Commensurability is Transitive Relation

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:

and:

Then:

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.