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    Latitudes through double altitudes revisited (Bowditch & Benj. Peirce)
    From: Joel Silverberg
    Date: 2007 Apr 18, 14:07 -0700

    
    I posted a notice on NavList earlier asking for assistance in
    deciphering the mathemtics behind the method for determining the
    latitude from measuring the altitude of the sun at two different times
    and noting the elapsed time behind the observations.
    
    Frank Reed was kind enough to direct me to a mid-nineteenth century
    treatise on spherical trigonometry written by Benjamin Peirce:  a
    friend, student and colleague of Nathaniel Bowditch (and Henry
    Ingersoll Bowditch, Nathaniel's son, according the the history of
    mathematics archives at the University of  St Andrews, Scotland).
    Thank you, Frank.   Peirce's treatise is a most remarkable resource,
    and I am looking forward to reading the work in its entirety over the
    summer.
    
    I have tried to organize, clarify,  illustrate, and describe Peirce's
    explanation of the derivation of Bowditch's method  and have provided
    a link to my notes below.
    
    I would also like to thank George Huxtable for providing a scanned
    copy of Cotter's explanation of Ivory's method.  The methods are
    virtually identical, but I found Cotter's diagrams and Cotter's
    notation to be extremely difficult to follow.  For me, at least, they
    clouded and obscured what was going on, rather than explained what was
    happening. I have not had the time to see if Ivory's original
    presentation to the Royal Society is on the web, but I'd be interested
    to see how Ivory explained things.
    
    I would also like to acknowledge the usefullness of the program
    "Spherical Easel" by David Austin and William Dickinson for preparing
    the diagrams that accompany this explanation.  This program is
    available as a java applet or as a downloadable program at no cost
    from http://merganser.math.gvsu.edu/easel/
    
    
    
    To see Bowditch's original Rules click on the links
    
         http://www.fer3.com/arc/img/bowditchrule.part1.jpg
         http://www.fer3.com/arc/img/bowditchrule.part2.jpg
    
    
    To see my illustrated explanation of Benjamin Peirce's Derivations of
    Bowditch's Rules click on
    
         http://www.fer3.com/arc/img/benjpeircederivations.pdf
    
    Sources:
    
    1825 edition of Nathaniel Bowditch's New American Practical Navigator
    (pages 247 through 249).
    
    1845 edition of Benjamin Peirce's Elementary Treatise on Plane and
    Spherical Trigonometry (p. 133,134)
    
    both accessible  via books.google.com
    
    Have fun,
       Joel
    
    
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