Book 10
113
Let A be a rational straight line and BD an apotome, and let the rectangle BD, KH be equal to the square on A, so that the square on the rational straight line A when applied to the apotome BD produces KH as breadth; I say that KH is a binomial straight line the terms of which are commensurable with the terms of BD and in the same ratio; and further KH has the same order as BD.
113
For let DC be the annex to BD; therefore BC, CD are rational straight lines commensurable in square only. [X. 73]
113
Let the rectangle BC, G be also equal to the square on A.
113
But the square on A is rational; therefore the rectangle BC, G is also rational.
113
And it has been applied to the rational straight line BC; therefore G is rational and commensurable in length with BC. [X. 20]
113
Since now the rectangle BC, G is equal to the rectangle BD, KH, therefore, proportionally, as CB is to BD, so is KH to G. [VI. 16]
113
But BC is greater than BD; therefore KH is also greater than G. [V. 16, V. 14]
113
Let KE be made equal to G; therefore KE is commensurable in length with BC.
113
And since, as CB is to BD, so is HK to KE, therefore, convertendo, as BC is to CD, so is KH to HE. [V. 19, Por.]
113
Let it be contrived that, as KH is to HE, so is HF to FE; therefore also the remainder KF is to FH as KH is to HE, that is, as BC is to CD. [V. 19]
113
But BC, CD are commensurable in square only; therefore KF, FH are also commensurable in square only. [X. 11]
