Kitap 10
103
And, if AE is commensurable in length with the rational straight line set out, CF is so also, if BE, then DF also, [X. 12] and, if neither of the straight lines AE, EB, then neither of the straight lines CF, FD. [X. 13]
103
Therefore CD is an apotome and the same in order with AB. Q. E. D.
PROPOSITION 104.
104
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.
104
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.
104
For, since AB is an apotome of a medial straight line, let EB be the annex to it.
104
Therefore AE, EB are medial straight lines commensurable in square only. [X. 74, 75]
104
Let it be contrived that, as AB is to CD, so is BE to DF; [VI. 12] therefore AE is also commensurable with CF, and BE with DF. [V. 12, X. 11]
104
But AE, EB are medial straight lines commensurable in square only; therefore CF, FD are also medial straight lines [X. 23] commensurable in square only; [X. 13] therefore CD is an apotome of a medial straight line. [X. 74, 75]
104
I say next that it is also the same in order with AB.
104
Since, as AE is to EB, so is CF to FD, therefore also, as the square on AE is to the rectangle AE, EB, so is the square on CF to the rectangle CF, FD.
