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Source — Euclid's Elements, encoded as an rrxiv paper — rrxiv

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Source — Euclid's Elements, encoded as an rrxiv paper

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V.8 Proposition V.8

Of unequal magnitudes the greater has to the same a greater ratio than the less has, and the same has to the less a greater ratio than it has to the 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

lines 74–74 in main.tex

58 files·489.1 KB·

· 58 files

← Claims Full view

V.8 Proposition V.8

Of unequal magnitudes the greater has to the same a greater ratio than the less has, and the same has to the less a greater ratio than it has to the 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

lines 74–74 in main.tex