Since the advent of 'derivatives' math historians and teachers assert that only by the agency of cartesian system and infinitesimal calculus modern mathematicians could create high-order algorithms for approximating roots, and that ancient were not able to do that because the only tool they had was Arithmetic: The science of Quantity.
However, beneath the light of the new extremely simple arithmetical methods for approximating roots with any desired convergence rate, those assertions become absolutely false, worst, they look truly egotistic and ridiculous, so it is clear all them will be compeled to edit all those statements in their books.
Of course, this new discovery seriously hits many egotistic spirits that used to control this area of human knowledge, sorry for that, but their tales on the god-like superiority of Infinitesimal calculus should finally come to an end. It is really striking to realize that only by the agency of the most simple Arithmetic one can achieve not only Newton's, Halley, Householder's, Bernoulli's, but many other new algorithms for approximating roots with any desired convergence rate, strinking and devastating indeed.
For those to use to appeal and level up their statements on references:
If this hurts so many egotistic authors, then: Sorry for that, that is the only thing I can do to help you, because to your disgrace, these new extremely simple high-order Arithmetical algorithms based on the Rational mean came to stay arround eternally.
Since, these new algorithms were brought to light in the net, no one initiated any other thread expressing inquires about square, cube roots, etc. Wonder why...