The Development of Geometry, I

Abstract

UNFETTERED BY TRADITION, ALGEBRA made rapid strides during the fifteenth and sixteenth centuries, while geometry, still felt by mathematicians to be in thrall to the towering achievements of the ancient Greeks, languished. But at the beginning of the seventeenth century geometry received a decisive stimulus through the injection of the methods of algebra. This was occasioned largely through the work of Fermat—in his Ad Locos Planos et Solidos Isagoge, “Introduction to Plane and Solid Loci”, written in 1629 but not published until 1679—and the philosopher-mathematician Descartes— in his La Géométrie, which appeared as an appendix to his seminal philosophical work Discours de la Methode of 1637. The major effect of the coordinate—also known as algebraic or analytic—geometry they created was to establish a correspondence between curves or surfaces and algebraic equations, thereby opening up geometric investigation to the powerful quantitative methods of the newly emerged algebra.