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