# Al-Biruni's Classic Experiment: How to Calculate the Radius of the Earth?

Al-Biruni a pioneering Muslim scientist figured out a truly remarkable and ingenious method to calculate the radius of the earth (and subsequently its circumference etc.). This 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 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 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 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:** Calculation of the height of a mountain requires three measurements.

- Angle of elevation of the mountain top at two different points lying on a straight line were measured using an astrolabe. Biruni probably had a much larger astrolabe then that illustrated on the right to ensure maximum accuracy close to two decimal places of a degree.
- The third being the distance between these two points 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:** 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 (sea or flat enough) 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 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.

**© 2013 StormsHalted**

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## 10 comments

Did you know that Al-biruni 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.

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.

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.

it is a wonderful palace to take information

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/Al-Biruni

Of course, as we now-a-days publish our work in English, during that era it was common to publish in Arabic. He was from North-East Iran and because Pakistan and Afghanistan were part of the country he had traveled to those places too.

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.

The equation is useless if you don't explain how do you get the ALPHA value?

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?

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