The least numbers of those which have the same ratio with them measure those which have the same ratio the same number of times, the greater the greater and the less the less.
与两数有相同比的最小两数，分别以相同倍数量尽这两数，即较大者量尽较大者，较小者量尽较小者。
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For let CD, EF be the least numbers of those which have the same ratio with A, B; I say that CD measures A the same number of times that EF measures B. Now CD is not parts of A. For, if possible, let it be so; therefore EF is also the same parts of B that CD is of A.
设CD、EF是与A、B有相同比的最小两数。假设CD不是A的一部分，而是A的几个部分。
[VII. 13 and Def. 20] Therefore, as many parts of A as there are in CD, so many parts of B are there also in EF. Let CD be divided into the parts of A, namely CG, GD, and EF into the parts of B, namely EH, HF; thus the multitude of CG, GD will be equal to the multitude of EH, HF.
则EF也是B的同样几个部分，因为CD是A的几部分，EF就是B的几部分（VII.13及定义20）。
Now, since the numbers CG, GD are equal to one another, and the numbers EH, HF are also equal to one another, while the multitude of CG, GD is equal to the multitude of EH, HF, therefore, as CG is to EH, so is GD to HF. Therefore also, as one of the antecedents is to one of the consequents, so will all the antecedents be to all the consequents. [VII. 12] Therefore, as CG is to EH, so is CD to EF.
将CD分为A的部分CG、GD，EF分为B的部分EH、HF，则CG、GD的个数等于EH、HF的个数。由于CG=GD，EH=HF，且个数相等，故CG:EH=GD:HF，从而CG:EH=CD:EF（VII.12）。
Therefore CG, EH are in the same ratio with CD, EF, being less than they: which is impossible, for by hypothesis CD, EF are the least numbers of those which have the same ratio with them. Therefore CD is not parts of A; therefore it is a part of it.
因此CG、EH与CD、EF有相同比，但比它们小，这与假设矛盾。故CD是A的一部分（VII.4），同理EF是B的一部分，且倍数相同。