# NavList:

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

**Re: Haversine formula**

**From:**Henry Halboth

**Date:**2008 Apr 8, 11:47 -0700

Further to my previous reply on the subject which was probably unnecessarily brief, I would add the following ... The Cosine-Haversine formula was advanced during the years in which logarithms were almost exclusively employed in solutions of the astronomical triangle, The basic formula for this solution was and still is the Sine-Cosine .... sin h = sin L x sin d + cos L x cos d x cos t and was originally used in the altitude intercept solution proposed by Marc St. Hillaire. In this formula, if the observed body be on the opposite side of the equator from the observer, the declination is considered as a negative angle. The functions of a negative angle are numerically the same as those of a similar positive angle, but, although the cosine of the negative angle is positive, the sine of the negative angle has a negative sign. Further, if logarithms be employed to perform the multiplications, each logarithmic sum must be changed to a natural number before the addition can be made. It was in order to simplify this work and to eliminate considerations relative to positive/negative quantities that the cosine-haversine formula was developed as ... hav Z = hav (L-d) + cos L x cos d x hav t and is most probably employed in the publication you are using. To use this latter formula, a table of logarithmic cosines, a table of natural haversines, and a table of logarithms are necessary. The time saving trick really was in combining the table for natural and logarithmic haversines, thereby dispensing with the need for the additional table of logarithms - as the logarithmic haversine is the logarithm of the number that is the natural haversine. In such a table, as appears in various editions of Bowditch and others, the logarithmic haversine is tabulated immediately beside the natural haversine, and greatly facilitates hand computation. The haversine function is 1/2 the versed sine, or 1/2 of unity minus the cosine. The cosine of an angle is never greater than unity. The cosine is negative for angles between 90 and 180 degrees, however the versed sine (1-cosine) is still positive. Since the versed sine is always positive, the haversine is always positive. The expression (L-d) is the difference between the observer's Latitude (L) and the declination (d) of the body observed. If L and d be of the same name, an actual subtraction is performed; if not, they are added. Whatever the name of the declination (d), the cosine is positive. Since the hour angle (t)is always positive, even when over six (6) hours, the expression cos L x cos d x hav t is always positive, so that this product is always added to hav (L-d). I have said that I am not calculator literate and I am not, however, it would seem unnecessary to consider haversines as a time saving device in their use and to revert to the basic formulae for the astronomical triangle solution. Regards, Henry --- hchwrote: > > Hi Alex, > > I am not particlarly calculator literate, however, a > realization that hav A = sin2 1/2A might be of help. > Haversines do seem to have fallen into disuse, but > most formulae employing them an be converted to the > use of sines. > > Regards, > > Henry > > > --- alexander.walster@arcor.de wrote: > > > > > Hello All, > > > > > > I have a book called "Practical Navigation for > > Second Mates" and it > > details the procedure for sight reduction using > > haversines. > > > > I am from the scientific calculator age and I was > > wondering if someone > > had a simple explanation on how haversines can be > > used? > > > > > > > > > > > > > > > ____________________________________________________________________________________ > You rock. That's why Blockbuster's offering you one > month of Blockbuster Total Access, No Cost. > http://tc.deals.yahoo.com/tc/blockbuster/text5.com > > > > ____________________________________________________________________________________ You rock. That's why Blockbuster's offering you one month of Blockbuster Total Access, No Cost. http://tc.deals.yahoo.com/tc/blockbuster/text5.com --~--~---------~--~----~------------~-------~--~----~ Navigation List archive: www.fer3.com/arc To post, email NavList@fer3.com To unsubscribe, email NavList-unsubscribe@fer3.com -~----------~----~----~----~------~----~------~--~---