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]
