Now, since A by multiplying itself has made C, and by multiplying B has made D,
therefore, as A is to B, so is C to D. [VII. 17]
Again, since A by multiplying B has made D,
and B by multiplying itself has made E,
therefore the numbers A, B by multiplying B have made the numbers D, E respectively.
Therefore, as A is to B, so is D to E. [VII. 18]
But, as A is to B, so is C to D;
therefore also, as C is to D, so is D to E.
And, since A by multiplying C, D has made F, G,
therefore, as C is to D, so is F to G. [VII. 17]
But, as C is to D, so was A to B;
therefore also, as A is to B, so is F to G.
Again, since A by multiplying D, E has made G, H,
therefore, as D is to E, so is G to H. [VII. 17]
But, as D is to E, so is A to B.
Therefore also, as A is to B, so is G to H.
And, since A, B by multiplying E have made H, K,
therefore, as A is to B, so is H to K. [VII. 18]
But, as A is to B, so is F to G, and G to H.
Therefore also, as F is to G, so is G to H, and H to K;
therefore C, D, E, and F, G, H, K are proportional in the ratio of A to B.
I say next that they are the least numbers that are so.
Complete text
Book I