# Commensurability of Elements of Proportional Magnitudes

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## Theorem

In the words of Euclid:

*If four magnitudes be proportional, and the first be commensurable with the second, the third will also be commensurable with the fourth; and, if the first be incommensurable with the second, the third will also be incommensurable with the fourth.*

(*The Elements*: Book $\text{X}$: Proposition $11$)

## Proof

Let $A$, $B$, $C$ and $D$ be four magnitudes in proportion:

- $A : B = C : D$

Let $A$ be commensurable with $B$.

Then from Ratio of Commensurable Magnitudes:

As $A : B = C : D$ it follows that:

From Magnitudes with Rational Ratio are Commensurable it follows that:

- $C$ is commensurable with $D$.

$\Box$

Let $A$ be incommensurable with $B$.

Then from Incommensurable Magnitudes have Irrational Ratio:

As $A : B = C : D$ it follows that:

From Magnitudes with Irrational Ratio are Incommensurable it follows that:

- $C$ is incommensurable with $D$.

$\blacksquare$

## Historical Note

This proof is Proposition $11$ 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