Product of Similar Plane Numbers is Square
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Theorem
In the words of Euclid:
- If two similar plane numbers by multiplying one another make some number, the product will be square.
(The Elements: Book $\text{IX}$: Proposition $1$)
Proof
Let $a$ and $b$ be similar plane numbers.
From Proposition $17$ of Book $\text{VII} $: Multiples of Ratios of Numbers:
- $a : b = a^2 : a b$
- there exists one mean proportional between $a$ and $b$.
- there exists one mean proportional between $a^2$ and $a b$.
From Proposition $22$ of Book $\text{VIII} $: If First of Three Numbers in Geometric Sequence is Square then Third is Square it follows that:
- $a b$ is square.
$\blacksquare$
Historical Note
This proof is Proposition $1$ of Book $\text{IX}$ of Euclid's The Elements.
It is the converse of Proposition $2$: Numbers whose Product is Square are Similar Plane Numbers.
Sources
- 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 2 (2nd ed.) ... (previous) ... (next): Book $\text{IX}$. Propositions