Around 240 BC, a librarian in Alexandria worked out how big the Earth is. He had no satellite, no ship that had sailed round the world, and no instrument fancier than a stick. What he had was one odd report about a well, one shadow, and a clear idea of what a shadow means.
Eratosthenes of Cyrene (about 276 to 194 BC) ran the great Library of Alexandria. Later writers, above all Cleomedes, tell the story of what he noticed. In Syene, the city we now call Aswan in southern Egypt, at noon on the summer solstice the sun shone straight down a well and lit the water at the bottom. Nothing cast a shadow. The sun was directly overhead.
In Alexandria, far to the north, the same moment gave a different picture. A vertical stick threw a short shadow. The sun was not overhead there. It stood a little to the south of straight up. Eratosthenes' own book, On the Measurement of the Earth, is lost, so we know the details only second-hand, from writers such as Cleomedes, Theon of Smyrna and Strabo.
The Sun is so far away that its rays reach us almost perfectly parallel. If the Earth were flat, a stick in Syene and a stick in Alexandria would both meet those parallel rays at the same angle and cast the same shadow. They did not. The only way to get two different shadow angles from parallel rays is for the ground to be curved, so that the two sticks point in different directions.
Better still, the angle of the shadow is the same as the angle between the two cities as seen from the centre of the Earth. A stick points straight away from the centre. Move round the globe by 7 degrees and the stick tips over by 7 degrees relative to the rays. Drag the city below and watch it happen.
Drag the second city around the globe. The top city is Syene, where the sun is directly overhead.
Here is the clever step. The angle he measured, about 7.2 degrees, is exactly one fiftieth of a full circle (360 divided by 7.2 is 50). So the distance between the two cities must be one fiftieth of the distance round the whole Earth. All he needed was that distance.
The figure that comes down to us is 5,000 stadia. It was probably taken from the records of travellers and surveyors rather than measured by Eratosthenes himself. Multiply by 50 and you get 250,000 stadia for the circumference of the Earth.
That depends on a question nobody can fully settle: how long was a stadion? Ancient Greek and Egyptian units varied, and historians still argue about it. If you take a stadion of about 157.5 metres, which is one commonly proposed value, the result is about 39,375 km. The modern figure is about 40,075 km round the equator, and about 40,008 km through the poles. That is within 2 per cent. If you take a longer stadion of about 185 metres, another figure that appears in the debate, the answer is about 46,250 km and you are 15 per cent too big. Choose your stadion below.
250,000 stadia multiplied by the length you choose.
The method was sound, but the inputs were rough. Syene is not exactly on the Tropic line, and it does not lie on exactly the same meridian as Alexandria; the real gap in longitude is about three degrees. The distance of 5,000 stadia is a round number, and so is 7.2 degrees, a suspiciously tidy fraction of a circle. At least one modern historian has argued that his angle was a little off. None of this ruins the idea. Even with rounded inputs, a stick and a well gave an answer within a few per cent of the truth, or within a sixth of it at the worst.
What matters is the pattern of thinking. He did not travel round the world. He found a place where the answer was visible in a shadow, turned an angle into a fraction, and scaled the fraction up. The same trick of measuring a small piece and multiplying is how surveyors and astronomers still work.
Next time the sun is high, put a stick in the ground and look at its shadow. You are looking at the curve of the whole planet.