Proposition V.6 — If two magnitudes be equimultiples of two magnitudes, and an…

Proof

With

a = mb

c = m d

, and subtractions

a^{'} = nb

c^{'} = n d

a - a^{'} = (m - n) b

and

c - c^{'} = (m - n) d

by Common Notion 3. If

m = n

the remainders are zero (equal); otherwise both remainders are

(m - n)

-fold of

b

d

respectively.

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Depends on (2)

V.5Proposition V.5If a magnitude be the same multiple of a magnitude that a subtracted part is of a subtracted part, the remainder also…
3Common notion 3If equals be subtracted from equals, the remainders are equal.

Required by (dependents) (1)

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

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