Book 9
13
For, if E is measured by another, and E measures D, that other will also measure D; so that it will also measure A [IX. 12], which is prime, though it is not the same with it: which is impossible.
13
Therefore A measures E.
13
And, since E measures D, let it measure it according to F.
13
I say that F is not the same with any of the numbers A, B, C.
13
For, if F is the same with one of the numbers A, B, C, and measures D according to E, therefore one of the numbers A, B, C also measures D according to E.
13
But one of the numbers A, B, C measures D according to some one of the numbers A, B, C; [IX. 11] therefore E is also the same with one of the numbers A, B, C: which is contrary to the hypothesis.
13
Therefore F is not the same as any one of the numbers A, B, C.
13
Similarly we can prove that F is measured by A, by proving again that F is not prime.
13
For, if it is, and measures D, it will also measure A [IX. 12], which is prime, though it is not the same with it: which is impossible; therefore F is not prime.
13
Therefore it is composite.
13
But any composite number is measured by some prime number; [VII. 31] therefore F is measured by some prime number.
13
I say next that it will not be measured by any other prime except A.
13
For, if any other prime number measures F, and F measures D, that other will also measure D; so that it will also measure A [IX. 12], which is prime, though it is not the same with it: which is impossible.
13
Therefore A measures F.
