Proposition V.7 — Equal magnitudes have to the same the same ratio, as also ha…

Proof

Let

a = b

and

c

arbitrary. For any equimultiples

ma

mb

a

b

, and

n c

c

, the test of V.5 succeeds vacuously because

ma = mb

. Therefore

a : c = b : c

and

c : a = c : b

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Depends on (2)

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…
1Common notion 1Things which are equal to the same thing are also equal to one another.

Required by (dependents) (1)

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…

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