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    Re: Graphs of Lunar Distances.
    From: George Huxtable
    Date: 2010 Sep 23, 12:32 +0100

    Douglas and I have had  some interesting and productive off-list
    discussion, about lunars and other matters, and he sent me his latest
    results before they were posted to Navlist, but I've been unable to comment
    back, until now.
    There are several things to say about those observations; a "text-book"
    example, from which lessons can be drawn.
    First, they are remarkably good, especially for an observer who is
    relatively new to the game of lunars. The scatter about the fit-line in his
    graphs would show a standard deviation of no more than 0.2' arc-minutes or
    so, at a guess (without actually calculating) from his graphs, after
    excluding that rogue point. I doubt if even the most seasoned observer
    would hope to produce anything better than that.
    Douglas' resulting deduction of GMT, to within a minute of time, is as good
    as anyone would expect from a lunar. Taken with a time-sight, that would
    give a longitude within 15 arc-minutes, which would content any mariner
    from the18th-century, before the days of chronometers, when the only
    alternative was dead-reckoning across an ocean. Those are observations that
    Douglas should be proud of, which would qualify him to sail with Cook, or
    for a post with the East India Company. The famous Longitude Prize asked
    for 30 arc-minutes overall precision, some of the expected error arising
    from deficiencies in the Moon position predictions of that era.
    Douglas is keen on plotting the observed lunar distance against time, and I
    agree with him about its usefulness in showing up anything going wrong,
    such as the point he has taken to be some sort of blunder, and excluded.
    Such a decision needs taking with some care, though. I think, in his
    position, I would have excluded that point too. It may have resulted of a
    misreading of the drum, by 1'. But it's a bit marginal; being less than 1'
    out-of-line with the others. Anyway, because it's only one point out of 8,
    the overall difference in the mean, between excluding and including it,
    would only be 0.1 arc-minutes, or about 12 seconds in the GMT; neither here
    nor there.
    How have the slopes of the plotted lines been chosen? If that's been simply
    done "by eye", to provide a fit that looks about right, the overall result
    can not be stitistically better than by simply averaging angles and times
    (after any exclusion has taken place). Lunar distance changes with time in
    a way that's very close to a straight line, over the usual duration of an
    observation, so simple averaging works well.  However, estimating the
    approximate mid-time of the observation, the appropriate slope can be
    pre-calculated, having taken account of all the clearing corrections in
    reverse. Then, having plotted that known slope-line, it can be adjusted up
    and down, strictly parallel with itself, to get the best fit to the points.
    This has the advantage that there's only one parameter to fit, not two, and
    the result should benefit from improved precision. In that way, only, can a
    graphical procedure improve on simple averaging.
    Squeezing all we can from those observations, it seems that two bodies, on
    opposite sides of the Moon, gave rise to opposite (but unequal) time
    errors, the interval between them being somewhat more than statistical
    scatter would lead us to expect. So it looks as if a bit of systematic
    error may be present. That's very likely due to the estimation of position
    of the Moon's limb with respect to the centre of the body, affected as it
    may well be by irradiation, due to brightness differences. This is where
    the observer's skill, which comes from experience, matters. And it's where
    modern observers have such an advantage, in being able to compare their
    observations with a known time-and-longitude. In the 18th century, that
    wasn't the case. Even when a known headland was sighted or harbour entered,
    its charted position, sometimes in latitude but frequently in longitude,
    would be imprecisely known and charted. Out at sea, there was nothing to
    judge the quality of their lunars by.
    I don't accept Douglas' conclusions about the impracticality of the lunar
    distance method at sea. Thousands of mariners, without chronometers, relied
    on it for nearly a century, having nothing better. Certainly, it was much
    more of a challenge than he faced in has garden, compared with the motion
    of a vessel and the sky obscured by sails.
    contact George Huxtable, at  george{at}hux.me.uk
    or at +44 1865 820222 (from UK, 01865 820222)
    or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    ----- Original Message -----
    From: "Douglas Denny" 
    Sent: Thursday, September 23, 2010 12:16 AM
    Subject: [NavList] Graphs of Lunar Distances.
    I have been in correspondence with George Huxtable about Lunars and their
    accuracy. I thought some input here might be of interest for the purposes
    of discussion; if it is of interest.
    I have spent some time for my amusement programming my HP50G calculator for
    almanac data and then moved on to Lunar distance clearing so naturally that
    stimulated my getting the sextant out to try it out again.  I played around
    with 'Lunars' many years ago and was not impressed then, getting poor
    results so was interested to see it better could be achieved being a little
    older and wiser :-) !
    I relatively quickly obtained Moon/Jupiter shot one evening some weeks ago,
    which gave a time result which was three minutes 'out' from true.  Last
    Monday I tried again with more care, and found the Moon/Jupiter combination
    was within six seconds of true, but the Moon/Vega combination was 51
    seconds 'out'.
    The main purpose of this short posting is to mention and promote the use of
    simple graphs of the observations for obtaining better accuracy than would
    otherwise be the case with simple averaging for the mean in a set of
    observations.  I know that statistical calculations will give better
    results too (at some considerably more trouble with calculations), but as I
    have always tried to promote the practical side of navigation here, it
    might be appropriate to mention it again with a recent example I have used
    Part of my text to George (to save my writing it again):-
    I took Moon/Jupiter and Moon/Vega sights.  Jupiter was 28 degrees to the
    left of the Moon, same level;  and Vega was 67 to the right and higher.
    I took several shots over some minutes each.   I dispensed with the
    shots by bubble sextant as I decided I am interested only in the accuracy
    considerations of the lunar distance itself,  and decided to use calculated
    (almanac) altitude values for the 'clearing' using my calculator programme
    (based on the rigorous formula mentioned in Cotter's book 'History of
    Nautical Astronomy').
    I then made graphs of the time / lunar distance observations.
    Graphing observations. I think, gives better accuracy than by just
    averaging shots. The change in lunar distance over say 15 or 20 minutes is
    indeed quite enough to be 'graphable'.
    Graphs are very powerful,  as they allow instantly to see the trend with
    eye and have an enormous advantage in being able to disallow any obvious
    rogue shots immediately. I think it is probably a better method than a
    'proper' statistical calculation (eg. least squares) because you can
    eliminate shots which would otherwise be included;   it is also simple, can
    be done quickly, possibly faster than a calculation procedure (unless
    programmed), and is a kind of 'analogue', visual, integration method for
    a few points (observations) -obtaining much better accuracy than any other
    method.  You will note in the graph of Vega/Moon this is shown quite well
    with one shot being 'way out' which is immediately discarded.
    The best result was the Moon/Jupiter combination giving a six second error
    only.  The Moon/Vega set was 51 seconds 'out' from absolute.
    This was probably because Jupiter is very bright, the LDist was smaller and
    the sextant easier to handle and adjust for co-incidence of object/Moon's
    This is an important note for practical lunar distances in my opinion,  as
    it is quite clear to me that stars are difficult to use unless they really
    are very bright. I had to use shades for the Moon for both Jupiter and
    Vega, but especially so for Vega, it being much less bright than Jupiter.
    My original negative conclusion about 'lunars' was confirmed with my recent
    attempt, where I obtained time within only three minutes for a single shot
    is therefore still valid in my mind;  i.e. 'lunars' are very difficult
    indeed to get anything like real accuracy for practical use of obtaining
    time down to seconds.
    The Moon only moves 30 minutes of arc in one hour against the background of
    stars;  i.e.  one minute of arc in two minutes of time.  That implies a
    necessary measurement accuracy of one second of arc measurement for a
    theoretical three seconds of time accuracy in the final result... and that
    ignoring all the errors and inaccuracies within the method... A very tall
    order indeed when at sea ! Impossible in fact.
    I cannot understand (and still do not believe) how it was possible at all
    at sea to use it as a practical method,  and still think it was most likely
    force-fed to mariners by the astronomers in the eighteenth century for the
    simple reason there was no other practical method at all available until
    Harrison. Also, because the astronomers were used to dry land and
    terra-firma (the more firma - the less terror), and they were used to using
    instruments which they knew were good for seconds of arc accuracy at any
    For today's use Lunars are great fun on dry land;  but I remain deeply
    sceptical about their use in any way as a practical method of navigation
    expertise when at sea.
    Douglas Denny.
    Chichester. England.
    I do hope there are no terrible typos in this as it is a bit long. Sorry if
    there are.
    Lunars. Observations:-
    Observer's position:
    Lat:  North 50deg-49.910
    Long: West 000deg-51.300
    Monday 20th IX 2010. All times GMT. (UTC)
    21Hr-50'-10" ....... 28deg-33'
         53-55                 30.5
         57-28                 31
         59-44                 29.8
    22Hr-01'-46"               28.7
         03-39                 27.9
         04-52                 27.3
         07-29                 27.1
    Later (not used):-
    22Hr-44'-36"         28deg-14.5
         46-28                 13.5
         48-10                 12.8
         49-16                 12.2
         50-50                 12.1
    22Hr-25'-15"         67deg-13'.0
         28-05                 13.5
         31-29                 13.7
         33-01                 14.5
         35-05                 13.1
         36-24                 13.8
         37-48                 14.1
         41-15                 14.5
    I used altitudes obtained from almanac data for the exact single time/lunar
    distance chosen from the graphs, for use in clearing the distance.
    I used Moon/Jupiter from the graph at 22Hr-01'-00" with observed LD to limb
    of 28deg-29'
    this, minus 14.7 Semi-Diam giving 28-14.3 observed LD. A calculated HP was
    The 'cleared' LD was:-  28deg-08.'756  giving an interpolated time using
    the Oliv Soft Lunar distance almanac of 22Hr-01'-6.4"
    i.e.  an error of 6.4"
    for the Moon/Vega combination:_
    I used 22Hr-40'-00" from the graph with LD Vega to limb of 67deg-14.4
    plus SD this time gives observed LD of 67deg-29.'1
    The 'cleared' LD was 66deg-58.'97
    which interpolated for time gives 22Hr-39'-8.1"
    i.e. an error of 51.9"

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