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

Theorem

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 $24$)

Proof

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.

$\blacksquare$

Historical Note

This proof is Proposition $24$ 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