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

**Re: Bygrave sight reduction by slide rule**

**From:**Paul Hirose

**Date:**2018 Jan 2, 14:44 -0800

On 2017-12-13 14:38, I wrote: > W = arctan(tan dec / cos LHA) > > If LHA is between 90 and 270, replace W with its supplement: W = 180 - W If LHA = 90 its cosine is zero and therefore the equation is undefined. In that case, alter LHA by an insignificant amount, say .001°. For example, lat = 50, dec = 40, LHA = 90. After you change LHA to 89.999, W = 89.99881 > If latitude and declination have same name > X = 90 - lat + W X = 130 to practical accuracy > Z = arc tan(tan LHA * cos W / cos X) Z = arc tan(tan 89.999 * cos 89.99881 / cos 130) At first that looks impossible on a slide rule, but tan 89.999 = 1 / tan .001, and cos 89.99881 = sin .00119. Thus the part in parentheses is equivalent to (sin .00119 / tan .001 / cos 130). The first quotient is easily and accurately computed on ST: 1) align indices on slide and body, 2) set cursor to 1.19 on ST, 3) set 1 on ST to cursor. The quotient is on D at the C index. Its characteristic is 0 (i.e., the value is in the range 0 - 10) if the angles are literally as set. The actual angles are both a thousand times less, so there's no net effect on the characteristic. The quotient is 1.19. Finish the equation: 4) set cursor to C index, 5) set 130 (alias black 40) on S to cursor, 6) set cursor to D index. Z = 61.65 at the cursor on T > Hc = arc tan(cos Z * tan X) Hc = 29.50 The errors in Z and Hc are respectively -.6 and .08 minutes, with respect to a solution computed for LHA exactly 90.