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

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V.12 Proposition V.12

If any number of magnitudes be proportional, as one of the antecedents is to one of the consequents, so will all the antecedents be to all the consequents.

Proof

Let

a_{i} : b_{i}

all equal

r

in the sense of Definition V.5. For any test multipliers

m

,

n

the sign of

m a_{i} - n b_{i}

is the same for every

i

; therefore the sign of

m \sum a_{i} - n \sum b_{i}

is the same too. By V.5 this is the equimultiples test for

\sum a_{i} : \sum b_{i} = r

.

lines 74–74 in main.tex