Proposition: Prop. 10.081: Construction of Second Apotome of Medial is Unique

(Proposition 81 from Book 10 of Euclid's “Elements”)

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


Modern Formulation

In other words,

\[\alpha^{1/4}-\frac{\beta^{1/2}}{\alpha^{1/4}} = \gamma^{1/4}-\frac{\delta^{1/2}}{\gamma^{1/4}}\] has only one solution: i.e., \[\gamma=\alpha\quad\text{ and }\quad \delta=\beta,\] where \(\alpha,\beta,\gamma,\delta\) denote positive rational numbers.


This proposition corresponds to [Prop. 10.44], 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