Proposition: Prop. 10.080: Construction of First Apotome of Medial is Unique

(Proposition 80 from Book 10 of Euclid's “Elements”)

Only one medial straight line, which is commensurable in square only with the whole, and contains a rational (area) with the whole, can be attached to a first apotome of a medial (straight line).


Modern Formulation

In other words, the first apotome of a medial straight line is unique, i.e. \[\alpha^{1/4} - \alpha^{3/4} = \beta^{1/4} - \beta^{3/4}\] has only one solution: i.e., \[\beta=\alpha,\] where \(\alpha,\beta\) denote positive rational numbers.


This proposition corresponds to [Prop. 10.43], with minus signs instead of plus 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