Least Ratio of Numbers
Theorem
In the words of Euclid:
(The Elements: Book $\text{VII}$: Proposition $33$)
Proof
Let $A, B, C$ be the given numbers, as many as we please.
$A, B, C$ are either coprime or not.
Let $A, B, C$ be coprime.
Then by Proposition $21$ of Book $\text{VII} $: Coprime Numbers form Fraction in Lowest Terms they are the least of those which have the same ratio with them.
Let $A, B, C$ not be coprime.
Then by Proposition $3$ of Book $\text{VII} $: Greatest Common Divisor of Three Numbers their GCD $D$ can be found.
As many times as $D$ measures each of $A, B, C$, let those be $E, F, G$ respectively.
That is:
- $D E = A, D F = B, D G = C$
By Proposition $20$ of Book $\text{VII} $: Ratios of Fractions in Lowest Terms, $E, F, G$ are in the same ratio with $A, B, C$.
Suppose $E, F, G$ are not the least of those which have the same ratio with $A, B, C$.
Let those numbers be $H, K, L$.
Then $H$ measures $A$ the same number of times that $K, L$ measure $B, C$.
Let this number of times be $M$.
We have that $H$ measures $A$ according to the units of $M$.
It follows from Proposition $16$ of Book $\text{VII} $: Natural Number Multiplication is Commutative that $M$ also measures $A$ according to the units of $H$.
For the same reason, $M$ also measures $B$ and $C$ according to the units of $K$ and $L$.
Therefore $M$ measures $A$.
That is:
- $A = H M$
For the same reason:
- $A = E D$
That is:
- $E D = H M$
Therefore by Proposition $19$ of Book $\text{VII} $: Relation of Ratios to Products:
- $E : H = M : D$
But $E > H$ and so $M > D$.
Also, $M$ measures $A$, $B$ and $C$.
But by hypothesis $D$ is the greatest common measure of $A, B, C$.
Therefore there cannot be any numbers less than $E, F, G$ which are in the same ratio with $A, B, C$.
Therefore $E, F, G$ are the least of those which are in the same ratio with $A, B, C$.
$\blacksquare$
Historical Note
This proof is Proposition $33$ 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