Multiples of Ratios of Numbers

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:

$B : C = D : E$

$\blacksquare$

Examples

Arbitrary Example

We have:

$12 : 15$ is the same ratio as $4 : 5$, and also the same ratio as $24 : 30$.

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
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): ratio
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): ratio