# Proportional Magnitudes have Proportional Remainders/Porism

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## Porism to Proportional Magnitudes have Proportional Remainders

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

From Proposition $19$ of Book $\text{V} $: Proportional Magnitudes have Proportional Remainders:

- $EB : FD = AB : CD$

From Proposition $16$ of Book $\text{V} $: Proportional Magnitudes are Proportional Alternately:

- $AB : BE = CD : FD$

Therefore magnitudes compounded are proportional.

But it was also proved that:

- $BA : AE = DC : CF$

and this is *convertendo*.

$\blacksquare$

## Historical Note

This proof is Proposition $19$ of Book $\text{V}$ of Euclid's *The Elements*.

It was suggested by Heiberg that this porism, along with its explanation, was not original to Euclid, but was a later interpolation, dating from a time before that of Theon.

## Sources

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