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