# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**Re: Index Card for Trig/Log Table**

**From:**Gary LaPook

**Date:**2009 Sep 11, 00:20 -0700

And we found that using the Bygrave formulas with a ten inch slide rule provides greater accuracy than when using the sine-cosine formulas. See: http://www.fer3.com/arc/m2.aspx?i=109019&y=200907 gl George Huxtable wrote: > A few comments about Greg's posting [9697]. > > He wrote- "I choose to drop the characteristics on the index card then keep > track of the decimal by approximation." > > Fair enough. That's a perfectly valid way to do the job, though it adds > quite a lot of complication; which he didn't mention when introducing us to > his procedure. That explains why he had to look up, and note down, the > cosines of the three angles, as well as the log-cosines, which would > otherwise have been sufficient. Then he can multiply them together, to get a > rough idea of the product. Simply estimating the product in-his-head should > suffice. That then allows him to put the decimal point in the right place > when the antilog gives the precise product-of-cosines. > > But isn't that a bit long-winded and unnecessary, when it can be bypassed, > simply by working the log calculations to include the integer before the > decimal place? > > That calls for writing down fully the log-cosines of the three angles, as > 9.91772, 9.99804, and 8.94030, as he has done below, summing them to give > 28.85606, as he has done below. Then discarding two tens from the result, to > arrive at 8.85606 (not 2.85060, as he has shown). Then the next step is to > use inverse logs, of 8.85606, to arrive at 0.07179, automatically. It's just > what such logs were invented for. > > > For LHA 85� : > appx(.07) appx(.05) > Log Cos Log Cos Sin (by slide rule) > Lat 34� 10' 9.91772 91772 .8274 .5616 > Dec 5� 26' 9.99804 99804 .9955 .0947 > MA 85� 00' 8.94030 94052 .0872 +.0532 > 28.85606 2.85628 ............. .0718 > Inv Log .07179 .07183 .1250 Inv Sin > Hc 7� 11' > =============================== > > Then the next step is to add to that the product of sines of lat and dec. In > this case, it can be seen that the two terms in this sum are of similar > magnitude, so a 1 in 1000 error (if that's accepted as reasonable, from a > slide rule) ends up as 1 in 2500 error in the sum, which would create an > overall error of well over a minute in the result, from that cause alone. > It's why slide rules are not really acceptable for such calculations. > Instead of multiplying those sines by a side rule, log sines could be > extracted instead, added, then the result found by an inverse log (antilog) > table. Then that has to be summed with the product-of-cosines, found > previously, and then the inverse sine obtained as before. > > Those log sin calculations could be completed in the time saved by not > having to work, and multiply, those cosines. And the result would not have > lost any of its precision. What I'm trying to point out is that in this case > the old ways were the best, and Greg hasn't saved time, but has lost > precision, by doing it differently. > > Of course, better ways of working that calculation, by logs, were developed > to avoid that awkward addition. > > George. > > contact George Huxtable, at george{at}hux.me.uk > or at +44 1865 820222 (from UK, 01865 820222) > or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK. > > > > > > > --~--~---------~--~----~------------~-------~--~----~ NavList message boards: www.fer3.com/arc Or post by email to: NavList@fer3.com To unsubscribe, email NavList-unsubscribe@fer3.com -~----------~----~----~----~------~----~------~--~---