If from a medial straight line there is subtracted a medial straight line which is commensurable with the whole in square only and which contains with the whole a rational rectangle, then the remainder is irrational; let it be called *first apotome of a medial* straight line.

From the medial straight line *AB* let there be subtracted the medial straight line *BC* which is commensurable with *AB* in square only and with *AB* makes the rectangle *AB* by *BC* rational.

I say that the remainder *AC* is irrational, and let it be called an apotome of a medial straight line.

Since *AB* and *BC* are medial, the squares on *AB* and *BC* are also medial. But 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 twice the rectangle *AB* by *BC* is also incommensurable with the remainder, the square on *AC*, since, if the whole is incommensurable with one of the magnitudes, then the original magnitudes are also incommensurable.

But twice the rectangle *AB* by *BC* is rational, therefore the square on *AC* is irrational, therefore *AC* is irrational. Let it be called a first apotome of a medial straight line.

Q.E.D.