| If two numbers are relatively prime, then the second is not to any other number as the first is to the second. | ||
| Let the two numbers A and B be relatively prime.
I say that B is not to any other number as A is to B. If possible as A is to B, let B be to C. | ||
| Now A and B are prime, primes are also least, 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, therefore A measures B as antecedent antecedent. | VII.21 | |
| But it also measures itself, therefore A measures A and B which are relatively prime, which is absurd.
Therefore B is not to C as A is to B. | ||
| Therefore, if two numbers are relatively prime, then the second is not to any other number as the first is to the second. | ||
| Q.E.D. | ||
Book IX Introduction - Proposition IX.15 - Proposition IX.17.