Newton's polynomial solver

The following short note describes a method of Newton for approximating roots of polynomial equations using slide rules. It first appears in Newton’s Waste Book, suggested by [9] around 1665. The method appears then to have been explained to John Collins in a letter of 1672, [7]. Collins was interested in finding the volume of liquid in a partially full barrel as a function of depth1. This was an important issue in “gauging” – the calculation of various taxes on liquids. Interestingly slide rules were particularly popular amongst customs officials, for whom they provided an instant volume calculation, by visual inspection, and therefore instantly settled the question of how much tax or duty was due. It is reported in [9] that this method was incorporated by Collins into a general method for solving cubic equations. Collins then wrote In Answer to Mon Leibnitz’s Letter about Solving a Cubick aequation by Plaine Geometry. This Answer was passed to Oldenburg who in the following letter, dated June 24 1675, quoted in [2, pg 21], but also in [5], communicated the ideas to Leibnitz.