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    Re: First sine table (Ptolemy)
    From: Frank Reed
    Date: 2009 Jan 24, 16:44 -0800

    George, you wrote:
    "Unless I've made an error, which is always possible, it shows that over 1800 
    years ago, there already existed a method that was more precise than what 
    Frank is suggesting, though the knowledge became lost."
    Yes, the method I described was not intended to be Ptolemy's method (which you 
    already described) and it was not intended to be the best approach. I was 
    merely trying to demonstrate the basic approach to generating a trig table 
    without using series expansions.
    And you wrote:
    "To increase the precision, Frank might (as he suggested) have to reduce his 
    starting value below 1�, which would then have entailed even more iterations, 
    each of which would have to be carried out to high precision and without 
    arithmetical errors"
    Right. But it's not as much work as you might imagine. Let's start with the 
    sine of one minute of arc. At that small angle, for some level of accuracy, 
    we can assume that the sine is equal to the arc. So sin(1')=(pi/180)/60. 
    Assuming we're in the early modern era, near the beginning of the history of 
    modern navigation, c.1700, anyone who needs to knows pi to at least fifteen 
    digits, but let's go with pi=3.141593. Then use the sine sum formula 
    and cos(a)=sqrt(1-sin(a)*sin(a))
    to take us from 1' to 5'. At that point we can jump to 10', then to 15' and 
    30', and from there to 1 degree. After that proceed as before, one degree, 
    and you have a very accurate table of sines and cosines. If each calculation 
    is done to nine digits accuracy, the final table is accurate to nearly seven 
    places. The largest errors are for sin/cos of 45 degrees which come out to 
    0.70710680 and 0.70710676 respectively where they should both be 0.70710678. 
    If we run the calculation up from 1 degree and down from 45, the results are 
    even better.
    You concluded:
    "Of course, Ptolemy had to perform similar tasks, without even the benefit of 
    modern decimal arithmetic."
    Yes, but he had grad students... er...slaves! er... same thing. It's a huge 
    amount of work to do this by hand, no doubt. Thank the gods for the printing 
    press and its information storage descendants so we never have to do it 
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