to develop the methods to work with polynomial equations of higher degree. It was Leonhard Euler who first applied the coordinate method in a systematic study of space curves and surfaces. Coordinates In analytic geometry, the plane is given a coordinate system, by which every point has a pair of real number coordinates. Similarly, Euclidean space is given coordinates where every point has three coordinates. The value of the coordinates depends on the choice of the initial point of origin. There are a variety of coordinate systems used, but the most common are the following: Cartesian coordinates (in a plane or space) The most common coordinate system to use is the Cartesian coordinate system, where each point has an x-coordinate representing its horizontal position, and a y-coordinate representing its vertical position. These are typically written as an ordered pair (x, y). This system can also be used for three-dimensional geometry, where every point in Euclidean space is represented by an ordered triple of coordinates (x, y, z). Polar coordinates (in a plane) In polar coordinates, every point of the plane is represented by its distance r from the origin and its angle θ, with θ normally measured counterclockwise from the positive x-axis. Using this notation, points are typically written as an ordered pair (r, θ). One may transform back and forth between two-dimensional Cartesian and polar coordinates by using these formulae: This system may be generalized to three-dimensional space through the use of cylindrical or spherical coordinates. Cylindrical coordinates (in a space) In cylindrical coordinates, every point of space is represented by its height z, its radius r from the z-axis and the angle θ its projection on the xy-plane makes with respect to the horizontal axis. Spherical coordinates (in a space) In spherical coordinates, every point in space is represented by its distance ρ from the origin, the angle θ its projection on the xy-plane makes with respect to the horizontal axis, and the angle φ that it makes with respect to the z-axis. The names of the angles are often reversed in physics. Equations and curves In analytic geometry, any equation involving the coordinates specifies a subset of the plane, namely the solution set for the equation, or locus. For example, the equation y = x corresponds to the set of all the points on the plane whose x-coordinate and y-coordinate are equal. These points form a line,