Relation of Ratios to Products

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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 $E = F$.

Euclid-VII-19.png

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 theorem is Proposition $19$ of Book $\text{VII}$ of Euclid's The Elements.


Sources