elem.5.6
若两个量分别是另两个量的等倍量，且从它们中减去的量也是后两个量的等倍量，则余量要么等于后两个量，要么是它们的等倍量。
AB、CD 分别是 E、F 的等倍量；AG、CH 也是 E、F 的等倍量。GB 是 AB 减 AG 的余量，HD 是 CD 减 CH 的余量。构造 K 使 KC = F，则 KH = CD，进而 GB 与 HD 与 E、F 之间也成等倍关系。
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For let two magnitudes AB, CD be equimultiples of two magnitudes E, F, and let AG, CH subtracted from them be equimultiples of the same two E, F; I say that the remainders also, GB, HD, are either equal to E, F or equimultiples of them. For, first, let GB be equal to E; I say that HD is also equal to F. For let CK be made equal to F.
设AB、CD分别是E、F的等倍量，AG、CH也是E、F的等倍量。
Since AG is the same multiple of E that CH is of F, while GB is equal to E and KC to F, therefore AB is the same multiple of E that KH is of F. [V. 2] But, by hypothesis, AB is the same multiple of E that CD is of F; therefore KH is the same multiple of F that CD is of F.
先设GB等于E，则HD等于F。作CK等于F。
Since then each of the magnitudes KH, CD is the same multiple of F, therefore KH is equal to CD. Let CH be subtracted from each; therefore the remainder KC is equal to the remainder HD. But F is equal to KC; therefore HD is also equal to F.
因AG是E的倍量与CH是F的倍量相同，且GB等于E、KC等于F，故AB是E的倍量与KH是F的倍量相同。
Hence, if GB is equal to E, HD is also equal to F. Similarly we can prove that, even if GB be a multiple of E, HD is also the same multiple of F.
由假设AB是E的倍量与CD是F的倍量相同，故KH与CD同为F的等倍量，因此KH等于CD。两边减去CH，得KC等于HD。但F等于KC，故HD等于F。类似可证若GB是E的倍量，则HD是F的同倍量。