I was reading about the history of Newton's Method. Newton used a cubic equation, $x^3 - 2x - 5 = 0$, to show the efficacy of his method around 1670. I was wondering that why Newton would choose this particular depressed cubic equation, one could have used the already existing Cardano's formula, an analytical/symbolic method, for a depressed cubic equations to get the real value of $x$. There are two additional roots of the given cubic equation which are complex and complex numbers hadn't entered the mainstream at that time.

It seems like Newton was more interested in coming up with an alternative way of solving problems and hence he decided to use an equation which was solvable using another existing method. Previously, the numerical techniques were used more in an ad hoc manner or out of necessity - either the solution was not obtainable using symbolic techniques or they lacked the proper mathematical tools at that time to come up with a symbolic formula. In this regard, Newton could be considered the first or one of the pioneers who laid down the foundation of numerical analysis, the "new analysis", as an alternative approach.

Do you agree with me to some extent? I'd appreciate it if you could correct me.

Later, in 1687 in his book Philosophiae NaturalisPrincipia Mathematica, Newton used his method of approximation to solve the nonpolynomial equation, $x-e \sin(x)=z$, also known as Kepler's problem, which doesn't have an analytical solution. In the equation, $z$ is the known mean anomaly, $e$ the eccentricity, and $x$ the eccentric anomaly to be determined.

10.2.6 Algebraic Nonintegrability of Ovals

Section 6, Book 1, of the Principia is devoted to the solution of the so-called Kepler problem. The problem consists in finding the area of a focal sector of the ellipse and is equivalent to the solution for x of the equation $x — e \sin x = z$ ($e$ and $z$ given). ...In Lemma 28, Section 6, Newton demonstrated that this problem cannot be resolved in finite algebraic terms (§13.3). In Proposition 30, Section 6, he showed that the determination of the position of a body orbiting in a parabolic trajectory (such that the area law is valid for the focus) is instead algebraic (see chapter 11). But to deal with the Kepler problem for elliptic trajectories, polynomial equations are not enough. In Section 6, Newton therefore illustrated how the roots of the Kepler’s equation can be determined via his method of successive approximations, which is to say, via infinite series (see §7.5, figure 7.10).2' Infinite series occur throughout the Principia (especially in the final Scholium to Section 13, Book 1: Proposition 45, Book 1: and Proposition 10, Book 2), and Newton might well have had this in mind when he stated that his work was based on new analysis. He also employed quadrature techniques, another key element of his new analysis.

[Isaac Newton on Mathematical Certainty and Method, Issue 4, By Niccolò Guicciardini, 2009; page #248]

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    $\begingroup$ I think not. Even as far as "Newton's" method, Newton was one in a line of many people after Cardano to approximate roots of polynomial equations numerically, and similar things were done before Cardano. Cardano's formula is of little use if one wants the numerical value (even square roots require numerical approximation) so symbolic formulas did not eliminate the "necessity", nor did their absence create it. Vieta and Napier, among others, developed numerical methods systematically before Newton, and many did after, but as a "field of study" numerical analysis is a 20th century item. $\endgroup$ – Conifold Jun 23 '20 at 5:01
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    $\begingroup$ It is not clear what this question really means. Numerical methods were practiced since the very ancient times, for example for extracting square and cubic roots with arbitrary precision, for approximating pi etc. Or think of computations in trigonometry and astronomy. Logarithms were invented long before Newton. In what sense was Newton the first ? $\endgroup$ – Alexandre Eremenko Jun 23 '20 at 12:10
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    $\begingroup$ The example you give yourself: Kepler actually invented a numerical method of solving Kepler's equation. And many other people invented their methods. Why give credit to Newton on the ground that he was one of them? $\endgroup$ – Alexandre Eremenko Jun 24 '20 at 13:09
  • $\begingroup$ Thank you, both of you! I better edit my question because I won't be able to clarify using the limited space. I'd do it in next couple of days. $\endgroup$ – PG1995 Jun 24 '20 at 20:49

Cardano's formula is useless if you want to really solve the cubic equation. In our culture we say that "solving an equation" means writing its solution in some closed form using a certain set of mathematical symbols. But this is just an agreement, and it was not always the case. If you need to solve an equation not to pass an exam, but for some practical purposes, you need to find an approximation of your solution. If you are going to use Cardano's formula you need to take cubic and square roots. But how do you take - say - a cubic root of 5 without a calculator? The easiest way is to use Newton's method to solve the equation $x^3=5$. Now to use Cardano formula you need to take two cubic roots and one square root, i.e. to use Newton methods three times. Why not just to use it for the original equation?

  • $\begingroup$ Thank you. I wouldn't say that Cardano's formula was altogether useless. At that time, there were extensive tables for square roors and cube roots available. I had this question in mind but didn't know how to properly reply to @Conifold and Alexandre Eremenko since both of them are 'gurus' and saying something which somewhat goes contrary to their viewpoint is not easy. Anyway, Newton could have chosen to use those tables instead of coming up with a method since the method also required some work and iterations to approximate the answer. Another main problem is that how one... Cont'd $\endgroup$ – PG1995 Jul 1 '20 at 7:51
  • $\begingroup$ defines a numerical method. For example, here hsm.stackexchange.com/questions/5612/…, a marine chronometer is presented as an example for the need of a numerical method. IMHO, the invention of marine chronometer was more of an ingenious engineering design feat. Here, hsm.stackexchange.com/a/5639/9341, Euler-Maclaurin formula is presented as another example of numerical method. IMHO, I don't really see it as a numerical method. Euler was able to solve Basel problem by 1735... Cont'd $\endgroup$ – PG1995 Jul 1 '20 at 7:52
  • $\begingroup$ and came up with a closed form solution by showing that the series converges to π²/6. So, anytime you sum up a only a limited number of terms without summing up all the infinite terms, does it become a numerical method? Then, what about the Taylor series because you also terminate the series after summing up a certain number of terms to come up with a Taylor polynomial? Also, what about the Fourier series? In Fourier series, you also don't sum up infinite number of sinusoids? Is Fourier series also a numerical method because you are approximating a function using a finite number of sinusoids? $\endgroup$ – PG1995 Jul 1 '20 at 7:53

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