elem.5.16
若四个量成比例，则它们也成交错比例。
若 A:B = C:D，则交错比例 A:C = B:D 也成立。下方 E、F、G、H 为对应等倍量。
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Let A, B, C, D be four proportional magnitudes, so that, as A is to B, so is C to D; I say that they will also be so alternately, that is, as A is to C, so is B to D. For of A, B let equimultiples E, F be taken, and of C, D other, chance, equimultiples G, H. Then, since E is the same multiple of A that F is of B, and parts have the same ratio as the same multiples of them, [V. 15] therefore, as A is to B, so is E to F.
设A、B、C、D成比例，即A比B等于C比D。取A、B的等倍量E、F，以及C、D的任意等倍量G、H。
But as A is to B, so is C to D; therefore also, as C is to D, so is E to F. [V. 11] Again, since G, H are equimultiples of C, D, therefore, as C is to D, so is G to H.
由V.15，A比B等于E比F；又A比B等于C比D，故C比D等于E比F。
[V. 15] But, as C is to D, so is E to F; therefore also, as E is to F, so is G to H. [V. 11] But, if four magnitudes be proportional, and the first be greater than the third, the second will also be greater than the fourth; if equal, equal; and if less, less.
由V.15，C比D等于G比H；结合上一步，得E比F等于G比H。
[V. 14] Therefore, if E is in excess of G, F is also in excess of H, if equal, equal, and if less, less. Now E, F are equimultiples of A, B, and G, H other, chance, equimultiples of C, D; therefore, as A is to C, so is B to D.
由V.14，若E大于G则F大于H，等于则等于，小于则小于；由V.定义5，得A比C等于B比D。