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    Re: Precision of sextant observations: taken from How many chronometers?
    From: Bill Morris
    Date: 2009 May 9, 22:24 -0700

    George Huxtable wrote:
    
    "We need to consider the instruments that were being used, throughout the
    lunar era, which would all be Vernier sextants, not micrometer types. I have
    such a sextant, which I haven't taken to sea. Using it on land, my old eyes
    have great difficulty, even with a good magnifier, perceiving which of three
    adjacent lines on the Vernier makes the best fit, even with really good
    illumination."
    
    and Henry wrote of "well calibrated modern instruments"
    
    For most of the period in which lunars were used in earnest, no calibration 
    service was available, at least in Britain. Kew Observatory offered the 
    service from about 1870 onwards, but calibrated only 49 instruments in the 
    first twenty years. Even then, the precision of the calibration could be no 
    better than the precision of the instrument itself. For example, an 
    instrument divided to 20 seconds could be calibrated only to that precision.
    
    George has mentioned the difficulty in actually reading the verniers and since 
    it is a rainy day, I have done some simple investigations to try to get some 
    idea of the reliability of a vernier reading in ideal circumstances. The 
    set-up is as shown in the attachment. The sextant sits on a levelling jig on 
    a surface table and an autocollimator is directed at the index mirror so that 
    its axis is at right angles to the mirror. A lamp and low power stereo 
    microscope are used to read the vernier at a magnification of x 20.
    
    I won't go into many details of the autocollimator. It is an optical device 
    that can measure the deviation of a projected light ray when it is reflected 
    back from a mirror. The instrument shown has a least division of 0.2 seconds 
    and in expert hands is said to be reliable to 0.3 seconds, given a high class 
    reflector and perfect viewing conditions. For my own part, I can very easily 
    detect a deviation of 3 seconds and I think it is safe to say that the 
    repeatability of a single reading would always fall inside that range.
    
    Using first the microscope and then the magnifier provided with the instrument 
    to set the vernier, I reset the sextant thirty times to the same reading and 
    took readings with the autocollimator on the index mirror to study the amount 
    of variation. The results do not depend on mirror adjustments, bearing 
    eccentricity and the like, though both instruments had recently been fully 
    stripped down, rebuilt and adjusted.
    
    The first sextant was a ladder framed one by Brandis and Son Inc. The "Inc" 
    places it after 1916 and its serial number places its manufacturing date 
    around the First World War. The vernier is divided to 20 seconds and its 
    simple plano-convex magnifier has a focal length of 25 mm, giving a nominal 
    magnification of x 10.
    
    The second sextant was a high class ladder frame sextant by Hughes and Son and 
    has a 1920 calibration certificate from the National Physical Laboratory 
    revealing no error across its range. The vernier is divided to 10 seconds and 
    the Ramsden magnifier magnifies x 7.5. Like the Brandis, its graduations are 
    sharp and easy to read, but are rather finer than the former's.
    
    The standard deviation is a measure of dispersion of the results around the 
    mean and in this context we can expect them to be normally distributed. That 
    is to say, about 68 percent of results will be found within 1 SD each side of 
    the mean  and 95 percent within 1.96 SDs each side of the mean. In the 
    results that follow, I give the rounded 68 and 95 percent ranges when using 
    the magnifier provided.
    
                    SD Microscope    SD Magnifier  2 SD range  3.92 x SD range
                                                      (68%)       (95%)
    
    Brandis             4                 11.7         23          46
    
    Hughes              4.3                4.0          8          16
    
    The better results of the Hughes instrument are probably due to the finer 
    graduations and possibly also to the colour correction and flatter field of 
    the magnifier. While repeatability of setting is not quite the same as 
    accuracy of reading of the vernier, this little experiment does give us some 
    insight into the latter. We could perhaps agree that when the Hughes sextant 
    is set to a nominal reading, there is a seventy percent chance that the 
    setting lies within 10 seconds of that chosen reading, and a ninety-five 
    percent chance that it lies within 16 seconds of it. 
    
    When it comes to reading a sextant, the results are critically dependent on 
    obtaining a correct index error. As I have shown in a previous post, 
    whichever method of ascertaining index error is used, there is quite a lot of 
    dispersion
    of the results, at least with micrometer instruments. For my own part, I doubt 
    that observations could be reliably and consistently made to within 10 
    arcseconds. 
    
    Bill Morris
    Pukenui
    New Zealand
    
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