Kitap 10
111
Let EF be the annex to it; therefore DF, FE are rational straight lines commensurable in square only, the square on DF is greater than the square on FE by the square on a straight line commensurable with DF, and DF is commensurable in length with the rational straight line DC set out. [X. Deff. III. 1]
111
Again, since AB is binomial, therefore DE is a first binomial straight line. [X. 60]
111
Let it be divided into its terms at G, and let DG be the greater term; therefore DG, GE are rational straight lines commensurable in square only, the square on DG is greater than the square on GE by the square on a straight line commensurable with DG, and the greater term DG is commensurable in length with the rational straight line DC set out. [X. Deff. II. 1]
111
Therefore DF is also commensurable in length with DG; [X. 12] therefore the remainder GF is also commensurable in length with DF. [X. 15]
111
But DF is incommensurable in length with EF; therefore FG is also incommensurable in length with EF. [X. 13]
111
Therefore GF, FE are rational straight lines commensurable in square only; therefore EG is an apotome. [X. 73]
111
But it is also rational: which is impossible.
111
Therefore the apotome is not the same with the binomial straight line. Q. E. D.
PROPOSITION 112.
