Proposition: Prop. 10.040: Side of Rational plus Medial Area is Irrational

(Proposition 40 from Book 10 of Euclid's “Elements”)

If two straight lines (which are) incommensurable in square, making the sum of the squares on them medial, and the (rectangle contained) by them rational, are added together then the whole straight line is irrational - let it be called the square root of a rational plus a medial (area) .


Modern Formulation

Thus, the square root of a rational plus a medial (area) has a length 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 \(\rho\).


This and the corresponding irrational with a minus sign1 whose length is expressible as

\[\sqrt{\frac{\sqrt{1+\rho^2}+\rho}{2\,(1+\rho^2)}} -\sqrt{\frac{\sqrt{1+\rho^2}-\rho}{2\,(1+\rho^2)}},\]

(see Prop. 10.77), are the positive roots of the quartic \[x^4-\frac 2{\sqrt{1+\rho^2}}x^2+ \frac{\rho^2}{(1+\rho^2)^2} = 0.\]

Proofs: 1

Proofs: 1 2 3 4
Propositions: 5 6 7

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


  1. which Euclid calls "that which makes with a rational (area) a medial whole" (see Prop. 10.77).