Descartes was educated in the Jesuit preparatory school of La Flèche and the University of Poitiers, taking a degree in law. He then spent two years in Paris where, outwardly living the life of a frivolous young gentleman, he began a serious study of mathematics. To see more of the world Descartes joined several armies as an unpaid volunteer; the brief intervals of tranquility during nine years of service provided him time to develop his mathematical and philosophic ideas. In 1628, Descartes decided to settle in Holland, where he remained for the next twenty years. There he wrote his great philosophic treatise on the scientific method, the Discours de la méthode (1637). (The still-quoted sentence, "I think, therefore I am," comes from the Discours.) In 1649, after much hesitation, Descartes accepted the invitation of the 22-year-old Queen Christina to come to Sweden as her private tutor. After only four months of winter tutoring sessions, always held at 5:00 in the morning in the ice-cold library, Descartes died of pneumonia.

The last of the three appendices to Descartes’s Discours was a 106-page essay entitled La géométrie. It provides the first printed account of what is now called analytic or coordinate geometry. The work exerted great influence after being published in a Latin translation along with explanatory notes. The Géométrie introduced many innovations in mathematical notation, most of which are still in use. With Descartes, small letters near the beginning of the alphabet indicate constants and those near the end stand for variables. He initiated the use of numerical superscripts to denote powers of a quantity, while occasionally writing aa for the second power, a². The familiar symbols +, -, and are also encountered in Descartes’s writing.

Descartes "algebrized" the study of geometry by shifting the focus from curves to their equations, allowing the tools of algebra, rather than diagrams, to be applied to the solution of various geometric problems. The Géométrie also treated one of the most important problems of the day, that of finding tangents to curves, by describing a procedure for constructing the normal to a curve at any point (the tangent is perpendicular to the normal). Another part of the work deals with matters in the theory of equations: Descartes states that x - a is a factor of a polynomial if and only if a is a root. He also notes that the maximum number of roots is equal to the degree of the polynomial.