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I saw this from Wikipedia.

The name "Newton's method" is derived from Isaac Newton's description of a special case of the method in De analysi per aequationes numero terminorum infinitas (written in 1669, published in 1711 by William Jones) and in De metodis fluxionum et serierum infinitarum (written in 1671, translated and published as Method of Fluxions in 1736 by John Colson).

However, it has no citations. When is its first use?

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  • $\begingroup$ The Babylonian method of extracting square roots is Newton's method... a solid few millennia before Newton. $\endgroup$ – vonbrand Feb 27 at 18:00
  • $\begingroup$ This was a very special case on Newton's general method. $\endgroup$ – Alexandre Eremenko Feb 28 at 1:15
  • $\begingroup$ But was it based on knowledge of calculus or on "hey, let's try something that seems to make sense" ? $\endgroup$ – Carl Witthoft Feb 28 at 14:47
  • $\begingroup$ @CarlWitthoft The latter; they averaged an overestimate of the square root with an underestimate (although you don't have to know which one is which). $\endgroup$ – J.G. Mar 1 at 7:03
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TL; DR. This is one of those cases where "first use" very much depends on what is meant. Depending on that, it can be ascribed to Babylonians (c. 1600 BC), al-Tusi (c. 1250), Briggs (1633), Newton (1669), Raphson (1690) or Simpson (1740).

The Babylonian rule (1800–1600 BC) for approximating square roots converted into modern notation gives the same formula as the "Newton's method", see Square Root Approximations in Old Babylonian Mathematics by Fowler and Robson. It does not come with even a vague anticipation of conceptions involved in the modern method. The same rule is described by Heron in Metrica (c. 50 AD), and is called "Heron's method" in older books. Heron also gives a related rule for cube roots, which does not reduce to the "Newton's method", suggesting that the connection is not too deep, see Remarks on Heron's cubic root iteration formula.

For the "warmer" part of history see Historical Development of the Newton-Raphson Method by Ypma:

"A method algebraically equivalent to Newton's method was known to the 12th century algebraist Sharaf al-Din al-Tusi [13], and the 15th century Arabic mathematician Al-Kashi used a form of it in solving $x^p- N = 0$ to find roots of $N$. In western Europe a similar method was used by Henry Briggs in his Trigonometria Britannica, published in 1633, though Newton appears to have been unaware of this."

Instead he worked from a more cumbersome perturbative method for solving polynomial equations laid out by Vieta in De numerosa potestatum (c.1600), with additional inspiration from a very old method of false position, a.k.a. Regula Falsi. It is close to what is now called the secant method. A simplified version of Vieta's method was published by van Schooten in 1646 and reproduced by Oughtred in Clavis Mathematica (1647), this was Newton's source.

Detailed remarks on it are found in Newton's unpublished 1664 notebook, and by 1669 he linearized Vieta's successive polynomials to produce something convertible into the modern version in De analysi per aequationes numero terminorum infinita (better known for the first formulation of the method of fluxions). But, as with Babylonians, Heron or al-Tusi, there is little evidence that "his" method was the modern one conceptually. Here is Ypma:

"Newton's tract... is the first recorded discussion by Newton of what we may recognize as an instance of the Newton-Raphson method (1.1), although the formulation differs considerably from the now conventional form, the computations are much more tedious than in the current formulation, and the method is given only in the context of solving a polynomial equation. No calculus is used in the presentation, and references to fluxional derivatives first appear later in that tract, suggesting that Newton regarded this as a purely algebraic procedure. In several other instances Newton is known to have used more traditional methods and notations in an effort to make his ideas more accessible to a wider audience, but there is no clear evidence that at that time he perceived this particular technique as an application of the calculus or derived it using the techniques of calculus."

The manuscript was not published until 1711, but private copies circulated earlier and the relevant content is reproduced in Wallis's Treatise of Algebra both Historical and Practical (1685). In Principia (1687) a similar procedure is applied to solving the Kepler's equation $x-e\sin x=M$, but again there is "no clear evidence that Newton associated his technique with the use of the calculus. There are numerous ways to derive this process that do not require the use of calculus".

Raphson simplified the technique further in 1690 by eliminating successive polynomials completely, and making the scheme iterative. He already felt that the resulting method was different from Newton's. However, the modern conception with derivatives (fluxions) does not appear until Simpson's Essays on Several Curious and Useful Subjects (1740), where he does not credit any predecessors, and explicitly contrasts his calculus based procedure to the previous algebraic ones.

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