Kitap 9
12
But A, E are prime, primes are also least, [VII. 21] and the least measure those which have the same ratio the same number of times, the antecedent the antecedent and the consequent the consequent; [VII. 20] therefore E measures A, as antecedent antecedent.
12
But, again, it also does not measure it: which is impossible.
12
Therefore E, A are not prime to one another.
12
Therefore they are composite to one another.
12
But numbers composite to one another are measured by some number. [VII. Def. 14]
12
And, since E is by hypothesis prime, and the prime is not measured by any number other than itself, therefore E measures A, E, so that E measures A.
12
[But it also measures D; therefore E measures A, D.]
12
Similarly we can prove that, by however many prime numbers D is measured, A will also be measured by the same. Q. E. D.
PROPOSITION 13.
13
If as many numbers as we please beginning from an unit be in continued proportion, and the number after the unit be prime, the greatest will not be measured by any except those which have a place among the proportional numbers.
13
Let there be as many numbers as we please, A, B, C, D, beginning from an unit and in continued proportion, and let A, the number after the unit, be prime; I say that D, the greatest of them, will not be measured by any other number except A, B, C.
13
For, if possible, let it be measured by E, and let E not be the same with any of the numbers A, B, C.
13
It is then manifest that E is not prime.
13
For, if E is prime and measures D, it will also measure A [IX. 12], which is prime, though it is not the same with it: which is impossible.
13
Therefore E is not prime.
13
Therefore it is composite.
13
But any composite number is measured by some prime number; [VII. 31] therefore E is measured by some prime number.
13
I say next that it will not be measured by any other prime except A.
