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A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
Altitudes, close to 90
From: Alexandre Eremenko
Date: 2004 Nov 24, 13:57 -0500
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.