If two numbers be prime to one another, the second will not be to any other number as the first is to the second.

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.

For, if possible, as A is to B, so let B be to C.

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.

But it also measures itself; therefore A measures A, B which are prime to one another: which is absurd.

Therefore B will not be to C, as A is to B. Q. E. D.