Book 10
41
And of these the square on DE is less than the square on EC; therefore the remainder, the rectangle AC, CB, is also less than the rectangle AD, DB, so that twice the rectangle AC, CB is also less than twice the rectangle AD, DB.
41
Therefore also the remainder, the sum of the squares on AC, CB, is greater than the sum of the squares on AD, DB. Q. E. D. 3. and which produce the types in question. The Greek is ποιουσῶν τὰ προκείμενα εἴδη, and I have taken εἴδη to mean types (of irrational straight lines), though the expression might perhaps mean satisfying the conditions in question.
PROPOSITION 42.
A binomial straight line is divided into its terms at one point only.
42
Let AB be a binomial straight line divided into its terms at C; therefore AC, CB are rational straight lines commensurable in square only. [X. 36 ]
42
I say that AB is not divided at another point into two rational straight lines commensurable in square only.
42
For, if possible, let it be divided at D also, so that AD, DB are also rational straight lines commensurable in square only.
42
It is then manifest that AC is not the same with DB.
42
For, if possible, let it be so.
42
Then AD will also be the same as CB, and, as AC is to CB, so will BD be to DA; thus AB will be divided at D also in the same way as by the division at C: which is contrary to the hypothesis.
42
Therefore AC is not the same with DB.
42
For this reason also the points C, D are not equidistant from the point of bisection.
