# Ratio of Products of Sides of Plane Numbers

## Theorem

In the words of Euclid:

*Plane numbers have to one another the ratio compounded of the ratios of their sides.*

(*The Elements*: Book $\text{VIII}$: Proposition $5$)

## Proof

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

Let $a = c d$ and $b = e f$.

From Proposition $4$ of Book $\text{VIII} $: Construction of Sequence of Numbers with Given Ratios we can find $g, h, k$ such that:

- $c : e = g : h$
- $d : f = h : k$

Let $d e = l$.

We also have that $a = c d$.

So from Proposition $17$ of Book $\text{VII} $: Multiples of Ratios of Numbers:

- $c : e = a : l$

But we also have:

- $c : e = g : h$

and so:

- $g : h = a : l$

Since:

- $e d = l$
- $e f = b$

from Proposition $17$ of Book $\text{VII} $: Multiples of Ratios of Numbers:

- $d : f = l : b$

But we also have:

- $d : f = h : k$

and so:

- $h : k = l : b$

But we have:

- $g : h = a : l$

and so from Proposition $14$ of Book $\text{VII} $: Proportion of Numbers is Transitive:

- $g : k = a : b$

But $g : k$ is the ratio compounded of the ratios of the sides of $a$ and $b$.

Therefore $a : b$ is the ratio compounded of the ratios of their sides.

$\blacksquare$

## Historical Note

This proof is Proposition $5$ 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