Proposition: Prop. 10.059: Root of Area contained by Rational Straight Line and Sixth Binomial

(Proposition 59 from Book 10 of Euclid's “Elements”)

If an area is contained by a rational (straight line) and a sixth binomial (straight line) then the square root of the area is the irrational (straight line which is) called the square root of (the sum of) two medial (areas).


Modern Formulation

If the rational straight line has unit length then this proposition states that the square root of a sixth binomial straight line is the square root of the sum of two medial area: i.e., a sixth binomial straight line has a length \[\sqrt{\alpha}+\sqrt{\beta}\] whose square root can be written \[\alpha^{1/4}\left(\sqrt{\frac 12+\frac{\delta}{2\sqrt{1+\delta^2}}}+\sqrt{\frac 12-\frac{\delta}{2\sqrt{1+\delta^2}}}\right),\] where \[\delta^2=\frac{\alpha-\beta}\beta.\] This is the length of the square root of the sum of two medial area (see [Prop. 10.41]).

Proofs: 1

  1. Lemma: Lem. 10.059: Sum of Squares on Unequal Pieces of Segment Is Greater than Twice the Rectangle Contained by Them

Proofs: 1 2
Propositions: 3

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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