2 Dimensional Geometry

Euclid, in 325 BC in 'The Elements', records his principles of geometry.

A set of 'points' and a set of 'lines' and a number of postulates and relationships between them. The most  important of these are;

  1. Through any two points there is a unique line
  2. A line can be continued indefinitely in either direction
  3. Around any point as centre, a circle of any radius can be drawn
  4. Any two right angles are equal
  5. Given a line and a point NOT on the line there is exactly one line through the point that does not intersect the given line
The last of the above, the 5 th. or Parallel Postulate (here in Playfairs' equivalent form) produced the most controversy, as many attempt were made to show that it followed from the other postulates.

In the 1820s two young and little known mathematicians, Bolyai in Hungary and Lobachevsky in Russia, showed that there is a perfectly good geometry (called non-Euclidean or Hyperbolic geometry) that shares all the initial assumptions of Euclidean geometry except the Parallel Postulate.
 

Poincaré disc model

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