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    Re: Haversine- how to derive it?
    From: Scott Owen
    Date: 2015 May 17, 14:11 -0500

    I'd like a copy, but why not just post it to "the list", that way anyone
    who wants it can get it.  Besides as long as the "list" continues online
    so will the table and reduction example.
    On 5/17/2015 9:31 AM, Hanno Ix wrote:
    > Frank,
    > respectfully, in my humble opinion you don't seem to understand the great,
    > unique
    > and practical advantages of the haversine. At least, you don't touch on
    > those.
    > hav(x) is
    > (i) *always positive*,
    > (ii)* no matter the sign of the angles x*
    > None of the other trig function have those fantastic characteristics which
    > greatly reduce the
    > error rate of human calculators.
    > You, as most other CelNav aficionados, may never do longhand math.
    > Therefore, these
    > advantages wouldn't matter to you. Also, for  the deduction of geometrical
    > relationships
    > the haversine is definitively the wrong tool.
    > There are other people, though, who don't, or cannot,  use electronic
    > calculators for
    > CelNav work. The reasons may include emergencies. For those, the haversine
    > is a
    > great blessing.
    > As for a haversine table: I have one that fits on just 2 pages, and I will
    > send a copy to
    > anyone who asks for it. It comes with a very simple sight reduction formula
    > plus example.
    > H
    > On Sat, May 16, 2015 at 4:06 PM, Frank Reed 
    > wrote:
    >> The use of the "haversine" name outside of historical discussions is
    >> mere jargon. There were a few decades in the early twentieth century when
    >> haversine tables were commonly used in navigation, but they are not
    >> important (they were also implicitly used, though not generally named as
    >> such, in many nineteenth century navigation solutions). The haversine of
    >> any angle x can be easily calculated:
    >>    hav(x)= [1 - cos (x)] / 2.
    >> Suppose you have been given an equation that is solved with a haversine
    >> result. In other words, hav(x)=something. You might think you need "hav"
    >> tables to look up the solution or maybe an "inverse hav" key on your
    >> calculator. You might imagine that you would work out the value of that
    >> "something" and then use the inverse haversine function to get x. But it's
    >> not necessary. This equation, hav(x)=something, is exactly equivalent to
    >>    cos(x) = 1 - 2·something.
    >> And obviously you can solve that with the usual cos tables or an "inverse
    >> cos" on a calculator. Historically, multiplying by 2 and subtracting from 1
    >> would have been considered a lot of work in a problem that had to be worked
    >> up over and over again. But those days are gone. Today the haversine is the
    >> trig function for the "navigator who has everything". If you are a "one of
    >> each" collector, then you'll want it for your stamp book. Otherwise, it has
    >> no value. You don't need, but it's certainly nice to know the jargon. And
    >> in the event the GPS ever goes down, you'll be able to jump up and say,
    >> "Haversines Ho!!".
    >> Incidentally, the name of the thing has a simple origin. The "versine" was
    >> historically defined as 1 minus the cosine, from "versus" for flipped, or
    >> more mnemonically, but still a little misleading, versine is the "reversed
    >> sine". But we want that quantity divided by two or "half versine" hence
    >> "haversine".
    >> Frank Reed
    >> View and reply to this message
    > View and reply to this message:
    > http://fer3.com/arc/m2.aspx/Haversine-how-derive-it-HannoIx-may-2015-g31329

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