NavList:
A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
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
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
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From: J Cora
Date: 2008 Apr 23, 10:32 0700
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 <george@huxtable.unet.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
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 offlist, 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 secondnature. 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 dyingaway 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.unet.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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