The square on the major straight line applied to a rational straight line produces as breadth the fourth binomial.

Let *AB* be a major straight line divided at *C*, so that *AC* is greater than *CB*, let *DE* be a rational straight line, and to *DE* let there be applied the parallelogram *DF* equal to the square on *AB* and producing *DG* as its breadth.

I say that *DG* is a fourth binomial straight line.

Make the same construction as shown before.

Since *AB* is a major straight line divided at *C*, therefore *AC* and *CB* are straight lines incommensurable in square which make the sum of the squares on them rational, but the rectangle contained by them medial.

Since the sum of the squares on *AC* and *CB* is rational, therefore *DL* is rational. Therefore *DM* is also rational and commensurable in length with *DE*.

Again, since twice the rectangle *AC* by *CB*, that is, *MF*, is medial, and it is applied to the rational straight line *ML*, therefore *MG* is also rational and incommensurable in length with *DE*.

Therefore *DM* is also incommensurable in length with *MG*. Therefore *DM* and *MG* are rational straight lines commensurable in square only. Therefore *DG* is binomial.

It is to be proved that it is a fourth binomial straight line.

In manner similar to the foregoing we can prove that *DM* is greater than *MG*, and that the rectangle *DK* by *KM* equals the square on *MN*.

Since the square on *AC* is incommensurable with the square on *CB*, therefore *DH* is also incommensurable with *KL*, so that *DK* is also incommensurable with *KM*.

But, if there are two unequal straight lines, and to the greater there is applied a parallelogram equal to the fourth part of the square on the less and deficient by a square figure, and if it divides it into incommensurable parts, then the square on the greater is greater than the square on the less by the square on a straight line incommensurable in length with the greater, therefore the square on *DM* is greater than the square on *MG* by the square on a straight line incommensurable with *DM*.

And *DM* and *MG* are rational straight lines commensurable in square only, and *DM* is commensurable with the rational straight line *DE* set out.

Therefore *DG* is a fourth binomial straight line.

Therefore, *the square on the major straight line applied to a rational straight line produces as breadth the fourth binomial.*

Q.E.D.