The following is as given in Sir Thomas L. Heath's translation, which can be found in the book The Thirteen Books of The Elements, Vol. 2.

Book I | Book II | Book IX |
---|---|---|

Definitions, Postulates, and Common Notions | Definitions | Proposition 20 |

Proposition 1, Proposition 3, | Proposition 14 | Proposition 36 |

Proposition 5, Proposition 6, | ||

Proposition 29, Proposition 47 |

*Prime numbers are more than any assigned multitude of prime numbers.*

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`.

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`.

For, if possible, let it be so.

Now `A`, `B`, `C` measure `DE`;

therefore `G` also will measure `DE`.

But it also measures `EF`.

Therefore `G`, being a number, will measure the remainder, the unit `DF`;

which is absurd.

Therefore `G` is not the same with an y one of the numbers `A`, `B`, `C`.

And by hypothesis it is prime.

Therefore the prime numbers `A`, `B`, `C`, `G` have been found which are more than the assigned multitude of `A`, `B`, `C`.

Q.E.D.