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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.
Geometria 1659
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 1. Geometric solution to a second-degree equation.
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).