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