# Proportional Magnitudes have Proportional Remainders

## Theorem

In the words of Euclid:

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

(*The Elements*: Book $\text{V}$: Proposition $19$)

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 $b$.

### Porism

In the words of Euclid:

*From this it is manifest that, if magnitudes be proportional*componendo*, they will also be proportional*convertendo*.*

(*The Elements*: Book $\text{V}$: Proposition $19$ : Porism)

## Proof

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

That is:

- $AB : CD = AE : CF$

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

We have that :$AB : CD = AE : CF$.

So from Proportional Magnitudes are Proportional Alternately we have that $BA : AE = DC : CF$.

From Magnitudes Proportional Compounded are Proportional Separated we have that $BA : EA = DF : CF$.

From Proportional Magnitudes are Proportional Alternately: $BE : DF = EA : FC$.

But by hypothesis $AE : CF = AB : CD$.

So by Equality of Ratios is Transitive $EB : FD = AB : CD$.

$\blacksquare$

## Historical Note

This proof is Proposition $19$ of Book $\text{V}$ 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 $\text{V}$. Propositions