Proposition: Prop. 10.046: Side of Rational Plus Medial Area is Divisible Uniquely

(Proposition 46 from Book 10 of Euclid's “Elements”)

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.


Modern Formulation

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

Thank you to the contributors under CC BY-SA 4.0!



Adapted from (subject to copyright, with kind permission)

  1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"

Adapted from CC BY-SA 3.0 Sources:

  1. Prime.mover and others: "Pr∞fWiki",, 2016