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    Re: H.R. Mills on setting/rising bodies
    From: Christian Scheele
    Date: 2010 Jan 25, 21:05 +0200

    Hello Robert,
    thank you very much for your quick reply and your elegant solution. I didn't 
    go "Duh", if only for the reason that I don't think I would have made your 
    sharp observation.
    Here's the rest of Mills deriviation, should the aforegoing have made you 
    curious, that is, assuming you are not already familiar with it - admittedly 
    a meagre reward for your effort, but I don't know what else I can offer right 
    Departing from where we left of at equation(4), quoting Mills:
    "By the sine relationship:" (Mills is referring to the sine rule relating to pherical triangles)
    "sin(HA)/cos(alt) = sin(Az)/cos(dec)
    sin x = {cos(dec).sin(Az)}/cos(dec)  (5)
    From...we know that
    cos(Az) = sin(dec)/cos(lat)  when the (alt) = O  (6)"
    [Mills is rearranging cos(Az) = {sin(dec) - sin(lat).sin(alt)}/cos(lat).cos(alt)]
    "so we can eliminate Az from (5) and (6) and so obtain an expression for x in terms of lat and dec.
    ..... by squaring and adding (5) and (6) we have
    sin^2(Az) + cos^2(Az) = 1"    ( i.e. Mills applies trig. identities)
    "and therefore,
    {sin^2(x).cos^2(dec)}/cos^2(lat) + sin^2(dec)/cos^2(lat) =  1
       cos^2(lat) = sin^2(x).cos^2(dec) + sin^2(dec)
         sin^2(x) = {cos^2(lat) - sin^2(dec)}/cos^2(dec)
                  = [cos^2(lat) - {1 - cos^2(dec)}]/ cos^2(dec)
               = {cos^2(dec) - 1 + 1}/ cos^2(dec)"
    The above line appears to contain another mistype: I'd say it should read
               = [{cos^2(dec) - 1}/ cos^2(dec)] + 1
    Mills proceeds:
    1 - sin^2(x) = cos^2(lat)/ cos^2(dec)"
    In my opinion there is another mistype in this line. I am referring to 
    "cos^2(lat)". It can't possibly have been put in deliberately. I'd say that 
    the term should in fact be (1 - cos^2(lat) so that the entire line of the 
    equation reads:
    1 - sin^2(x) = {1 - cos^2(lat)}/ cos^2(dec)
    Mills continues:
    "cos x = sin(lat)/ cos(dec)"
    I agree (most humbly).
    Mills then closes the section with this observation:
    "so the angle of rising or setting of a body depends only on lat and dec...."
    Kind regards and many thanks once again,
    Christian Scheele
    ----- Original Message -----
    From: berneckywr-mail@yahoo.com
    To: NavList@fer3.com
    Sent: Monday, January 25, 2010 2:09:24 AM GMT +02:00 Harare / Pretoria
    Subject: [NavList] Re: H.R. Mills on setting/rising bodies
    Hi Christian,
    You are right, Mills did drop a minus sign in equation (3).
    Getting to step 4 is not hard (I'm thinking you are going to think, "Duh"). 
    But it's just not easy to always be clear when writing about math.
    Here's one way to see what he did:
    cos(alt).d(alt)= cos(lat).cos(dec).sin(HA).d(HA) (3)
    Divide both sides of the equation by d(HA).cos(dec):
    cos(alt).d(alt)/(d(HA).cos(dec))= cos(lat).sin(HA)
    so that we can replace d(alt)/(d(HA).cos(dec)) on the left-hand side
    with sin(x) (from equation 2). This gives us
    cos(alt).sin(x)= cos(lat).sin(HA)
    or, solving for sin(x):
    sin(x)= cos(lat).sin(HA)/cos(alt)
    and, by equation 2,
    sin(x)= d(alt)/(d(HA).cos(dec))
    so we have two different expressions for sin(x), and that
    is what Mills is indicating with (4).
    Robert Bernecky
    Mystic CT
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