Book 10
60
And it is applied to the rational straight line DE; therefore DM is rational and commensurable in length with DE. [X. 20]
60
Again, since AC, CB are rational straight lines commensurable in square only, therefore twice the rectangle AC, CB, that is MF, is medial. [X. 21]
60
And it is applied to the rational straight line ML; therefore MG is also rational and incommensurable in length with ML, that is, DE. [X. 22]
60
But MD is also rational and is commensurable in length with DE; therefore DM is incommensurable in length with MG. [X. 13]
60
And they are rational; therefore DM, MG are rational straight lines commensurable in square only; therefore DG is binomial. [X. 36]
60
It is next to be proved that it is also a first binomial straight line.
60
Since the rectangle AC, CB is a mean proportional between the squares on AC, CB, [cf. Lemma after X. 53] therefore MO is also a mean proportional between DH, KL.
60
Therefore, as DH is to MO, so is MO to KL, that is, as DK is to MN, so is MN to MK; [VI. 1] therefore the rectangle DK, KM is equal to the square on MN. [VI. 17]
60
And, since the square on AC is commensurable with the square on CB, DH is also commensurable with KL, so that DK is also commensurable with KM. [VI. 1, X. 11]
60
And, since the squares on AC, CB are greater than twice the rectangle AC, CB, [Lemma] therefore DL is also greater than MF, so that DM is also greater than MG. [VI. 1]
