MATHS 460 - BALL STATE UNIVERSITY - SUMMER 2000
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Hyperbolic Geometry (or Lobachevskian Geometry)Nikolai Ivanovich LobachevskyBorn: 1 Dec 1792 in Nizhny Novgorod (was Gorky from 1932-1990), Russia  Lobachevskii's first
publication on hyperbolic geometry was published in the Kazan Messenger
in 1829. This is a paper from Kazan University in Russia and does not have a
wide spread publication. Since no one had heard of him, there was not anyone
that really wanted to listen. Finally he got his break and was published in
Crelle’s Journal in 1837. This being a French publication brought his thought
and ideas to a much larger audience, but they were not accepted immediately by
the readers. In 1840 Lobachevsky published a 61-page booklet entitled Geometrical
investigations on the theory of parallels, in which he further explain how his
new geometry works. In this work he derives a NEW parallel postulate.
Lobachevski made an assumption about parallel lines that differed from Euclid's and proceeded to draw out its consequences. This way of working cannot guarantee the consistency of one's findings, so strictly speaking they could not prove the existence of a new geometry in this way. He described a three-dimensional space different from Euclidean space, couching his findings in the language of trigonometry. The formulas he obtained were exact analogues of the formulas that describe triangles drawn on the surface of a sphere, with the usual trigonometric functions replaced by those of hyperbolic trigonometry. The functions hyperbolic cosine, written cosh, and hyperbolic sine, written sinh, are defined as follows:
They are called hyperbolic because of their use in describing the hyperbola. Their names derive from the evident analogy with the trigonometric functions, which Euler showed satisfy these equations:
The formulas were what gave their work the precision needed to give conviction in the absence of a sound logical structure. Lobachevski observed that it had become an empirical matter to determine the nature of space, Lobachevsky even went so far as to conduct astronomical observations, although these proved inconclusive (Encyclopedia Britannica). Here are a few of the effects of hyperbolic geometry.
The best way to understand hyperbolic geometry is to see it (Graves).
Select a link to continue.
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Lobachevsky's Parallel
Postulate – There exist two lines parallel to a
given line through a given point not on the line.
