Alternate Ratios of Equal Fractions

Theorem

As Euclid defined it:

If a (natural) number be a part of a (natural) number, and another be the same part of another, alternately also, whatever part or parts the first is of the third, the same part, or the same parts, will the second also be of the fourth.

(The Elements: Book VII: Proposition $9$)

Proof

Let the (natural) number $A$ be a part of the (natural) number $BC$, and another $D$ be the same part of another, $EF$, that $A$ is of $BC$.

We need to show that whatever part or parts $A$ is of $D$, the same part or parts is $BC$ of $EF$.

We have that whatever part $A$ is of $BC$, the same part $D$ is of $EF$.

Therefore, as many numbers as there are in $BC$ equal to $A$, so many also are there in $EF$ equal to $D$.

Let $BC$ be divided into the numbers equal to $A$, namely $BG, GC$.

Let $EF$ be divided into the numbers equal to $D$, namely $EH, HF$.

Thus the multitude of $BG, GC$ equals the multitude of $EH, HF$.

We have that $BG = GC$ and $EH = HF$.

Therefore whatever part or parts $BG$ is of $EH$, the same part or the same parts is $GC$ of $HF$ also.

So, from Book VII Proposition 5: Divisors Obey Distributive Law and Book VII Proposition 6: Multiples of Divisors Obey Distributive Law, we have that whatever part or parts $BG$ is of $EH$, the same part also, or the same parts, is the sum $BC$ of the sum $EF$.

But $BG = A$ and $EH = D$.

So whatever part or parts $A$ is of $D$, the same part or parts is $BC$ of $EF$ also.

$\blacksquare$

Historical Note

This is Proposition 9 of Book VII of Euclid's The Elements.