Proposition V.10 — Of magnitudes which have a ratio to the same, that which has…

Proof

Same contrapositive of V.8: if

a : c > b : c

then

a > b

, since if

a \leq b

then

a : c \leq b : c

by V.7 or V.8 applied in reverse.

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Depends on (2)

V.7Proposition V.7Equal magnitudes have to the same the same ratio, as also has the same to equal magnitudes.
V.8Proposition V.8Of unequal magnitudes the greater has to the same a greater ratio than the less has, and the same has to the less a…

Required by (dependents) (1)

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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