If Ratio of Square to Number is as between Two Squares then Number is Square

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Let $a, b, c, d \in \Z$ be integers such that:

$\dfrac a b = \dfrac {c^2} {d^2}$

Let $a$ be a square number.

Then $b$ is also a square number.

In the words of Euclid:

If two numbers have to one another the ratio which a square number has to a square number, and the first be square, the second will also be square.

(The Elements: Book $\text{VIII}$: Proposition $24$)


From Proposition $18$ of Book $\text{VIII} $: Between two Similar Plane Numbers exists one Mean Proportional:

$\tuple {c^2, c d, d^2}$

is a geometric sequence.

From Proposition $8$ of Book $\text{VIII} $: Geometric Sequences in Proportion have Same Number of Elements:

$\tuple {a, m, b}$

is a geometric sequence for some $m$.

We have that $a$ is a square number.

From Proposition $22$ of Book $\text{VIII} $: If First of Three Numbers in Geometric Sequence is Square then Third is Square:

$b$ is a square number.


Historical Note

This proof is Proposition $24$ of Book $\text{VIII}$ of Euclid's The Elements.