How did Newton & Raphson's versions of the N-R method differ?

Question

To quote Wikipedia,

Raphson's most notable work... contains a method, now known as the Newton–Raphson method... Newton had developed a very similar formula in his Method of Fluxions, written in 1671, but this work would not be published until 1736, nearly 50 years after Raphson's Analysis. However, Raphson's version of the method is simpler than Newton's, and is therefore generally considered superior. For this reason, it is Raphson's version of the method, rather than Newton's, that is to be found in textbooks today.

It's a shame we normally call it Newton's method, since if anything it should be named after the otherwise forgotten Raphson. But what was Newton's version of the formula? Raphson's is our modern formula,

$$x\mapsto x-\frac{f(x)}{f'(x)}.$$

Answer 1

According to Peter Deuflhard's A Short History of Newton's Method, Newton began by familiarising himself with the methods of Vieta, which had been simplified by Oughtred. (Vieta's method was already known to al-Kāshī, who published a method using Vieta's perturbation technique in 1427, though Vieta appears to have been unaware of al-Kāshī's work.)

As an example, he discussed the numerical solution of the cubic polynomial $$f(x) := x^3 - 2x - 5 = 0.$$ Newton first noted that the integer part of the root is $2$ setting $x_0 = 2$. Next, by means of $x = 2 + p$, he obtained the polynomial equation $$p^3+6p^2+10p-1 = 0.$$ He neglected terms higher than first order setting $p ≈ 0.1$. Next, he inserted $p = 0.1 + q$ and constructed the polynomial equation $$q^3 + 6.3q^2 + 11.23q + 0.061 = 0.$$ Again neglecting higher order terms he found $q ≈ −0.0054$. Continuation of the process one further step led him to $r ≈ 0.00004853$ and therefore to the third iterate $$x_3 = x_0 + p + q + r = 2.09455147.$$

Deuflhard continues

In 1690, Joseph Raphson (1648–1715) managed to avoid the tedious computation of the successive polynomials, playing the computational scheme back to the original polynomial; in this now fully iterative scheme, he also kept all decimal places of the corrections. He had the feeling that his method differed from Newton’s method at least by its derivation.

Note that neither Newton nor Raphson mention derivatives, even if, as Deuflhard notes :

Note that the relations $10p − 1 = 0$ and $11.23q + 0.061 = 0$ given above correspond precisely to $$p = x_1 − x_0 = −f(x_0)/f′(x_0)$$ and to $$q = x_2 − x_1 = −f(x_1)/f′(x_1).$$

Finally

In 1740, Thomas Simpson (1710–1761) actually introduced derivatives (‘fluxiones’) in his book ‘Essays on Several Curious and Useful Subjects in Speculative and Mix’d Mathematicks [No typo!], Illustrated by a Variety of Examples’. He wrote down the true iteration for one (nonpolynomial) equation and for a system of two equations in two unknowns thus making the correct extension to systems for the first time. His notation is already quite close to our present one (which seems to date back to J. Fourier).

Also worth noting is that, according to UBC Math

At first sight, the method Newton uses doesn’t look like the Newton Method we know. The derivative is not even mentioned, even though the same manuscript develops the Newtonian version of the derivative!

(My emphasis.)

Answer 2

Newton's method of successive approximation to roots of equations first appeared in print in 1685 in John Wallis's 'A Treatise of Algebra, both Historical and Practical' at pp.338-347, Chap.XCIV & XCV, 'A new method of extracting roots in Simple and Affected Equations' -- containing also Wallis's attribution to Newton. This presentation has been discussed by Niccolo Guicciardini, who has remarked that Newton's description 'is based on concrete examples ... a craft more than a theory': N Guicciardini (2009), 'Isaac Newton on Mathematical Certainty and Method'.

Newton published a further concrete example of the method in the 'Principia' (all editions, from 1687), at Book 1, Prop.31 (Scholium). This example was an application of the method, to solve an equation defining the orbital position of a planet, one version of Kepler's problem. The example was discussed in 1882 by John Couch Adams (Monthly Notices of the Royal Astronomical Society 43, pp.43-49), who pointed out that in the first edition of the Principia, Newton had tried to describe a refinement of the method, but had made a mistake (equivalent to forgetting to update one of the 'temporary variables' in the iteration); Newton corrected the description (as amended in the second and the third editions) with a simplified version of the quadratically-converging method. In the 1999 translation of Newton's Principia, the method appears at pp.514-515 (Isaac Newton, 'The Principia ... A new Translation', I Bernard Cohen, Anne Whitman, 1999).

Raphson's method of successive approximations to roots of equations was first published in his 'Analysis Aequationum Universalis', 1690, which went through at least three editions. (Neither Newton nor Raphson used any notation close to the modern " -f(x)/f'(x) " for the iterative corrections.)

Raphson's method has been discussed and compared with Newton's method by F Cajori among others (1911, American Mathematical Monthly, Vol.18, 29-32). Cajori showed how Raphson's method resembled that of Newton especially in that they both used a divisor for the next correction that evaluates, in effect, to (what in modern notation would be) f'(r), where r was the most recent corrected value. (But neither of them used a symbolic notation such as f'(r), they expressed each case of the quantity in an extended e.g. polynomial form.)

But where Newton had derived each successive step of approach to the root from a newly formed equation derived from the original one, Raphson, closer in that respect to the modern procedure, found each successive corrected value by substitution in the original equation. (Cajori also remarked that Newton's forerunners such as Vieta had used a different and generally non-equivalent principle for evaluating the relevant divisor. But in some specific example-cases, Vieta's value of the divisor had coincided with the value of Newton's divisor -- which had led some commentators mistakenly to assimilate Newton's method with Vieta's.)

Cajori gave a history of the methods, and in view of Raphson's important modification proposed the name 'Newton-Raphson' method as a more accurate attributive label than just the name 'Newton's method', which had been current. His suggestion, made over a century ago now, does seem to have taken hold, 'Newton-Raphson' is widely used to denote the modern version of the method.

Answer 3

Full details in my paper: Historical Development of the Newton-Raphson Method, published in SIAM Review in 1995. Deuflhard's paper is basically a summary of that work.