Al-Biruni's Classic Experiment: How to Calculate the Radius of the Earth
Abū Rayḥān Al-Bīrūnī, a pioneering Muslim scientist figured out a truly remarkable and ingenious method to calculate the radius of the earth (and subsequently its circumference). This method was very simple yet accurate, requiring just four measurements in all to be taken and then applying a trigonometric equation to arrive at the solution. What Biruni figured out with unprecedented accuracy and precision in the 10th century was not known to the west until the 16th century.
The need to calculate the size of earth was first felt when the Abbasid Caliphate spread far and wide from Spain till Indus river in modern day Pakistan. Muslims are required to pray facing the direction of the Kaaba and being far away from Kaaba does not spares one from this obligation. So no matter how far Muslims were from the Kaaba they needed to determine its exact direction to pray. To do this accurately they needed to know the curvature of the earth and knowing this demanded that they know the size of the earth. By the way the Caliph was also curious to know the size of his empire!
Abbasid Caliph Al-Mamun thus employed a team of renowned scholars of that time and assigned them the task of calculating the size of the earth. They started by finding the distance over which the sun's angle at noon changed by 1 degree, multiply it by 360 and you arrive at the circumference from which size can be deduced. They arrived at a value which was within 4% of the actual value. The problem with this method was that it was cumbersome to measure large straight line distances between two points in the heat of the desert and perhaps they only had to count paces to measure it.
Al-Biruni's Classic Method
Al-Biruni devised a more sophisticated and reliable method to achieve this objective.
To carry out his method Biruni only needed three things.
- An astrolabe.
- A suitable mountain with a flat horizon in front of it so that angle of depression of horizon could be accurately measured.
- Knowledge of trigonometry.
The first step in Biruni's method was to calculate the height of the mountain. This calculation utilizes three of the total four measurements required.
- First two being the angle of elevation of a mountain top at two different points lying on a straight line.
These were measured using an astrolabe. Biruni probably had a much larger astrolabe then that illustrated above to ensure maximum accuracy close to two decimal places of a single degree.
- The third measurement was the distance between these two points. This was perhaps found using paces.
These values were then computed with simple trigonometric techniques to find the height as shown in the figure above. This is a relatively simple and easy to understand problem, I even used to solve these types of problem back in school! Biruni used the following formula: (For the purpose of simplicity lengthy derivation is omitted.)
The second step in his method was to find the angle of dip or angle of depression of the flat horizon from the mountain top using the astrolabe in the same way. This being the fourth measurement. It can be further seen from the diagram that his line of sight from the mountain top to the horizon will make an angle of 90° with the radius.
And finally we come to the useful bit, the ingenuity of this method lies in how Biruni figured out that that the figure linking the earth’s center C, the mountaintop B, and the flat horizon S was a huge right triangle on which the law of sines could be made to yield the earth’s radius!
Now we can apply the law of sines to this triangle to find the radius R.
So Exactly how Accurate was Biruni ?
With his formula Biruni arrived at the value of the circumference of the earth within 200 miles of the actual value of 24,902 miles, that is less then 1% of error. Biruni's stated radius of 6335.725 km is also very close to the original value.
Criticism on Al-Biruni's Method
Some scholars have criticized Al-Biruni's method that it is not as remarkably accurate as claimed. While the mathematics generally seems correct and genuine at first place, scholars have expressed concern over the facts that:
- Measurements were converted from cubits to modern units to arrive at the quoted answer. It is therefore the conversion factor from cubits to modern units which is claimed to be ambiguous. It is also unclear as to which version of cubits Al-Biruni used.
- It is not possible to precisely measure the angle of depression of horizon due to the underlying physical phenomenon of refraction. Refraction may displace the image of horizon as viewed by an observer from distance (mountain top) from its actual position because of light passing through different layers of air.
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