If a medial area incommensurable with the whole is subtracted from a medial area, then two remaining irrational straight lines arise, either a second apotome of a medial straight line or a straight line which produces with a medial area a medial whole.

As in the foregoing figures, let there be subtracted the medial area *BD* incommensurable with the whole from the medial area *BC*.

I say that the side of *EC* is one of two irrational straight lines, either a second apotome of a medial straight line or a straight line which produces with a medial area a medial whole.

Since each of the rectangles *BC* and *BD* is medial, and *BC* is incommensurable with *BD*, therefore each of the straight lines *FH* and *FK* is rational and incommensurable in length with *FG*.

Since *BC* is incommensurable with *BD*, that is, *GH* with *GK*, therefore *HF* is also incommensurable with *FK*.

Therefore *FH* and *FK* are rational straight lines commensurable in square only. Therefore *KH* is an apotome.

If then the square on *FH* is greater than the square on *FK* by the square on a straight line commensurable with *FH*, while neither of the straight lines *FH* nor *FK* is commensurable in length with the rational straight line *FG* set out, then *KH* is a third apotome.

But *KL* is rational, and the rectangle contained by a rational straight line and a third apotome is irrational, and the side of it is irrational, and is called a second apotome of a medial straight line, so that the side of *LH*, that is, of *EC*, is a second apotome of a medial straight line.

But, if the square on *FH* is greater than the square on *FK* by the square on a straight line incommensurable with *FH*, while neither of the straight lines *HF* nor *FK* is commensurable in length with *FG*, then *KH* is a sixth apotome.

But the side of the rectangle contained by a rational straight line and a sixth apotome is a straight line which produces with a medial area a medial whole.

Therefore the side of *LH*, that is, of *EC*, is a straight line which produces with a medial area a medial whole.

Therefore, *if a medial area incommensurable with the whole is subtracted from a medial area, then two remaining irrational straight lines arise, either a second apotome of a medial straight line or a straight line which produces with a medial area a medial whole.*

Q.E.D.