If a square does not measure a square, then neither does the side measure the side; and, if the side does not measure the side, then neither does the square measure the square.

Let *A* and *B* be square numbers, and let *C* and *D* be their sides, and let *A* not measure *B.*

I say that neither does *C* measure *D.*

If *C* measures *D, A* also measures *B.* But *A* does not measure *B,* therefore neither does *C* measure *D.*

Next, let *C* not measure *D.*

I say that neither does *A* measure *B.*

If *A* measures *B,* then *C* also measures *D.* But *C* does not measure *D,* therefore neither does *A* measure *B.*

Therefore, *if a square does not measure a square, then neither does the side measure the side; and, if the side does not measure the side, then neither does the square measure the square.*

Q.E.D.