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)



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.


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.