Book 8
2
Let the ratio of A to B be the given ratio in least numbers; thus it is required to find numbers in continued proportion, as many as may be prescribed, and the least that are in the ratio of A to B.
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Let four be prescribed; let A by multiplying itself make C, and by multiplying B let it make D; let B by multiplying itself make E; further, let A by multiplying C, D, E make F, G, H, and let B by multiplying E make K.
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Now, since A by multiplying itself has made C, and by multiplying B has made D, therefore, as A is to B, so is C to D. [VII. 17]
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Again, since A by multiplying B has made D, and B by multiplying itself has made E, therefore the numbers A, B by multiplying B have made the numbers D, E respectively.
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Therefore, as A is to B, so is D to E. [VII. 18]
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But, as A is to B, so is C to D; therefore also, as C is to D, so is D to E.
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And, since A by multiplying C, D has made F, G, therefore, as C is to D, so is F to G. [VII. 17]
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But, as C is to D, so was A to B; therefore also, as A is to B, so is F to G.
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Again, since A by multiplying D, E has made G, H, therefore, as D is to E, so is G to H. [VII. 17]
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But, as D is to E, so is A to B.
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Therefore also, as A is to B, so is G to H.
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And, since A, B by multiplying E have made H, K, therefore, as A is to B, so is H to K. [VII. 18]
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But, as A is to B, so is F to G, and G to H.
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Therefore also, as F is to G, so is G to H, and H to K; therefore C, D, E, and F, G, H, K are proportional in the ratio of A to B.
