This article traces the historical development of the Newton-Raphson method for solving nonlinear algebraic equations, focusing on the contributions of Isaac Newton, Joseph Raphson, and Thomas Simpson. The method, which is often referred to as Newton's method or Newton-Raphson method, is used to find the roots of an equation \( f(x) = 0 \). The article highlights how Newton's formulation differed from the iterative process of Raphson, and that Simpson was the first to provide a general formulation applicable to nonpolynomial equations using fluxional calculus. Simpson's extension of the method to systems of equations is also discussed. The article begins by discussing the method of Viète, which was a precursor to the Newton-Raphson method, and Newton's early notes on this method. It then examines Newton's original presentation of his method, contrasting it with Raphson's iterative formulation for polynomial equations. The article also explores Newton's use of the secant method, which is similar to the Newton-Raphson method but does not involve derivatives. Raphson's work in 1690 is detailed, including his formula for solving polynomial equations and his iterative approach. The article notes that Raphson's method was a significant improvement over Newton's, as it did not require the generation of intermediate polynomials and retained all significant digits computed in successive iterations. Finally, the article discusses Simpson's general formulation for nonlinear equations using fluxional calculus and his extension of the method to systems of equations. The historical development of the Newton-Raphson method is traced through these contributions, highlighting the evolution of the method from its early algebraic forms to its modern iterative form.This article traces the historical development of the Newton-Raphson method for solving nonlinear algebraic equations, focusing on the contributions of Isaac Newton, Joseph Raphson, and Thomas Simpson. The method, which is often referred to as Newton's method or Newton-Raphson method, is used to find the roots of an equation \( f(x) = 0 \). The article highlights how Newton's formulation differed from the iterative process of Raphson, and that Simpson was the first to provide a general formulation applicable to nonpolynomial equations using fluxional calculus. Simpson's extension of the method to systems of equations is also discussed. The article begins by discussing the method of Viète, which was a precursor to the Newton-Raphson method, and Newton's early notes on this method. It then examines Newton's original presentation of his method, contrasting it with Raphson's iterative formulation for polynomial equations. The article also explores Newton's use of the secant method, which is similar to the Newton-Raphson method but does not involve derivatives. Raphson's work in 1690 is detailed, including his formula for solving polynomial equations and his iterative approach. The article notes that Raphson's method was a significant improvement over Newton's, as it did not require the generation of intermediate polynomials and retained all significant digits computed in successive iterations. Finally, the article discusses Simpson's general formulation for nonlinear equations using fluxional calculus and his extension of the method to systems of equations. The historical development of the Newton-Raphson method is traced through these contributions, highlighting the evolution of the method from its early algebraic forms to its modern iterative form.