Proportional Numbers are Proportional Alternately

Theorem

In the words of Euclid:

If four numbers be proportional, they will also be proportional alternately.

(The Elements: Book $\text{VII}$: Proposition $13$)

Proof

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

We need to show that $A : C = B : D$.

We have that $A : B = C : D$.

So from Book $\text{VII}$ Definition $20$: Proportional we have that whatever aliquot part or aliquant part $A$ is of $B$, the same aliquot part or aliquant part is $C$ of $D$.

So from Proposition $10$ of Book $\text{VII} $: Multiples of Alternate Ratios of Equal Fractions, whatever aliquot part or aliquant part $A$ is of $C$, the same aliquot part or aliquant part is $B$ of $D$.

Therefore from Book $\text{VII}$ Definition $20$: Proportional $A : C = B : D$.

$\blacksquare$

Historical Note

This proof is Proposition $13$ 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