# Magnitudes Proportional Compounded are Proportional Separated

## Theorem

In the words of Euclid:

*If magnitudes are proportional*componendo*, they will also be proportional*separando*.*

(*The Elements*: Book $\text{V}$: Proposition $17$)

That is:

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

## Proof

Let $AB, BE, CD, DF$ be magnitudes which are proportional *componendo*, so that $AB : BE = CD : DF$.

We need to show that they are proportional *separando*, that is, that $AE : EB = CF : DF$.

Let equimultiples $GH, HK, LM, MN$ be taken of $AE, EB, CF, FD$.

Let $KO, NP$ be other arbitrary equimultiples of $EB, FD$.

We have that $GH$ is the same multiple of $AE$ that $HK$ is of $EB$.

So from Multiplication of Numbers is Left Distributive over Addition $GH$ is the same multiple of $AE$ that $GK$ is of $AB$.

But $GH$ is the same multiple of $AE$ that $LM$ is of $CF$.

Therefore $GK$ is the same multiple of $AB$ that $LM$ is of $CF$.

Again, we have that $LM$ is the same multiple of $CF$ that $MN$ is of $FD$.

So from Multiplication of Numbers is Left Distributive over Addition $LM$ is the same multiple of $CF$ that $LN$ is of $CD$.

Therefore $GK, LN$ are equimultiples of $AB, CD$.

Again, we have that $HK$ is the same multiple of $EB$ that $MN$ is of $FD$.

Also, $KO$ is the same multiple of $EB$ that $NP$ is of $FD$.

So from Multiplication of Numbers is Right Distributive over Addition the sum $HO$ is also the same multiple of $EB$ that $MP$ is of $FD$.

We also have that $AB : BE = CD : DF$.

Also, we have that $GK, LN$ are equimultiples of $AB, CD$.

Also, we have that $EB, FD$ are equimultiples of $HO, MP$.

Therefore:

- $GK > HO \implies LN > MP$
- $GK = HO \implies LN = MP$
- $GK < HO \implies LN < MP$

Suppose $GK > HO$.

If we subtract $HK$ from each, we have that $GH > KO$.

But $GK > HO \implies LN > MP$.

Therefore $LN > MP$.

So if $MN$ is subtracted from each, we have that $LM > NP$.

So:

- $GH > KO \implies LM > NP$

Similarly we can prove that:

- $GH = KO \implies LM = NP$
- $GH < KO \implies LM < NP$

Also, we have that:

- $GH, LM$ are equimultiples of $AE, CF$
- $KO, NP$ are equimultiples of $EB, FD$

Therefore $AE : EB = CF : FD$.

$\blacksquare$

## Historical Note

This proof is Proposition $17$ of Book $\text{V}$ of Euclid's *The Elements*.

It is the converse of Proposition $18$ of Book $\text{V} $: Magnitudes Proportional Separated are Proportional Compounded.

## Sources

- 1926: Sir Thomas L. Heath:
*Euclid: The Thirteen Books of The Elements: Volume 2*(2nd ed.) ... (previous) ... (next): Book $\text{V}$. Propositions