Proposition V.16 — If four magnitudes be proportional, they will also be propor…

Proof

Let

a : b = c : d

. Test

a : c

against

b : d

with equimultiples

ma

m c

nb

n d

: by V.4 the original proportion lifts to

ma : mb = n c : n d

, and by V.15 to

ma : n c = mb : n d

. The sign of

ma - n c

matches the sign of

mb - n d

for all

m

n

, which by V.5 is the alternated proportion

a : c = b : d

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Depends on (4)

V.4Proposition V.4If a first magnitude have to a second the same ratio as a third to a fourth, any equimultiples whatever of the first…
V.11Proposition V.11Ratios which are the same with the same ratio are also the same with one another.
V.15Proposition V.15Parts have the same ratio as the same multiples of them taken in corresponding order.
V.5Definition V.5Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any…

Required by (dependents) (1)

V.19Proposition V.19If, as a whole is to a whole, so is a part subtracted to a part subtracted, the remainder will also be to the remainder…

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