I hope the following webpage on new roots solving algorithms based on the Arithmonic mean (a particularcase of the Rational Mean) could be of some interest for this audience:
References and links can be found at its home page:
Indeed, there are very good news here, specially, for young people because from now on, by means of simple arithmetic they will be able to learn at secondary school, among many other new simple algorithms, those which have been classed as the most advanced and superb analytical methods: Halleys, Newtons, Bernoullis and Householders.
Some authors have pointed out that "Arithmetic" was the main obstacle ancients should overcome in order to solve problems involving what we call nowadays "roots-solving methods of higher degree", and that such analytic algorithms could only be found, formulated and explained by agency of the modern Cartesian system and infinitesimal calculus. We can see now that ancients certainly had at hand the most simple arithmetical tool (The Rational Mean, The Fifth Arithmetical Operation) for solving all those problems involving higher degree equations. It is really striking to realize that since ancient times mathematicians could have easily carried out such an elemental operation and roots solving methods but --from all the evidences-- they didn't!.
Based on the extremely simple arithmetical processes and wonderful properties of Number shown in the book and its introductory web pages, it is so hard to realize these so simple arithmetical methods do not appear in any book on numbers since ancient times up to now.
Domingo Gomez Morin