When Euclid presented his axiomatic treatment of geometry, one of his assumptions, his fifth postulate, appeared to be less obvious or fundamental than the others. As it is now conventionally formulated, it asserts that there is exactly one parallel to a given line through a given point. Attempts to derive this from Euclid’s other axioms did not succeed, and, at the beginning of the 19th century, it was realized that Euclid’s fifth postulate is, in fact, independent of the others. It was then seen that Euclid had described not the one true geometry but only one of a number of possible geometries.
Zenos-paradox-illustrated-by-Achilles-racing-a-tortoiseFigure 1: Zeno’s paradox, illustrated by Achilles racing a tortoise.[Credits : Encyclopædia Britannica, Inc.]
Contrasting-triangles-in-Euclidean-elliptic-and-hyperbolic-spacesFigure 2: Contrasting triangles in Euclidean, elliptic, and hyperbolic spaces.[Credits : Encyclopædia Britannica, Inc.]
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