For, if possible, as A is to B, so let D be to E;
therefore, alternately, as A is to D, so is B to E. [VII. 13]
But A, D 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.
And, as A is to B, so is B to C.
Therefore B also measures C; so that A also measures C.
And since, as B is to C, so is C to D, and B measures C,
therefore C also measures D.
But A measured C; so that A also measures D.
But it also measures itself;
therefore A measures A, D which are prime to one another: which is impossible.
Therefore D will not be to any other number as A is to B.
Q. E. D.
Complete text
Book I