Proof
Apply the method of exhaustion: inscribe similar polyhedra (XII.17);
by XII.12 (similar cones) and the polyhedron's similar-pyramid
decomposition, the inscribed solids are in the triplicate ratio of
diameters. By X.1 the inscribed solids approach the spheres in
volume; the limit gives the result.
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Depends on (4)
- X.1Proposition X.1Two unequal magnitudes being set out, if from the greater there be subtracted a magnitude greater than its half, and…
- XII.8Proposition XII.8Similar pyramids which have triangular bases are in the triplicate ratio of their corresponding sides.
- XII.12Proposition XII.12Similar cones and cylinders are to one another in the triplicate ratio of the diameters in their bases.
- XII.17Proposition XII.17Given two spheres about the same centre, to inscribe in the greater sphere a polyhedral solid which does not touch the…
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