Proposition: Prop. 10.082: Construction of Minor is Unique

(Proposition 82 from Book 10 of Euclid's “Elements”)

Only one straight line, which is incommensurable in square with the whole, and (together) with the whole makes the (sum of the) squares on them rational, and twice the (rectangle contained) by them medial, can be attached to a minor (straight line).


Modern Formulation

In other words, \[\sqrt{\frac{1+\alpha}{2\sqrt{1+\alpha^2}}} - \sqrt{\frac{1-\alpha}{2\sqrt{1+\alpha^2}}} =\sqrt{\frac{1+\beta}{2\sqrt{1+\beta^2}}} - \sqrt{\frac{1-\beta}{2\sqrt{1+\beta^2}}}\] has only one solution: i.e., \[\beta=\alpha,\] where \(\alpha,\beta\) denote positive rational numbers.


This proposition corresponds to [Prop. 10.45], 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