circle of the sphere and which was as high as a sphere took up 1.5 times the space as the sphere. Archimedes wanted the image of a sphere within such a cylinder on his tomb.

Archimedes also developed a method for proving his solution to problems which mathematicians have considered important ever since. Basically, the method is to enumerate all possible solutions to the problems, eliminates unnecessary elements, and then rules them out one by one.