How was Cartesian coordinate system discovered?

It wasn't discovered - discovery is when you find something that was already there (such as America, Pluto, etc.). Invention is when something is created.

The Cartesian system was named for its inventor, Rene Descartes.

The idea was developed by Descartes in 1637 in two writings, and independently by Pierre de Fermat (Fermat did not publish his work).

In part two of his Discourse on Method, Descartes introduces the new idea of specifying the position of a point or object on a surface, using two intersecting axes as measuring guides. In La Geometries, he further explores the concept.

In mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane through two numbers, usually called the x-coordinate or abscissa and the y-coordinate or ordinate of the point. To define the coordinates, two perpendicular directed lines (the x-axis, and the y-axis), are specified, as well as the unit length, which is marked off on the two axes) Cartesian coordinate systems are also used in space (where three coordinates are used) and in higher dimensions.

Using the Cartesian coordinate system, geometric shapes (such as curves) can be described by algebraic equations, namely equations satisfied by the coordinates of the points lying on the shape. For example, the circle of radius 2 may be described by the equation x2 + y2 = 4 Cartesian means relating to the French mathematician and philosopher Rene Descartes (Latin: Cartesius), who, among other things, worked to merge algebra and Euclidean geometry. This work was influential in the development of analytic geometry, calculus, and cartography. The idea of this system was developed in 1637 in two writings by Descartes and independently by Pierre de Fermat, although Fermat did not publish the discovery.[1] In part two of his Discourse on Method, Descartes introduces the new idea of specifying the position of a point or object on a surface, using two intersecting axes as measuring guides.