Proposition V.8 — Of unequal magnitudes the greater has to the same a greater …

Proof

Let

a > b

. Pick a multiplier

n

such that

n (a - b) > c

(the Archimedean property of magnitudes, Definition V.4) and an

m

such that

m c

falls between

nb

and

na

. Then

na > m c

but

nb < m c

; by Definition V.7 this is precisely the assertion

a : c > b : c

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Depends on (2)

V.4Definition V.4Magnitudes are said to have a ratio to one another which are capable, when multiplied, of exceeding one another (the…
V.7Definition V.7When, of the equimultiples, the multiple of the first magnitude exceeds the multiple of the second, but the multiple of…

Required by (dependents) (3)

V.9Proposition V.9Magnitudes which have the same ratio to the same are equal to one another; and magnitudes to which the same has the…
V.10Proposition V.10Of magnitudes which have a ratio to the same, that which has a greater ratio is greater; and that to which the same has…
V.14Proposition V.14If a first magnitude have to a second the same ratio as a third to a fourth, and the first be greater than the third,…

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