A Newton quadrilateral is a convex quadrilateral inscribed in a circle whose one side is the diameter of the circle. Solving the problem of finding the diameter d of a circle that has an inscribed convex quadrilateral given by the three sides a; b, and c, Newton discovered the relation known as the Newton equation. Newton was not concerned with solving his own equation, however, he derived it in six different ways. Positive integer solutions of the Newton equation were found by P. Bachmann. All integer solutions of Newton’s equation were found by A. Oppenheim, and the formula that produces infinitely many rational solutions was found by N. Anning, who used an elegant relation among the cosines of the angles of a triangle. All rational solutions of Newton’s equation were found by I. A. Barnett using a simple and elegant method based on linear algebra. The same results of Barnett, M. Hajja achieved using the relation between the cosines of angles of a generalized triangle. In this thesis, a direct algebraic method is presented which gives all rational solutions. Also, the case of a non-convex Newton quadrilateral is presented, certain properties of the zeros of the Newton’s polynomial and its associated polynomials are established, the converse of the Newton result is presented and proved and the formula for the area of the Newton’s quadrilateral is given. The last chapter describes several examples where Newton’s polynomial and its associated polynomial appears.