Kitap 8
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Let A, B be cube numbers, and let C be the side of A, and D of B; I say that between A, B there are two mean proportional numbers, and A has to B the ratio triplicate of that which C has to D.
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For let C by multiplying itself make E, and by multiplying D let it make F; let D by multiplying itself make G, and let the numbers C, D by multiplying F make H, K respectively.
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Now, since A is a cube, and C its side, and C by multiplying itself has made E, therefore C by multiplying itself has made E and by multiplying E has made A.
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For the same reason also D by multiplying itself has made G and by multiplying G has made B.
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And, since C by multiplying the numbers C, D has made E, F respectively, therefore, as C is to D, so is E to F. [VII. 17]
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For the same reason also, as C is to D, so is F to G. [VII. 18]
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Again, since C by multiplying the numbers E, F has made A, H respectively, therefore, as E is to F, so is A to H. [VII. 17]
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But, as E is to F, so is C to D.
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Therefore also, as C is to D, so is A to H.
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Again, since the numbers C, D by multiplying F have made H, K respectively, therefore, as C is to D, so is H to K. [VII. 18]
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Again, since D by multiplying each of the numbers F, G has made K, B respectively, therefore, as F is to G, so is K to B. [VII. 17]
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But, as F is to G, so is C to D; therefore also, as C is to D, so is A to H, H to K, and K to B.
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Therefore H, K are two mean proportionals between A, B.
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I say next that A also has to B the ratio triplicate of that which C has to D.
