# NavList:

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

**Re: Bygrave formula accuracy on 10 inch slide rule**

**From:**Gary LaPook

**Date:**2009 Jul 7, 22:06 -0700

The rms of 1.8 minutes using the Bygrave method is much better than the 4.7' error level you found using the standard sine-cosine formulas and confirms what everyone has written about the accuracy of the Bygrave. As you point out, the longer scale on a Bygrave would result in an even higher level of accuracy. Since you found 1.8 with just a ten inch standard slide rule the Byrave, which has a scale 7.85 times longer than a ten inch rule, would produce much better accuracy and certainly no worse accuracy than the ten inch rule. So the claim of one or two minute accuracy with a Bygrave appears to be validated and is consistent with my own tests on my recreation of the Bygrave. gl On Jul 6, 3:19�pm, Paul Hirosewrote: > My computer simulation of the Bygrave sight reduction formulas, worked > on a 10 inch slide rule, had altitude accuracy of 1.8' and azimuth > accuracy of 2.0'. Those are the square roots of the mean squared errors. > > In a run of 500,000 random sight reduction problems, 95% of the > solutions were correct within plus or minus 3.7' in altitude, and 95% > were within plus or minus 4.2' in azimuth. > > The maximum altitude error seen by the program was 10.4'. Worst cases > always seem to occur around 40� - 50� altitude. Note that the Bygrave > solution reads altitude on the tangent scale, which is most compressed > at 45�. > > The maximum azimuth error seen by the program was 28.0'. Worst cases > occur at high altitudes. > > I was suspicious of the accuracies reported by the test program. They > seemed too good, so I worked six random problems (generated by the > program) by hand on a 10 inch rule. Altitude errors (minutes) were +1.1, > -5.2, 0.0, +1.1, -2.3, -.6. Azimuth errors were -.8, +.1, +.3, -1.3, > +1.1, -1.4. These results suggest the program's modeling of slide rule > errors is realistic. > > To operate the slide rule I wore reading glasses but did not use my > hands free magnifying glass, though it would have helped a good deal. > > My program generates each sight reduction problem from a random azimuth > and altitude, the latter being weighted so the simulated stars tend to > have constant density everywhere in the sky instead of packing closer > with increasing altitude. Altitudes less than 5� or greater than 80� are > rejected. > > A random observer latitude between 0� and 70� is generated in similar > fashion. > > Each azimuth, altitude, and latitude combination is converted to LHA and > declination. The sight reduction module converts these values back to > azimuth and altitude, injecting a random error in each slide rule > operation, then compares results to the correct values. > > Slide rule error is assumed to be .1% RMS per multiplication or division > (which involves two settings and one reading). The Bygrave azimuth > formula requires *three* settings and one reading, so for that > calculation I increase the error accordingly. > > I have modeled the Bygrave formulas on a standard slide rule, but not > the Bygrave rule itself. Its error should be in inverse proportion to > its scale length relative to a 10 inch rule. > > -- > I filter out messages with attachments or HTML. --~--~---------~--~----~------------~-------~--~----~ Navigation List archive: www.fer3.com/arc To post, email NavList@fer3.com To unsubscribe, email NavList-unsubscribe@fer3.com -~----------~----~----~----~------~----~------~--~---