>I don't see how taking the root of a number requires any >usage of RxR in that the solution can be carried out on >the original number line. So I am wondering how you come >to claim a conflict with the cartesian system?
Show me any ancient or modern _ARITHMETICAL_ method THAT IS, created by agency of Arithmetic (only the four basic arithmetical operations), not based neither on infinitesimal calculus, o any geometrical method within the cartesian system, for computing, say, the fifth root of 2, with convergence rate of seventh order (that is, the number of digits multiplies by 7 in each iteration)
You will realize that neither ancient nor modern mathematicians created such HIGH-ORDER ARITHMETICAL METHOD, and they were forced to invent the Infinitesimal Calculus, even though we can see now that ancient mathematicians (5000 B.C.) certainly had the elementary tools to achieve this, but they didn't, and that is desvastating.
Worst, modern mathematicians think that the only way to achive Householder's method, among others, was by puting aside Arithmetic and creating a new "superior" science: The infinitesimal calculus.
Specifically in page 44, there is an example, as well as in pages 19 and 34. Anyonw who knows a bit about root solving methods can inmediately realize what is going on in there.
I am sure you will not pretend that I must publish the entire book in the net for free, while a tenis player who has no responsability at all, earn lots of money for just hitting a ball.
I don't claim conflicts. I claim that there are other simpler (arithmetical) ways to find all those high-order methods that modern mathematicians achived after creating their infinitesimal calculus. That is, that ancient mathematicians had the tools to find that and many other new and better high-order things by just using ARITHMETIC.