Given two numbers, to investigate whether it is possible to find a third proportional to them.

Let A, B be the given two numbers, and let it be required to investigate whether it is possible to find a third proportional to them.

Now A, B are either prime to one another or not.

And, if they are prime to one another, it has been proved that it is impossible to find a third proportional to them. [IX. 16]

Next, let A, B not be prime to one another, and let B by multiplying itself make C.

Then A either measures C or does not measure it.

First, let it measure it according to D; therefore A by multiplying D has made C.

But, further, B has also by multiplying itself made C; therefore the product of A, D is equal to the square on B.

Therefore, as A is to B, so is B to D; [VII. 19] therefore a third proportional number D has been found to A, B.

Next, let A not measure C; I say that it is impossible to find a third proportional number to A, B.

For, if possible, let D, such third proportional, have been found.

Therefore the product of A, D is equal to the square on B.

But the square on B is C; therefore the product of A, D is equal to C.

Hence A by multiplying D has made C; therefore A measures C according to D.

But, by hypothesis, it also does not measure it: which is absurd.

Therefore it is not possible to find a third proportional number to A, B when A does not measure C. Q. E. D.