How a muslim scholar measured the radius of the Earth

          Ever wondered how was the size of the Earth measured? Well, here's a way how.
          I was quite amazed by the this documentary film entitled "Islam and Science" (part 2) released by BBC especially when Prof. Jim Al-khalili explained how a muslim scholar measured the radius of the Earth. Prof Al-khalili is a physics professor of University of Surrey. This film really suites him for he understands and can speak the Arabic language and it was advantageous for him because he had to delve into Muslim history and Arabic books.

          We, muslims, are required to face a direction towards Kaaba whenever we pray. And that leaves no excuse to anyone far from Kaaba. Wherever country a muslim resides, he should always find that direction. And this eventually led to the measuring of the Earth's radius.

Professor Al-khalili explained:

  "... A more reliable and sophisticated method for estimating the Earth's size was needed, and two centuries after Al-Ma'mun died, it came. What made it possible was a great leap of imagination. And the fact that, by 900 AD, much of the world's mathematical knowledge had been translated into Arabic, so scholars could scrutinize and improve on it. 

  Out of this obsession with scholarly learning came a true mathematical visionary - Abu Rayhan Muhammad Ibn Ahmad Al-Biruni. And like all Islamic scholars of the time, Al-Biruni was obsessed with the science and mathematics of the ancient Greeks, Babylonians and Indians. And because of the success of the Translation Movement, he had literally on his desk the great work on geometry by Euclid, Ptolemy's Almagest, the Indian text the Sindhind, and the famous work on algebra by Al-Khwarizmi."

Let me dissect this subject a little bit further. The way I see it, we need three things here:
1. astrolabe (this is just like a large protractor)
2. distance measuring device (in our present time, we can use the rolling Measuring wheel), and
3. knowledge on basic trigonometry.

So let’s try to follow what Birani did in measuring the radius of the Earth. Here we’ll use the most basic ideas in trigonometry. Okey, here we go. Find a fairly high mountain in which we can see the vast flat horizon, the sea. What I can never forget in this is that with few simple measurements around this mountain, you can measure the size of our whole world.

Here's the step made by Biruni.
Height of the mountain

1. Al Biruni’s first step was to work out the height of the mountain. He did this by going to two points at sea level and knowing the distance apart and them measuring the angles from these points to the mountain top. The two points and the mountain top must line in one straight line. In measuring the angle, he had to use an   Astrolabe.

Prof. Jim Al-khalili using the astrolabe in measuring the angle of inclination of the mountain top.

Angle of inclination from the two points are measured and the distance between them is denoted by d in this figure.

2. We may consider first the triangle on the right side (drawn by yellow lines). We may use the Sine Law to solve for the slanted side l.

3. Then, we consider the right triangle on the left. We can use the sine function to solve for the height h of the mountain. (Refer to the diagram above, the right triangle whose hypotenuse is the side l.)

Now, one more measurement is needed to get the size of the Earth.

Radius of the Earth

4. To get the last measurement, one must climb to the top of the mountain to measure the angle of the line of sight to the horizon as it dips below the horizontal.

As you can see from the diagram, the line connecting the mountain top and the point where the line of sight reaches the horizon is perpendicular to the line connecting center of the Earth and the the point where the line of sight reaches the horizon. And the distance between the center of the Earth and the mountain top is equal to the sum of the radius of the Earth and and the height of the mountain (h+R).

5. We now have a very huge right triangle where the hypotenuse is equal to the sum of the radius of the Earth and and the height of the mountain. 

and by using the identity

would eventually lead to the expression:

The accuracy of this measurement is remarkable achievement for someone 1000 years ago!

EXERCISE

If you wanna try the steps described above, here's a list of values.

The distance (d) between two points is 100 m. The angle of inclination from one point near the seashore is 24.5 degrees while the angle of inclination from the point nearer to the mountain is 26.5 degrees. From these three given values, you can calculate the height of the mountain.

The angle of declination is 0.5 degrees as measured when one is on the mountain top.

(These values were measured by Prof. Al-khalili. And following the steps described above obtains a value of the radius of the Earth far greater than the known value. Big error isn't it? Well, Prof. Al-khalili did the measurements only once. But the measurement of Al-Biruni was more accurate than this because he did it again and again.