If then A, B, C are in continued proportion, and the extremes of them A, C are prime to one another, it has been proved that it is impossible to find a fourth proportional number to them. [IX. 17]
Next, let A, B, C not be in continued proportion, the extremes being again prime to one another;
I say that in this case also it is impossible to find a fourth proportional to them.
For, if possible, let D have been found, so that, as A is to B, so is C to D, and let it be contrived that, as B is to C, so is D to E.
Now, since, as A is to B, so is C to D, and, as B is to C, so is D to E,
therefore, ex aequali, as A is to C, so is C to E. [VII. 14]
But A, C are prime, primes are also least, [VII. 21]
and the least numbers measure those which have the same ratio, the antecedent the antecedent and the consequent the consequent. [VII. 20]
Therefore A measures C as antecedent antecedent.
But it also measures itself;
therefore A measures A, C which are prime to one another: which is impossible.
Therefore it is not possible to find a fourth proportional to A, B, C.<*>
Complete text
Book I