For, if possible, let it be measured by E, and let E not be the same with any of the numbers A, B, C.

It is then manifest that E is not prime.

For, if E is prime and measures D, it will also measure A [[IX. 12](elem.9.12)], which is prime, though it is not the same with it: which is impossible.

Therefore E is not prime.

Therefore it is composite.

But any composite number is measured by some prime number; [[VII. 31](elem.7.31)] therefore E is measured by some prime number.

I say next that it will not be measured by any other prime except A.

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](elem.9.12)], which is prime, though it is not the same with it: which is impossible.

Therefore A measures E.

And, since E measures D, let it measure it according to F.

I say that F is not the same with any of the numbers A, B, C.

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.

But one of the numbers A, B, C measures D according to some one of the numbers A, B, C; [[IX. 11](elem.9.11)] therefore E is also the same with one of the numbers A, B, C: which is contrary to the hypothesis.

Therefore F is not the same as any one of the numbers A, B, C.

Similarly we can prove that F is measured by A, by proving again that F is not prime.

For, if it is, and measures D, it will also measure A [[IX. 12](elem.9.12)], which is prime, though it is not the same with it: which is impossible; therefore F is not prime.

Therefore it is composite.

But any composite number is measured by some prime number; [[VII. 31](elem.7.31)] therefore F is measured by some prime number.

I say next that it will not be measured by any other prime except A.

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](elem.9.12)], which is prime, though it is not the same with it: which is impossible.

Therefore A measures F.

And, since E measures D according to F, therefore E by multiplying F has made D.

But, further, A has also by multiplying C made D; [[IX. 11](elem.9.11)] therefore the product of A, C is equal to the product of E, F.

Therefore, proportionally, as A is to E, so is F to C. [[VII. 19](elem.7.19)]

But A measures E; therefore F also measures C.

Let it measure it according to G.

Similarly, then, we can prove that G is not the same with any of the numbers A, B, and that it is measured by A.

And, since F measures C according to G therefore F by multiplying G has made C.

But, further, A has also by multiplying B made C; [[IX. 11](elem.9.11)] therefore the product of A, B is equal to the product of F, G.

Therefore, proportionally, as A is to F, so is G to B. [[VII. 19](elem.7.19)]

But A measures F; therefore G also measures B.

Let it measure it according to H.

Similarly then we can prove that H is not the same with A.

And, since G measures B according to H, therefore G by multiplying H has made B.

But further A has also by multiplying itself made B; [[IX. 8](elem.9.8)] therefore the product of H, G is equal to the square on A.

Therefore, as H is to A, so is A to G. [[VII. 19](elem.7.19)]

But A measures G;< therefore H also measures A, which is prime, though it is not the same with it: which is absurd.

Therefore D the greatest will not be measured by any other number except A, B, C. Q. E. D.