The shadow on the well

Around 240 BCE a librarian in Alexandria measured the size of the Earth using a stick, a well and some arithmetic, and got an answer within a few per cent of the modern figure.

Eratosthenes had read that at noon on the summer solstice, in the city of Syene far to the south, sunlight reached the bottom of a deep well. That meant the sun was directly overhead there, casting no shadow. In Alexandria on the same day at the same hour, a vertical stick did cast a shadow, and from its length he calculated the angle of the sun: a little over seven degrees from vertical.

The step that makes this famous is the reasoning rather than the measurement. If the Earth were flat, the sun's rays, arriving effectively parallel, would strike both cities at the same angle. They did not. The difference in angle therefore had to come from the curvature of the surface between them. Seven degrees is roughly a fiftieth of a full circle, so the distance from Alexandria to Syene had to be about a fiftieth of the way round the Earth. He knew that distance from the records of surveyors who paced out routes for a living, multiplied by fifty, and had his circumference.

The error in his result comes mostly from the distance figure, not from the method. With a better survey the same arithmetic would have been nearly exact.

What the story demonstrates is not ancient genius so much as the power of a single well-chosen comparison. He did not need to travel, and he did not need better instruments. He needed two measurements taken at the same moment in two places, and the discipline to take the difference between them seriously.