MEASURING THE EARTH’S CIRCUMFERENCE WITH A STICK: THE ERATOSTHENES METHOD

With space probes, it is very easy to compute the dimensions of the Earth with great accuracy. The first measurements of the circumference of our planet, however, date back to the third century BC, thanks to the work of Eratosthenes who used a simple idea to obtain a result nothing short of incredible for the times in which he lived.

The Ancient Greeks were strongly convinced of the sphericity of the Earth, but despite this nobody before Eratosthenes had managed to devise a method to calculate its dimensions. To obtain his measurement, the Greek astronomer used only a gnomon, that is, a stick planted vertically in a flat ground, from which the shadow of it could be followed the movements of the Sun during the year.

Eratosthenes realized that in Siene (present-day Aswan, at the time an ancient Egyptian city) the Sun at noon on the summer solstice did not cast shadows. This is because Siene, indicated with S in the figure, is almost on the Tropic and therefore at the solstice the Sun is at the zenith. The stick planted in the ground in Siene on this specific day of the year therefore did not cast a shadow.

Eratosthenes then knew that Alexandria (A in the image) was north of Siene on the same meridian at a distance of 5000 stages. The astronomer realized that by measuring the shadow projected by a stick in Alexandria on the day of the summer solstice he would be able to obtain the angle subtended between the direction of the sun’s rays and the stick (the alpha angle). This angle, for geometric construction, was equal to that which separates Alexandria from Siene with respect to the center of the Earth. This angle turned out to be 7.2.

Knowing then the angle and the physical distance D between Siene and Alexandria through a simple proportion Eratosthenes was able immediately to calculate the terrestrial circumference, turned out to be 250 thousand stages, equivalent to 39 000 km, an incredibly similar value to the real one of 40 075 km. The slight difference between the two results is due to some factors, in particular to the fact that the shape of the Earth is not a perfect sphere as thought by Eratosthenes. Moreover, the distance between Siene and Alexandria is slightly different from that used by Eratosthenes and the two cities are not exactly on the same meridian. 

The Eratosthene method shows how from a simple idea you can get incredible results, not having any tools available, except that of your mind. 

Credit: via Wikipedia.

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