# NavList:

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

**Re: Azimuth and Declination formulae**

**From:**Henry Halboth

**Date:**2005 Jul 20, 11:55 -0400

Many thanks Lu - you saved me the trouble. There is probably no modern use to the formul;a posted - just, in may opinion, an interesting historical note and perhaps part of one's math education. On Tue, 19 Jul 2005 21:05:42 -0700 Lu Abelwrites: > Whoa on the haversines. It's not half of a sine, it's half of a > versine. > > A versine (x) = 1 - cos (x). Note that vers (x) has a range from 0 > to 2. > > Haversine (x) = vers (x) / 2. This just makes hav (x) have a range > from 0 to 1. > > The whole reason for versines and haversines was to allow sight > reductions to be done using logarithms (and therefore the requisite > multiplications become additions); but logs are not defined for > negative > numbers, hence the need to shift everything to have a positive > value. > > Versines and haversines can also be expressed in terms of sine > squared, > vers (x) = 2 sin^^2 (x/2). > > As a side note, the traditional formula for the great circle > distance > between two points breaks down into finding the difference between > two > nearly equal large quantities for small distances. This can produce > inaccurate answers because calculators and computers only carry out > calculations with a limited number of digits. The equivalent > haversine > formula is well behaved, subtracting two small numbers. Therefore > all > GPS's actually use the haversine formula for calculating the > distance > between two points. > > Lu Abel > > Peter Fogg wrote: > >>From: Henry C. Halboth > >>A bit more complicated, but generally employed with the Time Sight > is ... > >> > >>hav Z = sec ho x sec L x sin 1/2S - ho x sin 1/2S -L, where ... > >> > >>Z = azimith, named according to Latitude + meridian angle, E or W > >>ho = corrected altitude > >>L = Latitude > >>pd = polar distance > >>S = ho + L + pd > > > > > > Interesting. Presumably 'hav' stands for haversine, which I > vaguely recall > > is a half sine? And 'sec' is secant? I don't know what that is. > > > > What do I need to be able to use this formula? Scientific > calculators I > > have. Do I need tables of havesines and secants? > > > > What is the advantage of this formula? > > > > >