# Proportional Magnitudes are Proportional Alternately

## Theorem

In the words of Euclid:

*If four magnitudes be proportional, they will also be proportional alternately.*

(*The Elements*: Book $\text{V}$: Proposition $16$)

That is:

- $a : b = c : d \implies a : c = b : d$

## Proof

Let $A, B, C, D$ be four proportional magnitudes, so that as $A$ is to $B$, then so is $C$ to $D$.

We need to show that as $A$ is to $C$, then $B$ is to $D$.

Let equimultiples $E, F$ be taken of $A, B$.

Let other arbitrary equimultiples $G, H$ be taken of $C, D$.

We have that $E$ is the same multiple of $A$ that $F$ is of $B$.

So from Ratio Equals its Multiples we have that $A : B = E : F$

But $A : B = C : D$.

So from Equality of Ratios is Transitive it follows that $C : D = E : F$.

Similarly, we have that $G, H$ are equimultiples of $C, D$.

So from Ratio Equals its Multiples we have that $C : D = G : H$

So from Equality of Ratios is Transitive it follows that $E : F = G : H$.

But from Relative Sizes of Components of Ratios:

- $E > G \implies F > H$
- $E = G \implies F = H$
- $E < G \implies F < H$

Now $E, F$ are equimultiples of $A, B$, and $G, H$ are equimultiples of $C, D$.

Therefore from Book $\text{V}$ Definition $5$: Equality of Ratios:

- $A : C = B : D$

$\blacksquare$

## Historical Note

This proof is Proposition $16$ 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