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.
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.
Notes
This proposition corresponds to [Prop. 10.80], with plus signs instead of minus signs.
Table of Contents
Proofs: 1
Mentioned in:
Propositions: 1
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References
Adapted from (subject to copyright, with kind permission)
 Fitzpatrick, Richard: Euclid's "Elements of Geometry"
Adapted from CC BYSA 3.0 Sources:
 Prime.mover and others: "Pr∞fWiki", https://proofwiki.org/wiki/Main_Page, 2016