Transcript Non Euclidean Parallels

Non Euclidian Geometry
• Carl Friedrich Gauß (1777-1855)
First to believe in PP as additional axiom, but keeps his
findings unpublished until after death
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• János Bolyai (1802-1860)
Also studies non Euclidian geometry.
Publishes as appendix in his father’s book (1831)
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• 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 Crelle’s Journal für die Reine und
angewandte Mathematik (1842).
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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 halfline 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.