Multiples of Ratios of Numbers

Theorem

 * If a (natural) number by multiplying two numbers make certain numbers, the numbers so produced will have the same ratio as the numbers multiplied.

Proof
Let the number $A$ by multiplying the two numbers $B, C$ to make $D, E$.

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


 * Euclid-VII-17.png

We have that $A \times B = D$.

Therefore $B$ measures $D$ according to the units in $A$.

But the unit $F$ also measures $A$ according to the units in it.

Therefore $F$ measures $A$ the same number of times that $B$ measures $D$.

So from, $F : A = B : D$.

For the same reason, $F : A = C : E$.

Therefore also $B : D = C : E$.

So from, $B : C = D : E$.