Similar plane numbers have to one another the ratio which a square number has to a square number.

Let *A* and *B* be similar plane numbers.

I say that *A* has to *B* the ratio which a square number has to a square number.

Since *A* and *B* are similar plane numbers, therefore one mean proportional number *C* falls between *A* and *B.*

Take *D, E,* and *F,* the least numbers of those which have the same ratio with *A, C,* and *B*

Then the extremes of them *D* and *F* are square. And since *D* is to *F* as *A* is to *B,* and *D* and *F* are square, therefore *A* has to *B* the ratio which a square number has to a square number.

Therefore, *similar plane numbers have to one another the ratio which a square number has to a square number.*

Q.E.D.