If two similar plane numbers multiplied by one another make some number, then the product is square.

Let *A* and *B* be two similar plane numbers, and let *A* multiplied by *B* make *C.*

I say that *C* is square.

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

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

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

Since as many number fall in continued proportion between those which have the same ratio, therefore one mean proportional number falls between *D* and *C* also.

And *D* is square, therefore *C* is also square.

Therefore, *if two similar plane numbers multiplied by one another make some number, then the product is square.*

Q.E.D.

To illustrate this proposition, consider the two similar plane numbers *a* = 18 and *b* = 8, as illustrated in the Guide to VII.Def.21. According to VIII.18, there is a mean proportional between them, namely, 12. And the square of the mean proportional is their product, *ab* = 144.

Let *a* and *b* be the given similar plane numbers. Then there is a mean proportional between them (VIII.18). And, since *a* : *b* = *a*^{2}:*ab,* therefore there is also a mean proportional between *a*^{2} and *ab* (VIII.1). But since *a*^{2} is a square, therefore *ab* is also a square (VIII.22). Thus, the product of the original similar plane numbers is a square.