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

Tom M. Apostol and Mamikon A. Mnatsakanian
The American Mathematical Monthly
Vol. 120, No. 4 (April 2013), pp. 298-321
DOI: 10.4169/amer.math.monthly.120.04.298
Stable URL: http://www.jstor.org/stable/10.4169/amer.math.monthly.120.04.298
Page Count: 24
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New Balancing Principles Applied to Circumsolids of Revolution, and to n-Dimensional Spheres, Cylindroids,
and Cylindrical Wedges
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Abstract

Abstract Archimedes’ mechanical balancing methods led him to stunning discoveries concerning the volume of a sphere, and of a cylindrical wedge. This paper introduces new balancing principles (different from those of Archimedes) including a balance-revolution principle and double equilibrium, that go much further. They yield a host of surprising relations involving both volumes and surface areas of circumsolids of revolution, as well as higher-dimensional spheres, cylindroids, spherical wedges, and cylindrical wedges. The concept of cylindroid, introduced here, is crucial for extending to higher dimensions Archimedes’ classical relations on the sphere and cylinder. We also provide remarkable new results for centroids of hemispheres in n-space. Throughout the paper, we adhere to Archimedes’ style of reducing properties of complicated objects to those of simpler objects.

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