Ratios of Equal Magnitudes

Theorem

As Euclid defined it:

Equal magnitudes have to the same the same ratio, as also has the same to equal magnitudes.

(The Elements: Book V: Proposition $7$)

That is:

$a = b \implies a : c = b : c$

$a = b \implies c : a = c : b$

Proof

Let $A, B$ be equal magnitudes and let $C$ be any other arbitrary magnitude.

We need to show that $A:C = B:C$ and $C:A = C:B$.

Let equimultiples $D, E$ of $A, B$ be taken, and another arbitrary multiple $F$ of $C$.

We have that $D$ is the same multiple of $A$ that $E$ is of $B$, while $A = B$.

Therefore $D = E$.

But $F$ is another arbitrary magnitude.

Therefore:

$D > F \implies E > F$

$D = F \implies E = F$

$D < F \implies E < F$

We have that $D, E$ are equimultiples of $A, B$ while $F$ is another arbitrary multiple of $C$.

So from Book V Definition 5: Equality of Ratios, $A : C = B : C$.

With the same construction we can show that $D = E$, while $F$ is some other magnitude.

Therefore:

$F > D \implies F > E$

$F = D \implies F = E$

$F < D \implies F < E$

But $F$ is a multiple of $C$, while $D, E$ are equimultiples of $A, B$.

So from Book V Definition 5: Equality of Ratios, $C : A = C : B$.

$\blacksquare$

Historical Note

This is Proposition 7 of Book V of Euclid's The Elements.

This is the converse of Book V Proposition $9$: Magnitudes with Same Ratios are Equal.