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    Re: Haversine formula
    From: Gary LaPook
    Date: 2008 Apr 08, 09:25 -0700

    Gary LaPook writes:
    Thanks George, I needed that.
    George Huxtable wrote:
    >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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