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Muslims successfully repeated the Eratosthenes measurement by use of angles of shadows at noon at two locations far from each other.
Al-Biruni misinterpreted and fudged the results of his “angle to the horizon from a hill” method of calculating the circumference of the Earth because he did not know that air had a lower density at the top of a hill, causing considerable refraction of the light.
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Muslims successfully repeated the Eratosthenes measurement, but al-Biruni misinterpreted and fudged the results of his “angle to the horizon from a hill” method of calculating the circumference of the Earth.
ANGLE OF SUN AT NOON METHOD
In 820, al-Farghani was on a team that performed a measurement equivalent to the Eratosthenes 240 BC “angle of noontime shadow on the same day at different latitudes” calculation of the circumference of the Earth, but by measuring the angle of the sun with a portable instrument instead of looking at a shadow in a well as Eratosthenes had done.
The al-Farghani team measured the circumference of the Earth to an accuracy of a couple of percent, pacing out the distance between 2 locations that are hundreds of kilometers north-south of each other, and measuring the angle of shadows at noon on the same day at both locations. The measurement was being repeated because the exact conversion factor between the Roman stadia used by Eratosthenes and Muslim units of measure was not known.
Astrolabes or sextants were used by Arab caravans to determine latitude, but centuries earlier Ptolemy had used an astrolabe to measure latitude.
AL-BIRUNI ANGLE FROM HILLTOP TO HORIZON METHOD
Al-Bīrūnī used a much less accurate method to attempt to measure the circumference of the Earth, and then altered his test results to match the results others had gotten by a different method.
Jim Al-Khalili’s BBC documentary about the Islamic Golden Age “The Empire of Reason” repeated Muslim false claims that al-Biruni had calculated the circumference of the Earth to an accuracy of plus or minus 1%. What al-Biruni actually did was make calculations that (because he didn’t know that light is bent when it travels through air of different densities) were off by 8%, and he must have altered his data observations to obtain 39,833 kilometers, only 1% different from the result he was expecting — the then known circumference of the Earth of either 39,986 kilometers or 40,248 kilometers. (The known circumference had been calculated 200 years earlier in 820 by a team repeating the experiment the ancient Greek Eratosthenes had done of precisely measuring the angle of the shadow of the sun at noon at two different locations hundreds of kilometers north-south of each other, and precisely measuring the distance between the 2 locations.]
Al-Biruni ridiculed the team including al-Farghani that had gone on a long trek through the desert to make a calculation of the circumference of the Earth that al-Biruni mistakenly thought he could improve upon with only a simple climb up a hill with portable surveying instruments of plumb line and square, carried by his assistant.
PROBLEMS WITH AL-BIRUNI’S METHOD
As seen from a hilltop hundreds of meters high, the horizon over the sea is about 1 degree below horizontal. By using the height of the hill, the angle, and some geometry about triangles, he could calculate the distance to the horizon. Then he multiplied that by 360 degrees and divided by the angle to the horizon to calculate the circumference of the Earth.
ACCURACY OF AL-BIRUNI
We know (from Muslim attempts to measure what we now call the tilt of the Earth’s axis) that the error in attempts to measure angles typically had an error of about half a degree but sometimes may have had an error of only a tenth of a degree. Using a small portable device with a plumbline that his assistant carried up a hill, he would have had an error of at least a tenth of a degree.
Even if his primitive plumb line and square surveying instrument could measure angles to an accuracy of 0.1 degrees, he might at best have been able to obtain an accuracy of plus or minus ten percent. By averaging many observations, he could improve that accuracy somewhat.
According to wikipedia, standing on a hill hundreds of meters high results in a lens effect bending the light downward because of air being more dense at the bottom of the hill, causing an error (ignoring differences in temperature) of precisely 8% in the measurement of the distance to the horizon, regardless of the height of the hill.
Assuming he climbed up the hill in the morning on a nice sunny day and made his observations around noontime, the temperature on the hilltop heated up by the sun is unlikely to be much cooler than the temperature of the sea below, so the altitude effect on the density of air would not have been cancelled out by the temperature effect on air density. Denver, Mexico City, and Mt. Everest are known for their thin air, not thick air, even though they are cooler than nearby lower elevations.
An alternate version of the story, that he made his measurements over a flat plain rather than over a sea, would be even less accurate because a plus or minus 1% accuracy measurement of an angle to the horizon of only one degree would require the plain to have no trees and to be perfectly flat with a slope of no more than a hundredth of a degree, a condition likely to only be true of the hot desert salt flats of a dried lake bed. Desert salt flats are notorious for shimmering images of the horizon, making accurate measurements impossible.
Image by cmglee, David Monniaux, jimht at shaw dot ca, via Wikimedia Commons.
image credit https://commons.wikimedia.org/wiki/File:Eratosthenes_measure_of_Earth_circumference.svg#mw-jump-to-license