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    Towards a basis for Bruce Stark's Tables
    From: Fred Hebard
    Date: 2003 Jan 3, 02:35 -0500

    OK, some history of this major obsession of mine,
    
    I used to sail and do coastal piloting, and always wanted to do
    celestial, but have lived inland for 30 years.  Then I noticed the
    price of a Davis Mark 3 on the Celestaire site and bought it.  Then
    of course spent hundreds of dollars on sight reduction books when I
    couldn't get my position down to better than 20 miles, trying to
    figure out what was going on.  Thought maybe elevation above sea
    level was involved, but in my case that's a half mile difference in a
    6000 mile radius of the earth; elevation affects lunar parallax and
    refraction a bit, but not much more.  Finally got my position to
    about 2 miles once.  Decided I needed a better sextant.  Got a 1948
    Husun Mate on Ebay for a little more than $200 because the batteries
    had burst the handle; but the enamel was not noticeably chipped in
    any of the photos and all the parts were there.  Found I would have
    to remove the index arm to remove the handle for repair.  Delayed
    doing that.  Ran a lunar and  got to within 12 seconds of GMT!  Maybe
    that Husun was OK, but handle needed fixing.  So removed index and
    fixed handle, but feared destruction of the calibration due to a
    small blunder while removing index.  To check calibration, I would
    like some accurate interstellar distances, thus need to apply Borda's
    method to correct for refraction.  But how do all those tables in
    Bruce Stark's book work?  A far more abstruse question.
    
    Table K, one of the big ones, is log(haversine()).  Found a reference
    to Gauss' formulas at
    http://mathworld.wolfram.com/SphericalTrigonometry.html .  They are
    similar to Napier's formulas.  So one of those must be the formula
    for the Gaussian table.  It clearly is not the normal approximation
    to the binomial.
    
    I have gotten about as far as this as I have time, and sure would
    like to see Bruce Stark's method laid out in detail.  It appears to
    be a fairly straight-forward application of log haversines to
    spherical triangle trigonometry.  He appears to be using Borda's
    method for clearing the distance, and the standard method for
    computing intercelestial distances (with no correction for
    refraction).
    
    Here is as far as I've gotten with his method for computing
    intercelestial distances.  Let del(GHA) be the absolute value of the
    difference in GHA between the two bodies, del(dec) be the
    corresponding value for declinations.  ~ is the operator for finding
    the absolute value of a difference.  Mdec is the declination of one
    body and Sdec the declination of the second;  logs are to the base
    10; hav is haversine; archav its inverse; and Gauss are values from
    his Gaussian table:
    
    =archav{-Gauss[(log(hav(del(GHA))) + log(Mdec) + log(Sdec))
    ~log(hav(del(dec)))]
    + [log(have(del(GHA))) + log(Mdec) + log(Sdec)) ___or___ log(hav(del(dec)))]}.
    
    
    The ___or____ function chooses the lesser of the two values.
    
    So what is the Gaussian, and how do you hook all these guys together
    in standard mathematical notation?  Hopefully, this will help and
    inspire somebody, maybe Bruce, to lay this out!
    
    Thanks,
    
    Fred
    
    
    
    --
    --------------------------------------------------------------------------
    Frederick V. Hebard, PhD                      Email: mailto:Fred{at}acf.org
    Staff Pathologist, Meadowview Research Farms  Web: http://www.acf.org
    American Chestnut Foundation                  Phone: (276) 944-4631
    14005 Glenbrook Ave.                          Fax: (276) 944-0934
    Meadowview, VA 24361
    
    
    

       
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