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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

   

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