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    Re: Averaging lunars: was Lunars with SNO-T
    From: George Huxtable
    Date: 2004 Nov 1, 15:47 +0000

    In reply to Herbert Prinz's posting-
    >George Huxtable wrote:
    >> It may be interesting to read what Tobias Mayer had to say on this topic,
    >> [...]
    >Thanks to George Huxtable for posting and Steven Wepster for translating. While
    >we are at it, Note I, immediately preceding the quoted passage might also be of
    >interest. In an earlier message I had casually compared the repeating circle
    >with our averaging technique. This note reminds me that the two methods are
    >different in one respect.
    >In Note I, Mayer says that the final error of an observation is inverse
    >proportional to the number of individual distances that have been accumulated
    >into this observation. Therefore, if one assumes an instrument error of 2' to
    >3', six observations are plenty sufficient to get the error below 30".
    >In passing I remark that the goal for accuracy that Mayer sets himself is truly
    >down to earth, much less ambitious than that of some of our list members! But
    >my question is: Is his arithmetic right?
    >It is right, if we assume that the observer is perfect and the only purpose of
    >the averaging process is the elimination of instrument error and reading error
    >(due to limited resolution of the division of the limb). This is what Mayer had
    >in mind.
    >When we average with a sextant, we are in a different situation. We can never
    >get around the instrument error or the reading error with repeated readings.
    >Our assumption is that the observer is not perfect and each observation is
    >fraught with random error. This error is inverse proportional to the square
    >root of the number of observations.  Given the above parameters, six
    >observations would not be sufficient.
    >If an imperfect observer producing random errors had a perfect sextant and a
    >perfect repeating circle, the result of a repeated observation would be the
    >same on both instruments (statistically). The error would obey the inverse
    >square root law on both instruments. Mayer did not address this kind of error.
    >He did not have the appropriate error theory. Might he have thought that he had
    >divided random error by the number of observations, too? I don't know the
    First, I should explain about the translation from Latin. Steven Wepster
    kindly sent me the copy of the article, but didn't translate it, having an
    exaggerated notion of my classical abilities. In fact, I was slung out of
    the Latin class, for incompetence, best part of 60 years ago.
    However, through a friend of a friend, I have made contact with the tutor,
    Nigel Creese, and an interested mature-student, Claude Miller, of a U3A
    Latin class in Melbourne, Australia.
    U3A, or University of the Third Age, is a setup through which ripely-mature
    scholars and students offer mutual help in furthering their learning. It
    also exists in the UK, and likely elsewhere. Strength to its elbow, I say.
    Anyway, between the two, who claim to know nothing about navigation or its
    instruments, they have kindly produced a translation into English, into
    which I have been able to put some input, knowing something of the topic
    but being woefully ignorant in Latin. We are now getting near the end of
    the process of batting the text to and fro between us, and for better or
    worse, arriving at an agreement. Then I will be happy to post a copy of the
    translated text to Nav-l. There's also a goodish diagram.
    Now for Mayer's instrument itself, and Herbert's comments.
    I think that the biggest problem that Mayer was fighting, whether or not he
    was fully aware of it, was inequalities in scale division, which in those
    days was done by hand.
    The clever part of his design was this- Each measure of lunar distance used
    a different segment of the circular scale. One started where the previous
    one finished. The circle was marked with 720 degree divisions, around the
    360 degrees of the circle, just like the markings of a sextant are doubled.
    If lunar distances of, say, 90 degrees were being measured, then after 8
    such measurements the two arms, the index arm and the telescope arm, had
    "walked" their way right round the 720 degree markings of the perimeter. No
    matter how unevenly the circle had been divided, those errors simply had to
    average out, over a complete turn of 720 degrees.
    So the most precise way to work this instrument (though it's not a point
    that Mayer makes) would be to make as many lunar distance observations as
    needed to sum up to somewhere near 720 deg (or even 1440 deg, or some
    higher multiple, being careful to count the number of whole turns). It's
    never necessary to observe the intermediate readings: only the first and
    last, and then divide by the number of readings taken.
    An aspect of Mayer's instrument, which differs from the use of the quadrant
    or modern sextant, is this. In between each lunar distance observation,
    it's necessary to move the telescope arm  to a new "reset" position in
    which the mirrors are parallel (though no need to record it). This is just
    like zeroing the index arm on a modern sextant, by aligning the Moon, or a
    star, or any distant object, with its reflected image.
    In the case of a sextant, this index-check may be made once, or perhaps
    twice, before and after a set of lunars, so it's no more accurate than that
    one or two observations, and then subtracted from the lunar distances,
    which may be the more-accurate average of many.
    For Mayer's device, this "resetting" happens between each measured
    distance, and at a different point on the scale each time. Each is thus an
    independent measurement, and repeating it many times ensures that the
    uncertainty is correspondingly reduced, to being no greater than any error
    in the distance.
    Any inaccuracy in the scale-reading of Mayer's instrument is indeed divided
    by the number of observations, because there's no need to record and
    average the intermediate values, just the difference between first and
    The error that remains is any scatter in the ability of the observer to
    recognise the coincidence between star and Moon's limb, and clamp the arm
    into that position (and time that event): and similarly in the resetting
    operation. Those errors will add quadratically, and so will be reduced only
    as the square-root of the number of repeats.
    I understand that Mayer was really a surveyor and astronomer, and his
    instrument was a development of one he had invented for surveying. He had
    no experience of marine instruments, had never been to sea, and had never
    even seen the sea.
    There's an interesting recent paperback, "The measure of all things" about
    a survey of the length of the meridian line through France, to establish
    the length of the metre, made in the turmoil of the Revolution. The main
    instrument used was a surveying circle by the famed French navigator Borda,
    which appears to have followed Mayer's circle in some detail.
    contact George Huxtable by email at george---.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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