Let A and B be the given two numbers. It is required to investigate whether it is possible to find a third proportional to them.
Now A and B are either relatively prime or not. And, if they are relatively prime, it was proved that it is impossible to find a third proportional to them.
Next, let A and B not be relatively prime, and let B multiplied by itself make C. Then A either measures C or does not measure it.
First, let it measure it according to D, therefore A multiplied by D makes C. But, further, B multiplied by itself makes C, therefore the product of A and D equals the square on B.
Therefore A is to B as B is to D, therefore a third proportional number D has been found to A and B.
Next, let A not measure C.
I say that it is impossible to find a third proportional number to A and B.
If possible, let D be such third proportional. Then the product of A and D equals the square on B. But the square on B is C, therefore the product of A and D equals C.
Hence A multiplied by D makes 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 and B when A does not measure C.