Source — Euclid's Elements, encoded as an rrxiv paper

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V.21 Proposition V.21

If there be three magnitudes, and others equal to them in multitude, which taken two and two together are in the same ratio, and the proportion of them be perturbed, then if ex aequali the first be greater than the third, the fourth will also be greater than the sixth.

Proof

A perturbed version of V.20: with

a : b = e : f

and

b : c = d : e

the ratio chain still imposes the same ordering between

a

versus

c

and

d

versus

f

, by V.13 and V.14.

lines 74–74 in main.tex