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    Re: Measuring Dip in the 18th Century
    From: Brad Morris
    Date: 2013 Dec 28, 20:56 -0500

    Hi Alex

    Nicolas de Hilster, a member of this board, constructed back staffs and performed the analysis you desire.  He posted his results here and on his web page http://www.dehilster.info/index.php?doc=http://www.dehilster.info/instrumenten/davis-quadrant/index.html to get you started.

    I will maintain my "inability" to state a performance of a back staff.  I do not think the performance is germane to this discussion.  What is germane is the resolution, as you adroitly reference.  All three items required to determine the latitude from the LAT meridian crossing of the sun are in minutes.  In my view, it is unlikely that those minutes were dropped in the computation.  As to the usefulness of that resolution or accuracy to the arc minute?  Well, that's a debatable point.

    I do get Frank's point about the resolution being "empty".  We, in this age, suffer from excessive resolution constantly.  Our GPS devices show a ridiculous number of digits compared to actual results.  I just had that conversation with the chief scientist of an autonomous robotics company.  When I asked him how he compensated for the obvious wander and dither of the instantaneous GPS position , he told me that the resolution was so small that it didn't matter!  

    So did our ancient navigators engage in this same practice?  Perhaps, but the telling evidence remains.  The damning fact is that the declination table, the dip table and the instrument are all resolved to one arc minute.  Useful or not, I think it readily apparent that they used the resolution. 

    Debate the usefulness all you want, it must have been used.

    Brad



    On Dec 28, 2013 8:26 PM, "Alexandre Eremenko" <eremenko@math.purdue.edu> wrote:

    Dear Brad
    
    > It appears that the resolution of the backstaff and expected accuracy of
    > the back staff are being treated as if they are the same unit.
    
    I agree that the things are different.
    
    > The expected performance is not 1  arc minute, just as Frank says and
    > again
    > something George agreed with.
    
    I do not understand on what basis did they claim this.
    Did they experiment with backstaff replicas? Or did they mention some other
    people who did? Or some historical observations from known places
    on land which we can compare with exact data? I would be interested to see
    any references or any
    other reasons for this estimate.
    
    On purely theoretical grounds I do not see why backstaff cannot achieve
    1' accuracy on Sun altitudes, for example.
    
    The accuracy of 1'-2' was certainly possible to achieve in antiquity,
    as some Ptolemy's observations indicate. These observations were of
    different nature from taking the altitudes over the horizon though.
    
    Actually, I think that making a good backstaff replica would be very
    interesting...
    
    Alex.
    
    
     the   In practice, the observer simply could not
    > reliably repeat measurements of the same angle to 1 minute.  Frank has
    > offered 10 minutes, a figure which I will not dispute.
    >
    > I do agree with Alex's assertion, that because the resolution was to an
    > arc
    > minute, and that the early dip tables were to an arc minute, we can expect
    > that they desired to observe to that resolution.  Achieving that was
    > another matter.  It does not follow logically that because they were not
    > achieving 1 arc minute accuracy in practice that the early dip tables were
    > irrelevant, to those navigators. Those navigators had dip tables, back
    > staffs and declination tables to an arc minute and USED THEM THUSLY.  They
    > could not compare their results with double reflecting instruments.  They
    > could not know that they were only good to 10 arc minutes because they had
    > nothing to compare it to. (The same can be said of celestial navigation
    > just before GPS.  Nothing to compare it to.  Our modern fascination with
    > accuracy of celestial navigation is highly reliant on a better method, to
    > wit GPS.)  These early navigators no doubt followed practice as later ones
    > did, by rote application of the rules, with little further speculation as
    > to the statistical accuracy.  After all, they were tooling along at 6-8
    > knots and being good to 10 miles (10 arc minutes) was certainly plenty
    > good
    > enough.
    >
    > As greater understanding of the physics came about and instruments
    > improved, the results of celestial navigation improved.   At best, one
    > could say by two orders of magnitude.  At nominal, by one.  The modern
    > double reflecting sextant is resolved to 0.1 arc minutes (one order of
    > magnitude better in resolution) and dip tables are resolved to 0.1 arc
    > minutes (one order of magnitude in resolution).
    >
    > I'll leave it to the cognoscente to rate the expected accuracy of the
    > sextant.  Is that two orders of magnitude better than the back staff?
    >
    > Brad
    > ------------------------------
    >
    >
    > As I said, good backstaffs were divided to 1'.
    > In principle, this instrument can achieve the same accuracy
    > as a wooden octant without a telescope. These wooden octants
    > were also divided to 1'.
    >
    > However there is one serious problem: the "parallax of the backstaff",
    > the angle measured strongly depends on the position of your eye
    > with respect to the staff, and there is no simple remedy for this.
    >
    > Thomas Harriot, whom I mentioned in a previous message
    > has a table of corrections for the "parallax of a backstaff".
    > It is clear from his recommendations and tables that they were aiming at
    > 1'
    > accuracy for determining latitude by Sun sights.
    >
    > Therefore a dip table and refraction table WERE relevant.
    >
    > Alex.
    >
    >
    >> Here is a well-preserved backstaf:
    >> http://www.ebay.com/itm/231110131338?ssPageName=STRK:MEWAX:IT&_trksid=p3984.m1438.l2648
    >>
    >> (Very rare item on e-bay, and the price is corresponding to this:-)
    >>
    >> If you magnify photo 3, you see that one of the arcs is subdivided to 1'
    >> First every degree is divided into 6 parts, and then they are subdivided
    >> by
    >> the nonius into 10 parts each. Nonius is the grid on the arc.
    >>
    >> I've seen many backstaffs like this in museums but this is the first one
    >> on
    >> e-bay. The only missing parts are two movable sighting vans.
    >>
    >>>From the examples I saw in museums I conclude that dividing into
    >> single minutes was the usual practice.
    >>
    >> Alex.
    >>
    >>> Hi guys,
    >>>          Please help me understand the 'instrument time line'. What is
    >>> the angular resolution of a cross-staff or back-staff?
    >>> I always thought it was about +/- a degree, and if thats true why would
    >>> you care about 1 arc minuet?
    >>> I always thought the first sub degree instrument was the octant.
    >>> ~Greg
    >>>
    >>> On 12/27/2013 06:04 PM, Alexandre Eremenko wrote:
    >>>> ------------------------------------------------------------------------
    >>>>
    >>>> I am reading a book about Thomas Harriot (an outstanding British
    >>>> mathematician) In 1595, he wrote "Instructions for assisting the
    >>>> navigators of Sir Walter Raleigh's voyage to Guiana.
    >>>> The instructions contain the following dip table
    >>>> 1 3'
    >>>> 2 4'
    >>>> 3 5'
    >>>> 4 5'
    >>>> 5 6'
    >>>> 6 6'
    >>>> 7 7'
    >>>> 8 7'
    >>>> 9 8'
    >>>> 10 8',
    >>>> where the left column is the height in PASE's; according to the authr
    >>>> of the paper,
    >>>> 1 pase=5 feet.
    >>>>
    >>>> The paper also has a reference to an early Portuguese source: Pedro
    >>>> Nunes, De arte ratione navigandi libri due (Coimbra, 1546).
    >>>>
    >>>> The article I am reading is Jon V. Pepper, Harriot's earlier works on
    >>>> mathematical navigation: theory and practice.
    >>>>
    >>>> Gary: have you received my request sent off-the-list?
    >>>> ----------------------------------------------------------------
    >>>> NavList message boards and member settings: www.fer3.com/NavList
    >>>> Members may optionally receive posts by email.
    >>>> To cancel email delivery, send a message to NoMail[at]fer3.com
    >>>> ----------------------------------------------------------------
    >>>>
    >>>> : http://fer3.com/arc/m2.aspx?i=125973
    >>>>
    >>>
    >>>
    >>>
    >>> : http://fer3.com/arc/m2.aspx?i=125980
    >>>
    >>>
    >>>
    >>>
    >>
    >>
    >> : http://fer3.com/arc/m2.aspx?i=125985
    >>
    >>
    >>
    >>
    >
    >
    > : http://fer3.com/arc/m2.aspx?i=125986
    >
    >
    > : http://fer3.com/arc/m2.aspx?i=126001
    >
    >
    >
    >
    
    

    : http://fer3.com/arc/m2.aspx?i=126007

       
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