Proposition: Prop. 10.053: Construction of Sixth Binomial Straight Line

(Proposition 53 from Book 10 of Euclid's “Elements”)

To find a sixth binomial (straight line).


Modern Formulation

If the rational straight line has unit length then the length of a sixth binomial straight line is \[\sqrt{\alpha}+\sqrt{\beta},\]

where \(\alpha,\beta\) denote positive rational numbers.


This, and the sixth apotome, whose length according to [Prop. 10.90] is \[\sqrt{\alpha}-\sqrt{\beta},\] are the roots of the quadratic function \[x^2- 2\,\sqrt{\alpha}\,x+(\alpha-\beta)=0,\] where \(\alpha,\beta\) denote positive rational numbers.

Proofs: 1

  1. Lemma: Lem. 10.053: Construction of Rectangle with Area in Mean Proportion to two Square Areas

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