Importance
Non-Euclidean geometry is an example of a paradigm shift in the history of science. Before the models of a non-Euclidean plane were presented by Beltrami, Klein, and Poincaré, Euclidean geometry stood unchallenged as the mathematical model of space. Furthermore, since the substance of the subject in synthetic geometry was a chief exhibit of rationality, the Euclidean point of view represented absolute authority. Non-Euclidean geometry, though assimilated by learned investigators, continues to be suspect for those not having exposure to hyperbolic and elliptical concepts.
The discovery of the non-Euclidean geometries had a ripple effect which went far beyond the boundaries of mathematics and science. The philosopher Immanuel Kant's treatment of human knowledge had a special role for geometry. It was his prime example of synthetic a priori knowledge; not derived from the senses nor deduced through logic — our knowledge of space was a truth that we were born with. Unfortunately for Kant, his concept of this unalterably true geometry was Euclidean. Theology was also affected by the change from absolute truth to relative truth in mathematics that was a result of this paradigm shift.
The existence of non-Euclidean geometries impacted the "intellectual life" of Victorian England in many ways and in particular was one of the leading factors that caused a re-examination of the teaching of geometry based on Euclid's Elements. This curriculum issue was hotly debated at the time and was even the subject of a play, Euclid and his Modern Rivals, written by the author of Alice in Wonderland.
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