Numbers whose Product is Square are Similar Plane Numbers
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Theorem
In the words of Euclid:
- If two numbers by multiplying one another make a square, they are similar plane numbers.
(The Elements: Book $\text{IX}$: Proposition $2$)
Proof
Let $a$ and $b$ be natural numbers such that $a b$ is square.
From Proposition $17$ of Book $\text{VII} $: Multiples of Ratios of Numbers:
- $a : b = a^2 : a b$
We have that $a b$ and $a^2$ are both square.
By Square Numbers are Similar Plane Numbers they are similar plane numbers.
- there exists one mean proportional between $a^2$ and $a b$.
- there exists one mean proportional between $a$ and $b$.
The result follows from Proposition $20$ of Book $\text{VIII} $: Numbers between which exists one Mean Proportional are Similar Plane.
$\blacksquare$
Historical Note
This proof is Proposition $2$ of Book $\text{IX}$ of Euclid's The Elements.
It is the converse of Proposition $1$: Product of Similar Plane Numbers is Square.
Sources
- 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 2 (2nd ed.) ... (previous) ... (next): Book $\text{IX}$. Propositions