Kitap 7
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Let A, B, C, D be four numbers in proportion, so that, as A is to B, so is C to D; and let A by multiplying D make E, and let B by multiplying C make F; I say that E is equal to F.
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For let A by multiplying C make G.
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Since, then, A by multiplying C has made G, and by multiplying D has made E, the number A by multiplying the two numbers C, D has made G, E.
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Therefore, as C is to D, so is G to E. [VII. 17]
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But, as C is to D, so is A to B; therefore also, as A is to B, so is G to E.
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Again, since A by multiplying C has made G, but, further, B has also by multiplying C made F, the two numbers A, B by multiplying a certain number C have made G, F.
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Therefore, as A is to B, so is G to F. [VII. 18]
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But further, as A is to B, so is G to E also; therefore also, as G is to E, so is G to F.
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Therefore G has to each of the numbers E, F the same ratio; therefore E is equal to F. [cf. V. 9]
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Again, let E be equal to F; I say that, as A is to B, so is C to D.
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For, with the same construction, since E is equal to F, therefore, as G is to E, so is G to F. [cf. V. 7]
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But, as G is to E, so is C to D, [VII. 17] and, as G is to F, so is A to B. [VII. 18]
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Therefore also, as A is to B, so is C to D. Q. E. D.
PROPOSITION 20.
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The least numbers of those which have the same ratio with them measure those which have the same ratio the same number of times, the greater the greater and the less the less.
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For let CD, EF be the least numbers of those which have the same ratio with A, B; I say that CD measures A the same number of times that EF measures B.
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Now CD is not parts of A.
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For, if possible, let it be so; therefore EF is also the same parts of B that CD is of A. [VII. 13 and Def. 20]
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Therefore, as many parts of A as there are in CD, so many parts of B are there also in EF.
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Let CD be divided into the parts of A, namely CG, GD, and EF into the parts of B, namely EH, HF; thus the multitude of CG, GD will be equal to the multitude of EH, HF.
