Let it measure it according to H;
therefore E by multiplying H has made B.
But further A has also by multiplying itself made B; [IX. 8]
therefore the product of E, H is equal to the square on A.
Therefore, as E is to A, so is A to H. [VII. 19]
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.
But, again, it also does not measure it: which is impossible.
Therefore E, A are not prime to one another.
Therefore they are composite to one another.
But numbers composite to one another are measured by some number. [VII. Def. 14]
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.
[But it also measures D;
therefore E measures A, D.]
Similarly we can prove that, by however many prime numbers D is measured, A will also be measured by the same.
Q. E. D.