Proposition X.15 — If two commensurable magnitudes be added together, the whole…

Proof

a, b

have a common measure

d

, so

a + b = (m + n) d

d

. Converse: if

a + b

and

a

share a measure, then so does

b

by Common Notion 3.

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Depends on (2)

3Common notion 3If equals be subtracted from equals, the remainders are equal.
X.1Definition X.1Those magnitudes are said to be commensurable which are measured by the same measure, and those incommensurable which…

Required by (dependents) (2)

X.16Proposition X.16If two incommensurable magnitudes be added together, the whole will also be incommensurable with each of them; and if…
X.17Proposition X.17If there be two unequal straight lines, and to the greater there be applied a parallelogram equal to the fourth part of…

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