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    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
    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
    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.
    --- 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.
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