📜 Measuring the Earth with a Stick and a Shadow

Around 240 BC the Greek scholar Eratosthenes, head of the Library of Alexandria, heard that at noon on the summer solstice a well at Syene (modern Aswan) had no shadow at all — the Sun stood directly overhead. On the same day, in Alexandria far to the north, a vertical gnomon still cast a short shadow. Because sunlight arrives from so far away that its rays are effectively parallel, that difference in shadow angle could only be explained by the curvature of the Earth — and its size could be calculated from simple geometry, without leaving the ground.

How it works

Both cities sit close to the same meridian. The angle Δθ between the two gnomon shadows equals the difference in latitude between the cities — and, on a sphere, that same angle is subtended at the Earth's centre by the arc distance d between them. Since a full circle is 360°, the whole circumference follows from a single ratio:

• Δθ — the measured difference between the shadow angles at the two gnomons.
• d — the ground distance between the cities, measured along the meridian (Eratosthenes used the travel time of camel caravans).
• C = 360° / Δθ × d — the Earth's circumference.

His result — about 250,000 stadia — lands within a few percent of the true circumference of roughly 40,030 km, an extraordinary feat for a method that used nothing but sticks, shadows and geometry. Adjust the season slider to see the shadows change through the year, move Alexandria to change the city separation, and dial in a distance-measurement error to see how sensitive the ancient method was to that one uncertain input.