If two numbers multiplied by one another make a square number, then they are similar plane numbers.

Let *A* and *B* be two numbers, and let *A* multiplied by *B* make the square number *C.*

I say that *A* and *B* are similar plane numbers.

Multiply *A* by itself to make *D.* Then *D* is square.

Now, since *A* multiplied by itself makes *D,* and multiplied by *B* makes *C,* therefore *A* is to *B* as *D* is to *C.*

And, since *D* is square, and *C* is so also, therefore *D* and *C* are similar plane numbers.

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

But, if one mean proportional number falls between two numbers, then they are similar plane numbers, therefore *A* and *B* are similar plane numbers.

Therefore, *if two numbers multiplied by one another make a square number, then they are similar plane numbers.*

Q.E.D.

As an example to illustrate this proposition, take any square number, such as 20^{2} = 400. It can be factored as a product of two numbers in several ways. One such factorization is as *a* = 50 times *b* = 8. These two numbers have a mean proportional between them, namely, 20, so by VIII.20, they are similar plane numbers. (The actual shapes given by that proposition make 8 to be 2 by 4, and 50 to be 5 by 10.)

As in the last proof, this one can be shortened. When the product *ab* is a square, say *e*^{2}, then a mean proportional between *a* and *b* is *e.*