Kitap 13
11
And, since BF is quadruple of FK, therefore BK is five times KF; therefore the square on BK is twenty-five times the square on KF.
11
But the square on MK is five times the square on KF; therefore the square on BK is five times the square on KM; therefore the square on BK has not to the square on KM the ratio which a square number has to a square number; therefore BK is incommensurable in length with KM. [X. 9]
11
And each of them is rational.
11
Therefore BK, KM are rational straight lines commensurable in square only.
11
But, if from a rational straight line there be subtracted a rational straight line which is commensurable with the whole in square only, the remainder is irrational, namely an apotome; therefore MB is an apotome and MK the annex to it. [X. 73]
11
I say next that MB is also a fourth apotome.
11
Let the square on N be equal to that by which the square on BK is greater than the square on KM; therefore the square on BK is greater than the square on KM by the square on N.
11
And, since KF is commensurable with FB, componendo also, KB is commensurable with FB. [X. 15]
11
But BF is commensurable with BH; therefore BK is also commensurable with BH. [X. 12]
11
And, since the square on BK is five times the square on KM, therefore the square on BK has to the square on KM the ratio which 5 has to 1.
