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    Altitudes, close to 90
    From: Alexandre Eremenko
    Date: 2004 Nov 24, 13:57 -0500

    My previous message, Wed Nov 24 2004 - 13:12:24 EST
    contains an explanation (due to Hadley and Chauvenet)
    why measuring distances
    close to 180 deg is hard.
    
    The general mathematical principle is that
    "distances close to 180 cause the same troubles as
    altitudes close to 90".
    
    Everyone knows that measuring altitudes close to 90 is
    hard, however I have not seen the precise explanation of this
    in the books on navigation that I read.
    
    As I think this is of more relevance to
    practical navigators than my previous message,
    I include a quantitative explanation.
    
    The source of error is that you cannot determine precisely
    the point on the horizon which is exactly "below" the body.
    So, in general, you bring the body in touch with a "somewhat wrong"
    point at the horizon. Suppose that the distance
    bwteeen the correct point and the wrong one is e (measured
    in minutes). Then the formula for your error (in minutes) is
    
    err=(e^2/2)sin(1')cot h,
    
    where h is the altitude.
    
    When h is close to 90 deg, this error becomes large.
    
    Maskelyne (Phil. Trans., May 28 1772) does not give this
    formula, but discusses its practical consequences at length.
    Let me cite just few places.
    
    "Observers are commonly told, that in making the fore
    observation they should move the index to bring the Sun
    down to the part of the horizon directly beneath him, and turn
    the quadrant about upon the axis of vision...
    "I allow that this rule would be true, if a person could by sight
    certainly know the part of the horizon beneath the Sun; but,
    as this is impossible, the precept is incomplete.
    Moreover, in taking the Sun's altitude in or near the zenith, this
    rule entirely fails, and the best observers advise to
    hold the quadrant vertical, and turn one's SELF ABOUT UPON THE HEEL,
    stopping when the Sun glides along the horizon without cutting it:
    and it is certain that this is a good rule in this case, and
    capable with care of answering the intended purpose."
    
    Finally, a very short explanation for those
    "mathematically inclined":
    the distance, as a function on the sphere has one
    singularity (fails to be smooth), at 180 degrees;
    the altitude, as a function on the sphere has two
    singularities (fails to be smooth), at 90 and -90 degrees.
    
    Alex.
    
    
    

       
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