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    Re: Azimuth and Declination formulae
    From: Lu Abel
    Date: 2005 Jul 20, 07:35 -0700

    Peter Fogg wrote:
    >>From Lu Abel
    >>
    >>...The whole reason for versines and haversines was to allow sight
    >>reductions to be done using logarithms (and therefore the requisite
    >>multiplications become additions); but logs are not defined for negative
    >>numbers, hence the need to shift everything to have a positive value.
    >>
    >
    >
    > So in order to use haversines log tables are needed?
    
    The opposite:  In order to use logs (optional, but it makes multiplying
    4 or 5 digit numbers a hell of a lot easier!), you need to use haversines.
    
    I'm not the expert on the history of navigation like some of our other
    list members, but in historical order:
    
    1.  Sight reduction formulae (actually, spherical triangle formulae) --
    developed by Euclid and pals 2500 years ago.
    
    2.  Nautical Almanacs.  Prince Henry's institute in Portugal produced
    the first tables for the sun's declination in the mid-1400's; it
    wouldn't surprise me if Arab astronomers had produced much more data
    much earlier.  With the explosion of interest in astronomy over the next
    couple of centuries, it also wouldn't surprise me if almanacs much like
    today's existed by the end of the 17th century.
    
    3.  (Long pause -- like 3 century's worth)  Sight reduction tables.
    
    Bottom line:  for many of the great explorations of the 17th and 18th
    (and perhaps even 19th) centuries (and for simple ship-borne commerce,
    too!), navigators could take sights easily.  But sight reduction was
    difficult.  Without reduction tables (like our HO229 and its
    predecessors), navigators had to do the equivalent of what many of us do
    today -- run through the sight reduction formulae -- but without the
    benefit of a calculator.
    
    Multiplying multi-digit numbers ain't fun and, more important, is error
    prone.  Being able to ADD the logs of those two numbers is a lot quicker
    and simpler.  Hence haversines.
    
    Lu Abel
    
    
    

       
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