Numbers between which exists one Mean Proportional are Similar Plane
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Theorem
Let $a, b \in \Z$ such that the geometric mean is an integer.
Then $a$ and $b$ are similar plane numbers.
In the words of Euclid:
- If one mean proportional number fall between two numbers, the numbers will be similar plane numbers.
(The Elements: Book $\text{VIII}$: Proposition $20$)
Proof
Let the geometric mean of $a$ and $b$ be an integer $m$.
Then, in the language of Euclid, $m$ is a mean proportional of $a$ and $b$.
Thus:
- $\left({a, m, b}\right)$
is a geometric sequence.
From Form of Geometric Sequence of Integers:
- $\exists k, p, q \in \Z: a = k p^2, b = k q^2$
So $a$ and $b$ are plane numbers whose sides are:
- $k p$ and $p$
and
- $k q$ and $q$
respectively.
Then:
- $\dfrac {k p} {k q} = \dfrac p q$
demonstrating that $a$ and $b$ are similar plane numbers by definition.
$\blacksquare$
Historical Note
This proof is Proposition $20$ of Book $\text{VIII}$ of Euclid's The Elements.
Sources
- 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 2 (2nd ed.) ... (previous) ... (next): Book $\text{VIII}$. Propositions