Cartesian geometry: The Dutch contribution

Abstract

At the beginning of the seven teen th century , as Morris Kline has remarked, mathematics was still essentially a body of geometry with algebraic appendages. Geometry was beginning to take on a new significance, however, particularly in the light of recent developments in science and technology. The conic sections so beloved of the Greeks described the orbits of celestial bodies and the trajectories of projectiles; a study of curves in general was essential for work on optics, astronomy, and mechanics. The mathematical techniques for handling curves were still rudimentary, however, being based on classical methods of Euclid and Apollonius. What was needed, and what was first developed by Ren6 Descartes (1596-1650) and Pierre de Fermat (1601-1655), was a radical new approach to the mathematical description of curves. In essence this was the now familiar algebraic description by which curves are characterised by equations. Fermat published little during his lifetime, but Descartes set ou t his n e w approach to geometr ica l problems in La G~omdtrie, one of three important appendices to his philosophical magnum opus Discours de la mdthode pour bien conduire sa raison et chercher la vdritd dans les sciences (published in 1637). The study of curves was now transformed into the study of equations, and a geometrical problem could be attacked by converting it into an algebraic one, solving the algebraic problem, and finally interpreting the algebraic solution in the context of the original geometrical problem. Descartes and Fermat laid the basis of the modern, algebraic approach to g e o m e t r y a n approach that forms a leitmotiv running through Newtonian mechanics, n ine teen th-cen tury science, and even the Einsteinian revo!ut ion--but the term Cartesian geometry in its m o d e r n sense is someth ing of an anachronism: Descartes never expressed his ideas in the way now so familiar. For example, he continued to use