Ratio of Commensurable Magnitudes
Theorem
In the words of Euclid:
- Commensurable magnitudes have to one another the ratio which a number has to a number.
(The Elements: Book $\text{X}$: Proposition $5$)
Proof
Let $A$ and $B$ be commensurable magnitudes.
By definition of commensurable, some magnitude will measure them both.
Let $C$ be a common measure of $A$ and $B$.
Let $D$ be the number of times $C$ is the measure of $A$.
Let $E$ be the number of times $C$ is the measure of $B$.
Since:
and
it follows that the unit measures $D$ the same number of times $C$ measures $A$.
From Ratios of Fractions in Lowest Terms:
- $\dfrac C A = \dfrac 1 D$
and:
- $\dfrac A C = \dfrac D 1 = D$
Similarly, since:
and
it follows that the unit measures $E$ the same number of times $C$ measures $B$.
From Ratios of Fractions in Lowest Terms:
- $\dfrac C B = \dfrac 1 E$
So by Equality of Ratios Ex Aequali:
- $\dfrac A B = \dfrac D E$
Hence the result.
$\blacksquare$
Historical Note
This proof is Proposition $5$ of Book $\text{X}$ of Euclid's The Elements.
Sources
- 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 3 (2nd ed.) ... (previous) ... (next): Book $\text{X}$. Propositions