Let D, E, the least numbers of those which have the same ratio with A, C, be taken; [VII. 33]
therefore D measures A the same number of times that E measures C. [VII. 20]
Now, as many times as D measures A, so many units let there be in F;
therefore F by multiplying D has made A,
so that A is plane, and D, F are its sides.
Again, since D, E are the least of the numbers which have the same ratio with C, B,
therefore D measures C the same number of times that E measures B. [VII. 20]
As many times, then, as E measures B, so many units let there be in G;
therefore E measures B according to the units in G;
therefore G by multiplying E has made B.
Therefore B is plane, and E, G are its sides.
Therefore A, B are plane numbers.
I say next that they are also similar.
For, <*> since F by multiplying D has made A, and by multiplying E has made C,
therefore, as D is to E, so is A to C, that is, C to B. [VII. 17]
Again, <*> since E by multiplying F, G has made C, B respectively,
therefore, as F is to G, so is C to B. [VII. 17]
But, as C is to B, so is D to E;
therefore also, as D is to E, so is F to G.
And alternately, as D is to F, so is E to G. [VII. 13]
Therefore A, B are similar plane numbers; for their sides are proportional.
Q. E. D.
Complete text
Book I