If there be as many numbers as we please in continued proportion, and the extremes of them be prime to one another, the last will not be to any other number as the first to the second.
若有任意多个数成连比例，且其首末两项互质，则末项与任何其它数的比都不等于首项与第二项的比。
连比例 A、B、C、D，且首末 A、D 互质；假设 D:E = A:B，推得 A 量 D，矛盾。
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For let there be as many numbers as we please, A, B, C, D, in continued proportion, and let the extremes of them, A, D, be prime to one another; I say that D is not to any other number as A is to B. For, if possible, as A is to B, so let D be to E; therefore, alternately, as A is to D, so is B to E. [VII. 13] But A, D are prime, primes are also least, [VII. 21] and the least numbers measure those which have the same ratio the same number of times, the antecedent the antecedent and the consequent the consequent.
设A、B、C、D成连比例，且A与D互质。假设D与某数E的比等于A与B的比。
[VII. 20] Therefore A measures B. And, as A is to B, so is B to C.
由比例更比，得A与D的比等于B与E的比（VII.13）。因A、D互质，故它们是最小的（VII.21），最小数能量尽与之同比的数相同次数（VII.20），所以A量尽B。
Therefore B also measures C; so that A also measures C. And since, as B is to C, so is C to D, and B measures C, therefore C also measures D.
由于A比B等于B比C，故B量尽C，从而A也量尽C。又因B比C等于C比D，且B量尽C，故C量尽D。
But A measured C; so that A also measures D. But it also measures itself; therefore A measures A, D which are prime to one another : which is impossible.
但A量尽C，所以A也量尽D。而A也量尽自身，故A量尽互质的A和D，这是不可能的。因此假设不成立。