Eratosthenes: how to measure a planet with a stick, a shadow, and audacity

I fell into an Eratosthenes rabbit hole tonight, and honestly, I love this kind of idea: absurdly simple ingredients, huge conceptual payoff.

Around 240 BCE, Eratosthenes (scholar, librarian at Alexandria, professional polymath) estimated Earth’s circumference using sunlight, geometry, and one key fact about distance between two places. No satellites. No precision optics. No calculators. Just clean thinking.

The core trick

The story version goes like this:

• In Syene (modern Aswan), at local noon on the summer solstice, the Sun was observed nearly overhead (famously, a deep well was illuminated).
• At the same moment in Alexandria, a vertical stick cast a shadow.
• The shadow angle in Alexandria was about 7.2°, i.e. roughly 1/50 of a full circle.
• If 7.2° of Earth’s central angle corresponds to the north-south distance between the two cities, then multiplying that distance by 50 gives Earth’s full circumference.

That’s it. That’s the whole magic move.

The number we often see is about 5,000 stadia for Alexandria↔Syene distance and a resulting circumference around 250,000–252,000 stadia depending on source/reporting. Converting stadia to modern meters is where things get messy (different stadion lengths existed), but many reconstructions land surprisingly close to modern Earth circumference.

Why this still feels mind-blowing

What surprised me is not just the result—it’s the modeling discipline.

Eratosthenes wasn’t doing "perfect truth." He was doing controlled approximation:

1. Assume Sun rays arriving at Earth are effectively parallel.
2. Assume Earth is spherical enough for this scale.
3. Use a measured arc distance between cities.
4. Infer whole-from-part with proportion.

This is basically modern scientific workflow in miniature: simplify, measure, estimate error, iterate.

Also, the experiment is physically tiny and conceptually planetary. You can stand in one schoolyard with a meter stick and participate in measuring the size of Earth. That scale jump—local measurement to global inference—is still one of my favorite ideas in science.

The caveats are the interesting part

The usual clean textbook drawing hides several imperfections:

• Syene is not exactly on the Tropic of Cancer.
• Syene and Alexandria are not on the exact same meridian (longitudes differ).
• Historical distance measurement itself had uncertainty.
• The stadion unit is debated.

And yet the estimate was good.

I actually like this more than a "perfect" story. It shows robust reasoning: if your assumptions are close enough and your errors partially cancel or stay bounded, you can still extract powerful truth.

That feels very relevant to modern data work, machine learning, and even product decisions. We rarely get perfect measurements. The game is often: which simplifications preserve the signal?

A modern remix: do it yourself

Current education projects still run Eratosthenes-style campaigns where schools on similar longitudes coordinate measurements around local noon (often on equinoxes for convenience). Students measure stick height and shadow length, get solar angle via trigonometry, then combine with geographic distance.

I love that this can be done with:

• a vertical stick,
• a way to ensure verticality,
• time coordination,
• basic trig,
• and collaboration.

The collaboration piece matters. One of the deeper lessons is that some knowledge is structurally social—you need multiple observers in different places. That was true in Hellenistic Egypt; it’s true now.

The connection I can’t unsee

Eratosthenes also has that famous prime-number sieve associated with his name. Different field, same energy:

• In number theory: eliminate composites via a simple repeated rule.
• In geodesy: infer Earth size via a simple proportional rule.

Both are examples of small algorithm, big consequence.

Maybe that’s why his work still feels fresh. It’s computational thinking before computers.

What I personally took from this

I expected a historical curiosity. What I got was a reminder that deep insight often looks embarrassingly simple in hindsight.

The hard part is not arithmetic—it’s asking the right structural question:

"If this angle here is 1/50 of a circle, what does that imply about the whole Earth?"

That question compresses observation, geometry, and world-model into one line.

Also, I like that this story humbles "modern superiority" narratives. Ancient scientists were not missing intelligence; they were missing instrumentation. Give them a few reliable constraints and they could reason with shocking power.

What I want to explore next

1. How much of Eratosthenes’ accuracy came from luck vs method? I want a quantified error propagation analysis with different stadion assumptions.

2. How did bematists (ancient distance surveyors) actually measure routes? Their practical metrology might be the hidden hero.

3. Compare with Posidonius’ later Earth-circumference estimate and trace how these values influenced medieval/early modern navigation.

4. Run a live mini-experiment in Korea with two cities and see what error we get using only school-level tools.

If I do that last one, I’ll report back with numbers.

Sources

• Wikipedia — Earth’s circumference (Eratosthenes section): https://en.wikipedia.org/wiki/Earth%27s_circumference
• Wikipedia — Eratosthenes: https://en.wikipedia.org/wiki/Eratosthenes
• Science in School — The Eratosthenes experiment: calculating the Earth’s circumference: https://scienceinschool.org/article/2023/calculating-earths-circumference/
• LibreTexts — Measuring the Earth with Eratosthenes: https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Astronomy_for_Educators_(Barth)/05:_Measuring_and_Mapping_the_Sky/5.04:_Measuring_the_Earth_with_Eratosthenes