Commentary on Proposition X.45¶
Back to Proposition X.45.
If possible, let the major irrational straight line be divided into terms in two ways, viz. as $(x + y)$ and $(x' + y')$, where $x', y'$ are supposed to be nearer equality than $x, y$.
We have then, as in x. 42, 43,
$$(x^2 + y^2) - (x'^2 + y'^2) = 2x'y' - 2xy.$$
But, by hypothesis, $(x^2 + y^2), (x'^2 + y'^2)$ are both rational, so that their difference is rational.
Also, by hypothesis, $2x'y', 2xy$ are both medial areas; therefore the difference of two medial areas is a rational area: which is impossible. [x. 26]
Therefore etc.