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 medial rectangle, then the remainder is irrational; let it be called *second apotome of a medial* straight line.

From the medial straight line *AB* let there be subtracted the medial straight line *CB* which is commensurable with the whole *AB* in square only such that the rectangle *AB* by *BC* which it contains with the whole *AB*, is medial.

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

Set out a rational straight line *DI*. Apply *DE*, equal to the sum of the squares on *AB* and *BC*, to *DI* producing *DG* as breadth. Apply *DH*, equal to twice the rectangle *AB* by *BC*, to *DI* producing *DF* as breadth. Then the remainder *FE* equals the square on *AC*.

Now, since the squares on *AB* and *BC* are medial and commensurable, therefore *DE* is also medial.

And it is applied to the rational straight line *DI*, producing *DG* as breadth, therefore *DG* is rational and incommensurable in length with *DI*.

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

And it equals *DH*, therefore *DH* is also medial.

And it is applied to the rational straight line *DI*, producing *DF* as breadth, therefore *DF* is rational and incommensurable in length with *DI*.

Since *AB* and *BC* are commensurable in square only, therefore *AB* is incommensurable in length with *BC*. Therefore the square on *AB* is also 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*, therefore twice the rectangle *AB* by *BC* is incommensurable with the sum of the squares on *AB* and *BC*.

But *DE* equals the sum of the squares on *AB* and *BC*, and *DH* equals twice the rectangle *AB* by *BC*, therefore *DE* is incommensurable with *DH*. But *DE* is to *DH* as *GD* is to *DF*, therefore *GD* is incommensurable with *DF*.

And both are rational, therefore *GD* and *DF* are rational straight lines commensurable in square only. Therefore *FG* is an apotome.

But *DI* is rational, and the rectangle contained by a rational and an irrational straight line is irrational, and its side is irrational.

And *AC* is the side of *FE*, therefore *AC* is irrational.

Let it be called a second apotome of a medial straight line.

Q.E.D.