Parallel Postulate

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This is an AoPSWiki Word of the Week for Februrary 12-19

The Parallel Postulate is the fifth postulate in Euclid's geometry treatise, The Elements. It has been a source of controversy for centuries, and is the basis of Euclidean geometry.

Postulate

The parallel postulate states that through any line and a point not on the line, there is exactly one line passing through that point parallel to the line. This is far less simple and elegant, and more wordy, than any of the other postulates stated in The Elements, which has made it a source of controversy.

Controversy

Because it is so non-elegant, mathematicians for centuries have been trying to prove it. Many great thinkers such as Aristotle attempted to use non-rigorous geometrical proofs to prove it, but they always used the postulate itself in the proving. Several plausible-looking algebraic geometry proofs have also been suggested, but all have some fatal flaw. In the late nineteenth century, mathematicians began to question whether the postulate was even true. This led to the development of non-Euclidean geometries such as spherical geometry, elliptical geometry, hyperbolic geometry, and others.

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