I came across this method for approximating n-th roots in an arithmetic book from 1843. I would like to know what iterative method is being used to perform the approximation, and if there is an example in modern notation.
iterative method for approximating nth roots
Second part: https://i.stack.imgur.com/b4W9P.jpg
This is Halley's method with cubic order of convergence. It is the Newton method for the equation $$ g(x)=x^{(n+1)/2}-ax^{-(n-1)/2} $$ which can be obtained from the usual equation $f(x)=x^n-a$ by applying "flattening at the root", modifying $f$ by a factor so that $g''(r)=0$, which turns out to be dividing $f(x)$ by $\sqrt{|f'(x)|}$. Noting that constant factors can be omitted, this results here in a division by $x^{(n-1)/2}$.
Then $$ x-\frac{g(x)}{g'(x)}=x-\frac{x^{(n+1)/2}-ax^{-(n-1)/2}}{\frac{n+1}{2}\,x^{(n-1)/2}+a\frac{n-1}{2}x^{-(n+1)/2}} \\~\\ =\frac{x((n+1)x^n+(n-1)a)-2x(x^n-a)}{(n+1)x^n+(n-1)a} \\~\\ =x\frac{(n-1)x^n+(n+1)a}{(n+1)x^n+(n-1)a} $$