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.
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.
Euclid. Euclid's Elements. Sir Thomas Little Heath. New York. Dover. 1956.
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