Halley's Methods for Solving Equations

1. Halley's approximations. About 1694 Edmund Halley [1] became interested in some rules that were formulated by the French mathematician Thomas Fautet de Lagny in his book Methodes nouvelles et abregees pour l'extraction et l'approximation des racines which was published in Paris in 1692, a second edition appearing in 1697. Halley did not see the book himself but heard of the rules through a friend. Halley proceeded to test the rules and to extend them. He expressed the opinion that de Lagny's rule for finding (a5+b)l"5 is printed incorrectly in the first edition of his book. This rule, according to Halley's method of derivation, is that the root is given approximately by an expression which in modern notation is