Proportional Numbers are Proportional Alternately

Theorem

As Euclid defined it:

If four (natural) numbers be proportional, they will also be proportional alternately.

(The Elements: Book 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 VII Definition 20: Proportional we have that whatever part or parts $A$ is of $B$, the same part or parts is $C$ of $D$.

So from Book VII Proposition 10: Multiples of Alternate Ratios of Equal Fractions, whatever part or parts $A$ is of $C$, the same part or parts is $B$ of $D$.

Therefore from Book VII Definition 20: Proportional $A : C = B : D$.

$\blacksquare$

Historical Note

This is Proposition 13 of Book VII of Euclid's The Elements.