For let C by multiplying D make K, and let F by multiplying G make L.
Now, since C, D are in the same ratio with F, G, and K is the product of C, D,
and L the product of F, G, K, L are similar plane numbers; [VII. Def. 21]
therefore between K, L there is one mean proportional number. [VIII. 18]
Let it be M.
Therefore M is the product of D, F, as was proved in the theorem preceding this. [VIII. 18]
Now, since D by multiplying C has made K, and by multiplying F has made M,
therefore, as C is to F, so is K to M. [VII. 17]
But, as K is to M, so is M to L.
Therefore K, M, L are continuously proportional in the ratio of C to F.
And since, as C is to D, so is F to G, alternately therefore, as C is to F, so is D to G. [VII. 13]
For the same reason also, as D is to G, so is E to H.
Therefore K, M, L are continuously proportional in the ratio of C to F, in the ratio of D to G, and also in the ratio of E to H.
Next, let E, H by multiplying M make N, O respectively.
Now, since A is a solid number, and C, D, E are its sides, therefore E by multiplying the product of C, D has made A.
But the product of C, D is K; therefore E by multiplying K has made A.
For the same reason also H by multiplying L has made B.
Now, since E by multiplying K has made A, and further also by multiplying M has made N,
therefore, as K is to M, so is A to N. [VII. 17]
But, as K is to M, so is C to F, D to G, and also E to H;
therefore also, as C is to F, D to G, and E to H, so is A to N.
Again, since E, H by multiplying M have made N, O respectively,
therefore, as E is to H, so is N to O. [VII. 18]
But, as E is to H, so is C to F and D to G;
therefore also, as C is to F, D to G, and E to H, so is A to N and N to O.
Again, since H by multiplying M has made O, and further also by multiplying L has made B,
therefore, as M is to L, so is O to B. [VII. 17]
But, as M is to L, so is C to F, D to G, and E to H.
Therefore also, as C is to F, D to G, and E to H, so not only is O to B, but also A to N and N to O.
Therefore A, N, O, B are continuously proportional in the aforesaid ratios of the sides.
Complete text
Book I