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Proposition 18

Spheres are to one another in triplicate ratio of their respective diameters.
Let the ABC and DEF be spheres, and let BC and EF be their diameters. I say that the sphere ABC has to the sphere DEF the ratio triplicate of that which BC has to EF.
For, if the sphere ABC has not to the sphere DEF the ratio triplicate of that which BC has to EF, then the sphere ABC has either to some less sphere than the sphere DEF, or to a greater, the ratio triplicate of that which BC has to EF. First, let it have that ratio to a less sphere GHK.
Let DEF be about the same center with GHK. Inscribe in the greater sphere DEF a polyhedral solid which does not touch the lesser sphere GHK at its surface. | XII.17

Also inscribe in the sphere ABC a polyhedral solid similar to the polyhedral solid in the sphere DEF. Therefore the polyhedral solid in ABC has to the polyhedral solid in DEF the ratio triplicate of that which BC has to EF. | XII.17,Cor.
But the sphere ABC also has to the sphere GHK the ratio triplicate of that which BC has to EF, therefore the sphere ABC is to the sphere GHK as the polyhedral solid in the sphere ABC is to the polyhedral solid in the sphere DEF, and, alternately the sphere ABC is to the polyhedron in it as the sphere GHK is to the polyhedral solid in the sphere DEF. | V.16
But the sphere ABC is greater than the polyhedron in it, therefore the sphere GHK is also greater than the polyhedron in the sphere DEF. | V.14
But it is also less, for it is enclosed by it. Therefore the sphere ABC has not to a less sphere than the sphere DEF the ratio triplicate of that which the diameter BC has to EF. Similarly we can prove that neither has the sphere DEF to a less sphere than the sphere ABC the ratio triplicate of that which EF has to BC. I say next that neither has the sphere ABC to any greater sphere than the sphere DEF the ratio triplicate of that which BC has to EF.
For, if possible, let it have that ratio to a greater, LMN. Therefore, inversely, the sphere LMN has to the sphere ABC the ratio triplicate of that which the diameter EF has to the diameter BC.
But, since LMN is greater than DEF, therefore the sphere LMN is to the sphere ABC as the sphere DEF is to some less sphere than the sphere ABC, as was before proved. | XII.2,Lemma
Therefore the sphere DEF also has to some less sphere than the sphere ABC the ratio triplicate of that which EF has to BC, which was proved impossible. Therefore the sphere ABC has not to any sphere greater than the sphere DEF the ratio triplicate of that which BC has to EF. But it was proved that neither has it that ratio to a less sphere. Therefore the sphere ABC has to the sphere DEF the ratio triplicate of that which BC has to EF.
Therefore, spheres are to one another in triplicate ratio of their respective diameters.
Q.E.D.

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This proposition completes Book XII.

Although this is an important proposition, it is just the beginning of the study of volumes of spheres. The arguments given in this proof are fairly convincing that any two similar solids are to each other in triplicate ratio of their linear parts. One difficulty is defining just what similar solids are.

#### The volume of a sphere

Euclid proved in proposition XII.10 that the cone with the same base and height as a cylinder was one third of the cylinder, but he could not find the ratio of a sphere to the circumscribed cylinder.

In the century after Euclid, Archimedes solved this problem as well as the much more difficult problem of the surface area of a sphere. He showed that the ratio of the sphere to the cylinder is 4:3. Since the volume of the cylinder is proportional to its base and its height, that implies volumes of spheres, cylinders, and cones can be found in terms of areas of circles. In algebraic terms, if we let pi stand for the ratio of a circle to the square on its radius, then the volume of a cylinder of radius r and height h is pi r2h; the volume of an inscribed cone is pi r2h/3; and the volume of a sphere of radius r is 4 pi r3/3.

Book XII Introduction - Proposition XII.17 - Book XIII Introduction.

© 1996
D.E.Joyce
Clark University