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    Re: Image of Sextant Used by Worsley
    From: Bill Morris
    Date: 2009 Feb 26, 12:14 -0800

    Brad, You wrote:
    "Thanks very much for the data on the calibration of your Class A sextant from 
    NPL.  I do have some questions, if you don't mind.  
    "Was that bidirectional or unidirectional?  A bi-directional set runs up the 
    scale and then back down the scale, stopping in the same locations.   A 
    unidirectional data set merely marches up (or down the scale), stopping in 
    the same locations from the same direction.  For a micrometer type device, 
    then the unidirectional approach would be preferred, due to gear lash and 
    lost motion.  For a vernier device, the accuracy and repeatablity is not 
    affected by the gear lash in the same way, so I suggest a bidirectional 
    approach might be more representative.  That is, the vernier reading is a 
    function of the index arm on the arc.
    "How many bidirectional or unidirectional runs did your data consist of? " 
    I'm not sure that I understand the question. I think it's to do with backlash 
    (or gear lash). In a micrometer instrument, ideally the final motion should 
    be such as to load the thrust face of the worm shaft bearing, rather than the 
    anti-backlash spring (read my book, The Naked Nautical Sextant, for detailed 
    anatomy). For most sextants, this is in the direction of reducing readings. 
    In SNO-T/Freibergers and some Hughes sextants, it can be ignored as they rely 
    on bearings in which backlash can be adjusted out. In the Hughes sextant I 
    reported on, the vernier tangent screw is opposed by a spring-loaded plunger, 
    the same sort of pattern as used in theodolites. I always made the final 
    motion against the spring, in case the action of the spring was imperfect and 
    perhaps provided a little more motion to the index arm after setting.
    I reported the actual method of calibration in NavList post number 4356. In 
    essence, 15 degrees of the sextant scale are used as an optical caliper to 
    subdivide a straight line (180 degrees) twenty-four times, so that two 
    autocollimators end up with their axes at a nominal 7 1/2 degrees. This 
    involved forty-eight settings of the vernier, twenty four at 0 degrees and 
    twenty-four at 15 degrees. I can't think clearly enough about these matters 
    to decide whether I should divide the standard deviation of 5.1 (see below) 
    by the root of 24 or 48. The sextant error at 15 degrees is thus either 3.6 
    +/- 1
    or 3.6 +/-0.7. 
    The corrected angular separation of the autocollimators is then used to 
    calibrate each fifteen degree interval. I took each reading four times on the 
    way up and only once on the way back down (this is tedious work and requires 
    a lot of concentration). Of course, each fifteen degree interval could be 
    done in the same way as the first fifteen degrees, but as the process takes 
    about two hours of careful, plodding work, I ask to be excused the task.
    The least graduation on the Hilger and Watt autocollimators is 0.2 seconds. A 
    mis-setting of 1 second is easy to see. I have checked their calibration 
    using a small angle generator and slip gauges and am confident of their 
    accuracy to one second.
    I suppose there is a possibility of bias, favouring approach of the vernier 
    index more from one direction than the other, but  it would apply equally to 
    the readings at each end of the interval and can therefore probably be 
    ignored if the bias is truly systematic.
    You also wrote:
    "We need to differentiate between the accuracy of the data and the 
    repeatability of the data.   The accuracy would be the numerical average 
    while the repeatability at each point would be the statistical  3 sigma 
    evaluation of all data for one arc location."
    To try to answer your question, I re-set the vernier to zero thirty times and 
    recorded the variation with an autocollimator.
    n = 30; sample standard deviation = 5.1; standard error of the mean = 0.93.
    For the non-statistician (e.g. me), I think this means that in a series of 
    results, about 64 % can be expected to fall into the range +/- 5 and 95 % 
    into the range +/- 10 from the true value. If I'm wrong, I'm sure there are 
    plenty of physicists waiting to pounce!
    "I suspect that the NPL merely provides us the accuracy figure of merit and 
    not the repeatability, which would be affected by many factors, including 
    I'm sure you're right. The collimators at the NPL were set up using a 
    theodolite (we are not told with how many repetitions or the probable error 
    of setting) and the error was taken from the reading of the instrument being 
    tested. The temperature question was raised in the thread that followed from 
    post 4356 and I gave a simple illustration.
    (If this discussion continues, should the thread name perhaps be changed?)
    Bill Morris
    New Zealand
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