Proposition V.19 — If, as a whole is to a whole, so is a part subtracted to a p…

Proof

a : c = a^{'} : c^{'}

with

a^{'} < a

c^{'} < c

, apply V.17 (separation) to obtain

(a - a^{'}) : a^{'} = (c - c^{'}) : c^{'}

, and V.11 / V.16 to re-express as

(a - a^{'}) : (c - c^{'}) = a : c

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Depends on (2)

V.16Proposition V.16If four magnitudes be proportional, they will also be proportional alternately.
V.17Proposition V.17If magnitudes composed be proportional, they will also be proportional separando.

Required by (dependents) (1)

V.25Proposition V.25If four magnitudes be proportional, the greatest and least are greater than the remaining two.

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