Construction of Rational Straight Lines Commensurable in Square Only whose Square Differences Incommensurable with Greater
Jump to navigation
Jump to search
Theorem
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 $30$)
Proof
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:
- $x^2 = \dfrac {m^2} {m^2 + n^2} \rho^2$
or:
- $x = \dfrac {\rho} {\sqrt {1 + k^2} }$
for some rational $k$.
Then $\rho$ and $\dfrac {\rho} {\sqrt {1 + k^2} }$ fit the condition.
$\blacksquare$
Historical Note
This proof is Proposition $30$ of Book $\text{X}$ of Euclid's The Elements.
Sources
- 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 3 (2nd ed.) ... (previous) ... (next): Book $\text{X}$. Propositions