Science Puzzle
How Did Eratosthenes Measure the Earth?
Around 240 BC, Eratosthenes, the head of the great library at Alexandria in Egypt, heard that at noon on midsummer's day in Syene, far to the south, upright posts cast no shadow: the Sun was straight overhead. At the same moment in Alexandria, an upright post did cast a shadow, showing the Sun was 7.2 degrees away from straight overhead.
Syene was roughly 800 kilometres south of Alexandria. From these facts, Eratosthenes worked out the distance all the way round the Earth. What did he get?
The Answer
About 40,000 kilometres, very close to the real figure. 7.2 degrees is one fiftieth of a full circle, so the distance from Syene to Alexandria must be one fiftieth of the way round the Earth: 50 × 800 = 40,000 kilometres.
The key idea is that the Sun is so far away that its rays arrive at the Earth parallel to each other. If the Earth were flat, those parallel rays would strike every upright post at the same angle, and posts everywhere would cast the same shadow. They don't, because the ground curves. The difference between the angles at two places is exactly the angle between the posts themselves, measured at the centre of the Earth.
Eratosthenes measured distance in stadia, and he gave the distance as 5,000 stadia and the Earth's circumference as 250,000. Historians still argue about how long his stadium was, so we can't be sure exactly how close he got. Either way, with a stick, a shadow and some geometry, he measured the size of the whole planet more than 2,200 years ago, and his method was right.
The principle: Measuring a planet with geometry. Sunlight arrives in parallel rays, so the difference in the Sun's angle at two places shows what fraction of the way round the Earth they are apart.