Geometry, branch of mathematics concerned with spatial figures, their relationships, and deductive reasoning concerning these figures and relationships. Different geometric systems exist, each based on its own rules (axioms or postulates), components (objects), and self-consistent conclusions (theorems).
Euclidean geometry, the most familiar type of geometry, was developed by the Greek mathematician Euclid in 300 B.C. It was the first formalized deductive mathematical system, serving as a model for later systems. The axioms it is based on describe points, lines, and circles in a flat surface (plane). They also describe relationships among these objects. According to Euclidean geometry, a straight line may be drawn between any 2 points; a circle may be drawn with a point as its center and any given radius; and through any point outside a line 1 and only 1 parallel line can be formed. Non-Euclidean geometries are based on axioms and objects that differ in part or completely from those of Euclidean geometry. Euclidean geometry was the predominant geometric description for centuries. It was not until the 19th century that Nikolai Lobachevski, Janos Bolyai, and G.F.B. Riemann verified the existence of self-consistent systems based on all of Euclid's axioms except the one concerning parallel lines. Through deductive logic, Lobachevski proved the existence of geometric systems where more than one parallel line can be drawn. Riemann's non-Euclidean geometry is a system where no parallel lines exist. Navigation, based on a geometrical system concerned with relationships on the surface of a sphere rather than in a plane, is an example of Riemannian geometry. In it a straight line is defined as a “great circle” (a circle with its center and radius being the same as that of the sphere), and no 2 great circles are parallel.
Other geometries exist. Analytic geometry, established in 1637 by René Descartes, significantly enhanced the development of geometry by generalizing it through the application of algebra. As a result, figures can be specified relative to coordinate systems by sets of numbers or equations and problems can be solved using algebraic and geometric methods. Projective geometry, developed around 1820 by Jean-Victor Poncelet, was a modification of Euclidean space by including points at infinity. Perspective drawing uses these concepts. Topology, the most recent branch of geometry (dating from 1911 with the work of Dutch mathematician L.E.J. Brouwer), deals with geometric objects that remain unchanged upon deformation.