If two straight lines incommensurable in square which make the sum of the squares on them rational but the rectangle contained by them medial are added together, then the whole straight line is irrational; let it be called *major.*

Let two straight lines *AB* and *BC* incommensurable in square, and fulfilling the given conditions, be added together.

I say that *AC* is irrational.

Since the rectangle *AB* by *BC* is medial, therefore twice the rectangle *AB* by *BC* is also medial.

But the sum of the squares on *AB* and *BC* is rational, therefore twice the rectangle *AB* by *BC* is incommensurable with the sum of the squares on *AB* and *BC*, so that the sum of the squares on *AB* and *BC* together with twice the rectangle *AB* by *BC*, that is, the square on *AC*, is also incommensurable with the sum of the squares on *AB* and *BC*. Therefore the square on *AC* is irrational, so that *AC* is also irrational. Let it be called *major*.

Q.E.D.