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    Re: Ulugh Beg's sextant
    From: Herbert Prinz
    Date: 2006 Mar 31, 01:49 -0500

    A while ago Frank Reed asked about Ulugh Beg's star catalogue:
    
    >It strikes me that the accuracy in latitude should
    >be  better than that in longitude. No?
    >
    >
    I still owe Frank and the list an answer. It took me a while to dig out
    the sample data, hence the late response.
    
    Let's only talk about zodiacal stars, since the question becomes more
    complicated in mid-latitudes and actually meaningless close to the pole.
    The answer depends on how one has measured the stars, i.e what
    instrument and what procedure was used; whether you consider systematic
    or random errors; and which errors you deliberately filter out for the
    purpose of a meaningful comparison with other data. We know little about
    Ulugh Beg's methods, but we must assume that they are not too different
    from Ptolemy's. C.H.F. Peters investigation of his catalogue may serve
    as an example for what can happen.
    
    Peters plotted lon and lat errors of 349 zodiacal stars of Ptolemy's
    catalogue versus their longitude. This resulted in two sinusoidal
    curves. Longitude errors have an amplitude of ca 15' with extrema at 0
    and 180 deg and are nearly zero at lon 90 and 270.  Latitude errors
    have an amplitude of ca 20', with their extrema at 140 and 320 deg
    longitude. Peters comes to the conclusion that such systematic errors
    can be caused by a faulty erection of the instrument and/or faulty
    positions of the axes and circles of the armillary sphere. The theory
    sounds reasonable, unfortunately he does not develop it.
    
    Before this analysis, the given errors for longitude had already been
    pre-processed. All longitudes had another systematic error, a constant
    bias of 34.9'. Peters eliminated this by changing the epoch from the one
    given by Ptolemy by ca. 40 years (precession). Whether or not that was a
    valid thing to do, the example just shows how complex things get.
    
    Peter's diagram is in Peters and Knobel, Ptolemy's Catalogue of Stars,
    1915, p.6.
    
    >What is the origin of this myth that  the positions were accurate to a few
    >seconds of arc? A number of web sites quote  an article by "Kevin Krisciunas" on
    >this point. Maybe it's just that single  source.
    >
    >
    
    There are two basic misunderstandings:
    
    1. That the Fakhri sextant was used to measure the star positions. In
    all likelihood it wasn't. This instrument appears to have been designed
    primarily for sun observations. It is not clear whether it could have
    been used for the moon or planets, but certainly not for the higher
    magnitude stars in the catalogue. These must have been observed with an
    armillary like the one Ptolemy describes as his "astrolabe". I don't
    think we have any conclusive documentation on the observation methods in
    Samarkand. At least I could not find any hint to it in secondary sources
    within my language limitations. (Some newer research is published only
    in Russian, Turkish, etc.)
    
    2. That the radius of 40m, which admittedly allows a graduation on the
    arc down to 5" with a spacing of 1mm, automatically implies that
    readings can be taken at that resolution. We cannot even guess at the
    obtainable accuracy if we don't know more about the construction and
    mode of operation of the instrument. Did it work like like a gnomon,
    like a pinhole camera, were there sights of any kind? We don't know for
    sure. Say the device worked like a pinhole camera. If you can locate the
    center of the sun image within half a foot (a figure I pulled at random
    from a discussion of meridian lines in churches), you obtain an accuracy
    of 12'. Ulugh Beg's solar theory was much better than that, in fact
    better than Ptolemy's, so his instrument must have been capable of much
    more. But "a few seconds of arc" sounds ridiculous to me.
    
    I just want to add that I googled Krisciuna's web site and find his
    article in general very informative. It overlaps largely with chapter 2
    of his book, "Astronomical Centers of the World", Cambridge U.P., 1988,
    which is a good read for anybody interested in observatories, past and
    present. This is no pop-science, but a serious book written by a
    professional astronomer who has studied his subject. Where the chapter
    on Ulugh Beg is concerned, there are a few details where I would have
    wished for more clarity, or where I come to slightly different
    conclusions than the author.
    
    Herbert Prinz
    
    
    

       
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