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    Re: Still on LOP's
    From: George Huxtable
    Date: 2002 Apr 27, 22:37 +0100

    Clive Sutherland appears to have difficulty accepting the concept of the
    true position, in considering the question of the probability of the true
    position lying within the cocked hat, when he says-
    >we can never
    >find the True position!  Indeed I postulate that the true position is an
    >imaginary, zero dimensional point that can never  be found.
    Well, satellite navigation will give a vessel's position (according to some
    datum, which is a matter it's best not to go into just now) to sufficient
    accuracy such that errors in the known true position can be neglected, in
    this context of measuring bearings to landmarks.
    If that doesn't satisfy him, perhaps he will consider taking a round of
    bearings of landmarks from a well-known position, such as that of Greenwich
    Observatory (the best-known position that I can think of). I admit that the
    fact that the Observatory is based on land rather than at sea is a minor
    complication, but the principle remains.
    Then Clive says-
    >George asks me if �I believe the true position must be inside the cocked hat�,
    >My question to George is �Does he think it must not?�
    I will answer his question, even though he has avoided answering mine. No,
    I don't think the true position must not be inside the cocked hat, of
    course. If enough measurements are made, the true position will be inside
    the cocked hat in roughly 1 in 4 of those measurements, and outside it in 3
    in 4.
    Clive continued-
    >Some time ago  I saw an analysis as described by J.E.D.Williams below in a
    >navigation text book  and I think it was the Admiralty Manual of Navigation but
    >it could have been an RAF manual. I can�t  be sure as I don�t have either to
    I think this was most likely J.E.D.Williams book "From Sails to Satellites"
    that I referred to and quoted from earlier. It's a fascinating work to
    delve into and uncover a wide range of unexpected and original insights
    into navigation.
    I am grateful to Clive for drawing attention to the correspondence about
    the cocked hat in the "Journal of Navigation". (no longer J.Inst of Nav, as
    Clive referred to it). Note that this is the UK-published journal from the
    Royal Institute of Navigation, and not its American namesake.
    Indeed, after his reminder, I found those very volumes on my own
    bookshelves. They make interesting reading, in that recent discussions on
    this list echo, with uncanny closeness, several of the arguments that arose
    a decade ago. They were triggered by a contribution from Williams, which
    corresponds closely with the reference to the same matter in his later
    For anyone that would like to follow the matter up, it can be found in the
    following issues of the Journal of Navigation-
    Vol 44 no 2 (May 1991) pages 269-271
    Vol 44 no 3 (Sept 1991) page 433
    Vol 45 no 1 (Jan 1992) pages 143-147
    Vol 46 no 1 (Jan 1993) pages 132-137
    The Journal should be available in any good library with a maritime flavour.
    Contained in the third of those listed issues is a contribution by
    P.J.D.Gething, which refers to a paper by H.E.Daniels (1951), "The theory
    of position finding", in Journal of the Royal Statistical Society, 13B,
    186. He says " Daniels considered polygons of more than three position
    lines as well as triangles; his general result was that the largest closed
    polygon formed by n lines is a confidence region of probability given by-
    1 - n/(2^(n-1)) " (where ^ means "to the power of")
    I haven't read Daniels' paper.
    George Huxtable.
    George Huxtable, 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    Tel. 01865 820222 or (int.) +44 1865 820222.

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