Kitap 5
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And GH, LM are equimultiples of AE, CF, while KO, NP are other, chance, equimultiples of EB, FD; therefore, as AE is to EB, so is CF to FD.
17
Therefore etc. Q. E. D.
PROPOSITION 18.
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If magnitudes be proportional separando, they will also be proportional componendo.
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Let AE, EB, CF, FD be magnitudes proportional separando, so that, as AE is to EB, so is CF to FD; I say that they will also be proportional componendo, that is, as AB is to BE, so is CD to FD.
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For, if CD be not to DF as AB to BE, then, as AB is to BE, so will CD be either to some magnitude less than DF or to a greater.
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First, let it be in that ratio to a less magnitude DG.
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Then, since, as AB is to BE, so is CD to DG, they are magnitudes proportional componendo; so that they will also be proportional separando. [V. 17]
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Therefore, as AE is to EB, so is CG to GD.
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But also, by hypothesis, as AE is to EB, so is CF to FD.
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Therefore also, as CG is to GD, so is CF to FD. [V. 11]
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But the first CG is greater than the third CF; therefore the second GD is also greater than the fourth FD. [V. 14]
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But it is also less: which is impossible.
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Therefore, as AB is to BE, so is not CD to a less magnitude than FD.
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Similarly we can prove that neither is it in that ratio to a greater; it is therefore in that ratio to FD itself.
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Therefore etc. Q. E. D.
PROPOSITION 19
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If, as a whole is to a whole, so is a part subtracted to a part subtracted, the remainder will also be to the remainder as whole to whole.
