Construction of Rational Straight Lines Commensurable in Square Only whose Square Differences Incommensurable with Greater

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In the words of Euclid:

To find two rational straight lines commensurable in square only and such that the square on the greater is greater than the square on the less by the square on a straight line incommensurable in length with the greater.

(The Elements: Book $\text{X}$: Proposition $30$)


Let $\rho$ be a rational straight line.

Let $m$ and $n$ be natural numbers such that $m^2 + n^2$ is not square.

Let $x$ be such that:

$m^2 + n^2 : m^2 = \rho^2 + x^2$

and so:

$x^2 = \dfrac {m^2} {m^2 + n^2} \rho^2$


$x = \dfrac {\rho} {\sqrt {1 + k^2} }$

for some rational $k$.

Then $\rho$ and $\dfrac {\rho} {\sqrt {1 + k^2} }$ fit the condition.


Historical Note

This proof is Proposition $30$ of Book $\text{X}$ of Euclid's The Elements.