Equality of Ratios is Transitive
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Theorem
In the words of Euclid:
(The Elements: Book $\text{V}$: Proposition $11$)
That is:
- $A : B = C : D, C : D = E : F \implies A : B = E : F$
Proof
Let $A : B = C : D$ and $C : D = E : F$.
Of $A, C, E$ let equimultiples $G, H, K$ be taken.
Of $B, D, F$ let other arbitrary equimultiples $L, M, N$ be taken.
We have that:
- $A : B = C : D$
- $G, H$ are equimultiples of $A, C$
- $L, M$ are equimultiples of $B, D$
So:
- $G > L \implies H > M$
- $G = L \implies H = M$
- $G < L \implies H < M$
Also, we have that:
- $C : D = E : F$
- $H, K$ are equimultiples of $C, E$
- $M, N$ are equimultiples of $D, F$
So:
- $H > M \implies K > N$
- $H = M \implies K = N$
- $H < M \implies K < N$
Also, we have that:
- $G, K$ are equimultiples of $A, E$
- $L, N$ are equimultiples of $B, F$
Therefore $A : B = E : F$.
$\blacksquare$
Historical Note
This proof is Proposition $11$ of Book $\text{V}$ of Euclid's The Elements.
Sources
- 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 2 (2nd ed.) ... (previous) ... (next): Book $\text{V}$. Propositions