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    Re: Haversine formula
    From: Robert Eno
    Date: 2008 Apr 08, 09:02 -0400

    This is an excellent and lucid explanation. Thankyou. I was waiting for you to 
    come through on this topic.
    Quite a few years ago, I decided that I wanted to learn how to reduce sights 
    the "old fashioned" way. I was aware of versines and haversines but had (and 
    still to some extent do not) very little idea as to what exactly they are and 
    how to apply them.
    So I decided to trying something different.
    First I had to teach myself about logarithms. I learned this stuff in high 
    school but I was not a math whiz back then, so a lot of it bounced off of my 
    thick skull. So I had to re-learn it; only this time, with a strong sense of 
    purpose, which made the task easier than it was in high school.
    Next, I ended up spending hours noodling with numbers, making lots of mistakes 
    but eventually, I started to understand the nuances of logs and logs of trig 
    functions and the various cyclic patterns.
    And not knowing anything about haversines and versines, I ended up devising my 
    own method of overcoming the problem of the logarthims of negative sines and 
    cosines which allows me use the basic spherical trigonometric formulas to 
    reduce sights.
    Without getting into the nitty gritty, using the method I devised, one must 
    know which quandrant the sine or cosine is in. If the quandrant makes it a 
    negative number, I treat it as a postive number but place an asterix beside 
    the number so that when the final reckoning comes out -- that is, when you 
    add the logariths and convert the result into XX degrees and minutes of 
    angle, I know that result is a negative number.
    I know. I know. This sounds as clear as mud and I admit that it is somewhat 
    circuitous, but it works for me. Using this method, I can reduce a sight with 
    a small thin book of 5-place trig and log tables, a single sheet of paper and 
    a pencil. Takes me roughly 10 minutes to reduce a sight in this manner. And 
    oddly enough, I really enjoy spending time playing with numbers in this 
    manner. Which also tells me that I should seek professional help as soon as 
    I suppose I should really learn how to apply haversines.
    ----- Original Message -----
    From: George Huxtable 
    Date: Tuesday, April 8, 2008 4:57 am
    Subject: [NavList 4817] Re: Haversine formula
    > 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,
    > and
    > 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.
    > George.
    > 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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