← All stories

The Well at Syene

True story · 3 min read

Some time in the third century before the common era, a librarian in Alexandria heard something that bothered him.

Eratosthenes was in charge of the great Library, which meant he spent his days among scrolls from all over the known world. One of them, or perhaps a traveler's account, said that in a southern town called Syene, near the first cataract of the Nile, something odd happened once a year. At noon on the summer solstice, the sun shone straight down a deep well and lit the water at the bottom. For a moment, no upright object in Syene cast a shadow.

That was interesting by itself. But Eratosthenes knew what it implied about Alexandria, because on that very day in Alexandria upright things did cast a shadow. A pole in the ground made a short but clear one.

Why would the same sun, at the same hour, make a shadow in one town and none in the other?

Greek astronomers already knew the Earth was round, partly from its curved shadow on the Moon during eclipses. What Eratosthenes saw was that the shadows could be used to measure it. The sun is so far away that its light arrives in parallel lines. But the Earth is curved, so a pole in Alexandria and a pole in Syene do not point the same way. They tilt apart, like two matches stuck into an orange at different spots.

The shadow told him how far apart they tilted. Using a bowl-shaped sundial, he measured the angle in Alexandria as about one fiftieth of a full circle, a little over seven degrees.

After that it was just geometry. If seven degrees of curve lay between the two towns, then the distance between them was one fiftieth of the distance all the way around the Earth. He needed that distance. Professional surveyors who walked and counted paces for the kingdom had a figure. As it was reported to him, it was about five thousand stadia.

Five thousand times fifty is two hundred and fifty thousand.

That was his answer for the circumference of the Earth: roughly 250,000 stadia. Earlier Greek writers had quoted figures, but none came with a method anyone could check. His is the first we can follow, and he got it without leaving Egypt, with a shadow, a well and arithmetic.

There is an honest wrinkle. We do not know exactly how long his stadion was. Different units were in use, and depending on which one he meant, his answer comes out somewhere between about thirty-nine thousand and forty-six thousand kilometers. The true figure, around the poles, is about forty thousand. Depending on the unit, he was anywhere from about two percent under to about sixteen percent over, which is remarkable for a method this simple.

Several of his assumptions were not quite right. He took Syene to lie due south of Alexandria; it is off by a few degrees. He took Syene to sit right on the line where the noon sun is overhead; it is a little north of it. And the surveyors' five thousand stadia was a rounded figure. Part of his success was careful method, and part was luck, because these small mistakes roughly canceled out.

Eratosthenes found two places where the same cause produced different effects, and let the difference do the measuring.

Words in this story: circumference, solstice, parallel, assumption, stadion