For let C by multiplying itself make E, and let D by multiplying itself make G;
further, let C by multiplying D make F, and let C, D by multiplying F make H, K respectively.
Now it is manifest that E, F, G and A, H, K, B are continuously proportional in the ratio of C to D. [VIII. 11, 12]
And, since A, H, K, B are continuously proportional, and A measures B, therefore it also measures H. [VIII. 7]
And, as A is to H, so is C to D; therefore C also measures D. [VII. Def. 20]
Next, let C measure D;
I say that A will also measure B.
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Book I