Construction of Rational Straight Lines Commensurable in Square Only whose Square Differences Commensurable with Greater
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 commensurable in length with the greater.
(The Elements: Book $\text{X}$: Proposition $29$)
Lemma 1
In the words of Euclid:
- To find two square numbers such that their sum is also square.
(The Elements: Book $\text{X}$: Proposition $29$ : Lemma 1)
Lemma 2
In the words of Euclid:
- To find two square numbers such that their sum is not square.
(The Elements: Book $\text{X}$: Proposition $29$ : Lemma 2)
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 a straight line such that:
- $(1): \quad m^2 : \paren {m^2 - n^2} = \rho^2 : x^2$
Thus:
- $x^2 = \dfrac {m^2 - n^2} {m^2} \rho^2$
and so:
- $x = \rho \sqrt {1 - k^2}$
where $k = \dfrac n m$.
From $(1)$:
- $x^2 \frown \rho^2$
where $\frown$ denotes commensurability in length.
Thus $x$ is a rational straight line, but:
- $x \smile \rho$
where $\smile$ denotes incommensurability in length.
From $(1)$:
- $ m^2 : n^2 = \rho^2 : \rho^2 - x^2$
so that:
- $\sqrt {\rho^2 - x^2} \frown \rho$
and in fact:
- $\sqrt {\rho^2 - x^2} = k \rho$
where $k$ is a rational number.
$\blacksquare$
Historical Note
This proof is Proposition $29$ 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