Around 240 BC, a librarian in Alexandria measured the whole Earth without leaving Egypt. His tools: a stick, a shadow, a distance, and one very good assumption.

The clueA city where noon has no shadow

Eratosthenes of Cyrene (about 276–194 BC) ran the great Library of Alexandria. He knew a curious fact about Syene, the town we now call Aswan, far up the Nile to the south. At noon on the summer solstice, the Sun stood straight overhead there. Upright sticks cast no shadow at all. Later writers even told of a well whose water was lit all the way to the bottom.

In Alexandria, on the same day at the same moment, a vertical stick did cast a short shadow. Same Sun, same instant, different shadow. Eratosthenes realised that difference was the Earth's curve, made visible.

Try itWalk a city around the globe

Drag Alexandria along the Earth's surface. The Sun is so far away that its rays arrive parallel. Watch the shadow grow, and notice that the shadow angle at the stick is exactly the angle between the two cities at the Earth's centre.

The mathsOne fiftieth of a circle

At Alexandria, Eratosthenes found the Sun's rays were tilted from vertical by about one fiftieth of a full circle, roughly 7.2°. Here is the trick. Because the rays are parallel, a line through the stick and on to the Earth's centre crosses them at the same angle as it meets the line from Syene. So the angle in the shadow is also the slice of the Earth between the two cities.

If that slice is 1/50 of the circle, then the distance between the cities is 1/50 of the way around the world. The accepted distance was 5,000 stadia, reportedly paced out by professional step-counters called bematists.

5,000 stadia × 50 = 250,000 stadia

Later sources quote 252,000, possibly rounded so it divides neatly into 60ths. The method, not the exact number, is what made it famous. Eratosthenes' own book is lost; we know the method mostly from a simplified retelling by the astronomer Cleomedes.

The catchShadows alone can't prove the Earth is round

Flip the toggle above to Flat Earth, nearby Sun. A flat ground lit by a Sun only a few thousand stadia up produces exactly the same pair of shadows. The measurement cannot tell these two worlds apart.

What settled it was the assumption Eratosthenes brought with him: the Sun is very far away, so its rays are parallel. Greek thinkers already had good reasons to believe the Earth was a sphere. Aristotle, a century earlier, pointed to the round shadow Earth casts on the Moon during every lunar eclipse. Given a sphere and a distant Sun, the shadow gives you its size.

How close?It depends on what a stadion was

How good was 250,000 stadia? Nobody knows exactly, because the stadion was not one fixed length. Different Greek and Roman stadia ranged from roughly 157 to 185 metres. Pick one and compare with the real distance around the Earth through the poles, about 40,008 km:

He also made errors that partly cancel. Syene is not exactly on the Tropic of Cancer, and it is not due south of Alexandria but a few degrees of longitude to the east. Still, whichever stadion he meant, he landed within a few percent to about 15% of the truth, with a stick and a shadow.

Why it mattersThe first time anyone weighed up a planet

Before Eratosthenes, the size of the world was a guess. After him, it was a number with a method behind it that anyone could repeat. That is the real legacy: two observations, one assumption stated out loud, and a bit of geometry turned a shadow into a planet.

You can repeat it yourself. Schools in different cities still do, comparing the length of noon shadows on the same day and sharing their results.