# 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:**Greg Rudzinski

**Date:**2009 Sep 10, 04:30 -0700

George, I am using the trig log table like a slide rule and keeping track of the decimal point by rounding the cosines to a single digit and multiplying out an approximate product. Greg On Sep 10, 3:15�am, "George Huxtable"wrote: > Andres Ruiz commented, about Greg Rudzinski's index-card method > > For Lat = 34�10'=34.1667� > LOG(COS(RAD(Lat))) = -0.0823 > And table vlue is: > 100000*LOG(10*COS(RAD(Lat))) = 91771.9422 = 91772 > Where LOG is the logarithm to base 10 > > ====================== > > from George- > > There's nothing wrong here, in that a log of a number less than one, such > as -0.0823, is in navigator's convention turned into a positive value by > adding 10.0 to it, (equivalent to multiplying the original cosine by > 10,000,000,000. ), so it becomes 9.9177. Greg omits the integer preceding > the decimal point, just concerning himself with the "mantissa" part, which > becomes .9177, or actually, as Greg quotes an additional digit, .91772. > > ================================ > > But difficulties can arise with the simplifications and short-cuts that Greg > has adopted, in certain circumstances. > > For example, in [9685], Greg wrote- "The result is accurate to four places > even when the the SIN product is calculated by slide rule.". > > Only sometimes (most times, perhaps] will that be the case. Let's assume > that the multiplication can be done to one part in 1000 by slide-rule (is > that fair?). > > The formula used is - > > sin h = cos lat cos dec cos (LHA) + sin lat sin dec. > > In the case Greg showed, that second term, product-of-sines, is much smaller > than the product-of-cosines first term, so finding it to 1 part in 1000 �is > quite adequate for getting the overall sum to 1 part in 10,000. In many > cases where the Sun is observed from low latitudes that will be true, as sin > dec never exceeds 0.4. But if the first term happens to be small, such as it > will if LHA gets near to 90�, (with the Sun near 6am or 6pm, for longitude) > , or with a star of high declination, then the second term will dominate, > and then a slide-rule calculation will be insufficiently precise. > > ======================== > > Also, Greg seems to have ignored the whole-number prefix that precedes the > decimal point in his logs, a procedure which has worked for the example he > chose. For each of the log-cosines, there should be a 9 before the decimal > point, and when those logs are summed, they should come to 29.88205, not > 2.88205 as he indicated. By discarding two 10's, as the navigators-log > convention allows, we get it to 9.88205, and by looking up that number in > inverse logs, the result will be 0.7622, as shown. Greg has discarded his 2. > and in its place has somehow substituted a 9. before looking up that > antilog., which has happened to give the right answer. > > But if his sum-of-cosines had happened to be somewhat less, so the result > summed to 28.xxxxx rather than 29.xxxxx, then he would have had to search in > quite a different part of his antilog tables, the part commencing with > 8.xxxxx rather than with 9.xxxxx. So if he fails to take account of those > "characteristic" numbers, before the decimal point, he will sometimes get > into serious trouble. > > I invite Greg to try a calculation using a LHA of 85�, for which the cosine > is .08716, and the log cosine is either -1.05970 or, in navigator's parlance > (adding 10.0) 8.94030. Unless he takes account of that prefix being 8. > rather than 9., his answers will be very wrong. Those prefix integers need > to be accounted for, not just ignored, as Greg appears to have done. > > If I've misunderstood his procedure, perhaps he will put me right. > > George. > > contact George Huxtable, at �geo...@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 -~----------~----~----~----~------~----~------~--~---