Book 9
19
Now either they are not in continued proportion, and the extremes of them are prime to one another; or they are in continued proportion, and the extremes of them are not prime to one another; or they are not in continued proportion, nor are the extremes of them prime to one another; or they are in continued proportion, and the extremes of them are prime to one another.
19
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]
19
†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.
19
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.
19
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]
19
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]
19
Therefore A measures C as antecedent antecedent.
19
But it also measures itself; therefore A measures A, C which are prime to one another: which is impossible.
19
Therefore it is not possible to find a fourth proportional to A, B, C.†
19
Next, let A, B, C be again in continued proportion, but let A, C not be prime to one another.
19
I say that it is possible to find a fourth proportional to them.
19
For let B by multiplying C make D; therefore A either measures D or does not measure it.
19
First, let it measure it according to E; therefore A by multiplying E has made D.
19
But, further, B has also by multiplying C made D; therefore the product of A, E is equal to the product of B, C; therefore, proportionally, as A is to B, so is C to E; [VII. 19] therefore E has been found a fourth proportional to A, B, C.
