Eratosthenes’ Method (ca. 240 BCE)
He observed that:
- In Syene (modern-day Aswan, Egypt), at noon on the summer solstice, the Sun was directly overhead—vertical objects cast no shadow.
- In Alexandria, ~800 km north of Syene, vertical objects did cast a shadow at the same time.
By measuring the angle of the shadow in Alexandria (~7.2°), he inferred that this angle corresponds to 1/50th of a full circle (360°). Multiplying the distance between the cities by 50 gave an estimate for Earth’s circumference.
Why This Only Works on a Curved Surface
- On a Spherical Earth: The Sun’s rays are essentially parallel (due to its great distance). The difference in shadow angles at different latitudes is due to the curvature of the Earth. The angle difference matches the arc between the cities on a curved surface. This allows for accurate triangulation and leads to a consistent estimate of Earth’s size.
- On a Flat Earth: To produce different shadow angles on a flat surface, you must assume that the Sun is close and small, and its rays spread out like a lamp over a table (non-parallel). But this fails logically and geometrically:
- Inconsistent Shadow Angles: If the Earth were flat, the angular difference in shadows wouldn’t match the consistent ratio Eratosthenes observed across different distances. The trigonometric relationships break down completely unless the Earth curves.
- Varying Elevation of the Sun: A small, close Sun would cause very different shadow lengths at nearby locations due to perspective. Yet we don’t see such drastic differences in reality.
- Solar Angular Size Problem: The Sun would appear larger or smaller depending on how close you are to it on the flat Earth. But the Sun’s angular size is constant (~0.5°) everywhere on Earth.
- Impossible Light Path: The Sun would have to emit selectively angled rays to illuminate one location vertically while casting angled shadows elsewhere—contrary to how light behaves.
Simulation
See the interactive simulation below comparing shadow angles on a globe and flat earth model: