Proposition X.2 — If, when the lesser of two unequal magnitudes is continually…

Proof

A common measure would persist through anthyphairesis (Common Notion 3); non-termination of the algorithm thus implies no common measure exists.

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Depends on (3)

X.1Proposition X.1Two unequal magnitudes being set out, if from the greater there be subtracted a magnitude greater than its half, and…
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) (1)

X.3Proposition X.3Given two commensurable magnitudes, to find their greatest common measure.

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