Around 240 BC, in the great library at Alexandria, a Greek scholar called Eratosthenes heard about a well. In the southern town of Syene (modern Aswan), people said, on the day of the summer solstice the noon Sun shone straight down it and lit the water at the bottom. No shadow anywhere. Yet in Alexandria, on that same day, a stick still threw a clear shadow.
Most people would shrug. Eratosthenes saw a tape measure for the whole planet.
The Sun is so far away that its rays reach the Earth almost perfectly parallel. If the ground were flat, a stick in Alexandria and a stick in Syene would cast identical shadows at the same moment. They did not. So the ground is not flat: it curves away between the two towns, and the shadow tells you by how much.
Try it. The diagram below is a slice through the Earth. Sunlight arrives from above, and Syene sits at the top, where the stick points straight at the Sun. Drag the second stick around the globe and watch what its shadow does.
Drag the left stick (or tap anywhere on the left half of the globe). Arrow keys nudge it when it is focused.
Here is the key. The stick at Syene points at the centre of the Earth, and so does the stick at Alexandria. The angle between the two sticks at the centre equals the angle the Sun's rays make with the Alexandria stick, because the Sun's rays are parallel. The shadow is a protractor made of light.
In the story, Eratosthenes measured the Alexandria shadow at about one fiftieth of a full circle, 7.2 degrees. That is the same slice of the whole Earth. So the distance from Alexandria to Syene must be one fiftieth of the distance around the planet.
The distance was known from travel records, and checked by professional surveyors who paced out the roads: 5,000 stadia. Multiply by 50 and you get 250,000 stadia. Our source for all this is a later writer, Cleomedes, who retold the method in a popular book; Eratosthenes' own book on the subject is lost. A later Roman author, Pliny, quotes 252,000 stadia, a number that divides evenly by every whole number from 1 to 10, which suggests it was tidied up.
Try the same sum in the lab: drag the stick to a larger angle, and the same 5,000 stadia now means a much smaller Earth. A small angle means a big planet. Everything rests on measuring that shadow angle well.
It depends on what a stadion was. Greek cities used several lengths, and nobody is certain which one Eratosthenes meant. With a stadion near 157.5 metres, 250,000 stadia comes to about 39,400 km. The Earth's real circumference round the equator is about 40,075 km. With a 185-metre stadion the answer is about 46,000 km, roughly 15% too large. Flip the switch in the lab to see both.
Either way, a man with a stick, a travel record and some geometry got the size of the planet within a few percent, or at worst within about fifteen. And he did it knowing nothing of satellites.
The method is cleverer than it is exact. Syene is not quite on the Tropic of Cancer: it lies a little north of it, so the Sun was never exactly overhead. Syene is also not due south of Alexandria; it is about 3 degrees of longitude to the east. The real straight-line distance between the two is roughly 840 km. These errors partly cancelled out, and the lucky result is part of why the story sounds so neat.
But the idea needs none of that luck. Two shadows, one distance and a fraction are enough. You can still do it today: find a friend in a town a few hundred kilometres north or south of you, measure the shadows of two upright sticks at local noon on the same day, and divide.