Polynomial truncations

Extract

In elaborating John Ward’s method for the calculation of pi, Cohen and Shannon drew attention to Ward’s assumption that the smallest positive root of a polynomial equation can be approximated by a root of the truncated equation in which higher powers are ignored. (The interested reader can find full details in [1] and [2].) The mathematical basis of this is justified in a plausible style in this note, and it can be readily used by the interested teacher as a basis for further computational work with students. It is likely that Ward employed a Newtonian fluxional kind of analysis when he obtained the result, rather than the purely algebraic considerations employed here. The historical significance of Ward is that his calculation of pi was possibly the last attempt to use geometric methods. Its mathematical significance was his use of successive trisection rather than the traditional bisection of angles.