Book 8
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Let them be taken, and let them be M, N, O, P.
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It is now manifest that F by multiplying itself has made H and by multiplying H has made M, while G by multiplying itself has made L and by multiplying L has made P. [VIII. 2, Por.]
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And, since M, N, O, P are the least of those which have the same ratio with F, G, and A, C, D, B are also the least of those which have the same ratio with F, G, [VIII. 1] while the multitude of the numbers M, N, O, P is equal to the multitude of the numbers A, C, D, B, therefore M, N, O, P are equal to A, C, D, B respectively; therefore M is equal to A, and P to B.
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Now, since F by multiplying itself has made H, therefore F measures H according to the units in F.
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But the unit E also measures F according to the units in it; therefore the unit E measures the number F the same number of times as F measures H.
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Therefore, as the unit E is to the number F, so is F to H. [VII. Def. 20]
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Again, since F by multiplying H has made M, therefore H measures M according to the units in F.
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But the unit E also measures the number F according to the units in it; therefore the unit E measures the number F the same number of times as H measures M.
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Therefore, as the unit E is to the number F, so is H to M.
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But it was also proved that, as the unit E is to the number F, so is F to H; therefore also, as the unit E is to the number F, so is F to H, and H to M.
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But M is equal to A; therefore, as the unit E is to the number F, so is F to H, and H to A.
