A straight line commensurable with a medial straight line is medial.

Let *A* be medial, and let *B* be commensurable with *A*.

I say that *B* is also medial.

Set out a rational straight line *CD*. Apply the rectangular area *CE* to *CD* equal to the square on *A*, producing *ED* as breadth. Then *ED* is rational and incommensurable in length with *CD*. And apply the rectangular area *CF* to *CD* equal to the square on *B*, producing *DF* as breadth.

Since *A* is commensurable with *B*, therefore the square on *A* is also commensurable with the square on *B*. But *EC* equals the square on *A*, and *CF* equals the square on *B*, therefore *EC* is commensurable with *CF*.

And *EC* is to *CF* as *ED* is to *DF*, therefore *ED* is commensurable in length with *DF*.

But *ED* is rational and incommensurable in length with *DC*, therefore *DF* is also rational and incommensurable in length with *DC*.

Therefore *CD* and *DF* are rational and commensurable in square only.

But the straight line is medial on which the square equals the rectangle contained by rational straight lines commensurable in square only, therefore the side of the square equals the rectangle *CD* by *DF* is medial.

And *B* is the side of the square equal the rectangle *CD* by *DF*, therefore *B* is medial.

Therefore, *a straight line commensurable with a medial straight line is medial.*

Q.E.D.

From this it is clear that *an area commensurable with a medial area is medial.*

And in the same way as was explained in the case of rationals it follows regards medials, that a straight line commensurable in length with a medial straight line is called medial and commensurable with it not only in length but in square also, since, in general, straight lines commensurable in length are always commensurable in square also.

But, if any straight line is commensurable in square with a medial straight line, then if it is also commensurable in length with it, the straight lines are called, in this case too, medial and commensurable in length and in square, but, if in square only, they are called medial straight lines commensurable in square only.