# NavList:

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

**Bygrave sight reduction by slide rule**

**From:**Paul Hirose

**Date:**2017 Dec 13, 14:28 -0800

In February I offered a version of the Bygrave celestial navigation sight reduction formulas. Though valid for all combinations of latitude, declination, LHA, and altitude (including negative altitude), I was not happy with some deviations from standard practice. For instance, my azimuth angle was defined differently from the normal sense of Z. Here are a new set of rules which conform to the traditional conventions of celestial navigation. For convenience on the slide rule, ALL NEGATIVE SIGNS ARE IGNORED. For instance, tan -10° = -.176, but you calculate as if it were +.176. Similarly, arc tangents are always in the range 0 - 90°. It's not necessary to normalize LHA to a certain range. For example, 10° and 350° work equally well. This Bygrave variant is intended for the conventional straight slide rule. It requires that you know how to read and set the full range of possible angles on the trig scales. See my "360 degree slide rule trig" series: http://fer3.com/arc/m2.aspx/360-degree-slide-rule-trig-Hirose-nov-2016-g37154 http://fer3.com/arc/m2.aspx/360-degree-slide-rule-trig-Hirose-nov-2016-g37223 http://fer3.com/arc/m2.aspx/360-degree-slide-rule-trig-Hirose-feb-2017-g38237 FORMULAS AND RULES W = arctan(tan dec / cos LHA) If LHA is between 90 and 270, replace W with its supplement: W = 180 - W If latitude and declination have same name X = 90 - lat + W If contrary name X = 90 - lat - W In either case lat is always positive. If X is not 0 to 180, add or subtract 180 to make it so. If the adjustment is necessary, ALTITUDE IS NEGATIVE. Z = arc tan(tan LHA * cos W / cos X) If EXACTLY ONE of these is true altitude negative X less than 90 replace Z with its supplement: Z = 180 - Z Azimuth angle Z is reckoned in the conventional sense: zero at the north (south) pole if lat is north (south), and increasing east (west) if the body is east (west) of the meridian. Hc = arc tan(cos Z * tan X) Apply negative sign if applicable. There are two places where an arc tangent is (in some cases) replaced with its supplement. But if you ignore signs, tangents of supplementary angles are equal. E.g., tan 10° = tan 170°. Therefore you can read the supplement on the T scale without any calculation. ACCURACY TEST I worked these 10 examples on a 10 inch slide rule. The problems were generated at random by a program which distributes stars uniformly on the celestial sphere, and observer positions uniformly on the Earth (both hemispheres). It excludes altitudes greater than 80 or within 5° of zero, and latitudes greater than 70. 51.7575 lat -5.1860 dec 27.9717 LHA 5.86 W (not so easy to read on some rules) 32.382 X 147.86 Z (error = .113°) 28.25 Hc (error = -.009°) -1.1816 lat 47.0243 dec 213.7008 LHA 127.78 W 141.038 X (alt neg) 152.25 Z (error = -.032) -35.60 Hc (error = -.002) 22.1846 lat 43.1575 dec 333.0564 LHA 46.44 W 114.255 X 40.50 Z (error = .033) 59.37 Hc (error = .005) -47.9567 lat -58.8701 dec 296.4248 LHA 74.96 W 117.003 X 49.00 Z (error = -.012) 52.16 Hc (error = -.038) -11.1965 lat 30.3719 dec 304.7456 LHA 45.80 W 33.004 X 129.82 Z (error = -.058) 22.58 Hc (error = -.035) 40.7939 lat -3.3744 dec 259.7277 LHA 161.70 W 67.506 X (alt neg) 85.82 Z (error = -.005) -10.00 Hc (error = .025) -50.1163 lat -51.0710 dec 97.9351 LHA 96.37 W 136.254 X 47.78 Z (error = .039) 32.74 Hc (error = -.033) 65.9394 lat -17.9946 dec 9.2966 LHA 18.20 W 5.861 X 171.11 Z (error = -.024) 5.79 Hc (error = .021) That one required cosines of small angles. -28.7610 lat -35.6695 dec 233.0997 LHA 129.86 W 11.099 X (alt neg) 41.02 Z (error = -.099) -8.42 Hc (error = .035) -9.1664 lat -54.4142 dec 10.8386 LHA 54.86 W 135.694 X 8.76 Z (error = -.001°) 43.94 Hc (error = -.016°) Root mean squared altitude error = 2.5 arc minutes.