# Commensurability is Transitive Relation

## 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:

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:

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

- 1926: Sir Thomas L. Heath:
*Euclid: The Thirteen Books of The Elements: Volume 3*(2nd ed.) ... (previous) ... (next): Book $\text X$. Propositions