Proposition X.5 — Commensurable magnitudes have to one another the ratio which…

Proof

d

is a common measure of

a

b

, then

a = m d

b = n d

, so

a : b = m : n

(a ratio of integers).

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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…
X.1Definition X.1Those magnitudes are said to be commensurable which are measured by the same measure, and those incommensurable which…

Required by (dependents) (4)

X.6Proposition X.6If two magnitudes have to one another the ratio which a number has to a number, the magnitudes will be commensurable.
X.8Proposition X.8If two magnitudes have not to one another the ratio which a number has to a number, the magnitudes will be…
X.9Proposition X.9The squares on straight lines commensurable in length have to one another the ratio which a square number has to a…
X.11Proposition X.11If four magnitudes be proportional, and the first be commensurable with the second, the third also will be…

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