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