The square on the first bimedial straight line applied to a rational straight line produces as breadth the second binomial.

Let *AB* be a first bimedial straight line divided into its medials at *C*, of which medials *AC* is the greater. Let a rational straight line *DE* be set out, and let there be applied to *DE* the parallelogram *DF* equal to the square on *AB*, producing *DG* as its breadth.

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

Make the same construction as before.

Then, since *AB* is a first bimedial divided at *C*, therefore *AC* and *CB* are medial straight lines commensurable in square only, and containing a rational rectangle, so that the squares on *AC* and *CB* are also medial.

Therefore *DL* is medial. And it was applied to the rational straight line *DE*, therefore *MD* is rational and incommensurable in length with *DE*.

Again, since twice the rectangle *AC* by *CB* is rational, therefore *MF* is also rational.

And it is applied to the rational straight line *ML*, therefore *MG* is also rational and commensurable in length with *ML*, that is, *DE*. Therefore *DM* is incommensurable in length with *MG*.

And they are rational, therefore *DM* and *MG* are rational straight lines commensurable in square only. Therefore *DG* is binomial.

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

Since the sum of the squares on *AC* and *CB* is greater than twice the rectangle *AC* by *CB*, therefore *DL* is also greater than *MF*, so that *DM* is also greater than *MG*.

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

And the rectangle *DK* by *KM* equals the square on *MN*, therefore the square on *DM* is greater than the square on *MG* by the square on a straight line commensurable with *DM*. And *MG* is commensurable in length with *DE*.

Therefore *DG* is a second binomial straight line.

Therefore, *the square on the first bimedial straight line applied to a rational straight line produces as breadth the second binomial.*

Q.E.D.