The square root of a rational plus a medial (area) can be divided (into its component terms) at one point only. * Let $AB$ be the square root of a rational plus a medial (area) which has been divided at $C$, so that $AC$ and $CB$ are incommensurable in square, making the sum of the (squares) on $AC$ and $CB$ medial, and twice the (rectangle contained) by $AC$ and $CB$ rational [Prop. 10.40]. * I say that $AB$ cannot be (so) divided at another point.
In other words, \[\sqrt{\frac{\sqrt{1+\alpha^2}+\alpha}{2\,(1+\alpha^2)}} + \sqrt{\frac{\sqrt{1+\alpha^2}-\alpha}{2\,(1+\alpha^2)}}=\sqrt{\frac{\sqrt{1+\beta^2}+\beta}{2\,(1+\beta^2)}} + \sqrt{\frac{\sqrt{1+\beta^2}-\beta}{2\,(1+\beta^2)}}\] has only one solution: i.e., \[\beta=\alpha,\]
where \(\alpha,\beta\) denote positive rational numbers.
This proposition corresponds to [Prop. 10.83], with plus signs instead of minus signs.
Proofs: 1
Propositions: 1