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    Re: Haversine formula
    From: George Huxtable
    Date: 2008 Apr 8, 09:57 +0100

    Alexander Walster asked-
    | 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?
    Response from George.
    Today, versines  and haversines have little purpose. But in their day, the
    period when logarithms were used for all precise calculation, they had
    significant advantages over sines and cosines.
    First, what is a versine? Versine of an angle A is abbreviated as vers A,
    vers A = (1- cos A). It's as simple as that. And a haversine, or hav A, is
    just half that value.
    It's also true, as Henry Halboth has explained, that
    hav A = (sin A/2) squared, which has its uses.
    The main difficulty with calculating by logs is that the log of a negative
    number is meaningless. There are fiddles to get around the problem, as you
    will find used in Chauvenet. Usually, complex rules would be introduced into
    a calculation that used logs, to ensure that all quantities remained
    positive. The trouble in using sines and cosines with logs is that for
    angles less than zero, sines become negative, and for angles greater than
    90, cosines become negative.
    This led to recasting formulae that had used sines or cosines into using
    versines, which can never go negative. Just like a sine, versine of angle 0
    degrees is 0, and for 90 degrees, it's 1, though it's shape differs greatly
    from that of a sine curve between those values. However, above 90, the
    versine keeps on rising, until at 180 degrees, it's exactly 2; then it
    starts to fall again.
    Why was it called a versine? That has puzzled me. I have seen it "explained"
    in terrms of using a shortened form of "reversed sine", which doesn't make
    sense to me, because a versine isn't any sort of reversed sine.
    Anyway the awkwardness of a table of versines was that it varied from 0 up
    to a maximum of 2, whereas sines and cosines never exceeded 1. So you had to
    keep an eye on the digit to the left of the decimal point, which could
    always be omitted in tables of sines and cosines. The next step was simply
    to use halved versines instead, and these became known as haversines. You
    then had tables of hav A and also tables of log hav A.
    When pocket calculators and computers came in, tnen all the complications
    involved in twisting the trig to make it suitable for logs became
    unnecessary, and we could go back to using the simple basic formulae,
    breathing a sigh of relief. There's never any need for versines and
    haversines today, and they are never available as a separate function on
    calculators. But you can always get a haversine from a calculator using (1 -
    cos A) / 2, if you really need to.
    Today, haversines appear only within special tables that use logs, in a
    disguised form, such as in the sight reduction tables of Bennett's
    "Celestial Navigator". In that table, the middle column is actually a
    tabulation of
    -13,030 log hav LHA, and the right column is
    200,000 hav (lat ~ dec). [For completeness, the left column is
    -13,030 log cos (lat or dec)].
    My apologies, if all this is more than Alexander really needed to know.
    contact George Huxtable at george@huxtable.u-net.com
    or at +44 1865 820222 (from UK, 01865 820222)
    or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
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