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    Re: Azimuth and Declination formulae
    From: Lu Abel
    Date: 2005 Jul 30, 15:15 -0700

    Herbert Prinz wrote:
    > Lu Abel wrote:
    >
    >> 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.
    >
    >
    > Using the transformation (-a)*b = -(a*b), you can use whatever trig
    > formula you want. Besides, the trig functions are normally tabulated in
    > the first quadrant only, leading to the distinction of "cases" rather
    > than negative numbers.
    
    A big "DUH" and head slap over this one, especially since I am old
    enough that my early engineering education included using logarithms for
    calculation and negative numbers never stopped us!   I must have
    disconnected my brain as I repeated what is said with many, many
    explanations for the use of versines and haversines.
    
    >> 1.  Sight reduction formulae (actually, spherical triangle formulae) --
    >> developed by Euclid and pals 2500 years ago.
    >
    >
    > The first spherical triangle formula in Greek mathematics appears around
    > 100AD (Theorem of Menelaos.) The formulae for the right spherical
    > triangle we owe to the Arabs. We know the first explicit statement of
    > the cosine theorem (the basis for the GCD formula) from Regiomontanus.
    > It might surprise you to learn that he used only sines and versines in
    > the formulation. At that time the cosine was rarely used and its name
    > not yet standardized. The versine, on the other hand, is as old as Hindu
    > trigonometry. See also the archives for a message that I posted on this
    > subject on Oct 6, 2004.
    
    Damn, this list is filled with smart and knowledgeable people.  I stand
    corrected on all accounts, was just trying to give a quick impression of
    the problems early navigators faced.
    
    Lu Abel
    
    
    

       
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