Proportional Numbers are Proportional Alternately

Theorem

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

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

We need to show that $A : C = B : D$.


 * Euclid-VII-13.png

We have that $A : B = C : D$.

So from we have that whatever part or parts $A$ is of $B$, the same part or parts is $C$ of $D$.

So from, whatever part or parts $A$ is of $C$, the same part or parts is $B$ of $D$.

Therefore from $A : C = B : D$.