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.