NavList:
A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
Re: Azimuth and Declination formulae
From: Lu Abel
Date: 2005 Jul 30, 15:15 -0700
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