The parallel postulate, or parallel axiom, is the fifth and most controversial of Euclid's postulates set forth in the Greek geometer's great work, Elements.

 

One way to state it is as follows: If a is an arbitrary straight line and A a point which does not lie on a, then in the plane determined by a and A, there is just one straight line which passes through A and does not intersect a.

 

Consequences of the parallel postulate

1. If two parallel lines are intersected by a third, step angles and alternate angles are equal and conversely if step-angles and alternate angles are equal the two lines intersected by the transversal must be parallel.
2. The interior angles of a triangle add up to 180°.
3. If l and m are two straight lines which are both parallel to a third line n, then l and m themselves are parallel.
4. If l and m are two parallel lines, P a point on l, and if PF is the perpendicular from P onto m, then PF is defined as the distance between the parallel lines. The distance defined in this way is the same wherever P is chosen to lie on l.

 

The dawn of non-Euclidean geometry

To later mathematicians, the parallel postulate always seemed less obvious than the other four and many attempts were made to derive it from them, but without success. In 1823, Janos Bolyai and Nikolai Lobachevsky independently realized that entirely self-consistent types of non-Euclidean geometry could be created in which the parallel postulate doesn't hold. Karl Gauss had made the same discovery earlier but kept the fact secret.