Proportion of Numbers is Transitive

Theorem

As Euclid defined it:

If there be as many (natural) 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 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 $A : C = D : F$.

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

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

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

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

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

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

$\blacksquare$

Historical Note

This is Proposition 14 of Book VII of Euclid's The Elements.