The Newton-Raphson method, a powerful iterative technique for solving nonlinear equations, has its historical roots in the works of Isaac Newton, Joseph Raphson, and Thomas Simpson. Newton's original formulation, though algebraic, was later refined by Raphson and Simpson. Newton's method, expressed as $ x_{i+1} = x_i - f(x_i)/f'(x_i) $, was first described in his work "De analysi per aequationes numero terminorum infinitas" (1669), where he used binomial expansions to approximate solutions. However, Newton did not explicitly use calculus in this formulation, and the connection to fluxional calculus was established later. Raphson, in his 1690 work "Analysis aequationum universalis," provided a more direct and simplified version of the Newton-Raphson method. His formula, $ x = \frac{c + bg - g^3}{3g^2 - b} $, is a specific case of the Newton-Raphson iteration applied to cubic equations. Raphson's method is algebraic and does not require the use of calculus, making it more accessible for practical computations. His approach involved iteratively improving estimates of the solution without generating intermediate polynomials, a technique that was more efficient than Newton's method. Thomas Simpson further generalized the Newton-Raphson method in the 18th century, applying it to systems of nonlinear equations and extending it to non-polynomial equations using fluxional calculus. Simpson's work demonstrated the method's versatility and laid the groundwork for its widespread application in numerical analysis. The historical development of the Newton-Raphson method highlights the evolution from early algebraic techniques to more sophisticated iterative methods. Newton's original formulation, while foundational, was algebraic and did not explicitly use calculus. Raphson's method simplified the process, and Simpson's contributions extended the method's applicability to a broader range of equations. The method's quadratic convergence, characterized by the doubling of correct significant digits in each iteration, was recognized by Newton, who understood its efficiency and effectiveness in solving nonlinear equations. The historical context of these developments underscores the importance of iterative methods in numerical analysis and their enduring impact on modern computational techniques.The Newton-Raphson method, a powerful iterative technique for solving nonlinear equations, has its historical roots in the works of Isaac Newton, Joseph Raphson, and Thomas Simpson. Newton's original formulation, though algebraic, was later refined by Raphson and Simpson. Newton's method, expressed as $ x_{i+1} = x_i - f(x_i)/f'(x_i) $, was first described in his work "De analysi per aequationes numero terminorum infinitas" (1669), where he used binomial expansions to approximate solutions. However, Newton did not explicitly use calculus in this formulation, and the connection to fluxional calculus was established later. Raphson, in his 1690 work "Analysis aequationum universalis," provided a more direct and simplified version of the Newton-Raphson method. His formula, $ x = \frac{c + bg - g^3}{3g^2 - b} $, is a specific case of the Newton-Raphson iteration applied to cubic equations. Raphson's method is algebraic and does not require the use of calculus, making it more accessible for practical computations. His approach involved iteratively improving estimates of the solution without generating intermediate polynomials, a technique that was more efficient than Newton's method. Thomas Simpson further generalized the Newton-Raphson method in the 18th century, applying it to systems of nonlinear equations and extending it to non-polynomial equations using fluxional calculus. Simpson's work demonstrated the method's versatility and laid the groundwork for its widespread application in numerical analysis. The historical development of the Newton-Raphson method highlights the evolution from early algebraic techniques to more sophisticated iterative methods. Newton's original formulation, while foundational, was algebraic and did not explicitly use calculus. Raphson's method simplified the process, and Simpson's contributions extended the method's applicability to a broader range of equations. The method's quadratic convergence, characterized by the doubling of correct significant digits in each iteration, was recognized by Newton, who understood its efficiency and effectiveness in solving nonlinear equations. The historical context of these developments underscores the importance of iterative methods in numerical analysis and their enduring impact on modern computational techniques.