Book 8
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For, the ratios being given which C has to E and D to F, let the least numbers G, H, K that are continuously in the ratios C : E, D : F be taken, so that, as C is to E, so is G to H, and, as D is to F, so is H to K. [VIII. 4]
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And let D by multiplying E make L.
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Now, since D by multiplying C has made A, and by multiplying E has made L, therefore, as C is to E, so is A to L. [VII. 17]
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But, as C is to E, so is G to H; therefore also, as G is to H, so is A to L.
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Again, since E by multiplying D has made L, and further by multiplying F has made B, therefore, as D is to F, so is L to B. [VII. 17]
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But, as D is to F, so is H to K; therefore also, as H is to K, so is L to B.
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But it was also proved that, as G is to H, so is A to L; therefore, ex aequali, as G is to K, so is A to B. [VII. 14]
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But G has to K the ratio compounded of the ratios of the sides; therefore A also has to B the ratio compounded of the ratios of the sides. Q. E. D. 1, 5, 29, 31. compounded of the ratios of their sides. As in VI. 23, the Greek has the less exact phrase, compounded of their sides.
PROPOSITION 6.
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If there be as many numbers as we please in continued proportion, and the first do not measure the second, neither will any other measure any other.
6
Let there be as many numbers as we please, A, B, C, D, E, in continued proportion, and let A not measure B; I say that neither will any other measure any other.
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Now it is manifest that A, B, C, D, E do not measure one another in order; for A does not even measure B.
