If a rational and a medial are added together, then four irrational straight lines arise, namely a binomial or a first bimedial or a major or a side of a rational plus a medial area.

Let *AB* be rational, and *CD* medial.

I say that the side of the area *AD* is a binomial or a first bimedial or a major or a side of a rational plus a medial area.

For *AB* is either greater or less than *CD*.

First, let it be greater. Set out a rational straight line *EF* , apply to *EF* the rectangle *EG* equal to *AB*, producing *EH* as breadth, and apply to *EF* *HI*, equal to *DC*, producing *HK* as breadth.

Then, since *AB* is rational and equals *EG*, therefore *EG* is also rational. And it is applied to *EF*, producing *EH* as breadth, therefore *EH* is rational and commensurable in length with *EF*.

Again, since *CD* is medial and equals *HI*, therefore *HI* is also medial. And it is applied to the rational straight line *EF*, producing *HK* as breadth, therefore *HK* is rational and incommensurable in length with *EF*.

Since *CD* is medial, while *AB* is rational, therefore *AB* is incommensurable with *CD*, so that *EG* is also incommensurable with *HI.*

But *EG* is to *HI* as *EH* is to *HK*, therefore *EH* is also incommensurable in length with *HK*.

And both are rational, therefore *EH* and *HK* are rational straight lines commensurable in square only. Therefore *EK* is a binomial straight line, divided at *H*.

Since *AB* is greater than *CD*, while *AB* equals *EG* and *CD* equals *HI*, therefore *EG* is also greater than *HI*. Therefore *EH* is also greater than *HK*. The square, then, on *EH* is greater than the square on *HK* either by the square on a straight line commensurable in length with *EH* or by the square on a straight line incommensurable with it.

First, let the square on it be greater by the square on a straight line commensurable with itself.

Now the greater straight line *HE* is commensurable in length with the rational straight line *EF* set out, therefore *EK* is a first binomial.

But *EF* is rational, and, if an area is contained by a rational straight line and the first binomial, then the side of the square equal to the area is binomial. Therefore the side of *EI* is binomial, so that the side of *AD* is also binomial.

Next, let the square on *EH* be greater than the square on *HK* by the square on a straight line incommensurable with *EH*.

Now the greater straight line *EH* is commensurable in length with the rational straight line *EF* set out, therefore *EK* is a fourth binomial.

But *EF* is rational, and, if an area be contained by a rational straight line and the fourth binomial, then the side of the area is the irrational straight line called major. Therefore the side of the area *EI* is major, so that the side of the area *AD* is also major.

Next, let *AB* be less than *CD*. Then *EG* is also less than *HI*, so that *EH* is also less than *HK*.

Now the square on *HK* is greater than the square on *EH* either by the square on a straight line commensurable with *HK* or by the square on a straight line incommensurable with it.

First, let the square on it be greater by the square on a straight line commensurable in length with itself.

Now the lesser straight line *EH* is commensurable in length with the rational straight line *EF* set out, therefore *EK* is a second binomial.

But *EF* is rational, and, if an area is contained by a rational straight line and the second binomial, then the side of the square it is a first bimedial, therefore the side of the area *EI* is a first bimedial, so that the side of *AD* is also a first bimedial.

Next, let the square on *HK* be greater than the square on *HE* by the square on a straight line incommensurable with *HK*.

Now the lesser straight line *EH* is commensurable with the rational straight line *EF* set out, therefore *EK* is a fifth binomial.

But *EF* is rational, and, if an area is contained by a rational straight line and the fifth binomial, then the side of the square equal to the area is a side of a rational plus a medial area.

Therefore the side of the area *EI* is a side of a rational plus a medial area, so that the side of the area *AD* is also a side of a rational plus a medial area.

Therefore, *if a rational and a medial are added together, then four irrational straight lines arise, namely a binomial or a first bimedial or a major or a side of a rational plus a medial area.*

Q.E.D.