# Proposition: Prop. 10.018: Condition for Incommensurability of Roots of Quadratic Equation

### (Proposition 18 from Book 10 of Euclid's “Elements”)

If there are two unequal straight lines, and a (rectangle) equal to the fourth part of the (square) on the lesser, falling short by a square figure, is applied to the greater, and divides it into (parts which are) incommensurable [in length], then the square on the greater will be larger than the (square on the) lesser by the (square) on (some straight line) incommensurable (in length) with the greater. And if the square on the greater is larger than the (square on the) lesser by the (square) on (some straight line) incommensurable (in length) with the greater, and a (rectangle) equal to the fourth (part) of the (square) on the lesser, falling short by a square figure, is applied to the greater, then it divides it into (parts which are) incommensurable [in length].

### Modern Formulation

This proposition states that if $\alpha\,x-x^2=\frac{\beta^2}4,$ (where $\alpha=BC$, $x=DC$, and $\beta=A$), then $\alpha\quad\text{ and }\quad\sqrt{\alpha^2-\beta^2}$ are incommensurable when $\alpha-x\quad\text{ and }\quad x$ are incommensurable, and vice versa.

Proofs: 1

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

Thank you to the contributors under CC BY-SA 4.0!

Github:

non-Github:
@Fitzpatrick