Proposition: Prop. 10.043: First Bimedial Straight Line is Divisible Uniquely

(Proposition 43 from Book 10 of Euclid's “Elements”)

A first bimedial (straight line) can be divided (into its component terms) at one point only. * Let $AB$ be a first bimedial (straight line) which has been divided at $C$, such that $AC$ and $CB$ are medial (straight lines), commensurable in square only, (and) containing a rational (area) [Prop. 10.37]. * I say that $AB$ cannot be (so) divided at another point.


Modern Formulation

In other words, \[\alpha^{1/4} + \alpha^{3/4} = \beta^{1/4} + \beta^{3/4}\] has only one solution: i.e., \[\beta=\alpha,\] where \(\alpha,\beta\) denote positive rational numbers.


This proposition corresponds to [Prop. 10.80], 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