Extracting square roots by hand

I recall feeling a curious pleasure when I first encountered the Maclaurin series for sine and cosine. With them I was able to compute for myself, by hand, sines and cosines of various angles to any degree of accuracy I wished. Perhaps part of that delight was the feeling of independence. Rather than requiring a table of values someone else prepared, or using a calculator, I could manufacture on my own any value I needed to any accuracy I needed. Curiosity was also part. Constants…