# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

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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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