Parallel Postulate

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The Parallel Postulate is the fifth postulate in Euclid's geometry treatise, The Elements it is also the axiom in third group of axioms in Hilbert's Foundations of Geometry. It has been a source of controversy for centuries, and is the basis of Euclidean geometry.


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. In Hilbert's Foundations of Geometry, the parallel postulate states In a plane there can be drawn through any point A, lying outside of a straight line a, one and only one straight line which does not intersect the line a. This straight line is called the parallel to a through the given point A. In The Elements, the parallel postulate states That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. 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.


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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