A Fresh Look at the Method of Archimedes

  title={A Fresh Look at the Method of Archimedes},
  author={T. Apostol and M. Mnatsakanian},
  journal={The American Mathematical Monthly},
  pages={496 - 508}
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… Expand
Solids Circumscribing Spheres
Both these and deeper results hold, not only for the tetrahe-dron or any polyhedron that circumscribes a sphere, but for more general solids called circumsolids (defined in section 4), whose faces can be curved as well as planar. Expand
New Balancing Principles Applied to Circumsolids of Revolution, and to n-Dimensional Spheres, Cylindroids, and Cylindrical Wedges
New balancing principles including a balance-revolution principle and double equilibrium, that go much further than those of Archimedes are introduced, 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. Expand
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In 1733 the French naturalist Georges Louis Leclerc, Comte de Buffon, posed and solved the following problem in geometric probability: when a needle of length L is dropped onto a wooden floorExpand
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Two higher-dimensional generalizations of this invariant ratio of the catenary curve are developed and it is found that each invariant ratios identifies a class of hypersurfaces connected to classical objects from differential geometry. Expand
Thinking like Archimedes with a 3D printer
This work uses technology from the 21st century to trace intellectual achievements from the 3rd century BC of Archimedes and illustrates ideas of a mathematician who is known both for his for his mechanical inventions as well as his breakthroughs in geometry and calculus. Expand
Newton’s Shell Theorem via Archimedes’s Hat Box and Single-Variable Calculus
Summary Newton’s shell theorem asserts that the net gravitational force between a point particle and a sphere with uniform mass density is the same as the force in the situation where the sphere isExpand
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Two incongruent solids with remarkable properties are shown: a slice of a solid hemispherical shell with inner radius r and outer radius R cut by a plane parallel to the equator and a cylindrical shell with the same radii and altitude h. Expand
The Method of Punctured Containers
We introduce the method of punctured containers, which geometrically relates volumes and centroids of complicated solids to those of simpler punctured prismatic solids. This method goes to the heartExpand
Archimedes, Gauss, and Stein
We discuss a characterization of the centered Gaussian distribution which can be read from results of Archimedes and Maxwell, and relate it to Charles Stein's well-known characterization of the sameExpand


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