# 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