Non-Euclidean geometriesNon-Euclidean geometry refers usually to geometrical spaces where the parallel postulate is false. They include elliptic geometry, where the sum of the angles of a triangle is more than 180°, and hyperbolic geometry, where this sum is less than 180°. Their introduction in the second half of 19th century, and the proof that their theory is consistent (if Euclidean geometry is not contradictory) is one of the paradoxes that are at the origin of the foundational crisis in mathematics of the beginning of 20th century, and motivated the systematization of axiomatic theories in mathematics.
Non,-Euclidean geometriesNon-Euclidean geometry refers usually to geometrical spaces where the parallel postulate is false. They include elliptic geometry, where the sum of the angles of a triangle is more than 180°, and hyperbolic geometry where this sum is less than 180°. Their introduction in the second half of 19th century, and the proof that their theory is consistent (if Euclidean geometry is not contradictory) is one of the paradoxes that are at the origin of the foundational crisis in mathematics of the beginning of 20th century, and motivated the systematization of axiomatic theories in mathematics.