At noon on the longest day, the sun was directly overhead in Syene, so a vertical stick there cast no shadow. At the same moment in Alexandria, a stick cast a shadow whose angle Eratosthenes measured at about one fiftieth of a full circle, a little over seven degrees.
If the sun’s rays are parallel, that angle is the angle between the two cities as seen from the centre of the Earth. So the distance between them is one fiftieth of the way round.
The distance was known from the time it took to walk it: about 5,000 stadia. Fifty times that gave 250,000 stadia for the whole circumference. Depending on the length of a stadion, that is within a few percent of the true figure, about 40,000 kilometres.
1. The Idea Behind It
The whole calculation is a fraction of a circle scaled up: if one fiftieth of the way round is 5,000, the way round is 250,000. A student who can find a whole from a part has the maths Eratosthenes used.
What he added was the courage to believe the geometry over the size of the thing being measured.