How Eratosthenes Estimated Earth’s Size

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The speakers explain the classic Eratosthenes calculation: compare the Sun’s direction at two locations, estimate the distance between them and scale up by the fraction of a circle. The geography and parallel-ray assumptions are approximate.

1Parallel raysNearly parallel sunlight provides a common direction at both locations.
2A central angleThe solar angle approximates the central angle between the locations.
3Scaling the distanceA measured distance and the fraction of a circle yield an estimated circumference.

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Nearly parallel sunlight provides a common direction at both locations.
The solar angle approximates the central angle between the locations.
A measured distance and the fraction of a circle yield an estimated circumference.

トピックMathematics

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Sunlight and the circumference

A simple geometric model turns two observations into an estimate.

The speakers recall the account of Eratosthenes measuring Earth: at Syene in Egypt, sunlight could shine nearly straight down a well around the summer solstice, while the Sun was not directly overhead in Alexandria.

The geometric model assumes Earth is approximately spherical and the Sun is far enough away that its rays arriving at the two locations are nearly parallel.

Around the summer solstice, the subsolar latitude is near the Tropic of Cancer. The story treats Syene as close enough to that latitude for the Sun to be almost overhead at local noon; this is an approximation.

In Alexandria, the Sun makes an angle of about seven degrees from the local vertical. With the simplifying geometry, that angle approximates the angle at Earth’s center between the two locations.

The measured arc is roughly seven out of 360 degrees of a full circle. Divide the estimated distance between the locations by that fraction to estimate Earth’s circumference, then divide by 2π to obtain its radius. The result depends on the distance estimate and geographic approximations.