Kitap 10
91
Again, since DH is medial and is equal to LO, therefore LO is also medial.
91
Since then LO is medial, while NO is rational, therefore LO is incommensurable with NO.
91
But, as LO is to NO, so is LP to PN; [VI. 1] therefore LP is incommensurable in length with PN. [X. 11]
91
And both are rational; therefore LP, PN are rational straight lines commensurable in square only; therefore LN is an apotome. [X. 73]
91
And it is the side of the area AB; therefore the side of the area AB is an apotome.
91
Therefore etc.
PROPOSITION 92.
92
If an area be contained by a rational straight line and a second apotome, the side of the area is a first apotome of a medial straight line.
92
For let the area AB be contained by the rational straight line AC and the second apotome AD; I say that the side of the area AB is a first apotome of a medial straight line.
92
For let DG be the annex to AD; therefore AG, GD are rational straight lines commensurable in square only, [X. 73] and the annex DG is commensurable with the rational straight line AC set out, while the square on the whole AG is greater than the square on the annex GD by the square on a straight line commensurable in length with AG. [X. Deff. III. 2]
92
Since then the square on AG is greater than the square on GD by the square on a straight line commensurable with AG, therefore, if there be applied to AG a parallelogram equal to the fourth part of the square on GD and deficient by a square figure, it divides it into commensurable parts. [X. 17]
