elem.5.23
若有三个量及与之个数相等的另三个量，它们两两成同一比例且比例形式为扰动比例，则它们也成等距比例。
扰动同比传递：A:B = E:F、B:C = D:E，则 A:C = D:F。G、H、K、L、M、N 为对应等倍量。
正文图形由校订坐标生成；点、线、角、圆可与证明和问答联动。
Let there be three magnitudes A, B, C, and others equal to them in multitude, which, taken two and two together, are in the same proportion, namely D, E, F; and let the proportion of them be perturbed, so that, as A is to B, so is E to F, and, as B is to C, so is D to E; I say that, as A is to C, so is D to F. Of A, B, D let equimultiples G, H, K be taken, and of C, E, F other, chance, equimultiples L, M, N. Then, since G, H are equimultiples of A, B, and parts have the same ratio as the same multiples of them, [V. 15] therefore, as A is to B, so is G to H. For the same reason also, as E is to F, so is M to N.
取A、B、D的等倍量G、H、K，及C、E、F的任意等倍量L、M、N。
And, as A is to B, so is E to F; therefore also, as G is to H, so is M to N. [V. 11] Next, since, as B is to C, so is D to E, alternately, also, as B is to D, so is C to E. [V. 16] And, since H, K are equimultiples of B, D, and parts have the same ratio as their equimultiples, therefore, as B is to D, so is H to K.
由A:B=E:F及V.15，得G:H=M:N。
[V. 15] But, as B is to D, so is C to E; therefore also, as H is to K, so is C to E. [V. 11] Again, since L, M are equimultiples of C, E, therefore, as C is to E, so is L to M. [V. 15] But, as C is to E, so is H to K; therefore also, as H is to K, so is L to M, [V. 11] and, alternately, as H is to L, so is K to M. [V. 16] But it was also proved that, as G is to H, so is M to N.
由B:C=D:E，经更比得B:D=C:E；再取等倍量得H:K=L:M。
Since, then, there are three magnitudes G, H, L, and others equal to them in multitude K, M, N, which taken two and two together are in the same ratio, and the proportion of them is perturbed, therefore, ex aequali, if G is in excess of L, K is also in excess of N; if equal, equal; and if less, less. [V. 21] And G, K are equimultiples of A, D, and L, N of C, F. Therefore, as A is to C, so is D to F.
由V.21，G、H、L与K、M、N成扰动比例，故若G大于L则K大于N，等等；因此A:C=D:F。