Definition: Def. 10.04: Rational and Irrational Magnitudes in Square

And let the square on the assigned straight line be called rational. And (let areas) commensurable with it (also be called) rational. But (let areas) incommensurable with it (be called) irrational, and (let) their square-roots1 (also be called) irrational - the sides themselves, if the (areas) are squares, and the (straight lines) describing squares equal to them, if the (areas) are some other rectilinear (figure).2

Modern Formulation

(not yet contributed)

Proofs: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
Propositions: 23

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. The square-root of an area is the length of the side of an equal area square. (translator's note). 

  2. The area of the square on the assigned straight line is unity. Rational areas are expressible as \(k\). All other areas are irrational. Thus, squares whose sides are of rational length have rational areas, and vice versa (translator's note).