Accounts, some of which had been published previously, on the iterative algorithm called Newton's method, are assembled into a readily accessible whole, including instructive excerpts from original sources: Viète (who influenced Newton); the secant method in Newton's `Waste Book'; several letters and other writings by Newton; Raphson's book on polynomial equations of 1690, with comments by Halley; Simpson in his `Essays' of 1740, and his (little known) extension of the method to systems of nonlinear equations in several variables; and the unified general treatment by Lagrange in 1798. Useful bibliography.