Newton's Method by Newton

Around 1670, Newton devised a method to approximate roots of equations using

(1)	x³ - 2x - 5 = 0

as an example. Arguing that there is a root near x = 2, he makes a substitution x = 2 + p and thus obtains an equation for p with a root near 0:

(2)	p³ + 6p² + 10p - 1 = 0.

Since p is expected to be small, he find its estimate by dropping the terms p³ and 6p². The resulting equation 10p - 1 = 0 yields an estimate p = 1/10 giving an estimate x = 2 + .1 = 2.1 for a root of the original equation. He is thus able to repeat the substitution step, now starting with (apparently) a better estimate x = 2.1 + q. The estimate for q is -0.0054 which gives a new approximation for x: 2.0946. Further steps may be taken if a better estimate is required.

What has this to do with Newton's method as we know it today?

The substitution x = 2 + p was a computational device. What Newton actually did was finding the Taylor series of f(x) = x³ - 2x - 5 around x = 2. Let's check that this is indeed so:

f(2) = -1, f '(2) = (3x² - 2)|_{x = 2} = 10, f ''(2) = (6x)|_{x = 2} = 12, f '''(2) = 6. It this follows that

	f(x)	= -1 + (x - 2)×10 + (x - 2)²×12 / 2 + (x - 3)³×6/3!
		= -1 + p×10 + p²×6 + p³

so that (1) is indeed equivalent to (2). Removing the terms from the Taylor series of orders higher than 2 leaves the first order approximation to the function which is the line tangent to the graph. So Newton was essentially trail blazing for the method now known by his name, however the form of what he did was quite different from what we are doing nowadays. He never presented his method as an iterative process but rather as a procedure for improving already existing approximation.

Joseph Raphson (1648-1715) was the first to describe the process as successive approximations, although his interest was still restricted to polynomial equations. Thomas Simpson (1710-1761) in 1740 formulated the method in its present day form as a method for solving general non-linear equations.

References

1. T. Gowers (ed.), The Princeton Companion to Mathematics, Princeton University Press, 2008, pp. 109-110

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