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.
若任意多个成连比例的数是与它们有相同比的最小数，则它们的两端互素。
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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. 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.
设A、B、C、D是成连比例且与它们有相同比的最小数，断言A与D互素。
Let them be taken, and let them be L, M, N, O. Now, since E, F are the least of those which have the same ratio with them, they are prime to one another.
取与A、B、C、D有相同比的最小数E、F，则E、F互素。
[VII. 22] 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] 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.
由E、F自乘得G、K，再乘得L、O，故G、K互素，L、O互素。
And L, O are prime to one another.
因A、B、C、D是与它们有相同比的最小数，且L、M、N、O也是同比最小数且个数相等，故A=L，D=O，而L、O互素，所以A、D互素。