Proposition V.4 — If a first magnitude have to a second the same ratio as a th…

Proof

By definition V.5, the original proportion gives an equimultiple relation across all multipliers. Substituting

ma

for

a

and

m c

for

c

throughout simply rescales the test multipliers; the test itself still passes for the rescaled magnitudes.

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Depends on (2)

V.3Proposition V.3If a first magnitude be the same multiple of a second that a third is of a fourth, and if equimultiples be taken of the…
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) (2)

V.16Proposition V.16If four magnitudes be proportional, they will also be proportional alternately.
V.18Proposition V.18If magnitudes separated be proportional, they will also be proportional componendo.

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