Ratios of Equal Magnitudes

Theorem

In the words of Euclid:

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

(The Elements: Book $\text{V}$: Proposition $7$)

That is:

$a = b \implies a : c = b : c$

$a = b \implies c : a = c : b$

Porism

In the words of Euclid:

From this it is manifest that, if any magnitudes are proportional, they will also be inversely proportional.

(The Elements: Book $\text{V}$: Proposition $7$ : Porism)

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 $\text{V}$ 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 $\text{V}$ Definition $5$: Equality of Ratios, $C : A = C : B$.

$\blacksquare$

Historical Note

This proof is Proposition $7$ of Book $\text{V}$ of Euclid's The Elements.
It is the converse of Proposition $9$ of Book $\text{V} $: Magnitudes with Same Ratios are Equal.

Sources

• 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 2 (2nd ed.) ... (previous) ... (next): Book $\text{V}$. Propositions