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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.

> 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.

Herbert Prinz

   

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