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 added together rational, but the rectangle contained by them medial, then the remainder is irrational; let it be called *minor.*

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

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

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

But the sum of the squares on *AB* and *BC* is rational, therefore the square on *AC* is irrational. Therefore *AC* is irrational.

Let it be called minor.

Q.E.D.