The Development of Analytic Geometry
Overview
The fundamental idea of analytic geometry, the representation of curved lines by algebraic equations relating two variables, was developed in the seventeenth century by two French scholars, Pierre de Fermat and René Descartes. Their invention followed the modernization of algebra and algebraic notation by François Viète and provided the essential framework for the calculus of Isaac Newton and Gottfried Leibniz. The calculus, in turn, would become an indispensable mathematical tool in the development of physics, astronomy, and engineering over the next two centuries.
Background
The relationship between geometry and algebra has evolved over the history of mathematics. Geometry reached the greater degree of maturity sooner. The Greek mathematician Euclid (335-270 B.C.) was able to organize a great many results in his classic book, the Elements. Algebra was a far less organized body of ideas, drawing on Babylonian, Egyptian, Greek, and Hindu sources, and dealing with problems ranging from commerce to geometry. Until the Renaissance, geometry might be used to justify the solutions to algebraic problems, but there was little thought that algebra would shed light on geometry. This situation would change with the adoption of a convenient notation for algebraic relationships and the development of the concept of mathematical function that it permitted.
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