NavList:
A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
Using logs. wa Re: Haversine formula
From: George Huxtable
Date: 2008 Apr 8, 21:14 +0100
From: George Huxtable
Date: 2008 Apr 8, 21:14 +0100
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. --~--~---------~--~----~------------~-------~--~----~ Navigation List archive: www.fer3.com/arc To post, email NavList@fer3.com To , email NavList-@fer3.com -~----------~----~----~----~------~----~------~--~---