Construction of Rational Straight Lines Commensurable in Square Only whose Square Differences Commensurable with Greater/Lemma 1
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.
(The Elements: Book $\text{X}$: Proposition $29$ : Lemma 1)
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:
and:
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.
Sources
- 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 3 (2nd ed.) ... (previous) ... (next): Book $\text{X}$. Propositions