# NavList:

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

**H.R. Mills on setting/rising bodies**

**From:**Christian Scheele

**Date:**2010 Jan 24, 23:07 +0200

I am busy working through H.R. Mills' "Positional Astronomy and Astro-Navigation Made Easy" (First edition 1978). I can highly recommend this book which certainly lives up to its title. Is anybody else familiar with it? There is section called "Angles of stars as they rise and set, with respect to the horizon". (Obviously, a mostly theoretical topic as "real navigators" might be quick to point out to me). I find one step in Mill's representation of a deriviation of a general formula for these angles confusing. Perhaps I have missed something. Mills presents his model, illustrated with an opening diagram. It shows three circles, each representing the same celestial body at different altitudes, against a horizon line. Placed one after the other in a straight line path that cuts the horizon at an acute angle, they touch each other tangententially so that the lower limb of the highest circle and the upper limb of the lowest circle graze the horizon, while the remaining circle in between them transits the horizon. The diagram is completed by verticals, horizontals and parallels relative to the straight line track of the three circles and are marked d(alt), d(Az), and d(HA).(cos{dec}) respectively. The angle x between the horizontal line and the parallel relative to the straight line track of the three circles, i.e. the angle between d(alt) and d(HA).(cos{dec}) represents the angle of rising and setting of the body signified by the three cirlces. Departing from this model, Mills derives a formula for this angle x so that cos x = sin (lat)/ cos (dec). I won't go into all the steps. Mills begins his exposition of the deriviation by stating two well-known formulas he uses as premises: sin(alt) = sin(lat).sin(dec) + cos(lat).cos(dec).cos(HA) (1) sin x = d(alt)/(d{HA}.cos{dec}) (2) Mills then differentiates (1) to yield: cos(alt).d(alt)= cos(lat).cos(dec).sin(HA).d(HA) (3) So far so good. But isn't d(cos{HA})/d(HA) = -sin(HA) and not sin(HA)? (Although it might not be important if the term is squared in a later step). Or is this a mistype? From (2) and (3) Mills yields: sin x = d(alt)/(d{HA}.cos{dec}) = (cos{lat}.sin{HA})/cos(alt) (4) How did Mills get to (4)? How did he arrive at d(HA) = cos(alt) and d(alt) = cos(lat).cos(dec).sin(HA)? Did I miss a basic rule governing differential calculus? Please get back to me if you find this interesting. I haven't scoured the internet yet for an enlightening review. Christian Scheele Cape Town