Between two Similar Plane Numbers exists one Mean Proportional

Theorem

In the words of Euclid:

Between two similar plane numbers there is one mean proportional number, and the plane number has to the plane number the ratio duplicate of that which the corresponding side has to the corresponding side.

(The Elements: Book $\text{VIII}$: Proposition $18$)

Proof

Let $m$ and $n$ be similar plane numbers.

Then for some $p_1, p_2, q_1, q_2 \in \Z$ such that $p_1 < p_2$ and $q_1 < q_2$:

$m = p_1 p_2$

$n = q_1 q_2$

such that:

$\dfrac {p_1}{q_1} = \dfrac {p_2}{q_2}$

Thus let:

$r := p_1 q_2 = q_1 p_2$

So:

\(\ds \dfrac m r\)

\(=\)

\(\ds \dfrac {p_1 p_2} {p_1 q_2}\)

\(\ds \)

\(=\)

\(\ds \dfrac {p_2}{q_2}\)

and:

\(\ds \dfrac r n\)

\(=\)

\(\ds \dfrac {q_1 p_2} {q_1 q_2}\)

\(\ds \)

\(=\)

\(\ds \dfrac {p_2}{q_2}\)

Thus by definition, $\paren {m, r, n}$ is a geometric sequence.

By definition, $r$ is a mean proportional between $m$ and $n$.

Thus as:

$\dfrac {p_1}{q_1} = \dfrac m r = \dfrac r n = \dfrac {p_2}{q_2}$

it follows that $m$ is in duplicate ratio to $n$ as their sides.

$\blacksquare$

Historical Note

This proof is Proposition $18$ 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