PROPOSITION 18.
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](elem.9.16)]

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.