Book 12
18
But, inasmuch as LMN is greater than DEF, therefore, as the sphere LMN is to the sphere ABC, so is the sphere DEF to some less sphere than the sphere ABC, as was before proved. [XII. 2, Lemma]
18
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.
18
Therefore the sphere ABC has not to any sphere greater than the sphere DEF the ratio triplicate of that which BC has to EF.
18
But it was proved that neither has it that ratio to a less sphere.
18
Therefore the sphere ABC has to the sphere DEF the ratio triplicate of that which BC has to EF. Q. E. D.
