How I derive a sequence of decimal approximations to square root of 2 to five-place accuracy using the chinese square root algorithm

I don't know what the "Chinese algorithm" is, but check out the following for various methods of calculating square roots. The digit-by-digit method may be what you're asking about - see the example for calculating \sqrt 2 about one third of the way down the page.

Methods of computing square roots - Wikipedia, the free encyclopedia (http://en.wikipedia.org/wiki/Methods_of_computing_square_roots)