Commensurability is Transitive Relation

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Theorem

In the words of Euclid:

Magnitudes commensurable with the same magnitude are commensurable with one another also.

(The Elements: Book $\text{X}$: Proposition $12$)


Proof

Let $A$ and $B$ be magnitudes which are both commensurable with another magnitude $C$.

As $A$ is commensurable with $C$, from Ratio of Commensurable Magnitudes:

$A$ has to $C$ the ratio which a number has to a number.

Let it have the ratio which $D$ has to $E$, where $D$ and $E$ are numbers.


As $C$ is commensurable with $B$, from Ratio of Commensurable Magnitudes:

$C$ has to $B$ the ratio which a number has to a number.

Let it have the ratio which $F$ has to $G$, where $F$ and $G$ are numbers.


From Construction of Sequence of Numbers with Given Ratios, let the numbers $H$, $K$, $L$ be assigned the ratios:

$D : E = H : K$
$F : G = K : L$

So:

\(\ds A : C\) \(=\) \(\ds D : E\)
\(\ds D : E\) \(=\) \(\ds H : K\)
\(\ds \leadsto \ \ \) \(\ds A : C\) \(=\) \(\ds H : K\) Equality of Ratios is Transitive

and:

\(\ds C : B\) \(=\) \(\ds F : G\)
\(\ds F : G\) \(=\) \(\ds K : L\)
\(\ds \leadsto \ \ \) \(\ds C : B\) \(=\) \(\ds K : L\) Equality of Ratios is Transitive

Then:

\(\ds C : B\) \(=\) \(\ds K : L\)
\(\ds A : C\) \(=\) \(\ds H : K\)
\(\ds \leadsto \ \ \) \(\ds A : B\) \(=\) \(\ds H : L\) Equality of Ratios Ex Aequali

That is, $A$ has to $B$ the ratio which a number has to a number.

From Magnitudes with Rational Ratio are Commensurable:

$A$ and $B$ are commensurable with each other.

$\blacksquare$


Historical Note

This proof is Proposition $12$ of Book $\text{X}$ of Euclid's The Elements.


Sources