Mathematics—Analytic Geometry

Analytic geometry began in 1637 with Rene Descartes and Pierre de Fermat.(1) Both Fermat’s Ad locos planos et solidos isagoge (Introduction to Plane and Solid Loci, 1637) and Descartes’s La geometrie (Geometry), which is an essay accompanying his Discours de la methode (Discourse on Method, 1637), presented the same basic techniques of relating algebra and geometry. The difference is Descartes published his version and Fermat did not; therefore, Descartes’s influence in developing analytic geometry is great and Fermat’s is negligible. As part of Descartes’s Geometry, he gave an exposition of analytic geometry, which is a method of representing geometric figures with algebraic equations that made many previously unsolvable problems solvable. An essential innovation to create the algebraic equations was Descartes’s use of coordinates to locate a point in two or three dimensions (called Cartesian coordinates from the second syllable of Descartes’s name). This enabled the techniques of algebra and later calculus to be used to solve geometrical problems, since every point had a precise numerical value (called analytic geometry). Before this, as in the works of the Greeks, a point on a plane was merely assigned a letter or other symbol, which limited its explanatory potential.(2)

 

 

He also introduced the conventions of representing known numerical quantities with a, b, c, etc., unknowns with x, y, z, etc., and squares, cubes, and other powers with numerical superscripts, as in x², x3, and so on, which made algebraic notation much clearer than before.

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From: Richard A. Denholm, Mathematics: Man’s Key To Progress (Sacramento, 1970), pp. 37-39

 

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