Euclid, Elements
Thomas L. Heath, Sir Thomas Little Heath, Ed.

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], 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] 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]

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] 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]

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] 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]