While algebra had been applied to geometry since the time of Archemedes, René Descartes (1596-1650) was the first to recognize that a point in a plane could be found if its distances, (x,y), from two fixed lines perpendicular to each other, were given. He found that while an infinite number of xs and ys could satisfy the equation f(x,y) = 0, the valid xs and ys could be used to illustrate geometrical properties. This became known as analytic geometry (often abbreviated anal. geometry), and became the source for the Cartesian Coordinate System, used so often today in graphs everywhere (i.e., math studies, and the media). He claimed that this could be extended to be used on three coordinates, but never explored it himself.

His most influential mathematical work was his Geometrie, published in 1637, as an appendix to his Discourse on Method. Along with analytic geometry discussed above, he contributed many conventions used today, such as using the letters at the beginning of the alphabet for known values, and letters at the end of the alphabet for unknown values. He introduced the system of induces used today, something that was hinted at by previous authors. He was first to recognize the advantages to placing all the unknown variables to one side of an equation. He used the current conventions used for finding the limits of equations with both positive and negative numbers.

While I will be using the mathematical practices introduced by Descartes for the rest of my life, I greatest credit him for his Meditations on the First Philosophy. While the only idea he came up with that was not later disputed was "Cogito ergo sum"—"I think, therefor I am," he brought many of the conventions used in mathematical proof to philosophy in an attempt to prove the existence of the external world. His method of proof in philosophy laid the foundation for all Modern Philosophy (and has since been abandoned by Postmodern Philosophers). My study of his methods and ideas now called Cartesian Dualism, dramatically altered the manner in which I think and evaluate the world.