Ratios of Fractions in Lowest Terms
Theorem
As Euclid defined it:
- The least (natural) numbers of those which have the same ratio with them measure those which have the same ratio the same number of times, the greater the greater and the less the less.
(The Elements: Book VII: Proposition $20$)
Proof
Let $CD, EF$ be the least (natural) numbers of those which have the same ratio with $A, B$.
We need to show that $CD$ measures $A$ the same number of times that $EF$ measures $B$.
Suppose $CD$ is parts of $A$.
Then from Book VII Proposition 13: Proportional Numbers are Proportional Alternately and Book VII Definition 20: Proportional, $EF$ is also the same parts of $B$ that $CD$ is of $A$.
Therefore as many parts of $A$ as there are in $CD$, so many parts of $B$ are there also in $EF$.
Let $CD$ be divided into the parts of $A$, namely $CG, GD$ and $EF$ into the parts of $B$, namely $EH, HF$.
Thus the multitude of $CG, GD$ will be equal to the multitude of $EH, HF$.
We have that the numbers $CG, GD$ are equal to one another, and the numbers $EH, HF$ are also equal to one another,
Therefore $CG : EH = GD : HF$.
So from Book VII Proposition 12: Ratios of Numbers is Distributive over Addition, as one of the antecedents is to one of the consequents, so will all the antecedents be to all the consequents.
Therefore $CG : EH = CD : EF$.
Therefore $CG, EH$ are in the same ratio with $CD, EF$ being less than they.
This is impossible, for by hypothesis $CD, EF$ are the least numbers of those which have the same ratio with them.
So $CD$ is not parts of $A$.
Therefore from Book VII Proposition 4: Natural Number Divisor or Multiple of Divisor of Another, $CD$ is a part of $A$.
Also, from Book VII Proposition 13: Proportional Numbers are Proportional Alternately and Book VII Definition 20: Proportional, $EF$ is the same part of $B$ that $CD$ is of $A$.
Therefore $CD$ measures $A$ the same number of times that $EF$ measures $B$.
$\blacksquare$
Historical Note
This is Proposition 20 of Book VII of Euclid's The Elements.
