Prime numbers are more than any assigned multitude of prime numbers.
素数的个数多于任意给定的素数集合。
素数无穷：取已知素数 A、B、C 的最小公倍数 DE，在 E 处加单位 EF 得 DF；DF 要么是新素数，要么有不同于 A、B、C 的素因子 G。
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Let A, B, C be the assigned prime numbers; I say that there are more prime numbers than A, B, C. For let the least number measured by A, B, C be taken, and let it be DE; let the unit DF be added to DE. Then EF is either prime or not. First, let it be prime; then the prime numbers A, B, C, EF have been found which are more than A, B, C.
设A、B、C为给定的素数，取被A、B、C整除的最小数DE。
Next, let EF not be prime; therefore it is measured by some prime number. [VII. 31] Let it be measured by the prime number G. I say that G is not the same with any of the numbers A, B, C.
在DE上加单位DF得EF。若EF是素数，则已找到A、B、C、EF四个素数，多于原集合。
For, if possible, let it be so. Now A, B, C measure DE; therefore G also will measure DE. But it also measures EF.
若EF不是素数，则它被某个素数G整除。
Therefore G, being a number, will measure the remainder, the unit DF: which is absurd. Therefore G is not the same with any one of the numbers A, B, C. And by hypothesis it is prime.
若G与A、B、C之一相同，则G整除DE，又整除EF，从而整除差DF（单位），矛盾。故G是不同于A、B、C的新素数。