Eratosthenes: Measuring the Earth's dimensions

Hi!

Today we are going to talk about one of the most astonishing accomplishments in the history of science: The calculation of the measures of the Earth. This event took place in the ancient Greece by the astronomer, philosopher, mathematician and geographer Eratosthenes, approximately in the year 250 BC.

Everything began in the city of Syene, Egypt (now known as Aswan), when Eratosthenes heard about a well in which the sun was reflected in the water at noon in the solstice of summer, June 21. For this to happen the sunlight beams should reach the surface of the Earth completely perpendicularly. As a consequence of this fact, that day at that time there were no shadows.

On the same day and at the same time, but in the city of Alexandria, Eratosthenes realized that a tower cast a shadow. By parsing this fact he realized that this was possible due to the spherical shape of the Earth. A simple sketch will explain why.

The only thing he had to do to calculate the dimensions of the Earth, was to obtain two data. First of all, the angle formed by the projected shadow in the city of Alexandria and secondly the distance between the two mentioned cities. The first one turned out to be 7⁰ 12’. It is not clear how he was able to obtain that value, but a theory says that he measured the length of the shadow and the height of the tower, and with the proportion between these two measurements and by using a tangent value table he managed to calculate the angle. The second datum he needed was measured by a servant by counting the steps from one city to the other. As a result the distance was 800 km altogether.

The sketch shows a geometrical construction with the values before obtained. By likeness the angle formed by a beam of sunlight and the tower of Alexandria  must be the same as the angle formed in the center of a circle (the Earth) between a sunlight which passes for Syene and an imaginary vertical line originating from the tower in Alexandria (see the sketch). This means that an angle of 7º 12’  is equivalent to a length of 800 km. If the Earth, as a sphere, has a ratio of 360º, it is very easy to calculate the length of the Earth by using a simple proportion.

The most recent and accurate value is 40.091 km. The difference between both data is just 0,2%, which is insignificant. It’s stunning how accurate the value given by Eratosthenes is, considering the few means exited by that age.

Finally, in order to find the radius of the Earth, we just have to use the formula of the length of a circle and use the datum before given.

Again, the value considered real or exact is 6.378 km, really close to the Eratosthenes’ datum.

It’s, at least, very curious how simple it is to find the dimensions of the earth, it is necessary two values and do a very easy calculation, and even more interesting how exact the result can be. It is without any doubt a historic landmark.

I hope you have enjoyed this article. Any comment would be well received. By the way, you can also read this article in Spanish Bye folks!

Bibliography:

• www.wikepedia.com
• www.khanacademy.org/partner-content/big-history-project/solar-system-and-earth/knowing-solar-system-earth/a/eratosthenes-of-cyrene