# Descartes

René Descartes (1596 – 1650), philosopher and mathematician, born in La Haye in France. At the age of eight, he started studying logic, physics, metaphysics, and mathematics and other subjects at a Jesuit school. He took part in the Thirty Years' War in Europe. In a dream one winter night in 1619 he is said to have had a vision about a method for attaining an absolutely certain and complete knowledge of nature. The theory was to be constructed on the same foundations as Euclid's Elements, with definitions, axioms, theorems, and proofs. The first proposition was: ”Je pense, donc je suis” (I think, therefore I am). With this as a point of departure, a series of new truths could be put forward. The method was published in Discours de la Méthode 1637. The book also contains sections on optics, meteors, and mathematics. In 1649 Descartes was summoned to Sweden by Queen Christina as her private tutor in philosophy.  The following year he died of pneumonia in Stockholm.

## Cartesian mathematics

La Geometrie is the mathematical part of Discours de la Méthode (1637). It was written in French. It became known to other mathematicians only after it was translated into Latin in 1649, with the title of Geometria. It was published again in 1659. Both Newton and Leibniz were inspired by Descartes' theories.

La Geometrie consists of three chapters (called books 1-3). In the first, Descartes solves second-degree equations geometrically by finding the intersecting points of a circle and a line. In the second, he determines the tangents of curves by finding a circle with the same tangent in the point of tangency. In the third chapter, he solves, among other things, higher-degree equations by finding intersecting points between circles and parabolas. What is original in Descartes is that he solves problems algebraically and geometrically. He introduces the algebra we use today, with x, y, z as unknown values and a, b, c, … for known ones.

From Geometria 1659. Book 2. Determining of a tangent to a curve through A and C. PC can be conceived of as a radius in a circle. Line FC is a tangent common to the circle and the curve. Book 3. Solution to a fourth-degree equation in the intersections between a parabola and a circle.

The curves are described with algebra in a system of coordinates, where only the horizontal axis can be seen clearly. Descartes is regarded as one of the pioneers of the coordinate system. The other pioneer is a contemporary Frenchman, Pierre de Fermat (1601 – 1665).