Numbers prime to one another are the least of those which have the same ratio with them.
互质的两个数是与它们有相同比的所有数中最小的。
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Let A, B be numbers prime to one another; I say that A, B are the least of those which have the same ratio with them. For, if not, there will be some numbers less than A, B which are in the same ratio with A, B. Let them be C, D.
设A、B互质，假设存在小于A、B的C、D与A、B有相同比。
Since, then, the least numbers of those which have the same ratio measure those which have the same ratio the same number of times, the greater the greater and the less the less, that is, the antecedent the antecedent and the consequent the consequent, [VII. 20] therefore C measures A the same number of times that D measures B. Now, as many times as C measures A, so many units let there be in E.
由第七卷命题20，最小同比数C、D分别量尽A、B相同次数，设该次数为E。
Therefore D also measures B according to the units in E. And, since C measures A according to the units in E, therefore E also measures A according to the units in C. [VII. 16] For the same reason E also measures B according to the units in D.
因此C量尽A的次数为E，故E量尽A的次数为C；同理E量尽B的次数为D。
[VII. 16] Therefore E measures A, B which are prime to one another: which is impossible. [VII. Def. 12] Therefore there will be no numbers less than A, B which are in the same ratio with A, B.
于是E量尽互质的A、B，与第七卷定义12矛盾，故假设不成立。