Eratosthenes measured Earth's size from two shadows

By the 3rd century BCE, educated Greeks accepted the Earth was a sphere, but nobody had a way to say how big it was — the obvious method, circumnavigating or otherwise physically traversing the whole planet, was wildly beyond the technology and organization of the era. Eratosthenes, running the Library of Alexandria, had access to travelers' reports and geographic records but no instrument capable of surveying a curve that large directly.

He had one useful fact on hand: at noon on the summer solstice in Syene (modern Aswan), sunlight reportedly reached straight to the bottom of a deep vertical well, meaning the sun sat directly overhead there at that exact moment. Nobody before him had thought to pair that isolated travelers' curiosity with a second, ordinary measurement taken somewhere else at the same moment.

On the same solstice noon, Eratosthenes measured the angle of the shadow cast by a vertical stick (a gnomon) in Alexandria and found it about 7.2 degrees off vertical — Syene's well proved the sun was at zero degrees there. Since Alexandria and Syene lay roughly on the same meridian, that 7.2-degree gap was exactly the curvature of the Earth between the two cities, one-fiftieth of a full circle; multiplying that fraction by the distance between them, paced out by professional bematists at about 5,000 stadia, gave the Earth's total circumference.

Two local shadow angles at the same solstice noon on one meridian: the 7.2-degree difference, one-fiftieth of a circle, times the paced distance, gave the circumference.

Sunlight arrives parallel, so angle differences between two points at the same time come only from surface curvature.

A 7.2-degree angle difference is one-fiftieth of a circle, giving a simple fraction to multiply by distance.

Using a well-known zero-angle point and a local stick measurement avoided any need to traverse the planet.

When a quantity is too large to measure directly, find two cheap readings that differ only because of that quantity, and compute it from their difference.

Eratosthenes' result of roughly 250,000 stadia converts, depending on which ancient stadion length is used, to somewhere between about 39,000 and 46,000 km against a true equatorial circumference of 40,075 km — a result within a few percent of correct, obtained with a stick, a well, and a distance paced out on foot.