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    Re: Impossible lunar example
    From: George Huxtable
    Date: 2010 Aug 30, 13:57 +0100

    In one of his postings under this thread on 29 Aug, Frank quotes "Nauticus"
    as writing, on 20/12/178-
    "...(examples 3d and 4th, pages 35 and 36!) Two triangles solved, in which
    the observed distances are considerably greater than the respective zenith
    distances of the objects observed. In the first, one side is greater than
    the other two by above three degrees; in the other, nearly six degrees."
    Nauticus' arithmetic seems at fault here. Example 3rd does indeed show an
    impossible excess of 3º, but 4th on page 36 is simply our old friend, with
    an excess of over 13º, not, "nearly six degrees". It appears twice, because
    Maskelyne tackled the same example by two different procedures (getting
    identical answers), the other being on page 31, as Nauticus discovered over
    his Christmas.
    And Maskelyne has simply accepted those errors in his later "errata"
    statement, as indeed he had to do. His statement that "No person can be
    puzzled or misled" is somewhat implausible, in spite of Frank's
    capital-letter endorsement. Any discerning student learns how useful it can
    be to sketch out such problems, at least in his mind, as an aid to
    understanding. If he tried that with this one he would come to a full stop.
    Treated as a rote-learning procedure, as much navigation unfortunately was,
    in those days, it can be worked through just the same, whether or not it's
    a real triangle, and will give some sort of answer. But in this case, that
    answer is devoid of any physical meaning, as was the original question. Of
    all subjects, celestial navigation is rooted in the real world, and not
    some mathematical abstraction into never-never land.
    In another posting on the same day, Frank addressed me as follows-
    "... you're wrong. REAL observations might not yield closed triangles and
    yet they can still yield excellent navigational data. I remember a few
    years back when Alex Eremenko and I first pointed this out to you. You were
    shocked then, but you seemed to accept it (then). Have you regressed
    Frank twists my words from 2004. I do not recall agreeing that such
    observations "can still yield excellent observation data". What he has
    recently quoted is the following passage-
    "What has surprised me (and intrigued Frank) is that the above expression
    continues to give a value for D in circumstances that are QUITE IMPOSSIBLE,
    in that the lunar distance is such that there's no value of azimuth
    (between 0 and 180) between Sun and Moon that can accomodate such a lunar
    distance. In those circumstances, although any attempt to deduce that
    azimuth would fail, the expression for D still seems to work, and gives
    some sort of result.
    When the numbers input to the equation correspond to azimuths in range 0 to
    180, then the result D has a simple physical meaning, the true lunar
    distance D. In other situations, is there any physical meaning we can
    attach to D?
    I find it interesting that although Frank and I are intrigued about this
    matter, our resident mathematician, Alex, takes it in his stride, as only
    to be expected. I have a lot to learn, it seems."
    I have indeed agreed that in those circumstances the clearance procedure
    yields "some sort of result" for lunar distance: which it does.
    Furthermore, the correction seems to show no discontinuity when crossing
    the boundary, at azimuth = 180º, between the real world and fairyland. That
    does not give the result any physical reality, however.
    Responding to Antoine, Frank wrote, discussing Maskelyne's example 3, with 
    a 3-degree excess-
    With the Moon 88 degrees high, the Sun 5 degrees high, and the measured
    lunar at 90 degrees, we have another case where the triangle doesn't close
    but in this case the difference is not as large and it is COMPLETELY
    ACCEPTABLE as a real observation. ... try clearing this latter example and
    then try clearing it again with a different value for the Moon's altitude
    (lowered by a few degrees). You will discover that there is no error at all
    despite the seemingly "impossible" geometry.
    Let's imagine Frank as navigator of an East Indiaman in the late 18th 
    century. An apprentice brings him the observations of a lunar, 
    corresponding to Maskelyne's example 3. Frank corrects for index, 
    semidiameters, and dip, then notices that the two zenith distances add up 
    to less than the lunar distance by 3 degrees; an impossible state of 
    affairs. It's quite clear that something is seriously wrong, by 3 degrees 
    at least. Never mind, he says, we can just clear the lunar anyway, it's 
    quite insensitive to Moon altitude.
    But how could he tell that the discrepancy was in the Moon altitude? It 
    could have been an error in the star altitude, or in the lunar distance, 
    that gave rise to the impossible lunar?
    Given an impossible state of affairs, the only course is to reject the 
    whole set and send the apprentice out for another try. That's the lesson 
    any mariner should draw from Maskelyne's impossible examples.
    contact George Huxtable, at  george@hux.me.uk
    or at +44 1865 820222 (from UK, 01865 820222)
    or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.

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