Proportional Numbers have Proportional Differences

Theorem

In the words of Euclid:

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

(The Elements: Book $\text{VII}$: 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 $\text{VII}$ Definition $20$: Proportional we have that whatever aliquot part or aliquant part $AB$ is of $CD$, the same aliquot part or aliquant part is $AE$ of $CF$.

So from:

Proposition $7$ of Book $\text{VII} $: Subtraction of Divisors obeys Distributive Law

and:

Proposition $8$ of Book $\text{VII} $: Subtraction of Multiples of Divisors obeys Distributive Law

$EB$ is the same aliquot part or aliquant part of $FD$ that $AB$ is of $CD$.

So by Book $\text{VII}$ Definition $20$: Proportional $EB : FD = AB : CD$.

$\blacksquare$

Historical Note

This proof is Proposition $11$ 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