# Multiples of Ratios of Numbers

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## Theorem

In the words of Euclid:

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

(*The Elements*: Book $\text{VII}$: Proposition $17$)

## Proof

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

We need to show that:

- $B : C = D : E$

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 Book $\text{VII}$ Definition $20$: Proportional:

- $F : A = B : D$

For the same reason:

- $F : A = C : E$

Therefore also:

- $B : D = C : E$

So from Proposition $13$ of Book $\text{VII} $: Proportional Numbers are Proportional Alternately:

- $B : C = D : E$

$\blacksquare$

## Historical Note

This proof is Proposition $17$ 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