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