New Balancing Principles Applied to Circumsolids of Revolution, and to n-Dimensional Spheres, Cylindroids, and Cylindrical Wedges

The sphere and circumscribing cylinder engraved on Archimedes’ tombstone commemorate his landmark discovery that their volumes and surface areas are related by the same constant ratio 2/3. He discovered the volume relation and many other geometrical results by mechanical balancing. In particular, he found the volume of a cylindrical wedge by introducing the special balancing shown in Figures 1a and 1b. First, he balances lengths of horizontal chords of a triangle and a semicircular disk with respect to a vertical axis through its center as in Figure 1a. He then builds two-dimensional regions from these chords, the right triangle and semicircular disk in Figure 1b, which now are in area balance with each