GEOMETRY

branch of mathematics that deals with the properties of space and objects in space. It is one of the oldest branches of mathematics, having arisen in response to such practical problems as those found in surveying, and its name is derived from Greek words meaning Earth measurement. Eventually it was realized that geometry need not be limited to the study of flat surfaces (plane geometry) and rigid three-dimensional objects (solid geometry) but that even the most abstract thoughts and images might be represented and developed in geometric terms. In several ancient cultures, there developed a form of geometry suited to the relationships between lengths, areas, and volumes of physical objects. This geometry was codified in Euclid's Elements in about 300 BC on the basis of 10 axioms and postulates, from which several hundred theorems were proved by deductive logic. The Elements epitomized the axiomatic-deductive method for many centuries. Analytic geometry, in which algebraic notation and procedures are used for the description of geometric objects, was introduced by Ren Descartes in 1637. Soon thereafter, Isaac Newton, Leonhard Euler, and others extended this idea to the study and classification of conic sections and other families of plane curves and the solution of problems involving them. These studies founded algebraic geometry, which reached its full development at the hands of Max Noether of Germany late in the 19th century and Corrado Segre and Federigo Enriques of Italy early in the 20th. In the late 18th century, Gaspard Monge of France elaborated analytic geometry into descriptive geometry, a rational system for depicting three-dimensional objects such as buildings and machines by means of coordinated views from three perpendicular directions. The related problem of perspective in painting became the basis of a further extension, namely, projective geometry. This branch of mathematics, systematized by Victor Poncelet of France by 1822, deals with those properties of geometric figures that are not altered by projection. It provided the proper context for two celebrated theorems that had been proved by Girard Desargues and Blaise Pascal in about 1640. The application of the concepts of mathematical analysiscontinuity and limitto the study of geometry was undertaken in about 1820 by Carl Friedrich Gauss in connection with practical problems of surveying and geodesy. Gauss initiated the field of differential geometry by providing analytic expressions for the length of arc and the curvature of plane curves and extending his results to the curvature of surfaces. From 1854 onward, his student Bernhard Riemann generalized these ideas to spaces of any number of dimensions; the resulting geometric concepts were adopted by Albert Einstein in the 20th century in framing his general theory of relativity. Riemann's researches made him one of the principal founders of non-Euclidean geometry. Around 1830 Nikolay Lobachevsky and Jnos Bolyai had published the results of their independent investigations of the consequences of replacing Euclid's parallel postulate with the stipulation that through a given point not on a given line, two parallels to the line can be drawn. (Gauss had examined this possibility but published nothing on it.) Riemann offered another alternative: there is no parallel. After 1870 non-Euclidean geometry was further generalized and unified with projective geometry by Felix Klein of Germany and Sophus Lie of Norway. Topology, the youngest and most sophisticated branch of geometry, focuses on the properties of geometric objects that remain unchanged upon deformation: a doughnut and a teacup are topologically equivalent. The continuous development of topology dates from 1911, when the Dutch mathematician L.E.J. Brouwer introduced methods generally applicable to the topic.