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

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

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V.15 Proposition V.15

Parts have the same ratio as the same multiples of them taken in corresponding order.

Proof

If

A = m B

and

C = m D

, group the

m

copies of

B

and

D

in parallel. Each parallel pair

(B_{i}, D_{i})

satisfies

B_{i} : D_{i} = B : D

(identity), and V.12 sums these to give

A : C = B : D

.

lines 74–74 in main.tex