Similar Plane Numbers have Same Ratio as between Two Squares

Theorem

In the words of Euclid:

Similar plane numbers have to one another the ratio which a square number has to a square number.

(The Elements: Book $\text{VIII}$: Proposition $26$)

Proof

Let $a$ and $b$ be similar plane numbers.

From Between two Similar Plane Numbers exists one Mean Proportional, there exists a mean proportional $m$ between them.

By definition of mean proportional:

$\left({a, m, b}\right)$

is a geometric sequence.

From Form of Geometric Sequence of Integers:

$\exists k, p, q \in Z: a = k p^2, b = k q^2, m = k p q$

Thus:

$\dfrac a b = \dfrac {k p^2} {k q^2} = \dfrac {p^2} {q^2}$

Hence the result.

$\blacksquare$

Historical Note

This proof is Proposition $26$ 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