If there be as many numbers as we please in continued proportion, and the first do not measure the second, neither will any other measure any other.
若有任意多个数成连比例，且第一个数不能量尽第二个数，则其中任何数也不能量尽任何其他数。
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Let there be as many numbers as we please, A, B, C, D, E, in continued proportion, and let A not measure B; I say that neither will any other measure any other. Now it is manifest that A, B, C, D, E do not measure one another in order; for A does not even measure B. I say, then, that neither will any other measure any other.
设A、B、C、D、E成连比例，且A不能量尽B。显然这些数不能依次量尽彼此，因为A已不能量尽B。
For, if possible, let A measure C. And, however many A, B, C are, let as many numbers F, G, H, the least of those which have the same ratio with A, B, C, be taken.
假设A能量尽C。取与A、B、C同比的最小数组F、G、H（根据VII.33）。
[VII. 33] Now, since F, G, H are in the same ratio with A, B, C, and the multitude of the numbers A, B, C is equal to the multitude of the numbers F, G, H, therefore, ex aequali, as A is to C, so is F to H. [VII. 14] And since, as A is to B, so is F to G, while A does not measure B, therefore neither does F measure G; [VII. Def. 20] therefore F is not an unit, for the unit measures any number.
由于F、G、H与A、B、C同比且个数相等，根据VII.14，A比C等于F比H。又因A比B等于F比G，且A不能量尽B，故F不能量尽G（根据VII.定义20），因此F不是单位，因为单位能量尽任何数。
Now F, H are prime to one another. [VIII. 3] And, as F is to H, so is A to C; therefore neither does A measure C.
根据VIII.3，F与H互质。而F比H等于A比C，故A不能量尽C。