Proposition V.12 — If any number of magnitudes be proportional, as one of the a…

Proof

Let

a_{i} : b_{i}

all equal

r

in the sense of Definition V.5. For any test multipliers

m

n

the sign of

m a_{i} - n b_{i}

is the same for every

i

; therefore the sign of

m \sum a_{i} - n \sum b_{i}

is the same too. By V.5 this is the equimultiples test for

\sum a_{i} : \sum b_{i} = r

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Depends on (3)

V.1Proposition V.1If there be any number of magnitudes whatever which are, respectively, equimultiples of any magnitudes equal in…
V.2Proposition V.2If a first magnitude be the same multiple of a second that a third is of a fourth, and a fifth also be the same…
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) (5)

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