Abstract

The purpose of this chapter is to give a brief outline of Newton’s methods for “squaring” a curve, which in Leibnizian terms one would call “integrations.” These methods are rarely considered by scholars, even by Newton scholars, with the exception of those, who like George—the dedicatee of this volume—are familiar with the “technical” Newton. My purpose here is not to address the specialists in the history of seventeenth-century mathematics, but rather to offer a reader-friendly primer in Newton’s “quadrature” techniques. I will not help the reader by adapting the notation to our standards: I will strictly adhere to Newton’s notation. But I will try to make things as simple as possible by choosing the most elementary examples. It is often stated that Newton’s mathematical methods were “geometrical.” But once Newton associated an equation to a curve, he could proceed by relying mostly upon symbolic manipulation. This algebraic aspect of Newton’s mathematics should be better known. I hope that the readers non familiar with this side of Newtonian mathematics will derive some instruction, and even some intellectual pleasure, by scratching the surface of the treasure trove of Newton’s algebraic quadrature techniques.