If from a rational straight line there is subtracted a rational straight line commensurable with the whole in square only, then the remainder is irrational; let it be called an *apotome.*

From the rational straight line *AB* let the rational straight line *BC*, commensurable with the whole in square only, be subtracted.

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

Since *AB* is incommensurable in length with *BC*, and *AB* is to *BC* as the square on *AB* is to the rectangle *AB* by *BC*, therefore the square on *AB* is incommensurable with the rectangle *AB* by *BC*.

But the sum of the squares on *AB* and *BC* is commensurable with the square on *AB*, and twice the rectangle *AB* by *BC* is commensurable with the rectangle *AB* by *BC*.

And, inasmuch as the sum of the squares on *AB* and *BC* equal twice the rectangle *AB* by *BC* together with the square on *CA*, therefore the sum of the squares on *AB* and *BC* is also incommensurable with the remainder, the square on *AC*.

But the sum of the squares on *AB* and *BC* is rational, therefore *AC* is irrational. Let it be called an apotome.

Q.E.D.