Book 10
78
Now, since the sum of the squares on AB, BC is medial and is equal to DE, therefore DE is medial.
78
And it is applied to the rational straight line DI, producing DG as breadth; therefore DG is rational and incommensurable in length with DI. [X. 22]
78
Again, since twice the rectangle AB, BC is medial and is equal to DH, therefore DH is medial.
78
And it is applied to the rational straight line DI, producing DF as breadth; therefore DF is also rational and incommensurable in length with DI. [X. 22]
78
And, since the squares on AB, BC are incommensurable with twice the rectangle AB, BC, therefore DE is also incommensurable with DH.
78
But, as DE is to DH, so also is DG to DF; [VI. 1] therefore DG is incommensurable with DF. [X. 11]
78
And both are rational; therefore GD, DF are rational straight lines commensurable in square only.
78
Therefore FG is an apotome. [X. 73]
78
And FH is rational; but the rectangle contained by a rational straight line and an apotome is irrational, [deduction from X. 20] and its side is irrational.
78
And AC is the side of FE; therefore AC is irrational.
78
And let it be called that which produces with a medial area a medial whole. Q. E. D.
PROPOSITION 79
79
To an apotome only one rational straight line can be annexed which is commensurable with the whole in square only.
79
Let AB be an apotome, and BC an annex to it; therefore AC, CB are rational straight lines commensurable in square only. [X. 73]
