Geometry

A branch of mathematics concerned with relationships in space, defined in terms of points, lines, angles, planes, surfaces, and solids. The original application of the term was to measurements on the Earth’s surface—a practical art developed by the Egyptians that was theorised when Thales imported it into the Greek world, its foundations clarified by Euclid’s Elements (fourth century b.c.). Euclid used deductive logic to construct a series of theorems—further augmented by such writers as Apollonius and Archimedes— based on five axioms that seemed intuitively indubitable (although the fifth axiom, which relates to parallel lines, seemed less indubitable than the rest). Significant augmentations of geometric technique followed the reintroduction of Euclid’s work to Europe in the Renaissance, the most important being Rene´ Descartes’ development of an ‘‘analytical geometry’’ that permitted the graphical representation of algebraic relationships. Euclidean geometry continued to be considered a definitive description of spatial properties even though notions of ‘‘absolute space’’ were repeatedly challenged by idealistic notions of space as an artefact of perception. Immanuel Kant’s proposal that space was a necessary a priori construct deflected attention away from the possibility that actual space might be significantly different from perceived space—or, at least, from the possibility of ever finding out if that were the case.

When mathematicians began developing ‘‘non-Euclidean’’ geometries based on the variance of Euclid’s fifth postulate in the early nineteenth century, suspicion was aroused that real space might be non-Euclidean. Carl Friedrich Gauss and Nikolai Ivanovich Lobatchevsky both made measurements of actual triangular relationships in the hope of finding a discrepancy in the sum of their angles that would offer evidence of a curvature in actual space. The formulation of an abstract philosophy of geometry by David Hilbert, which refused to