第8卷命题 8 · 连比例数插入定理

Let the numbers C, D fall between the two numbers A, B in continued proportion with them, and let E be made in the same ratio to F as A is to B; I say that, as many numbers as have fallen between A, B in continued proportion, so many will also fall between E, F in continued proportion. For, as many as A, B, C, D are in multitude, let so many numbers G, H, K, L, the least of those which have the same ratio with A, C, D, B, be taken; [VII. 33] therefore the extremes of them G, L are prime to one another. [VIII. 3] Now, since A, C, D, B are in the same ratio with G, H, K, L, and the multitude of the numbers A, C, D, B is equal to the multitude of the numbers G, H, K, L, therefore, ex aequali, as A is to B, so is G to L.

设数C、D插入两数A、B之间成连比例，且E与F的比等于A与B的比。

[VII. 14] But, as A is to B, so is E to F; therefore also, as G is to L, so is E to F. But G, L are prime, primes are also least, [VII. 21] and the least numbers measure those which have the same ratio the same number of times, the greater the greater and the less the less, that is, the antecedent the antecedent and the consequent the consequent.

取与A、C、D、B有相同比的最小数组G、H、K、L，则G与L互质。

[VII. 20] Therefore G measures E the same number of times as L measures F. Next, as many times as G measures E, so many times let H, K also measure M, N respectively; therefore G, H, K, L measure E, M, N, F the same number of times. Therefore G, H, K, L are in the same ratio with E, M, N, F.

由等比传递，G与L的比等于E与F的比；因G、L互质且最小，故G、L分别量尽E、F相同次数。

[VII. Def. 20] But G, H, K, L are in the same ratio with A, C, D, B; therefore A, C, D, B are also in the same ratio with E, M, N, F. But A, C, D, B are in continued proportion; therefore E, M, N, F are also in continued proportion.

设H、K分别量尽M、N相同次数，则G、H、K、L与E、M、N、F有相同比，从而E、M、N、F成连比例。