If as many numbers as we please beginning from an unit be in continued proportion, and the number after the unit be prime, the greatest will not be measured by any except those which have a place among the proportional numbers.
若从单位起任意多个数成连比例，且单位后的数为素数，则最大数不能被任何不在该等比数列中的数整除。
从单位起连比例 A、B、C、D（A 为质数），D 不被 A、B、C 以外的数量度；反证中引入 E、F、G、H 形成矛盾。
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Let there be as many numbers as we please, A, B, C, D, beginning from an unit and in continued proportion, and let A, the number after the unit, be prime; I say that D, the greatest of them, will not be measured by any other number except A, B, C. For, if possible, let it be measured by E, and let E not be the same with any of the numbers A, B, C. It is then manifest that E is not prime. For, if E is prime and measures D, it will also measure A [IX. 12], which is prime, though it is not the same with it: which is impossible. Therefore E is not prime. Therefore it is composite. But any composite number is measured by some prime number; [VII. 31] therefore E is measured by some prime number. I say next that it will not be measured by any other prime except A. For, if E is measured by another, and E measures D, that other will also measure D; so that it will also measure A [IX. 12], which is prime, though it is not the same with it: which is impossible. Therefore A measures E.
设从单位起的连比例数A、B、C、D，A为素数。假设D被E整除，且E不同于A、B、C。
And, since E measures D, let it measure it according to F. I say that F is not the same with any of the numbers A, B, C. For, if F is the same with one of the numbers A, B, C, and measures D according to E, therefore one of the numbers A, B, C also measures D according to E. But one of the numbers A, B, C measures D according to some one of the numbers A, B, C; [IX. 11] therefore E is also the same with one of the numbers A, B, C: which is contrary to the hypothesis. Therefore F is not the same as any one of the numbers A, B, C. Similarly we can prove that F is measured by A, by proving again that F is not prime. For, if it is, and measures D, it will also measure A [IX. 12], which is prime, though it is not the same with it: which is impossible; therefore F is not prime. Therefore it is composite. But any composite number is measured by some prime number; [VII. 31] therefore F is measured by some prime number.
E非素数，否则E将整除A（IX.12），与A为素数矛盾。故E为合数，被某素数整除，该素数必为A，否则该素数整除D进而整除A，矛盾。
I say next that it will not be measured by any other prime except A. For, if any other prime number measures F, and F measures D, that other will also measure D; so that it will also measure A [IX. 12], which is prime, though it is not the same with it: which is impossible. Therefore A measures F. And, since E measures D according to F, therefore E by multiplying F has made D. But, further, A has also by multiplying C made D; [IX. 11] therefore the product of A, C is equal to the product of E, F. Therefore, proportionally, as A is to E, so is F to C. [VII. 19] But A measures E; therefore F also measures C. Let it measure it according to G. Similarly, then, we can prove that G is not the same with any of the numbers A, B, and that it is measured by A. And, since F measures C according to G therefore F by multiplying G has made C.
设E整除D得F，则F不同于A、B、C。类似可证F非素数，且被A整除。由A·C = D = E·F，得A:E = F:C，故F整除C，设得G。
But, further, A has also by multiplying B made C; [IX. 11] therefore the product of A, B is equal to the product of F, G. Therefore, proportionally, as A is to F, so is G to B. [VII. 19] But A measures F; therefore G also measures B. Let it measure it according to H. Similarly then we can prove that H is not the same with A. And, since G measures B according to H, therefore G by multiplying H has made B. But further A has also by multiplying itself made B; [IX. 8] therefore the product of H, G is equal to the square on A. Therefore, as H is to A, so is A to G. [VII. 19] But A measures G;< therefore H also measures A, which is prime, though it is not the same with it: which is absurd.
由A·B = C = F·G，得A:F = G:B，故G整除B，设得H。由A·A = B = G·H，得H:A = A:G，故H整除A，与A为素数矛盾。