Proposition: Prop. 10.077: That which produces Medial Whole with Rational Area is Irrational

(Proposition 77 from Book 10 of Euclid's “Elements”)

If a straight line, which is incommensurable in square with the whole, and with the whole makes the sum of the squares on them medial, and twice the (rectangle contained) by them rational, is subtracted from a(nother) straight line then the remainder is an irrational (straight line). Let it be called that which makes with a rational (area) a medial whole.


Modern Formulation

Thus, that which makes with a rational (area) a medial whole is a straight line, whose length is expressible as

\[\sqrt{\frac{\sqrt{1+\rho^2}+\rho}{2\,(1+\rho^2)}} -\sqrt{\frac{\sqrt{1+\rho^2}-\rho}{2\,(1+\rho^2)}},\] for some positive rational number \(\rho\). See also [Prop. 10.40].

Proofs: 1

Proofs: 1 2 3 4
Propositions: 5 6

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