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    Re: Biruni and the radius of the Earth by dip
    From: Frank Reed
    Date: 2011 Jan 6, 20:22 -0800

    George Huxtable, you wrote:
    "And one part in 3500 of WHAT?"

    Sorry to confuse you. I believe my previous posts were quite clear on this point: one part in 3500 of the ratio (or in other words, when converted, about one minute of arc --normal visual acuity). So if the two points of the level are separated by 10 meters, that's about 10m/3500 or about 3mm in the measurement of the vertical distance down from the level line.

    You asked:
    "Is Frank (and John Huth too) suggesting that a flexible pipe and a pair of
    vertical glass tubes be used, then, to provide a suitable level, to measure
    to an arc-minute?"

    Yes, I am.

    You added:
    "Not, of course, that such technology was available to al-Biruni."

    That's right. John has said that he was interested in trying this with his students and so we can set aside some historical accuracy in favor of ease of assembly. John's original suggestion of a trough of water is fundamentally the same and would have been available to Biruni. Build yourself a narrow trough ten meters long and put two sticks in the water with the same length above the water. The tops of the sticks make an excellent level. Then you just measure the distance down to the visible horizon on the further stick as seen from the top of the first (add a sight peep there to ensure that the observer's eye is in exactly the right spot). That technology would have been available in the 11th century.

    You asked:
    "I wonder whether either of them has ever tried to make, and fill, and use, such a levelling device?"

    Yep. Sure have. I think the longest I set up was around 5 meters from end to end, but 10 meters does not strike me as a big step. This used to be a common DIY project for leveling but laser and digital levels of various sorts seem to have replaced water levels, even though they can be much more accurate than the high-tech alternatives. Here's a little web page, originally written eleven years ago, on setting up a water level:
    I was amused to see that the author of this page references "Bowditch" regarding the curvature of the Earth. The comment at the bottom suggests adding a little detergent to the water to reduce the meniscus. That's a good idea. Also, it may help to add some dye to the water for contrast.

    You wrote:
    "The problem is to avoid entraining air-bubbles, which will then collect at any humps in the tube, unbalance the water-columns, and result in a false reading."

    That's right, but it's easy to deal with.

    And you wrote:
    "If it was so simple in practice, buiders would use such devices in place of spirit-levels and theodolites."

    Well, just because builders and surveyors have better professional tools available doesn't mean that a simple tool like this doesn't work.

    You also wrote:
    "If we're still discussing Biruni, as the threadname implies"

    The exact details of Biruni's observations are certainly interesting, but we've moved on slightly to a discussion of how this might be done by students today replicating the principles of Biruni's observations and respecting the general properties of the equipment available a thousand years ago, rather than duplicating every exact detail.

    Of Biruni's original observations, you added:
    "If an astrolabe was used for the observation" etc.

    But is there any original documentation actually stating that Biruni used an astrolabe? This seems to be more of a supposition by some commentators --in other words, no more than a guess. John has already dismissed this idea, and I see no reason to insist that Biruni must have used an astrolabe. It would be a poor choice for this observation. There are, in fact, excellent ways of measuring small angles near the horizontal, just as above, that would have made this a practical operation. The limiting factor is not the angular accuracy, though it contributes some error. The uncertainty in the height of the mountain, as you also described, is a bigger factor. Considerably greater than both of these issues is the systematic problem of refraction. Even if we minimize errors in the measured angle and in the measured height of the observing point, the observed dip yields a radius for the Earth that is roughly 16% or 1000km too large (assuming we don't know about terrestrial refraction and we treat it as a pure geometry problem with rays of light travelling in straight lines).

    George, you concluded:
    "As a consequence, any similarity between his Earth-radius and its true value was the result of a happy accident."

    Yes, I agree. In case, you missed it, I wrote three days ago:
    "claiming that Biruni's estimate of the Earth's radius was accurate to 36 km, well, if it was, he got plain lucky and his observational error just by chance cancelled out the very large systematic error intrinsic to the method."


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