elem.5.15
若一线段是另一线段的同倍量，且另一线段是第三线段的同倍量，则第一线段与第三线段之比等于第二线段与第四线段之比。
AB、DE 是两段；GH、KL 各是 AB、DE 的同倍量。C、F 标注每段所属的量。AB:DE = GH:KL。
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For let AB be the same multiple of C that DE is of F; I say that, as C is to F, so is AB to DE. For, since AB is the same multiple of C that DE is of F, as many magnitudes as there are in AB equal to C, so many are there also in DE equal to F.
设AB是C的同倍量，DE是F的同倍量，即AB包含与C相等的量，DE包含与F相等的量，且个数相同。
Let AB be divided into the magnitudes AG, GH, HB equal to C, and DE into the magnitudes DK, KL, LE equal to F; then the multitude of the magnitudes AG, GH, HB will be equal to the multitude of the magnitudes DK, KL, LE. And, since AG.
将AB分为AG、GH、HB，各等于C；将DE分为DK、KL、LE，各等于F。则AG、GH、HB的个数等于DK、KL、LE的个数。
GH, HB are equal to one another, and DK, KL, LE are also equal to one another, therefore, as AG is to DK, so is GH to KL, and HB to LE. [V. 7] Therefore, as one of the antecedents is to one of the consequents, so will all the antecedents be to all the consequents; [V. 12] therefore, as AG is to DK, so is AB to DE.
由于AG、GH、HB彼此相等，DK、KL、LE也彼此相等，故AG比DK等于GH比KL，也等于HB比LE（第五卷命题7）。
But AG is equal to C and DK to F; therefore, as C is to F, so is AB to DE.
因此，一个前项与一个后项之比等于所有前项之和与所有后项之和之比（第五卷命题12），即AG比DK等于AB比DE。但AG等于C，DK等于F，故C比F等于AB比DE。