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