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

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.*

Q.E.D.