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Q.E.D.

16 17 archImedes theorem mysteries of the sphere Archimedes proved that the volume of a sphere is two thirds that of the smallest cylinder containing it, and that its surface area is the same as that of the hollow cylinder. So taken was the philosopher with these relationships that he had a sphere and its surrounding cylinder inscribed on his tombstone. Opposite we use Cavalieris principle see page 10 to derive the formula 4 3 p r 3 for the volume of a sphere of radius r, and thereby confirm Archimedes first discovery. For some real magic, project every point of the sphere, except the poles, on to another point on the cylinder, as shown below. Then it can be proved that any patch on the sphere gets projected on to a patch of equal area on the cylinder. If we then take the patch to be the whole sphere, it follows that its image is the cylinder, implying Archimedes second discovery. If we replace the sphere by a globe, project it on to the cylinder, and then slice the cylinder open, we find ourselves with a highly useful equalarea map of the Earth.
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