# Construction of Rational Straight Lines Commensurable in Square Only whose Square Differences Commensurable with Greater/Lemma 1

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## Lemma to Construction of Rational Straight Lines Commensurable in Square Only whose Square Differences Commensurable with Greater

In the words of Euclid:

To find two square numbers such that their sum is also square.

## Proof

Let $a$ and $b$ be two natural numbers such that $a > b$.

Let $a$ and $b$ be either both even or both odd.

From:

Even Number minus Even Number is Even

and:

Odd Number minus Odd Number is Even

their difference $c = a - b$ is even.

Let $d = \dfrac c 2$.

Let $a$ and $b$ be similar plane numbers.

By definition of similar plane numbers it is noted that $a$ and $b$ may both be square.

Then by Square of Sum less Square:

$a b + d^2 = \paren {a - d}^2$

From Product of Similar Plane Numbers is Square, $a b$ is a square number.

So two square numbers $a b$ and $d^2$ have been found whose sum is square.

Similarly:

$a b = \paren {a - d}^2 - d^2$

from which it is noted that two square numbers $\paren {a - d}^2$ and $d^2$ have been found whose difference is square.

$\blacksquare$

## Historical Note

This proof is Proposition $29$ of Book $\text{X}$ of Euclid's The Elements.