If two numbers have to one another the ratio which a square number has to a square number, and the first is square, then the second is also a square.

Let the two numbers *A* and *B* have to one another the ratio which the square number *C* has to the square number *D,* and let *A* be square.

I say that *B* is also square.

Since *C* and *D* are square, *C* and *D* are similar plane numbers. Therefore one mean proportional number falls between *C* and *D.*

And *C* is to *D* as *A* is to *B,* therefore one mean proportional number falls between *A* and *B* also. And *A* is square, therefore *B* is also square.

Therefore, *if two numbers have to one another the ratio which a square number has to a square number, and the first is square, then the second is also a square.*

Q.E.D.