NavList:

John H., you wrote:
"Then, if he used a long trough of water, that would be a good reference for horizontal. Then, just measure the dip angle from that. I'm trying to figure out how long a trough of water would be required to get the precision quoted in Wikipedia."

First, the accuracy quoted by the Wikipedia article here:
http://en.wikipedia.org/wiki/Biruni
is impossible for this method. See below. But there would still be some interesting possibilities for a student experiment...

How about instead of using a trough, you take two identical graduated cylinders with drains at the bottom ends (standard glassware) and connect the drains with a long length of rubber tubing. Now fill with water from both ends, get the bubbles out of the tubing, and adjust the heights of the cylinders until the water fills to exactly the same graduation level on each cylinder. The tops of the cylinders are now level relative to gravity. Much easier than building a trough! Fix them in place, and the water can be drained out. Note that the distance between the cylinders is only practically limited by the length of tubing available. You could do ten meters easily. There's really no reason to go to greater distance unless you're using a small telescope. Then you sight along the cylinders, aligning the tops visually, and take note of the position of the distant horizon relative to the sighting line. An assistant could easily raise or lower some sort of pointer affixed to the further cylinder until it's right by the apparent horizon. Then you read off the dip as the distance down the side of the further graduated cylinder. The limiting resolution of the human eye is about one part in 3500, so you could visually see the position on that second cylinder to an accuracy of about 3 millimeters, assuming it's 10 meters away.

Angles ARE ratios. If the distance of the apparent horizon below the horizontal is 25 centimeters at a distance of 10 meters, then the dip angle is 0.025 (+/- 0.0003 for observers with good visual acuity and no magnification). And of course the short equation for the geometric dip is dip=sqrt(2*h/R) so we can easily calculate the radius of the Earth by turning it around.

But these observations have significant uncertainties. If we observe from a mountain 2km high, the uncertainty in the observed angle, limited to visual acuity, would imply an uncertainty in the calculated radius of the Earth of about 150 km. The uncertainty in the altitude of the mountain contributes in direct proportion, so if the altitude is wrong by 5% (100 meters for a 2km mountain) then the calculated radius of the Earth would also be wrong by about 5% or some 300km.

There is a much more serious systematic problem with this technique for measuring the radius of the Earth. Rays of light travelling from the horizon to the observer are bent by refraction. And this is the BIG problem with any claims for accuracy of this method. The geometric dip is significantly different from the refracted dip because light rays curve as they travel from the denser air near the surface at the horizon to less dense air higher up at the observing location. For observers in the lower few kilometers of the atmosphere, the refracted dip is about 92% of the expected geometric dip. Even if we measure the dip angle with perfect exactness and if we know the altitude of the mountain with perfect accuracy, the dip that we measure CANNOT yield an accurate value for the radius of the Earth unless we use a detailed theory of refraction AND we know a great deal about the temperature at various levels in the atmosphere. In other words, this is a physics problem, not a geometry problem. In fact, the calculated radius would be systematically too large by about 16% or approximately 1000km if we ignore refraction. To make matters worse, the refraction is quite variable so one day the difference might be 750km and another day it might be 2000km. The apparent horizon relative to a carefully laid out local level rises and falls with the weather. If there is a moderate temperature inversion, light rays will curve at a rate equal to the curvature of the Earth, and the appparent horizon can coincidence with the geometric level --thus proving that the Earth is flat now and then :).

As for the Wikipedia article 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. But what's more likely here is that we're seeing the usual phenomenon of, putting it politely, "exuberant" Wikipedian nationalists who write up their favorite local or national hero as the greatest hero in the history of all history.

-FER

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