Book 8
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If as many numbers as we please in continued proportion be the least of those which have the same ratio with them, the extremes of them are prime to one another.
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Let as many numbers as we please, A, B, C, D, in continued proportion be the least of those which have the same ratio with them; I say that the extremes of them A, D are prime to one another.
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For let two numbers E, F, the least that are in the ratio of A, B, C, D, be taken, [VII. 33] then three others G, H, K with the same property; and others, more by one continually, [VIII. 2] until the multitude taken becomes equal to the multitude of the numbers A, B, C, D.
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Let them be taken, and let them be L, M, N, O.
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Now, since E, F are the least of those which have the same ratio with them, they are prime to one another. [VII. 22]
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And, since the numbers E, F by multiplying themselves respectively have made the numbers G, K, and by multiplying the numbers G, K respectively have made the numbers L, O, [VIII. 2, Por.] therefore both G, K and L, O are prime to one another. [VII. 27]
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And, since A, B, C, D are the least of those which have the same ratio with them, while L, M, N, O are the least that are in the same ratio with A, B, C, D, and the multitude of the numbers A, B, C, D is equal to the multitude of the numbers L, M, N, O, therefore the numbers A, B, C, D are equal to the numbers L, M, N, O respectively; therefore A is equal to L, and D to O.
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And L, O are prime to one another.
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Therefore A, D are also prime to one another. Q. E. D.
PROPOSITION 4.
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Given as many ratios as we please in least numbers, to find numbers in continued proportion which are the least in the given ratios.
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Let the given ratios in least numbers be that of A to B, that of C to D, and that of E to F; thus it is required to find numbers in continued proportion which are the least that are in the ratio of A to B, in the ratio of C to D, and in the ratio of E to F.
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Let G, the least number measured by B, C, be taken. [VII. 34]
