Measuring the Earth (Greece) - wiki: the discovery

Eratosthenes had heard from travelers that in Syene, in southern Egypt, on the day of the summer solstice at noon the sun was so high in the sky that its reflex fell down a well and that a stick (gnomon in Greek) planted vertically in the ground made ​​no shadow on the ground. In the city of Alexandria, it was not the same: looking at the obelisk, Eratosthenes discovered that the day of the summer solstice at noon the obelisk had a shadow, same as a gnomon planted vertically into the ground. Eratosthenes was thus faced with a terrible conundrum: on the same day, at noon, no shadow in Syene and a shadow in Alexandria. Why? Two logical hypotheses were available then for Eratosthenes:

  • First hypothesis: the Earth is flat, the Sun is very close, and, thus, its rays strike the ground at different angles.
  • Second hypothesis: the Sun is far away from Earth and, therefore, its rays are nearly parallel. If the surface of the Earth is not flat but curved, then the difference in shadow between Syene and Alexandria can be explained. And if the Earth's surface is curved, the Earth itself has to be round. Eratosthenes chose this second hypothesis.

Many philosophers had already advanced this hypothesis in the past. Pythagoras had pointed out that a sailor perched on the mast of his boat could see farther away than the horizon of a seaman standing on the deck. Similarly, Aristotle had also noticed that during the lunar eclipse the shadow of the Earth was round, which seemed to indicate that the Earth was not flat. Eratosthenes eventually supplied the irrefutable evidence, thanks to his sense of observation and the power of his intellect.

If the Earth is round, two gnomons, planted one in Alexandria and one in Syene, must intersect at the center of the Earth, once we imagine to extend them. Eratosthenes reflects that, since the Sun's rays are parallel, the angle "alpha" in the center of the Earth between the obelisk of Alexandria and the well of Syene must be equal to the angle made by the shadow of the obelisk of Alexandria. The angles "alpha" are the two interior angles formed by a transversal crossing parallel lines – do you remember that? It should awake a souvenir of your primary school years!

Thales of Miletus lends a hand

From that bright geometric reasoning, everything comes together very fast for Eratosthenes, who, in 205 BC, decides to measure the angle "alpha". To do this, Eratosthenes uses the theorem attributed by oral tradition to one of the seven sages of ancient Greece, Thales of Miletus. For Thales of Miletus too had wandered in Egypt, where he was said to have managed to calculate the height of the Great Pyramid of Cheops, without measuring it directly. Thales would have noticed that the ratio between the size of a gnomon and its shadow was exactly the same as the ratio of the shadow of the pyramid and its height. This finding was handed down to posterity as the "theorem of similar triangles" – that you learned in elementary school under the name of “Thales' theorem”. This is probably the first mathematical theorem in the history of humankind.

Eratosthenes’ breathtaking demonstration

 Armed with this precious theorem, Eratosthenes can then tackle the measurement of the circumference of the Earth. He discovers that on the day of the summer solstice in Alexandria, at noon, the angle "alpha" is 7 degrees - equivalent to the fiftieth part of the circumference of a circle. Based on the length of the travel of camel caravans, he has estimated that Syene and Alexandria are separated by 5,000 Egyptian stages (or 157.5 m). All Eratosthenes has got left to do is to multiply 5000 by 50 stages to get to the result that the Earth’s circumference is 250,000 stadia (or 39 735 km). Give or take 10%, this is actually the circumference of the Earth. Impressive ... The error comes mainly from the fact that the exact distance Syene-Alexandria is not 5000 stadia, or 787.5 kilometers, but 701 km. But hey, we won’t quibble!

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