Commensurability is Transitive Relation

Theorem

In the words of Euclid:

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

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.

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

$\blacksquare$

Historical Note

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