If two numbers be prime to one another, the second will not be to any other number as the first is to the second.
若两数互质，则第二数不能与任何其他数成第一数比第二数之比例。
两数 A、B 互质；假设存在 C 使 A:B = B:C，由互素+最小数性质推 A 量 B，矛盾。
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For let the two numbers A, B be prime to one another; I say that B is not to any other number as A is to B.
设两数A、B互质，假设存在数C使得A比B等于B比C。
For, if possible, as A is to B, so let B be to C.
因A、B互质，故它们是最小的具有该比例的数（第七卷命题21）。
Now A, B are prime, primes are also least, [VII. 21] and the least numbers measure those which have the same ratio the same number of times, the antecedent the antecedent and the consequent the consequent; [VII. 20] therefore A measures B as antecedent antecedent.
最小数能量尽与其有相同比例的数相同次数，前项量尽前项，后项量尽后项（第七卷命题20）。
But it also measures itself; therefore A measures A, B which are prime to one another: which is absurd.
因此A量尽B（作为前项量尽前项），但A也量尽自身，故A量尽互质的A、B，矛盾。