AlBiruni's Classic Experiment: How to Calculate the Radius of the Earth
Abū Rayḥān AlBīrūnī, 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 AlMamun 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.
AlBiruni's Classic Method
AlBiruni 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.
First Step
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.)
Second Step
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 AlBiruni's Method
Some scholars have criticized AlBiruni'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 AlBiruni 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.
Questions & Answers
How do we calculate the angle of height of a hill?
The angle of elevation of the hill is not calculated, it is measured with the help of Astrolabe.
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© 2013 StormsHalted
Comments
The trigonometry is quite insightful. And he proof very clear.
The comments are however, bizarre. 500 yrs ago the shape and size of the earth was already well known. The shape has been known for millennia. Heliocentric v geocentric, Copernicus and Galileo, was in debate, not the shape and size pretty accurately given the tech of the times.
However, without accurate values for, or even existence of, refraction the radius would be over stated, by a lot; on the order of 15%. And the angle to the horizon is extremely sensitive. An astrolabe, without optics, it’s hard to believe the angle could be measured better than 1/10 of degree.
Still the history shows, the shape and size of the earth was known to spherical, and big, and that is a huge accomplishment.
With a theodolite today, and knowledge of refraction, and the method attributed to Biruni the radius can easily be measured to 0.25%.
My hopefully correct error analysis tells, that to get the radius of the earth to 1% accuracy you have to measure the drop angle to the horizon to this amount of accuracy. The error due to standard refraction (k = 0.14) in his configuration (horizon at about 62 km) is about 8%. Did he know about refraction or was it a coincidence to get the radius with such an accuracy?
Anyway: an estimate of R to 10% would not be bad. In no case is the earth flat.
refraction_angle_correction = arcsin( k * distance / diameter_earth ) = 0.04 degrees
Lacho your comment shows your limited level on knowledge: "Excuse me... didn't Biruni lived like a 1200 y. ago ? And didn't the globe idea became popular only 500 y. ago ? So, I guess... the question is, who is lying here, because not only that islam claim the Earth is flat, but also... ...". Nothing in your statement correct, apart from Birun's name and roughly how long ago he lived! Islam claims the earth is flat! There may be some people who might claim but 'Islam' per se makes no such claim! Besides, during the age we are talking about many mathematical and scientific advances were being made in the Islamic world since orthodoxy, the killer of new ides, had not set in, as yet. The acceptance of new ideas and the effort to better them as paramount in the minds of these people. If in doubt, read the following:
George Saliba Islamic Science And The Making Of The European Renaissance
archive.org/details/GeorgeSalibaIslamicScienceAndTheMakingOfTheEuropeanRenaissanceTransformationsStu
Arabic mathematics : forgotten brillia
wwwgroups.dcs.stand.ac.uk/history/HistTopics/Arabic_mathematics.html
Assessing Arabic contributions to the sciences
math.ucr.edu/~res/math153/history06a.pdf
All guys are from my land UZBEKISTAN though none knows about it much due to USSR. We were the best Scientist of the world but after Russian colonization everything became too unattractive here.
In the advancement of all aspects of astronomy and geography, stands out the dominating figure of Abu Raihan Muhammad ibniAhmad AlBiruni (9731048AD) who was one of the very greatest scientists of Islam. Once he wrote, I do not scorn to accept truth from whatever source I can find. Having a command over the Sanskrit language he exploited the best sources of Indian sciences including mathematics, astronomy and chronology. He described the earth, its axis and its movements, and threw much light on the Indian contribution to the general geography of India. The teory of the movement of th earth was apparently borrowed by AlBiruni from Arya Bhatta, who in the fifth century AD suggested that the earth revolved round the Sund and rotated on its axis. His QanunalMas’udi (compiled in 1030 AD) is the most important work on astronomy which is largely based o Indian astronomical ideas. He also utilized the knowledge of Indian geology and minerology. Besides, AlBiruni translated Surya Siddhanta of Varahamihira. To this period also belonged the celebrated scientist, astronomer and physician Abu Ali Sina (AH 428/1036AD) who advanced the knowledge of physics and astronomy with the help of Indian and Greek sources.
Y’aqubi (AH 234/897AD) the famous Arab historian refers to Indian achievements in these words:
The Indians are men of science and thought. They surpass all other people in every science; their judgement on astronomical problems is the best. In the science of medicine their ideas are highly advanced. There are a large number of books which deal with their principles. And they have a large number of other books which are too many to be mentioned. (source: India's contribution to world thought and culture. Chapter name: India's contribution to Arab civilization. Page no. 583)


So interesting. I watched a documented film that included an abstracted information on the Great scientist. Then, I did not understand the Scientist's methodology, then I tried to find a way to calculate earth radius. I have studied mathematics till the end of high school and some further during BSc studies. Then, I thought. I have found a simpler method with a simpler formula. The method is different from that used by the Great Iranian Scientist Abu Reyhan Biruni.
Excuse me... didn't Biruni lived like a 1200 y. ago ? And didn't the globe idea became popular only 500 y. ago ? So, I guess... the question is, who is lying here, because not only that islam claim the Earth is flat, but also... there was no indication of the Earth as a globe 1200 y. ago ?
I find this particularly interesting since I've been looking at flat earthers lately, and the number of flat earthers has been growing.
I do have to respond to Alan, though. The Church knew very well that the earth was round. It's a myth that medieval thinkers thought the earth was flat, spread by Enlightenmentera thinkers who wanted to contrast their enlightened and humanistic age with the "dark ages" that came before. We've gotten a lot of notions from them that persist to this day.
But that makes the modern flat earthers even more of a riddle, since it's a form of biblical literalism for them; young earth creationism just isn't literal enough.
I wonder if the Foshay Tower is tall enough to reasonably reproduce AlBiruni's measurement. I will try.
Albiruni did first to calculation of the Earth's in Tilla Jogian  The highest peak in the Eastern Salt Range in Province of Punjab Pakistan
Iran is the modern name for Persia.
I am fascinated that even in the 10th Century, people knew the Earth was not flat!
Could it be that we have all been mislead by the Church of Rome?
The equation is useless if you don't explain how do you get the ALPHA value?
Actually, no one has said that Biruni was from Pakistan. He carried out his calculations while in India. This part of the country is in modern day Pakistan.
Interestingly that people from Pakistan are trying to say that Biruni was from Pakistan. He was neither Arab or from Pakistan but he was a Persian scientist. Please read https://en.wikipedia.org/wiki/AlBiruni
Of course, as we nowadays publish our work in English, during that era it was common to publish in Arabic. He was from NorthEast Iran and because Pakistan and Afghanistan were part of the country he had traveled to those places too.


it is a wonderful palace to take information
Fascinating account. That 10th century scholars even had the confidence to think they could determine the radius of the earth is astounding. One suggestion: your first statement of the law of sines had me befuddled since it uses A and C to represent both angles and points. Other than that, it was a clear and very interesting presentation.
Pakistan is a country I have visited and like very much and the people too. I have been doing some research into the area and there is a lot to learn. So much has happened in your country over the centuries. It has a rich history and you can be proud of this.
Did you know that Albiruni did this work in what is now Pakistan? He used the mountain at Nandana for his calculations and he studied Sanskrit at the ancient university at Katas Raj. All these places are near the Khewra Salt Mine in the mountains of the salt range.
I have been to the salt mine but did not know the history of the area at the time. As you live in Karachi you could make a trip there.
I look forward to reading more of your hubs.
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