Proposition: Prop. 10.076: Minor is Irrational

Euclid's Formulation

If a straight line, which is incommensurable in square with the whole, and with the whole makes the (squares) on them (added) together rational, and the (rectangle contained) by them medial, is subtracted from a(nother) straight line then the remainder is an irrational (straight line). Let it be called a minor (straight line). * For let the straight line $BC$, which is incommensurable in square with the whole, and fulfils the (other) prescribed (conditions), have been subtracted from the straight line $AB$ [Prop. 10.33]. * I say that the remainder $AC$ is that irrational (straight line) called minor.


Modern Formulation

Thus, a minor 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 rational number \(\rho\). See also [Prop. 10.39].

Proofs: 1

Corollaries: 1
Proofs: 2 3 4 5 6 7 8 9 10
Propositions: 11 12 13 14 15 16 17 18

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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