The story
The terminology was introduced in 1871 by Felix Klein, who classified some non-Euclidean geometries as hyperbolisch (“hyperbolic”), elliptisch (“elliptical”), and parabolisch (“parabolic”).
Etymology from Wiktionary, CC BY-SA 4.0.
hyperbolic geometry
Today it means: A type of geometry that rejects the parallel postulate. Given a straight line L and a point p not on L, more than one straight line can be drawn through p without intersecting L. Full definition →