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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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