# Relation of Ratios to Products

## Theorem

In the words of Euclid:

*If four numbers be proportional, the number produced from the first and fourth will be equal to the number produced from the second and third; and, if the number produced from the first and fourth be equal to the number produced from the second and third, the four numbers will be proportional.*

(*The Elements*: Book $\text{VII}$: Proposition $19$)

That is:

- $a : b = c : d \iff ad = bc$

## Proof

Let $A, B, C, D$ be four (natural) numbers in proportion, so that $A : B = C : D$.

Let $A \times D = E$ and $B \times C = F$.

We need to show that $E = F$.

Let $A \times C = G$.

Then $A \times C = G$ and $A \times D = E$.

So from Proposition $17$ of Book $\text{VII} $: Multiples of Ratios of Numbers:

- $C : D = G : E$

But we have:

- $C : D = A : B$
- $A : B = G : E$
- $A \times C = G$
- $B \times C = F$

Then from Proposition $18$ of Book $\text{VII} $: Ratios of Multiples of Numbers:

- $A : B = G : F$

Further, we have that:

- $A : B = G : E$

Thus:

- $G : F = G : E$

So from Proposition $9$ of Book $\text{V} $: Magnitudes with Same Ratios are Equal:

- $E = F$

$\Box$

Now suppose that $E = F$.

We need to show that:

- $A : B = C : D$

Using the same construction, from Proposition $7$ of Book $\text{V} $: Ratios of Equal Magnitudes:

- $G : E = G : F$

But from Proposition $17$ of Book $\text{VII} $: Multiples of Ratios of Numbers:

- $G : E = C : D$

Then from Proposition $18$ of Book $\text{VII} $: Ratios of Multiples of Numbers:

- $G : F = A : B$

So:

- $A : B = C : D$

$\blacksquare$

## Historical Note

This proof is Proposition $19$ of Book $\text{VII}$ 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 $\text{VII}$. Propositions