Proportional Numbers have Proportional Differences

Theorem

As Euclid defined it:

If, as a whole is to a whole, so is a (natural) number subtracted to a (natural) number subtracted, the remainder will also be to the remainder as whole to whole.

(The Elements: Book VII: Proposition $11$)

That is:

$a : b = c : d \implies \left({a - c}\right) : \left({b - d}\right) = a : b$

where $a : b$ denotes the ratio of $a$ to $b$.

Proof

As the whole $AB$ is to the whole $CD$, so let the $AE$ subtracted be to $CF$ subtracted.

We need to show that $EB : FD = AB : CD$.

We have that :$AB : CD = AE : CF$.

So from Book VII Definition 20: Proportional we have that whatever part or parts $AB$ is of $CD$, the same part or parts is $AE$ of $CF$.

So from Book VII Proposition 7: Subtraction of Divisors Obeys Distributive Law and Book VII Proposition 8: Subtraction of Multiples of Divisors Obeys Distributive Law, $EB$ is the same part or parts of $FD$ that $AB$ is of $CD$.

So by Book VII Definition 20: Proportional $EB : FD = AB : CD$.

$\blacksquare$

Historical Note

This is Proposition 11 of Book VII of Euclid's The Elements.