To find numbers in continued proportion, as many as may be prescribed, and the least that are in a given ratio.
求给定个数且具有给定比的最小连比例数。
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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. 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. 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] 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. Therefore, as A is to B, so is D to E. [VII. 18] But, as A is to B, so is C to D; therefore also, as C is to D, so is D to E.
设A与B为给定比的最小两数，要求四个连比例数。令A自乘得C，A乘B得D，B自乘得E；再令A乘C、D、E得F、G、H，B乘E得K。
And, since A by multiplying C, D has made F, G, therefore, as C is to D, so is F to G. [VII. 17] But, as C is to D, so was A to B; therefore also, as A is to B, so is F to G. Again, since A by multiplying D, E has made G, H, therefore, as D is to E, so is G to H. [VII. 17] But, as D is to E, so is A to B. Therefore also, as A is to B, so is G to H. And, since A, B by multiplying E have made H, K, therefore, as A is to B, so is H to K.
由VII.17，A:B = C:D；由VII.18，A:B = D:E，故C:D = D:E。
[VII. 18] But, as A is to B, so is F to G, and G to H. 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. I say next that they are the least numbers that are so. For, since A, B are the least of those which have the same ratio with them, and the least of those which have the same ratio are prime to one another, [VII. 22] therefore A, B are prime to one another. And the numbers A, B by multiplying themselves respectively have made the numbers C, E, and by multiplying the numbers C, E respectively have made the numbers F, K; therefore C, E and F, K are prime to one another respectively. [VII. 27] But, if there be as many numbers as we please in continued proportion, and the extremes of them be prime to one another, they are the least of those which have the same ratio with them.
由VII.17，A乘C、D得F、G，故C:D = F:G，而C:D = A:B，故A:B = F:G。同理，A:B = G:H，且由VII.18，A:B = H:K。
[VIII. 1] Therefore C, D, E and F, G, H, K are the least of those which have the same ratio with A, B. Q. E. D. PORISM.
因A、B互素（VII.22），由VII.27，C与E、F与K分别互素；由VIII.1，C、D、E和F、G、H、K是具有A:B比的最小连比例数。