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On the Sphere and Cylinder is a work that was published by Archimedes in two volumes c. 225 BC.[1] It most notably details how to find the surface area of a sphere and the volume of the contained ball and the analogous values for a cylinder, and was the first to do so.[2]

Contents

The principal formulae derived in On the Sphere and Cylinder are those mentioned above: the surface area of the sphere, the volume of the contained ball, and surface area and volume of the cylinder. In his work, Archimedes showed that the surface area of a cylinder is equal to:

A = 2 \pi r^2 + 2 \pi r h = 2 \pi r ( r + h ).\,

and that the volume of the same is:

V = \pi r^2 h. \,[3]

On the sphere, he showed that the surface area is four times the area of its great circle. In modern terms, this means that the surface area is equal to:

4\pi r^2.\,

The result for the volume of the contained balled stated that it is two-thirds the volume of a circumscribed cylinder, meaning that the volume is

\frac{4}{3}\pi r^3.

Archimedes was particularly proud of this latter result, and so he asked for a sketch of a sphere inscribed in a cylinder to be inscribed on his grave. Later, Roman philosopher Marcus Tullius Cicero discovered the tomb, which had been overgrown by surrounding vegetation.[4]

The argument Archimedes used to prove the formula for the volume of a ball was rather involved in its geometry, and many modern textbooks have a simplified version using the concept of a limit, which, of course, did not exist in Archimedes' time. Archimedes used an inscribed half-polygon in a semicircle, then rotated both to create a conglomerate of frustums in a sphere, which he then determined the volume of.[5]

Notes

References

  • Dunham, William (1990), Journey Through Genius (1st ed.), John Wiley and Sons, ISBN 0-471-50030-5  
  • Dunham, William (1994), The Mathematical Universe (1st ed.), John Wiley and Sons, ISBN 0-471-53656-3  
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