Book 10
44
But EG is equal to the squares on AC, CB, and HK is equal to twice the rectangle AC, CB; therefore EG is incommensurable with HK, so that EH is also incommensurable in length with HN. [VI. 1 , X. 11 ]
44
And they are rational; therefore EH, HN are rational straight lines commensurable in square only.
44
But, if two rational straight lines commensurable in square only be added together, the whole is the irrational which is called binomial. [X. 36 ]
44
Therefore EN is a binomial straight line divided at H.
44
In the same way EM, MN will also be proved to be rational straight lines commensurable in square only; and EN will be a binomial straight line divided at different points, H and M.
44
And EH is not the same with MN.
44
For the squares on AC, CB are greater than the squares on AD, DB.
44
But the squares on AD, DB are greater than twice the rectangle AD, DB; therefore also the squares on AC, CB, that is, EG, are much greater than twice the rectangle AD, DB, that is, MK, so that EH is also greater than MN.
44
Therefore EH is not the same with MN. Q. E. D.
PROPOSITION 45.
45
A major straight line is divided at one and the same point only.
45
Let AB be a major straight line divided at C, so that AC, CB are incommensurable in square and make the sum of the squares on AC, CB rational, but the rectangle AC, CB medial; [X. 39 ] I say that AB is not so divided at another point.
