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    Re: Problem with a sextant
    From: George Huxtable
    Date: 2006 Apr 25, 20:55 +0100

    Alex quoted Wales, Cook's astronomer on his second voyage, as follows-
    | "It must be owned there is yet something
    | in the constitution of this Quadrant very disagreable,
    | and not easily to be accounted for. Sometimes, many months
    | together, the longitudes deduced from observations made about the same
    | time with my two sextants would not differ more than 10
    | or 15 miles, and very seldom so much; after which
    | the longitudes, so deduced, would begin to differ, and the difference
    | would gradually increase, sometimes more than a degree and an half:
    | In little time it would again decrease, and soon after the observations
    | would agree as well as ever.
    | It will be readily supposed, that no means were left untried by me
    | to discover the ause of this strange aberration; but all
    | my endeavours were ineffectual;
    | and I mention the circumstance to induce some person,
    | more skilful in mechanics, to attempt it".
    It seems like he had a serious problem. A degree and a half of longitude 
    corresponds to about 3 arc-minutes of divergence between
    the two instruments. That's a lot.
    A possible explanation might be collimation error. If the telescope of one 
    sextant was badly aligned, to be out of the plane of the
    instrument, that would give rise to an error which became greater at greater 
    lunar distances. And the sign of the resulting
    longitude error would switch, between a waxing moon and a waning one. In that 
    way one might expect a divergence between longitudes
    taken by the two instruments, which varied systematically over the month. 
    Could that be what's behind Wales' observations?
    How could he be sure which of the two was in error? Presumably he could take a 
    long series of observations from a static known
    location. A likely place would be at Ship Cove in Queen Charlotte Sound, at 
    the North end of New Zealand's South Island, to which
    Cook would return again and again. And then one might expect that a good 
    instrument would give a constant longitude when lunars were
    taken, and a bad instrument would give a varying answer.
    But it wasn't quite that simple. The Almanac itself could be significantly in 
    error, by rather more than half an arc-minute, an
    error that would vary cyclically over a lunar month. This problem was 
    investigated, for a voyage made a few years later,  by
    Nicholas A Doe in the "Journal of Navigation", vol 48 no3, Sept 1995, 
    pages374-388, in a paper "Captain Vancouver's longitudes
    1792". In which case, even a perfect instrument would show cyclic variations in the resulting longitude.
    That question, of systematic errors in the lunar distances in the Almanac, 
    could be of some interest to maritime history. At a
    particular moment, all navigators using lunar distance, wherever they might 
    be, would be similarly affected: even those using the
    French "Connaissance du Temps", which took its predictions from Greenwich.. 
    So, in retrospect, positions recorded by those
    navigators could now be corrected to some extent for those known errors.
    The Moon's predicted ecliptic latitude and longitude were recorded in the 
    almanac, to the nearest arc-second, at 12-hour intervals.
    From these, the lunar distances to chosen fixed stars were deduced, at 3-hour 
    intervals by interpolation, and we can probably assume
    that that part of the operation was done precisely. With modern precise 
    knowledge of the Moon's orbit, its ecliptic lat and long can
    now be predicted, and is readily available from several sources. It would be a 
    service if someone would survey the almanacs of the
    lunar-distance era, and tabulate the day-by-day errors in those predictions. 
    That would, until recently, have presented problems, in
    getting access to those old and precious almanacs. But now, scanned copies 
    have recently become available on line, and
    would provide an interesting and useful exercise for any student to correlate 
    modern predictions with the contemporary ones, without
    even stirring from his keyboard. Anyone interested in such a project?
    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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