From: "Alec Mihailovs" <alec@mihailovs.com>
Another approach is using a polynomial approximation of sin. Linear approximation sin(x) = x is not very good. For x=pi/6 it gives 1/2=pi/6 and pi=3. However, cubic approximation sin(x) = x + Ax^3 is already pretty good (quadratic term is not present because sin is odd).
That reminds me of a question I've wanted to ask here. I have fun conceptual design for a difference engine that uses paper tape and computes one digit (of one order of difference) at a time. But I would like to have a good example to run. Didn't they actually used to use the method of differences to calculate trig & log tables by hand? If so, how did they calculate the initial numbers for each page (say) without having to do more work than it takes to calculate the entries separately? Do you just calculate the first n entries manually, then take the differences, differences of differences, etc., or is there a more numerical-methods way? --Steve