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    Re: Summary of Bowditch Table 15
    From: George Huxtable
    Date: 2005 Jan 27, 22:47 +0000

    Jim Thompson is getting closer to the marrow in the bone.
    I had written ...
    >> ...as the height-difference H-h increases as the observer
    >> descends his mast, then the increasing vertical angle corresponds to the
    >> same distance d as before, because that hasn't changed. The distance d is
    >> NOT the distance from the observer's horizon to the object. That was
    >> someone's misunderstanding; Trevor's, perhaps, that has since been
    >> recanted. d is the distance between the observer and the object.
    >Is that:
    >(1) The distance over the curve of the sea, between the observer's ship's
    >hull and the waterline at the base of the lighthouse?
    >(2) The length of the line from the oberver's eye to the top of the
    Response from George-
    I don't know. Does it matter? If you imagined a (floating) measuring tape
    wrapped around the surface of the sea, from hull to lighthouse waterline,
    and unwrapped it to stretch between eye and top of lighthouse instead,
    wouldn't its ends first start moving in a vertical direction? And noting
    the small elevation angles that the table concerns itself with, is there
    going to be an appreciable difference in distance between Jim's cases (1)
    and (2)? I doubt it; but to check it needs a thorough analysis from first
    principles of the basis of table 15 (9), which I haven't made, and don't
    propose to make. Others are welcome!
    >Let me take a stab now at the title of Table 15 (9), which reads, "Distance
    >by Vertical Angle Measured Between Sea Horizon and Top of Object Beyond Sea
    >Horizon".  In plain English with some useful commas placed as a result of
    >your mental hand-holding down this path of lateral thinking, I now believe
    >that the title translates to, "Distance between the observer and an object,
    >determined by measuring the vertical angle between the observer's sea
    >horizon and the top of the object, when the object lies beyond the sea
    >horizon".  Did the penny drop, finally?
    Hmmm. Jim's rewritten title removes one confusion, in that it can no longer
    be interpreted as "Distance ... beyond sea horizon", which I see now is
    what has caused much anguish.
    However, I don't think that final phrase "when the object lies beyond the
    sea horizon", or its shorter equivalent in Bowditch " ... Beyond Sea
    Horizon" is necessary at all, in that (it seems to me) Table 15 (9) could
    be applied just as well to an object that is closer than the distance of
    the sea horizon. It's just that if the object is indeed closer than the
    horizon, then its sea-base can be seen, in which case a different
    expression or Table, that's (arguably) simpler, and doesn't rely on known
    dip, could be used instead. But that doesn't invalidate the use of Table 15
    (9) in those circumstances, does it? Indeed, if you look at the range of
    calculated distances, tabulated in Table 15 (9), it's clear that they go
    down to values of less than a mile: much closer, indeed, than a horizon
    will ever be!
    But where I think the title of table 15 is very misleading is in the phrase
    "Distance ... determined by measuring the vertical angle between the
    observer's sea horizon and the top of the object ..." without mentioning
    that before entering the table that angle has first to be corrected for
    dip. So that it's no longer "the vertical angle between the observer's sea
    horizon and the top of the object", but the angle between the true
    horizontal and the top of the object. Somewhere, on each page of the table,
    that point should have been noted, in rather large letters, within the
    heading or elsewhere. It's omission is inexcusable, and is likely to
    mislead, into significant errors in the result.
    Going back to Jim's earlier posting under threadname "Bowditch Table 15",
    it is satisfying that what we thought was yet another error in Bowditch's
    expression for the basis of Table 15 (9) was no more than Jim's
    mis-transcription. So Bowditch is vindicated, at least to some extent. At
    his third try, Bowditch has got the expression correct!
    contact George Huxtable by email at george@huxtable.u-net.com, by phone at
    01865 820222 (from outside UK, +44 1865 820222), or by mail at 1 Sandy
    Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.

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