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    Re: Coordinates on Cook's maps
    From: George Huxtable
    Date: 2007 Apr 19, 19:56 +0100

    Alex's latest mailing takes us into deepish waters.
    
    | On Tue, 17 Apr 2007, George Huxtable wrote:
    |
    | > Cook had metal sextants, with large radius, and Vernier scales, so their
    | > scales could be read rather precisely. But they were divided by hand,
    | > machine division being a decade or two into the future, so systematic
    errors
    | > in the scale calibration were the big problem. I suppose they could have
    | > been corrected by a programme of measuring star-to-star distances, but I
    | > have no idea whether that was ever done.
    |
    | These statements raise several interesting issues.
    |
    | Let me share my own experience (and I will appreciate if
    | other list members add to this).
    |
    | 1. Reading a vernier scale to 0.1' is a tricky business,
    | especially at night. This can be very annoying and time consuming
    | but this cannot be a serious additional source of errors
    | I mean no more than 10"-15" if you read with maximal care,
    | even on a 7 inch arc.
    
    I agree. Remember, these early instruments were MUCH bigger than the
    7-inch-radius arc that Alex mentions. On the other hand, they seem not to
    have been fitted with a magnifier, unless a separate hand-held lens was
    used.
    
    | 2. That sextants were divided by hand. It would be very interesting
    | to know how accurately were they divided.
    
    I agree that it would, but I don't know if that has ever been studied. I
    could try to get an answer to that question from my contacts at Greenwich.
    
    |  I do not see a priori reasons,
    | why machine division should be more precise. Were not the dividing
    | machines themselves divided by hand?
    
    No, it was an "automatic" process, though a very time-consuming one, to make
    a master wormwheel. You needed an accurately-pitched worm to start with
    (well, two, one with notches to cut the teeth on the wormwheel, to be
    replaced by an identically-pitched smooth worm to use when the wheel had
    been made). Cutting those teeth on the wheel presents real problems, when
    you think about it. You might think of starting with a light "milling" cut
    around the edge of the wheel, then work your way round the wheel, turn after
    turn, going deeper and deeper. But that won't work, because as you go
    deeper, the radius gets less, so the required pitch of the worm would be
    less. Instead, you have to cut a few teeth, over a short sector, to full
    depth, by going to and fro over that little arc, then extend it by another
    short arc, and so on. Chapman, in "Dividing the Circle", says that Ramsden's
    wheel, of 40 inches diameter, carried 2160 teeth, at 10' spacing. The moment
    of truth occurs when you get back to your starting point; has there been an
    exact number of 2160 teeth, at 10' intervals, or not? Any mismatch AT ALL
    between the first and last tooth is a disaster!
    
    My friend and neighbour, and occasional Navlist contributor, Clive
    Sutherland, has made a serious study of Ramsden's dividing techniques, and
    understands much better than I do.
    
    Then to divide a sextant arc, the arc was clamped, precisely centred, into
    place on the wheel, and each time the operator pressed a treadle, a diamond
    cutter would scribe a mark, and the wheel would notch around to the next
    position. Once the master-wheel had been made, it was a precise and
    repeatable process.
    
    Dividing by hand called for the sort of construction we used in geometry
    classes at school, but using a precise beam compass fitted with magnifying
    lenses. First, by drawing a chord exactly equal to the radius, you can
    easily generate an equilateral triangle, with angle 60 degrees. Halving an
    angle with the compass is easy, so two halvings takes us to 15 degrees. But
    from that point, to get down to degrees, calls for division by 3 and by 5,
    and that's a really tricky operation, involving much trial-and-error. It
    called for a sharp eye and a steady hand.
    
    Bird was the technician that was most famous for the precision of his
    divisions. Were arcs reliably hand-divided to better than half a minute or
    so? I just don't know; but I doubt it. Chapman states- "Using a Vernier, or
    diagonal scale, the early Hadley octants could scarce be trusted within a
    minute of arc in 1760, whilst by the 1800s, Troughton was producing brass
    sextants of 8 in., that could be read to 5" by a practised eye." But he is
    describing there the precision of reading, and saying little about the
    overall precision of calibration.
    
    | More serious matter is that at that time they were not apparently
    | "certified". (The Kew observatory certificates begin
    | somewhere in XIX century). And it is not clear to me
    | how those early sextant were really
    | tested. And whether they were tested.
    
    I don't know that either. One could readily compare one instrument against
    another, at spot values, by comparing horizontal-angle spacings between
    known distant landmarks, such as tree-trunks. Even the 19th-century checks
    for Kew certificates involved first setting up a series of collimators by
    comparing with a standard theodolite, so they did not relate to an absolute
    angle. How you got an absolute calibration right, in the first place, is a
    mystery to me.
    |
    | 3. Concerning the star-distance method of testing,
    | I remember your own question, George, on the old list
    | several years ago: "Has anyone succeded in finding the arc error
    | of a modern sextant with star distances?" (I cite from memory).
    | Nobody ever replied that s/he succeeded:-)
    | I failed, despite my serious efforts to so this over 3 or 4 years.
    | I very seriously doubt that star distances permit you
    | to test your arc to 0.1' or 0.2' accuracy. I could never do this.
    | Same applies to IC from stars btw. It is only the constancy of my SNO
    | index error that permits me to determine it reliably by very many
    | observations.
    | My Lunars are better than star distances.
    |
    | 4. When speaking of high precision Lunars
    | (I am talking of 0.2' accuracy
    | or so) I see two main difficultiues:
    | a) How to achieve a precise touch of the two objects in
    | your field of view. This seems to require years of continuous practice.
    | It took me 3 years of frequent observations to achieve the results I
    | recently posted. (My sight is considered normal by doctors and I am 50+
    | years old). The errors here are NOT random, and cannot be eliminated
    | by averaging. The observer just has to feel how the picture should look
    | then the objects really touch. And feel this under very different
    | conditions of brightness, sextant position and the loo
    | of the objects themselves. This is what I find hardest.
    | One needs reasonably good optics. (I find the optics on my old
    | sextants just terrible
    | in comparison with the SNO inverting scope).
    | b) Sextant rigidity. This (together with the optics)
    | was probably a major
    | problem
    | of old sextants. Again, I find my modern SNO far superior to the
    | old vernier C. Plath in this respect. There is a lot of discussion of
    | rigidity in XVIII century literature, and one of Cook's associates
    | complains that his sextant has unpredictably changing instrumental
    | error which he cannot explain.
    | Same problem I experienced for several
    | years, and I still don't know whether this was an instrumental error
    | or a personal error, but it looks that by now most of it is
    | eliminated. Though if one looks at the distribution of my
    | errors produced in a previous message by George, you see that it is
    | asymmetric, it is not a normal distribution, and it is slightly
    | biased to the positive side.
    |
    | U suppose that the main reason why they eventually switched to smaller
    | frames (6 and 1/2 to 7 and 1/2 inches) was rigidity problems.
    | It is probably impossible to make a brass frame of 9 inches radius
    | rigid enough, and so that you could still lift your sextant without
    | an assistant. I also suspect that modern lightweight alloys
    | (duralumin) permitted to make much more rigid frames.
    |
    | Anyway, the conclusions from my limited experience are
    | that
    | a) achieving 0.2 accuracy of a measurement is VERY hard
    | b) averaging does not always help (because systematic personal
    | and instrumental errors dominate the overall error).
    | Of course, averaging helps to some extent (as seen from my statistics,
    | for example)
    | c) on the question whether the best XVIII sextants were better
    | or not than those which are available nowadays I still have no
    | definite answer. Here by "better" I mean "better ultimate accuracy".
    | There is no doubt that modern sextants are much more convenient
    | in use. I used to think that the old ones were better
    | but now I don't think this anymore.
    
    There are many aspects to consider there.
    
    How might they have calibrated such instruments, around Cook's time? Well,
    star-to-star distances are one way, in spite of the difficulties that Alex
    reports. It depended on those star distances having been precisely
    determined beforehand, by some more-accurate observatory instrument, so it
    wasn't an absolute method. Could a sextant be clamped with its frame in the
    plane of the equator, and a near-equatorial star, such as Sirius, or the
    Sun, observed, as it moved across the sky, timed with an observatory clock
    (with corrections for refraction)? Perhaps, in the end, there was no better
    method than relying on the division of an arc by geometrical methods, to as
    high a precision as possible.
    
    Alex rightly stresses the importance of sextant rigidity. For ordinary
    altitude navigation, up from the horizon, the sextant is always in the same
    posture, and flexing due to gravity isn't an issue. For lunars, where the
    sextant may need to be held any way up, gravity-flexure really mattered.
    
    In "The charts and views of Captain Cook's Voyages", vol 2, Hakluyt Society
    1992, Andrew David (editor, who was a Hydrographer)) has this to say, on
    page XX-
    
    "The astronomers were each supplied with a 15-inch brass sextant by Jesse
    Ramsden and a similar one by Peter Dollond...  The Ramsden sextant supplied
    to Wales had been used by him on his voyage to and from Hudson Bay and he
    therefore already knew its value. It was cut from a solid plate of hammered
    brass about one ninth of an inch thick, supported on the back with
    perpendicular "edge bars" secured firmly by screws, passing through the
    frame of the sextant into the bars themselves, with the index arm similarly
    strengthened. Wales noted that the massiveness of these bars made the
    instrument rather heavy, but added that he never met one that preserved its
    figure, plane and adjustments so well. He went on to advocate that, as these
    properties were so essential, the weight of the sextant should never be
    reduced, adding that once he had got used to it, he did not find the weight
    to be inconvenient. The Dollond sextants ... were constructed in a similar
    way, but were less massive ..."
    
    Those 15-inch sextants were not the largest, either. The first sextant, made
    by Bird to the design of Campbell, was all of 18 inches radius, and its
    weight was relieved by a ball-joint at the top of a wooden staff, which
    fitted into a socket in the observer's leather belt.
    
    George.
    
    contact George Huxtable at george---.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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