Non Euclidian Geometry

1
Non Euclidian Geometry

• Carl Friedrich Gauß (1777-1855)
• First to believe in PP as additional axiom, but
  keeps his findings unpublished until after death
• János Bolyai (1802-1860)
• Also studies non Euclidian geometry. Publishes
  as appendix in his fathers book (1831)
• Nikolai Lobachevsky (1792-1856)
• First to publish on non Euclidian geometry
• On the principles of Geometry in Kazan Messenger
  (1829).
• Geometrische Untersuchungen zu Theorie der
  Parallellinien in Crelles Journal für die Reine
  und angewandte Mathematik (1842).

2
Definition of Parallels

• Gauß If the coplanar straight lines AM, BN do
  not intersect each other, while on the other
  hand, every straight line through A between AM
  and AB cuts BN, then AM is said to be parallel to
  BN.
• Bolyai The directed half-line BN is parallel
  with the directed half-line AM if the
  counterclockwise rotation of the half-lines from
  BA around B results in the half-line BN, which
  does not intersect AM.
• Lobachevsky All straight lines which in a plane
  go out from a point can, with reference to a
  given straight line in the same plane, be divided
  into two classes - into cutting and non-cutting.
  The boundary lines of the one and the other class
  of those lines will be called parallel to the
  given line.