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The study of the properties of shapes – including lines, curves, and surface – and of spaces. Today it divides the following main areas: Many of these divisions have grown up only in the last few hundred years.

The name geometry is a reminder of its earliest use – for the measurement of land and materials. The Babylonian and Egyptian civilizations thus gained great empirical knowledge of elementary geometric figures, including how to construct a right-angled triangle. The Greek philosophers transformed this practical art into an intellectual pastime through which they sought access to the secrets of nature. About 300 BC Euclid collected together and added to the Greek rationalization of geometry in his Elements. Later Alexandrian geometers began to develop trigonometry. The revival of interest in life-like painting in the Renaissance led to the development of projective geometry, though its is to the philosopher-scientist Decartes that we owe the invention of the algebraic (coordinate) geometry which allows algebraic functions to be represented geometrically. The next new branch of geometry to be developed followed fast upon the invention of calculus: differential geometry.

The greatest upset in the history of geometry came in the 19th century. Men such as Karl Gauss, Nikolai Lobachevsky, and Janós Bolyai began to question the Euclid parallel-lines axiom and discovered hyperbolic geometry, the first non-Euclidean geometry. The elliptical geometry of Bernhard Riemann aided Albert Einstein in the development of the general theory of relativity.

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