If from a straight line there is subtracted a straight line which is incommensurable in square with the whole, and which with the whole makes the sum of the squares on them medial but twice the rectangle contained by them rational, then the remainder is irrational; let it be called *that which produces with a rational area a medial whole.*

From the straight line *AB* let there be subtracted the straight line *BC* which is incommensurable in square with *AB* and fulfills the given conditions.

I say that the remainder *AC* is the irrational straight line aforesaid.

Since the sum of the squares on *AB* and *BC* is medial, while twice the rectangle *AB* by *BC* is rational, therefore the sum of the squares on *AB* and *BC* is incommensurable with twice the rectangle *AB* by *BC*. Therefore the remainder, the square on *AC*, is also incommensurable with twice the rectangle *AB* by *BC*.

And twice the rectangle *AB* by *BC* is rational, therefore the square on *AC* is irrational. Therefore *AC* is irrational. Let it be called that which produces with a rational area a medial whole.

Q.E.D.