Elementary hyperbolic geometry was born in 1903 when Hilbert provided,
using the end-calculus to introduce coordinates, a first-order axiomatization
for it by adding to the axioms for plane absolute geometry a hyperbolic parallel
axiom stating that ”Through any point P not lying on a line l there are two
rays r1 and r2 ,not belonging to the same line, which do not intersect l, and
such that every ray through P contained in the angle formed by r1 and r2 does
intersect l” [1, pp. 34]. The hyperbolic geometry is a non-Euclidean geometry
Elementary hyperbolic geometry was born in 1903 when Hilbert provided,
using the end-calculus to introduce coordinates, a first-order axiomatization
for it by adding to the axioms for plane absolute geometry a hyperbolic parallel
axiom stating that ”Through any point P not lying on a line l there are two
rays r1 and r2 ,not belonging to the same line, which do not intersect l, and
such that every ray through P contained in the angle formed by r1 and r2 does
intersect l” [1, pp. 34]. The hyperbolic geometry is a non-Euclidean geometry
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