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    Re: Using logs. wa Re: Haversine formula
    From: J Cora
    Date: 2008 Apr 23, 10:32 -0700

    I was interested in this topic; after searching
    google books found this book


    incase the link doesnt work the search was
    six place logarithmic tables   and the result
    shows webster wells as  publisher

    at page 38 is the start of log tables for sin, cos and tan
    and what was done was to add the number 10 to the
    log10 result so no logarithms are negative.  I havent
    actually tried to use these tables as yet but I am
    guessing that this is another workaround as discussed
    in this thread?

    The fun part for me was writing a small program
    to generate the figures shown in the book

    On Tue, Apr 8, 2008 at 1:14 PM, George Huxtable <george@huxtable.u-net.com> wrote:

    Robert Eno wrote-

    |...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.


    Brilliant! Robert has devised for himself a method of dealing with the
    problem which corresponds exactly with that used by William Chauvenet, in
    "Spherical and Practical Astronomy", nearly 150 years ago. That was what I
    was thinking of when I wrote, in Navlist 4807, "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

    The only difference is that when treating a negative number as a positive
    one, Chauvenet prefixed with a letter "n", to show that the original
    quantity it corresponded to was negative, where Rob marks it with an
    asterisk. This flagging has nothing to do with the sign of the log itself,
    which might be positive or negative. At the end of the log calculation,
    which in itself will always give a positive answer when converted from logs
    to real numbers, that flag is used to show that the final answer has to be
    switched to become a negative one.

    Indeed, Chauvenet takes it a bit further. In a long calculation, which might
    contain several terms to be multiplied or divided (so the logs are
    correspondingly added or subtracted), more than one of those terms might be
    a negative one. And of course, two negatives, mutiplied or divided, make a
    positive. So Chauvenet's rule seems to be that you count how many terms have
    been marked with an n, and only change the sign of the result if there is an
    odd number of such flaggings. He doesn't explain that explicitly, but that's
    how it seems to work.


    Alexande Walster, who asked the original question in [4807], has written a
    nice note to me off-list, which includes this-

    "Some books of mine ... assume some prior knowledge of Logs and Natural
    Logs - mathematical devices I know by name only."

    My schooldays were long before calculators, so we had to use logs and
    antilogs for all our precise calculations, and to us the method became
    almost second-nature. But I can see that generations will have been educated
    since then that were never introduced to logs at all. If anyone is
    interested in old navigation methods, it will be necessary to learn
    something about using logs, which played a vital role in the navigator's

    Alexander mentions Natural Logs, but I can advise him to forget about them.
    These are useful in the context of the growth and decay of natural
    processes, such as radioactivity, biological growth, the dying-away of
    sound. But the logs used as a calculating tool are decimal logs, or logs to
    the base 10, and it's always those, not natural logs, that are being
    referred to in a navigational context.

    I wonder how many list members find, when they look at old texts, that their
    understanding is hampered by such a gap in their knowledge, about logs.
    Perhaps, on this list, we might do something to help. And those that did use
    logs at school will notice that there are distinct differences with the
    usage of navigational logs. That was something that threw me for quite a
    long time until I worked out what was going on.


    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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