Proposition: Prop. 10.039: Major is Irrational

(Proposition 39 from Book 10 of Euclid's “Elements”)

If two straight lines (which are) incommensurable in square, making the sum of the squares on them rational, and the (rectangle contained) by them medial, are added together then the whole straight line is irrational - let it be called a major (straight line). * For let the two straight lines, $AB$ and $BC$, incommensurable in square, and fulfilling the prescribed (conditions), be laid down together [Prop. 10.33]. * I say that $AC$ is irrational.


Modern Formulation

Thus, a major straight line has a length expressible as \[\sqrt{\left(1+\frac{\rho}{\sqrt{1+\rho^2}}\right)\frac 12} + \sqrt{\left(1-\frac{\rho}{\sqrt{1+\rho^2}}\right)\frac 12}\]

for some positive rational number \(\rho\).


The major and the corresponding minor, (see Prop. 10.76), whose length is expressible as

\[\sqrt{\left(1+\frac{\rho}{\sqrt{1+\rho^2}}\right)\frac 12} - \sqrt{\left(1-\frac{\rho}{\sqrt{1+\rho^2}}\right)\frac 12}\]

are the positive roots of the quartic \[x^4-2\,x^2+ \rho^2/(1+\rho^2) = 0.\]

Proofs: 1

Corollaries: 1
Proofs: 2 3 4 5 6 7 8
Propositions: 9 10 11 12 13 14

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