If four numbers be proportional, the number produced from the first and fourth will be equal to the number produced from the second and third; and, if the number produced from the first and fourth be equal to that produced from the second and third, the four numbers will be proportional.
若四个数成比例，则第一个与第四个数之积等于第二个与第三个数之积；反之，若第一个与第四个数之积等于第二个与第三个数之积，则这四个数成比例。
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Let A, B, C, D be four numbers in proportion, so that, as A is to B, so is C to D; and let A by multiplying D make E, and let B by multiplying C make F; I say that E is equal to F. For let A by multiplying C make G. Since, then, A by multiplying C has made G, and by multiplying D has made E, the number A by multiplying the two numbers C, D has made G, E.
设A、B、C、D成比例，即A比B等于C比D。令A乘D得E，B乘C得F。需证E等于F。
Therefore, as C is to D, so is G to E. [VII. 17] But, as C is to D, so is A to B; therefore also, as A is to B, so is G to E. Again, since A by multiplying C has made G, but, further, B has also by multiplying C made F, the two numbers A, B by multiplying a certain number C have made G, F.
令A乘C得G。因A乘C得G，乘D得E，故A乘C、D得G、E。由VII.17，C比D等于G比E。又C比D等于A比B，故A比B等于G比E。
Therefore, as A is to B, so is G to F. [VII. 18] But further, as A is to B, so is G to E also; therefore also, as G is to E, so is G to F. Therefore G has to each of the numbers E, F the same ratio; therefore E is equal to F.
又因A乘C得G，B乘C得F，故A、B乘同一数C得G、F。由VII.18，A比B等于G比F。但A比B也等于G比E，故G比E等于G比F。因此G与E、F各有相同比，由V.9得E等于F。
[cf. V. 9] Again, let E be equal to F; I say that, as A is to B, so is C to D. For, with the same construction, since E is equal to F, therefore, as G is to E, so is G to F. [cf. V. 7] But, as G is to E, so is C to D, [VII. 17] and, as G is to F, so is A to B.
反之，设E等于F。需证A比B等于C比D。同上构造，因E等于F，故G比E等于G比F（V.7）。但G比E等于C比D（VII.17），G比F等于A比B（VII.18），故A比B等于C比D。