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!

Get wealthy to get wealthier yet

Few days ago while scanning through the morning news I came across an article in Le Temps. The article was discussing the increasing income inequalities in the US and presented the data as a nice graph. The graph referenced the (previously unknown to me) site The World Top Income Database.

After an (unsuccessful) attempt to contain my curiosity , I decided to try out the database for myself and plotted the following nice graphs using the web-sites practical Graphics interface.

Fig 1: Comparison of the income share of the super-rich.

Fig 2: Income shares of rich (r.), super-rich (b.), and super-super-rich (g.).

Fig 3: Average incomes of rich, super-rich, and super-super-rich.

The conclusions we can draw from the graphs above are quite remarkable. First of all, while the share of the total income of the super-rich (top 1%) seems to be quite stable in France and Japan, it seems to follow a different "law" in English speaking Australia and the US. The growth of the income of the super-rich seems to be the most accentuated in the US, where since 1986 it has been growing larger within the income pie.

From the Fig. 2 and 3 we can draw more conclusions. While the incomes of the rich (top 10%) are growing steadily, the incomes of the super-rich are growing even faster. According to Le Temps article, although during 1990s economic expansion (significant) 45% of total growth in income was captured by the super-rich, during the presidency of George Bush (2002-2007) this number rose to a whopping 65%. On top of that the super-super-rich (top 0.1%) have been doing better yet. Between 1990-2005 they claimed 2% of the 2.3% increase in the total share of the super-rich.

Well, to conclude I cite the quote from the article attributed to Ernest Hemingway:

The super-rich are very different from the rich. They have even more money.

Have a great Friday!