1. Non Euclidian Geometry
Carl Friedrich Gauß ( )
First to believe in PP as additional axiom, but keeps his findings unpublished until after death
János Bolyai ( )
Also studies non Euclidian geometry. Publishes as appendix in his father’s book (1831)
Nikolai Lobachevsky ( )
First to publish on non Euclidian geometry
On the principles of Geometry in Kazan Messenger (1829).
Geometrische Untersuchungen zu Theorie der Parallellinien in Crelle’s 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.