Book 8
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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.
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For the same reason also, as the unit E is to the number G, so is G to L and L to B.
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Therefore, as many numbers as have fallen between A, B in continued proportion, so many numbers also have fallen between each of the numbers A, B and the unit E in continued proportion. Q. E. D.
PROPOSITION 10.
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If numbers fall between each of two numbers and an unit in continued proportion, however many numbers fall between each of them and an unit in continued proportion, so many also will fall between the numbers themselves in continued proportion.
