  At this time, Descartes discovered and conveyed to Beeckman the fundamental insight that makes analytic geometry possible: the technique for describing lines of all sorts by using mathematical equations involving ratios between lengths. Descartes himself did not foresee replacing geometrical constructions with algebraic formulas; rather, he viewed geometry as the basic mathematical science and he considered his algebraic techniques to provide a powerful alternative to actual compass-and-ruler constructions when the latter became too intricate. When, in the nineteenth century, algebra and analysis took precedence over geometry, the rectilinear coordinate system of algebraic geometry came to be called “Cartesian coordinates” in honor of Descartes' discovery.