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    Re: First sine table (Ptolemy)
    From: George Huxtable
    Date: 2009 Jan 24, 11:54 -0000

    Frank gave a recipe for generating sine and cosines (copied below), starting 
    with an approximation for sine and cos of 1�, which give me, from a 
    calculator, starting values of .01745329252 and .9938476797, respectively.
    I wondered how accurate that recipe might be, when compared with the 
    predictions for sines implicit in Ptolemy's table of chords, as decribed in 
    my posting [7137], which dated back to around 150 AD, and laid out in detail 
    I have used Frank's recipe to calculate sin 2� and cos 2�, and then 
    reiterated to obtain, from those, a value for sin 4� by the same method, 
    which has given-
    sin 4�=.06976001. Compare this with the true value of-
    sin 4�=.06975647 and with Ptolemy's value, implicit in chord of arc 8�-
    sin 4�=.06975694. The error in Ptolemy's value is much less, and I suggest 
    that this was not just a numerical fluke.
    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. 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, using long-arithmetic without benefit of calculators.
    Of course, Ptolemy had to perform similar tasks, without even the benefit of 
    modern decimal arithmetic.
    contact George Huxtable, at  george@hux.me.uk
    or at +44 1865 820222 (from UK, 01865 820222)
    or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    ----- Original Message ----- 
    Sent: Saturday, January 24, 2009 7:23 AM
    Subject: [NavList 7149] Re: First sine table (Ptolemy)
    Jim, you asked:
    "Were infinite series used in the initial creation of tables?"
    No, not the early tables.
    One way to generate tables is to use the trigonometric identities for sums 
    and differences of angles. Consider this: we know some exact values, e.g. 
    sine and cosine of 0�, 30�, 45�, 60�, 90�. If you have arbitrarily good 
    values for the square root of two and the square root of three, then you 
    have arbitrarily good values for the trig functions of those angles. Now we 
    can also calculate the sine of one degree to fairly good accuracy using a 
    variety of methods (see below). That's our anchor. And the cosine of one 
    degree is sqrt(1-sin^2(1)) so we calculate that one by hand as accurately as 
    possible. Then how do we get the sine and cosine of 2 degrees? Use the 
    sin(a+b) = sin(a)*cos(b) + sin(b)*cos(a),
    cos(a+b) = cos(a)*cos(b) - sin(a)*sin(b),
    which in this case would be
    sin(2�) = sin(1�)*cos(1�) + sin(1�)*cos(1�),
    cos(2�) = cos(1�)*cos(1�) - sin(1�)*sin(1�).
    And then the sine and cosine of 3 degrees? Just build on the previous 
    sin(3�) = sin(2�)*cos(1�) + sin(1�)*cos(2�),
    cos(3�) = cos(2�)*cos(1�) - sin(2�)*sin(1�).
    And so on...
    And note that you really only need to calculate from 0 to 45 degrees. After 
    that, you just re-label the table using sin(x) = cos(90-x) etc.
    Next we look up all of these values in our table of logarithms (let's assume 
    someone has already provided us with that) and then we can get a table of 
    tangents rather quickly from tan(a) = inverselog[logsin(a)-logcos(a)].
    These are the general sorts of calculations that were used. There are lots 
    of tricks for speeding things up and reducing error. For example, we can get 
    exact values for the sine and cosine of 15 degrees using half-angle 
    identities. And from there, the sine/cosine of 7.5 degrees. Then going back 
    to the sum identities, we get "exact" values for any angle that is a 
    multiple of 7.5 degrees.
    Finally, up at the top I mentioned that we need the sine of 1 degree. At 
    some level of accuracy the sine of a small angle is equal to the arc. That 
    sin(x degrees) = x*pi/180, for a small angle. For a low accuracy table, you 
    could assume that this holds for x=1�. For a higher accuracy table, you 
    would assume that this holds for x=1'. Then you build up arcminute by 
    arcminute until you get a better value for the sine of 1 degree and proceed 
    again as above.
    Navigation List archive: www.fer3.com/arc
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