Proposition V.15 — Parts have the same ratio as the same multiples of them take…

Proof

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

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Depends on (2)

V.12Proposition V.12If any number of magnitudes be proportional, as one of the antecedents is to one of the consequents, so will all the…
V.2Definition V.2The greater is a multiple of the less when it is measured by the less.

Required by (dependents) (1)

V.16Proposition V.16If four magnitudes be proportional, they will also be proportional alternately.

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