If two numbers be prime to one another, and numbers fall between them in continued proportion, then, however many numbers fall between them in continued proportion, so many will also fall between each of them and an unit in continued proportion.
若两数互质，且它们之间有若干数成连比例，则它们之间有多少个数成连比例，每个数与单位之间也有同样多个数成连比例。
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Let A, B be two numbers prime to one another, and let C, D fall between them in continued proportion, and let the unit E be set out; I say that, as many numbers as fall between A, B in continued proportion, so many will also fall between either of the numbers A, B and the unit in continued proportion. For let two numbers F, G, the least that are in the ratio of A, C, D, B, be taken, three numbers H, K, L with the same property, and others more by one continually, until their multitude is equal to the multitude of A, C, D, B. [VIII. 2] Let them be taken, and let them be M, N, O, P. It is now manifest that F by multiplying itself has made H and by multiplying H has made M, while G by multiplying itself has made L and by multiplying L has made P.
设A、B互质，C、D在它们之间成连比例，取单位E。取最小数F、G，其比等于A、C、D、B的比。
[VIII. 2, Por.] And, since M, N, O, P are the least of those which have the same ratio with F, G, and A, C, D, B are also the least of those which have the same ratio with F, G, [VIII. 1] while the multitude of the numbers M, N, O, P is equal to the multitude of the numbers A, C, D, B, therefore M, N, O, P are equal to A, C, D, B respectively; therefore M is equal to A, and P to B. Now, since F by multiplying itself has made H, therefore F measures H according to the units in F. But the unit E also measures F according to the units in it; therefore the unit E measures the number F the same number of times as F measures H.
再取三个数H、K、L，以及更多一个的数，直到个数等于A、C、D、B的个数，设为M、N、O、P。则F自乘得H，乘H得M；G自乘得L，乘L得P。
Therefore, as the unit E is to the number F, so is F to H. [VII. Def. 20] Again, since F by multiplying H has made M, therefore H measures M according to the units in F. But the unit E also measures the number F according to the units in it; therefore the unit E measures the number F the same number of times as H measures M. Therefore, as the unit E is to the number F, so is H to M.
由于M、N、O、P是与F、G有相同比的最小数组，且A、C、D、B也是与F、G有相同比的最小数组，且个数相等，故M、N、O、P分别等于A、C、D、B，即M=A，P=B。
But it was also proved that, as the unit E is to the number F, so is F to H; therefore also, as the unit E is to the number F, so is F to H, and H to M. But M is equal to A; therefore, as the unit E is to the number F, so is F to H, and H to A. For the same reason also, as the unit E is to the number G, so is G to L and L to B.
因F自乘得H，故单位E与F的比等于F与H的比；又F乘H得M，故单位E与F的比等于H与M的比。因此单位E与F的比等于F与H的比，也等于H与A的比。同理，单位E与G的比等于G与L的比，也等于L与B的比。