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## Search Loci: Convergence:

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An expert problem solver must be endowed with two incompatible qualities, a restless imagination and a patient pertinacity.

In Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1969.

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# Archimedes' Method for Computing Areas and Volumes

## Proposition 2 of The Method

You can see the balancing of the cone and sphere with the cylinder dynamically by clicking on Archimedes' Proposition 2 of The Method.

A static version of this balancing appears below.

Since this balance holds for every cross section, it follows that if the entire sphere and cone are moved so that their centers of gravity are at H, they will exactly balance the cylinder, left where it is. By symmetry, the center of gravity of the cylinder is at K, so by the law of the lever,

 (Volume of Sphere + Volume of Cone) × |AH|  = (Volume of Cylinder) × |AK|

Since |AH| = 2|AK|, it follows that

 2(Volume of Sphere + Volume of Cone) = Volume of Cylinder

or

 Volume of Sphere = 1 2 (Volume of Cylinder) – Volume of Cone

Finally, using the fact that the volume of the cone is 1/3 the volume of the cylinder, we find that the volume of the sphere is 1/6 the volume of the cylinder. Now since the cylinder's height and radius are both 2r (where r is the radius of the sphere), the volume of the cylinder is

$$\pi$$·(radius)2·(height) = $$\pi$$·(2r)2·(2r)=8$$\pi$$r3,

so the volume of the sphere is

 V = 1 6 ·8$$\pi$$r3= 4$$\pi$$r3 3

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