Newton's Method for Resolving Affected Equations

Summary

During the 300 years since Newton and Leibniz began disputing which of them had discovered the calculus, debates have continued over the credit due to Newton for various scientific and mathematical achievements. Recent research by Nick Kollerstrom [11] has led him to credit Thomas Simpson (1710–1761) with the first discovery and publication in 1740 [18] of what is now called Newton's method. William Dunham [8] has pointed out the irony that Newton, who “bitterly resented people's getting credit for results they did not originally discover,” is credited with a method of approximation that “in its full generality seems to be due to” Simpson.

The debate over priority for Newton's method may now be settled, but almost forgotten in the discussion is that Newton presented his method for approximating real roots side by side with a similar method for writing y in terms of x when y is implicitly defined in terms of x by a polynomial equation—a so-called “affected equation.” This second “Newton's method” is an important tool in modern algebraic geometry and, although it is more subtle than his method for approximating roots, it can be understood by precalculus students.

Richard S. Westfall [22] highlights how Newton used his method for resolving affected equations to integrate algebraic equations:

… in the mid-1660s, Newton was working toward a general method of squaring curves, as they put it then; let us say “integration” for simplicity.