Spheres are to one another in the triplicate ratio of their respective diameters.
两球体积之比等于其直径之比的立方。
Let the spheres ABC, DEF be conceived, and let BC, 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 will have 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 conceived about the same centre with GHK, let there be inscribed in the greater sphere DEF a polyhedral solid which does not touch the lesser sphere GHK at its surface, [XII. 17] and let there also be inscribed 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, Por.] But the sphere ABC also has to the sphere GHK the ratio triplicate of that which BC has to EF; therefore, as the sphere ABC is to the sphere GHK, so is the polyhedral solid in the sphere ABC to the polyhedral solid in the sphere DEF; and, alternately, as the sphere ABC is to the polyhedron in it, so is the sphere GHK to the polyhedral solid in the sphere DEF.
假设球ABC与球DEF的体积比不等于直径BC与EF之比的立方，则球ABC与某个小于或大于球DEF的球之比等于该立方比。
[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. 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.
先设球ABC与小于球DEF的球GHK之比等于BC与EF之比的立方。以同一中心作球DEF，在其内内接一个不触及球GHK表面的多面体，并在球ABC内内接一个相似多面体。
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, 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.
由XII.17及推论，球ABC内多面体与球DEF内多面体之比等于BC与EF之比的立方。但球ABC与球GHK之比也等于该立方比，因此球ABC与球GHK之比等于两多面体之比。
[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.
由比例换位，球ABC与其内多面体之比等于球GHK与球DEF内多面体之比。但球ABC大于其内多面体，故球GHK也大于球DEF内多面体，然而球GHK被该多面体包围，矛盾。类似可证球ABC与大于球DEF的球之比也不可能等于该立方比。