Relation of Ratios to Products

Theorem

As Euclid defined it:

If four (natural) 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 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 Book VII Proposition 17: 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 Book VII Proposition 18: Ratios of Multiples of Numbers, $A : B = G : F$

Further, we have that $A : B = G : E$.

Thus $G : F = G : E$.

So from Book V Proposition 9: 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 Book V Proposition 7: Ratios of Equal Magnitudes, $G : E = G : F$.

But from Book VII Proposition 17: Multiples of Ratios of Numbers, $G : E = C : D$.

Then from Book VII Proposition 18: Ratios of Multiples of Numbers, $G : F = A : B$.

So $A : B = C : D$.

$\blacksquare$

Historical Note

This is Proposition 19 of Book VII of Euclid's The Elements.