A Fresh Look at the Method of Archimedes

Abstract

1. INTRODUCTION. A spectacular landmark in the history of mathematics was the discovery by Archimedes (287–212 B.C.) that the volume of a solid sphere is twothirds the volume of the smallest cylinder that surrounds it, and that the surface area of the sphere is also two-thirds the total surface area of the same cylinder. Archimedes was so excited by this discovery that he wanted a sphere and its circumscribing cylinder engraved on his tombstone, even though there were many other great accomplishments for which he would be forever remembered. He made this particular discovery by balancing slices of a sphere and cone against slices of a larger cylinder, using centroids and the principle of the lever, which were also among his remarkable discoveries. The volume ratio for the sphere and cylinder can be derived from first principles without using levers and centroids (see [5]). This simpler and more natural method, presented in sections 2 and 3, paves the way for generalizations. Section 4 introduces a family of solids circumscribing a sphere. Cross sections of each solid cut by planes parallel to the equatorial plane are disks bounded by similar n-gons that circumscribe the circular cross sections of the sphere. We call these solids Archimedean globes in honor of Archimedes, who treated the case n = 4. The sphere is a limiting case, n →∞ . Each globe is analyzed by dividing it into wedges with two planar faces and one semicircular cylindrical face. In fact, Archimedes discussed (both mechanically and geometrically) volumes of wedges of this type. Figure 1 shows the top view of examples of globes with n = 3, 4, 6, and the limiting sphere.