Proposition: Prop. 10.047: Side of Sum of Two Medial Areas is Divisible Uniquely

(Proposition 47 from Book 10 of Euclid's “Elements”)

The square root of (the sum of) two medial (areas) can be divided (into its component terms) at one point only. * Let $AB$ be [the square root of (the sum of) two medial (areas)] which has been divided at $C$, such that $AC$ and $CB$ are incommensurable in square, making the sum of the (squares) on $AC$ and $CB$ medial, and the (rectangle contained) by $AC$ and $CB$ medial, and, moreover, incommensurable with the sum of the (squares) on ($AC$ and $CB$) [Prop. 10.41]. * I say that $AB$ cannot be divided at another point fulfilling the prescribed (conditions).


Modern Formulation

In other words, \[\beta^{1/4}\sqrt{\frac{1+\alpha}{2\sqrt{1+\alpha^2}}}+\beta^{1/4}\sqrt{\frac{1-\alpha}{2\sqrt{1+\alpha^2}}}=\gamma^{1/4}\sqrt{\frac{1+\delta}{2\sqrt{1+\delta^2}}} + \gamma^{1/4}\sqrt{\frac{1-\delta}{2\sqrt{1+\delta^2}}}\] has only one solution: i.e., \[\delta=\alpha\quad\text{ and }\quad\gamma=\beta,\]

where \(\alpha,\beta,\gamma,\delta\) denote positive rational numbers.


This proposition corresponds to [Prop. 10.84], 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