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    Re: Biruni and the radius of the Earth by dip
    From: John Huth
    Date: 2011 Jan 7, 19:07 -0500
    In terms of distance measurements - some metrology is as crude as traveler's reports.   On the other hand, there were professional pacers.   Recall that a (Roman) mile is a thousand paces.   The problem in this historical record is that there is no mention of whether professional pacers were used or simply traveler reports were used for the long baselines in the al-Farghani measurement.   

    For a shorter baseline, I imagine pacing could have been employed.   I tested 150 students and derived a mean of 980 paces per mile, which is fairly close to the Roman mile (I forget the calculated uncertainty - but with some effort, I can dig it up).   Again, we don't know that al-Biruni really carried out this measurement, but if we stipulate he did, then someone pacing between the two sites would give the most precise value, I'd think.  The next step would be chains, which I'm assuming one could use for a relatively short baseline, but I'm fairly certain this technique wasn't employed until far later.

    As a footnote to George - I wouldn't give this to students as a project if I didn't first test it out myself.   I'm not claiming any a priori knowledge of the precision I can get with an hydraulic level.   I am, however, curious as how accurately one can measure a small angle with the technology available to someone in the 11th century.   I work with a fairly clever guy who organizes the lecture demos.   We've already discussed this and we're trying out a few schemes.    If they look plausible, we'll take it up to the observation deck of the John Hancock tower and see how we fare.

    On Fri, Jan 7, 2011 at 1:23 PM, Marcel Tschudin <marcel.e.tschudin@gmail.com> wrote:
    I'm questioning myself why Biruni measured the height of the mountain
    in order to obtain the radius of the earth? Not knowing about
    refraction he must have thought of a geometry as shown in Fig. 2 of
    the Gomez paper. If he knew that the circumference of a circle is 2 r
    pi corresponding to 360 degrees and if he was able to measure
    distances in the plain he also could have proceeded as follow:
    1) He decides from which hilltop he intends to measure the dip of the
    apparent horizon in the plain.
    2) He searches the tangent point S in the plain where the hilltop just
    touches the apparent horizon. In Fig. 2 he looks from S to B assuming
    a straight line whereas in reality it is a refracted one.
    3) He measures the distance between S and the base A of the hilltop
    knowing that this would be the length of the arc between those two
    points. If the hilltop was about 320m above the plain, then A would
    have been around 68km from S. (I neglect here that the plain was
    likely not completely flat.)
    4) He climbs the hilltop and measures the refracted dip of the
    apparent horizon to be 34 or 35 moa imagining that he would measure
    the angle theta in Fig. 2.
    5) He can now calculate the radius of the circle where 34 moa
    corresponds to an arc length of 68km.
    The question here is (appart from so many others) how distances of
    around 70km could be measured in those days?


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