Analytic Geometry

Preview

The first field of mathematics to benefit from the new language of equations was geometry. Around 1630, both Fermat and Descartes realized that geometric problems could be translated into algebra by means of coordinates, and that many problems could then be routinely solved by algebraic manipulation. The language of equations also provides a simple but natural classification of curves by degree. The curves of degree 1 are the straight lines; the curves of degree 2 are the conic sections; so the first “new” curves are those of degree 3, the cubic curves. Cubic curves exhibit new geometric features—cusps, inflections, and self-intersections—so they are considerably more complicated than the conic sections. Nevertheless, Newton attempted to classify them, and in doing so he discovered that cubic curves, when properly viewed, are not as complicated as they seem. We will find our way to the “right” viewpoint in Chapters 8 and 15. In the meantime we discuss another theorem that depends on the “right” viewpoint: B´ezout’s theorem, according to which a curve of degree m always meets a curve of degree n in mn points.