## Proposition 110

 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. X.22 Since BC is incommensurable with BD, that is, GH with GK, therefore HF is also incommensurable with FK. VI.1 X.11 Therefore FH and FK are rational straight lines commensurable in square only. Therefore KH is an apotome. X.73 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. X.Def.III.3 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. X.93 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. X.Def.III.6 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. X.96 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.
(Forthcoming)