Measuring Earth's Radius à la al-Biruni.

Recently I got interested in the story of a great Central Asian polymath Abu Rayhan al-Biruni (known as Albironius in the west). Biruni lived in the 10th and 11th centuries AD and contributed to numerous scientific disciplines that I am not going to list here, but which can be easily found on the web.

The way I approached the topic was via the book Brilliant Biruni by M. Kamiar. Except for a few supremacist statements embedded, the book is very easy to read and provides an interesting account of Biruni's life.

One thing I particularly enjoyed was the way Biruni applied abstract geometry developed by the Greeks to the real world problems. In particular, Biruni provided a very elegant way of calculating the earth's radius.

Biruni's method proceeded in two steps. First, Biruni chose a mountain and measured its height by choosing two points at the sea level with known distances between them (Fig 1.)

Fig 1: Measuring the height of a mountain

Suppose that we know the angles and the distance d in the Fig 1, then we can easily determine the height with the following formula

Eq 1: Computing the height

In practice, Biruni used an astrolab to measure the angles. Then, Biruni climbed the mountain and used the astrolab again to measure the angle to the horizon (Fig 2.)

Fig 2: Measuring the earth's radius

Then finally he could compute the radius via the following formula

Eq 2: Computing the radius

In 10th century Beruni obtained the number 6339.6 km, which is remarkably close to the equatorial radius of 6378 km.

Note that an astrolab is just another name for an inclinometer, which can be easily assembled at home by combining a protractor, a string, a straw, and some weight [click here].

So, here it is a brilliant application of trigonometry by a famous mathematician whose works played a significant role in our development. To conclude I quote a well known joke for mathematicians:

Mathematicians never die - they only lose some of their functions.

Talk to you soon!