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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


--- hch  wrote:

>
> 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

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