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.