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    Re: Using logs. wa Re: Haversine formula
    From: Henry Halboth
    Date: 2008 Apr 23, 17:06 -0700

    Hi,
    
    I came across the referenced publication some time ago
    while searching for six (6) place logarithmic tables.
    
    Although they can certainly be derived, the
    significant problem with this book is the absence of
    tabulated logarithmic Secants and Cosecants. These
    values were provided to six (6) places in many of the
    older publications and were necessary as a matter of
    convenience in navigational calculations, particularly
    Lunars.
    
    I, for years, used a copy of Norie's Tables, 1904
    Edition, which included all necessary tables to six
    (6) places, and for which I have not been able to find
    a suitable replacement. Before the advent of
    calculators it was the "way to go".
    
    Regards,
    
    Henry
    
    
    --- coralline algae  wrote:
    
    > I was interested in this topic; after searching
    > google books found this book
    >
    >
    
    http://books.google.com/books?hl=en&id=NmkPAAAAYAAJ&dq=six+place+logarithmic+tables+++&printsec=frontcover&source=web&ots=Fk9RFTEZeL&sig=wnJErSklEh3_7DJObTwi2aWTzTY
    >
    > 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
    > 
    > 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
    > > Chauvenet."
    > >
    > > 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
    > > craft.
    > >
    > > 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.
    > >
    > > 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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