If four straight lines be proportional, the parallelepipedal solids on them which are similar and similarly described will also be proportional; and, if the parallelepipedal solids on them which are similar and similarly described be proportional, the straight lines will themselves also be proportional.
若四条线段成比例，则其上相似且相似放置的平行六面体也成比例；反之，若其上相似且相似放置的平行六面体成比例，则线段本身也成比例。
Let AB, CD, EF, GH be four straight lines in proportion, so that, as AB is to CD, so is EF to GH; and let there be described on AB, CD, EF, GH the similar and similarly situated parallelepipedal solids KA, LC, ME, NG; I say that, as KA is to LC, so is ME to NG. For, since the parallelepipedal solid KA is similar to LC, therefore KA has to LC the ratio triplicate of that which AB has to CD.
设AB、CD、EF、GH四条线段成比例，即AB比CD等于EF比GH。在它们上作相似且相似放置的平行六面体KA、LC、ME、NG。
[XI. 33] For the same reason ME also has to NG the ratio triplicate of that which EF has to GH. [id.] And, as AB is to CD, so is EF to GH.
由于KA与LC相似，根据XI.33，KA与LC的比是AB与CD之比的立方。同理，ME与NG的比是EF与GH之比的立方。
Therefore also, as AK is to LC, so is ME to NG. Next, as the solid AK is to the solid LC, so let the solid ME be to the solid NG; I say that, as the straight line AB is to CD, so is EF to GH.
因为AB比CD等于EF比GH，所以KA比LC等于ME比NG。
For since, again, KA has to LC the ratio triplicate of that which AB has to CD, [XI. 33] and ME also has to NG the ratio triplicate of that which EF has to GH, [id.] and, as KA is to LC, so is ME to NG, therefore also, as AB is to CD, so is EF to GH.
反之，若KA比LC等于ME比NG，则由于KA与LC的比是AB与CD之比的立方，ME与NG的比是EF与GH之比的立方，因此AB比CD等于EF比GH。