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    Re: Lunars using Bennett
    From: George Huxtable
    Date: 2008 Apr 4, 15:23 +0100

    D. Walden's note about using Bennett's tables to work lunars was
    interesting, but misleading. There would be nothing wrong with the notion,
    in principle, if the tables were much more precise than they are. But.
    
    There were two conclusions for the assiduous reader to pick out.
    
    1. The remarkably close agreement between the corrected lunar distance
    between Moon and Sun centres, calculated from Bennett's tables by Walden as
    46 deg 08', and that calculated more precisely by Frank Reed's online
    calculator to be 46 deg 07.4', a discrepancy between them of only 0.6
    arc-minutes.
    
    That apparent agreement is entirely fortuitous. As Alex has pointed out,
    each pass through a Bennett solution of a navigational triangle involves a
    potential error of the order of 1 arc-minute. In the method Walden
    describes, four such passes are called for, in various directions. What's
    more, each prediction for position of each body involves summing several
    numbers, each presented no more precisely than to the nearest minute.
    Refraction corrections, parallax corrections, and semidiameters are given
    only to the nearest minute. And these potential errors sum up; not
    arithmetically but quadratically, as statistical errors generally do.
    
    D.Walden can reach no conclusions about potential errors in using the
    Bennett tables for that purpose, until he has checked out enough predictions
    to see what the resulting scatter amounts to (say, 10; at least 4, anyway).
    His one-off "bull's eye" signifies no more than I would achieve if I scored
    a bull's eye at my first throw at a dartboard. It would be a lucky accident;
    no more than that. I predict that he will see an overall range of scatter of
    3 to 4 minutes or so. That would render it pretty useless for lunar
    distances, the deduced longitudes covering a range of longitudes getting on
    for 2 degrees..
    
    So come on, D Walden, spend a bit more time with your tables, and let's see
    what scatter you come up with.
    
    By the way, there's an entry "HP = 58", in a line listing Sun data, which
    should presumably apply to the Moon instead. I've no reason to think it's
    been applied incorrectly in practice, but it may confuse readers. I can't
    check out any predictions from the tables, as my own copy expired in 2007.
    
    2. The enormous discrepancy between the predictions, above, and the reported
    lunar distance measurement, between centres of Moon and Sun, of 46 deg 30'.
    This is stated as "Error in lunar - 26' ", but actually seems to correspond
    to a difference of  22 or 21.6 minutes. As a consequence of this, Walden
    writes- "surprisingly, final watch errors may be within 3 minutes or so".
    Not so, however. Lunar distance changes only slowly, at about 30' in each
    hour. So a difference of 21.6, 22, or 26 minutes in lunar distance calls for
    an error in watch timing of best-part of an hour, not "within 3 minutes or
    so"! It would correspond to a position error of some 11 to 13 degrees of
    longitude.
    
    And if that's really the case, it should then be necessary to redo the lunar
    distance calculations with a better estimate for time, because all the
    predicted positions will have been so grossly in error.
    
    But where does this supposed watch error, and this reported lunar distance
    comes from? Walden doesn't tell us whether he is reporting an observation
    that was actually made, or just contrived out of his head. I presume it was
    contrived, solely because of the round-number whole degrees that were used
    for latitude and longitude. And if so, and if the lunar distance and the
    time were no more than invented ones, then does the discrepancy between
    so-called observation and calculation have any significance at all? I think
    not.
    
    Walden may like to know that Bennett has written a paper proposing use of
    his tables for lunars, in exactly the same manner as he has, which appeared
    in "Navigator's Newsletter", issue 79, spring, 2003. It was titled "Lunar
    Distances - A Simple and Concise Solution" (though it was neither simple nor
    concise, occupying more than 5 A4 pages). Unfortunately, it contained many
    errors and also misprints, some (but not all) of which were corrected in the
    next issue.
    
    More seriously, it dressed up discrepancies that appeared, between
    observation and calculation, as due to an erroneous watch, out by 8 minutes
    of time. However, analysis of the detailed results has proved that the watch
    was not at fault. Presumably, then, at least part of that error in the
    result, which amounted to about 4 arc-minutes in the lunar distance, was due
    to using Bennett''s book of tables for a purpose for which they were not
    designed, and for which they were nowhere near sufficiently accurate.
    
    By the way, in that paper Bennett described a way of adapting use of his
    tables to deal with lunar distances greater than 90 degrees, which escaped D
    Walden.
    
    George.
    
    contact George Huxtable at george@huxtable.u-net.com
    or at +44 1865 820222 (from UK, 01865 820222)
    or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    
    
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