A straight line commensurable with an apotome of a medial straight line is an apotome of a medial straight line and the same in order.

Let *AB* be an apotome of a medial straight line, and let *CD* be commensurable in length with *AB*.

I say that *CD* is also an apotome of a medial straight line and the same in order with *AB*.

Since *AB* is an apotome of a medial straight line, let *EB* be the annex to it.

Then *AE* and *EB* are medial straight lines commensurable in square only.

Let it be contrived that *AB* is to *CD* as *BE* is to *DF*. Then *AE* is also commensurable with *CF*, and *BE* with *DF*.

But *AE* and *EB* are medial straight lines commensurable in square only, therefore *CF* and *FD* are also medial straight lines commensurable in square only.

Therefore *CD* is an apotome of a medial straight line.

I say next that it is also the same in order with *AB*.

Since *AE* is to *EB* as *CF* is to *FD*, therefore the square on *AE* is to the rectangle *AE* by *EB* as the square on *CF* is to the rectangle *CF* by *FD*.

But the square on *AE* is commensurable with the square on *CF*, therefore the rectangle *AE* by *EB* is also commensurable with the rectangle *CF* by *FD*.

Therefore, if the rectangle *AE* by *EB* is rational, then the rectangle *CF* by *FD* is also rational, and if the rectangle *AE* by *EB* is medial, the rectangle *CF* by *FD* is also medial.

Therefore *CD* is an apotome of a medial straight line and the same in order with *AB*.

Therefore, *a straight line commensurable with an apotome of a medial straight line is an apotome of a medial straight line and the same in order.*

Q.E.D.