A straight line commensurable in length with an apotome is an apotome and the same in order.

Let *AB* be an apotome, *B* and let *CD* be commensurable in length with *AB*.

I say that *CD* is also an apotome and the same in order with *AB*.

Since *AB* is an apotome, let *BE* be the annex to it, therefore *AE* and *EB* are rational straight lines commensurable in square only.

Let it be contrived that the ratio of *BE* to *DF* is the same as the ratio of *AB* to *CD*. Then one is to one as are all to all. Therefore the whole *AE* is to the whole *CF* as *AB* is to *CD*.

But *AB* is commensurable in length with *CD*, therefore *AE* is also commensurable with *CF*, and *BE* with *DF*.

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

Now since *AE* is to *CF* as *BE* is to *DF*, therefore, alternately, *AE* is to *EB* as *CF* is to *FD*. And the square on *AE* is greater than the square on *EB* either by the square on a straight line commensurable with *AE* or by the square on a straight line incommensurable with it.

If then the square on *AE* is greater than the square on *EB* by the square on a straight line commensurable with *AE*, then the square on *CF* is also greater than the square on *FD* by the square on a straight line commensurable with *CF*.

And, if *AE* is commensurable in length with the rational straight line set out, then *CF* is also; if *BE*, then *DF* also; and, if neither of the straight lines *AE* nor *EB*, then neither of the straight lines *CF* nor *FD*.

But, if the square on *AE* is greater than the square on *EB* by the square on a straight line incommensurable with *AE*, then the square on *CF* is also greater than the square on *FD* by the square on a straight line incommensurable with *CF*.

And, if *AE* is commensurable in length with the rational straight line set out, then *CF* is also; if *BE*, then *DF* also; and, if neither of the straight lines *AE* nor *EB*, then neither of the straight lines *CF* nor *FD*. Therefore *CD* is an apotome and the same in order with *AB*.

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

Q.E.D.