Parallel Lines Postulate

Also known as Euclid's Fifth Postulate.

The oddest of the axioms proposed by EuclidOfAlexandria's Elements for describing the plane. There are many equivalent formulations of it (equivalent, that is, given the other axioms). The usual one, which wasn't actually the original, is:

Given any line, and any point not on the line, there's another line on which the point does lie and which doesn't meet the given line at all; furthermore, there is only one such line.

For quite some time, many mathematicians (including Euclid, who added it as a postulate after being unable to prove it) thought that the ParallelLinesPostulate was derivable from the other axioms (in that it should be a theorem), and looked for proof. Others insisted it was indeed a fundamental axiom. What actually occurred caused a scandal in the mathematical community; it was shown that consistent geometries could be constructed by altering the parallel lines postulate.

In other words, it's an axiom but not a required one for a consistent system.

Altering the ParallelLinesPostulate leads to the various NonEuclidean geometries, including, but not limited to, elliptic geometry and hyperbolic geometry. AnalyticGeometry? in two dimensions exhibits this postulate as a consequence of the definition of parallelism as having the same slope (including infinity as a slope).

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CategoryMath