Over 2,000 years ago, Eratosthenes calculated the size of the Earth using shadows and geometry—an experiment later memorably explained by Carl Sagan.

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Contents

• Carl Sagan’s famous demo
• What Eratosthenes set up
• The key measurement
• Where the distance came from
• From ratio to circumference
• What, exactly, is a “stadion”?
• How close was he?
• Assumptions and caveats
• Try it yourself today

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Carl Sagan’s famous demo

Most people first hear of Eratosthenes through Carl Sagan’s TV show Cosmos. Sagan told the story of how, more than 2,000 years ago, a librarian with a good idea, a long baseline, and some simple geometry managed to weigh the Earth—so to speak. At noon on the summer solstice, the Sun stood overhead in Syene (now Aswan, Egypt). But in Alexandria, far to the north, a stick cast a shadow. That small difference opened the door to measuring a giant sphere.¹

What Eratosthenes set up

The trick was simple but brilliant. If sunlight is essentially parallel (a safe assumption), then the angle of a shadow at one city equals the angle of Earth’s surface between that city and another. Eratosthenes knew that in Syene, the Sun shone straight down a deep well at noon, but in Alexandria it didn’t. The angle of the shadow there was the key.²

The key measurement

At noon, Alexandria’s shadow gave an angle of about 7.2°. That’s one-fiftieth of a full circle. If Alexandria was one-fiftieth of the way around the world from Syene, then multiply their distance by fifty, and you get the whole circumference.³

Where the distance came from

Eratosthenes didn’t walk the Nile with a tape measure. Distances were already known thanks to surveyors called bematists, who measured by pacing or by wheel. From these records, the journey between Alexandria and Syene was put at about 5,000 stadia.²⁴

From ratio to circumference

Multiply 5,000 stadia by 50 and you get roughly 250,000 stadia. Eratosthenes rounded to 252,000, likely because it was a neat number that divided easily in calculations. By modern reckoning, that comes tantalizingly close to Earth’s actual size.⁵

What, exactly, is a “stadion”?

A stadion was a unit of length, but not a fixed one. Different regions used slightly different versions. If we take the “Egyptian” stadion of around 155–160 m, Eratosthenes’ result comes within a few percent of the real circumference. Using a longer Greek stadion pushes the number higher. That fuzziness is why historians still argue about his accuracy.⁶

How close was he?

If the shorter stadion is right, then Eratosthenes estimated Earth’s circumference at about 39,000–40,300 km. The true polar circumference is ~40,008 km. That’s astoundingly close—especially considering the tools: shadows, distances paced along a river, and some sharp thinking.⁵⁶

Assumptions and caveats

• Parallel rays: The Sun is far enough that its rays are essentially parallel.³
• Same meridian: Syene and Alexandria aren’t exactly on a north–south line, and the Nile road wasn’t a perfect straight shot.²⁴
• Syene on the tropic: It’s close, but not a perfect alignment. Still, it was “good enough” for remarkable accuracy.²

Try it yourself today

You can repeat Eratosthenes’ experiment with two sticks, two cities, and a bit of coordination. Schools and astronomy clubs often do this as a group project. All it takes is noting the shadow angle at the same time and comparing distances. It’s a wonderful way to see how simple observations can reach cosmic conclusions.³⁷

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References

1. Carl Sagan, “Cosmos” clip explaining Eratosthenes — youtube.com/watch?v=G8cbIWMv0rI ↩︎
2. Cleomedes, On the Circular Motions of the Celestial Bodies — 1891 Teubner edition on Internet Archive: archive.org/details/kleomedouskyklik00cleo ↩︎ ↩︎ ↩︎ ↩︎
3. NOAA Education — “The History of Geodes: Global Positioning Tutorial” (explains Eratosthenes’ formula and assumptions): oceanservice.noaa.gov/education/tutorial_geodesy/geo02_hist.html ↩︎
4. Heath, Thomas L., A History of Greek Mathematics, Vol. I (Oxford 1921) — Cornell scan: archive.org/details/cu31924008704219 ↩︎ ↩︎
5. Encyclopaedia Britannica — “Eratosthenes”: britannica.com/biography/Eratosthenes ↩︎ ↩︎
6. Roller, Duane W., Eratosthenes’ Geography (Princeton, 2010) — Google Books listing: books.google.com/…/Eratosthenes_Geography ↩︎ ↩︎
7. “The Eratosthenes experiment: calculating the Earth’s circumference” — Science in School classroom activity (PDF): scienceinschool.org/…/Issue-63-Eratosthenes.pdf ↩︎