Proportion of Numbers is Transitive

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Theorem

In the words of Euclid:

If there be as many numbers as we please, and others equal to them in multitude, which taken two and two are in the same ratio, they will also be in the same ratio ex aequali.

(The Elements: Book $\text{VII}$: Proposition $14$)


Proof

Let there be as many (natural) numbers as we please, $A, B, C$, and others equal to them in multitude, $D, E, F$, which taken two and two are in the same ratio, so that:

$A : B = D : E$
$B : C = E : F$

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

Euclid-VII-14.png

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

So from Proposition $13$ of Book $\text{VII} $: Proportional Numbers are Proportional Alternately, it follows that $A : D = B : E$.

Similarly, we have $B : C = E : F$.

So again from Proposition $13$ of Book $\text{VII} $: Proportional Numbers are Proportional Alternately, it follows that $B : E = C : F$.

Putting them together, we get $A : D = C : F$.

Finally, again from Proposition $13$ of Book $\text{VII} $: Proportional Numbers are Proportional Alternately, it follows that $A : C = D : F$.

$\blacksquare$


Historical Note

This theorem is Proposition $14$ of Book $\text{VII}$ of Euclid's The Elements.


Sources