Eratosthenes · a stick, a shadow, and one summer noon
A man measured the whole Earth using nothing but a stick and a shadow — without ever leaving his city.
Parallel sun rays strike two sticks on Earth's curved surface at different angles. The 7.2° shadow difference at the surface equals the 7.2° arc at Earth's centre — giving the full circumference.
Eratosthenes of Cyrene, chief librarian at Alexandria, heard something striking: on the summer solstice at noon, a vertical stick in Syene (modern Aswan) cast no shadow at all — sunlight fell straight down a deep well. But in Alexandria, 800 km to the north, the same stick cast a shadow at 7.2°.
If Earth were flat, both sticks would see the same sun at the same angle — no difference. The difference could only mean Earth's surface is curved, and the two cities lie at different angles to the incoming light. The 7.2° shadow angle equals the central angle between the two cities as seen from Earth's centre. And 7.2° is exactly 1/50th of 360° — so the walk from Alexandria to Syene covers 1/50th of the way around the world.
all the way around the Earth = 50 × the walk between the two cities
That multiplier — ×50 — is the discovery. And notice what it does not depend on: any particular unit. Measure the walk in Greek stadia and the answer comes out in stadia — his surveyors called it 5,000 stadia, so he announced 250,000 stadia. Measure the same walk in kilometres and the answer comes out in kilometres. The geometry doesn't care what a unit is; it only ever says fifty of whatever that walk was.
No spacecraft. No GPS. Just two sticks, one sunny day, and the geometry of a sphere. Every rung of the cosmic ladder builds on this first measurement.
⚖ So… was he right? (careful — there is a trap here)
The tempting next line is: "a stadion was about 160 metres, so 250,000 stadia ≈ 40,000 km — almost exactly right!" That line hides two tricks, and honesty is the whole point of this ladder.
Nobody knows exactly how long his stadion was. Ancient sources disagree — and some modern estimates of the stadion are themselves worked out by assuming Eratosthenes was accurate. Using those to check him is circular: the answer was baked into the ruler before the test began.
"40,000 km" was never a target he luckily hit. When the metre was invented in 1793, it was defined as one ten-millionth of the distance from the North Pole to the equator — which makes Earth's circumference 40,000 km by definition of the metre, not by measurement coming out that way. (Later, better surveys showed the original definition was slightly off — that is why today's figure is 40,008 km. The unit was already in use and it would be impractical to change it, the Earth itself never changed, so the number is not 40.000 anymore.)
So the honest question is not "did his number match ours?" but "is the geometry right?" — and that we can check today with no stadia anywhere. The straight north–south distance from Alexandria to Aswan is about 790 km. Apply his own ×50: ≈ 39,600 km, about 1% under the true 40,008 km — and that 1% traces to his rounded angle and the two cities not lying exactly north–south. The method is sound: run it today with precise instruments and the correct answer simply falls out. That is what "he measured the Earth" really means.
One more honesty note: Eratosthenes' own book is lost. The story reaches us second-hand, mainly through a textbook written some 300 years later — so the suspiciously tidy numbers (exactly 7.2°, exactly 5,000 stadia) are probably a teacher's rounded retelling of a messier real measurement.
⚑ Assumptions that made it work
The Sun is so far away its rays arrive perfectly parallel. If the Sun were nearby, its light would fan out — the shadow difference would reflect the Sun's angle in the sky, not Earth's curvature. Parallel rays require the Sun to be at an enormous distance. This was assumed, not proven — and it quietly implied that Earth moves in a vast orbit around the Sun, which in turn meant nearby stars should show a tiny wobble as Earth moves. Nobody could detect that wobble for another 2,000 years, and its absence was used as the main argument against a moving Earth.
Earth is a perfect sphere. The geometry only works cleanly for a sphere. Earth is slightly flattened at the poles, but the error is small. Eratosthenes had no proof of sphericity — it was an accepted philosophical position.
Syene and Alexandria lie on the same north–south meridian. They don't quite — Alexandria is about 3° west of Syene. This small error was unknown to him.
The walk between the cities is accurately known. He relied on professional pacers called bematists, who put it at 5,000 stadia. Any error in that baseline scales the final answer directly — the ×50 is only as good as the walk it multiplies.
Try it yourself measure the Earth with a friend, two sticks and one sunny day
You need one friend far away, due north or due south of you, and you need to know how far apart you are. At an agreed moment you both measure the same thing: how far the Sun leans away from straight up.
Setting up the stick
Choose flat, open ground that will still be sunlit at the agreed time, and use a stick about a metre tall — shorter than that and the shadow is too stubby to measure well.
Get it truly upright. Tie a small weight to a string and hold it beside the stick: the string hangs perfectly vertical, so line the stick up with it. Check from two directions at right angles — a stick that looks straight from the front can still lean sideways.
Measure the height that actually casts the shadow — from the ground to the very top, not the buried part.
Mark the shadow tip at the agreed instant with a pebble, then measure from the foot of the stick to that pebble. The shadow creeps visibly, so mark first and measure afterwards.
Your two sticks do not need to be the same height. Only the shape of each triangle matters, never its size.
Turning the shadow into an angle
The stick and its shadow form a right-angled triangle, and the angle you want sits at the top of the stick — between the stick and the ray of sunlight running down to the shadow's tip. Two ways to read it:
By drawing (no maths needed). On squared paper draw the stick as a vertical line and the shadow as a horizontal line at its foot — say 1 cm on paper for every 10 cm you measured. Join the top of the stick to the tip of the shadow. Put a protractor at the top corner and read the angle. Anyone can do this.
By calculator. Divide the shadow by the height and press the inverse tan key (tan⁻¹ or atan). A 1 m stick with a 26 cm shadow gives 0.26 → about 14.6°.
Putting the two together
Subtract the two angles, see how many times that difference fits into 360°, and multiply by the distance between you.
circumference = distance between you × 360° ÷ (your angle − their angle)
Why it works
The Sun is so far away that its rays reach both of you travelling in the same direction — parallel. So if the Sun leans differently in your two skies, it is not the Sun that changed: it is the two of you who are tilted differently, because you stand on different parts of a curved Earth. The angle between two upright sticks is exactly the angle of the slice of Earth between them. Measure that slice, and you know what fraction of the whole planet lies between you — which turns two sticks and a known walk into the size of a world.
Why farther apart is better
Your protractor is good to about half a degree, however careful you are. That error stays the same size no matter where you stand — but the angle you are measuring grows with distance, at roughly 1° for every 111 km north or south:
100 km apart → only about 0.9° to measure. A ±0.5° slip is more than half your signal: the answer could come out twice too big or half too small. Nearly hopeless.
500 km apart → about 4.5°. The same slip is ~11% — an answer in the right ballpark.
1,000 km apart → about 9°. Now the slip costs only ~6%. This is roughly Eratosthenes' baseline, and roughly his accuracy.
That is the lesson every rung of this ladder repeats: you cannot make your instrument perfect, so make the thing you are measuring bigger. It is exactly why the distance to a star (rung 5) had to wait for a baseline as wide as Earth's entire orbit.
⚑ Three things that catch people out
Only north–south counts. Walking east or west adds nothing to the angle. If your friend is off to one side, take just the north–south part of the gap from a map — otherwise your answer comes out too big.
Match the moment. On the same north–south line, your local noons happen at the same instant, so "at noon" and "at the same time" mean the same thing. If you are not aligned, each measure at your own local noon instead — that removes the east–west problem entirely.
Usually subtract — but sometimes add. The equator itself changes nothing, and it makes no difference which hemisphere either of you is in. What matters is the spot where the Sun stands straight overhead, which drifts between the tropics with the seasons. If that spot is outside the pair of you, both shadows point the same way and you subtract. If it falls between you, your shadows point in opposite directions and you must add. And if one of you casts no shadow at all, that person's angle is simply 0° — the well at Syene, all over again.
What to expect: an answer of roughly the right size rather than a perfect one — exactly Eratosthenes' situation too. Landing on 35,000 or 45,000 km is not a failure: it means you personally established that the Earth is a few tens of thousands of kilometres around, without leaving your garden.
🏛️The Great Library of Alexandria was just opening — Eratosthenes was its chief librarian! His job: collect every book ever written, from every country in the world.
⚔️Rome and Carthage were fighting huge wars, with war elephants marching across the Alps — one of history's most dramatic military campaigns.
🇨🇳At the exact same moment, on the other side of the planet, workers were building the first Great Wall of China.
👗People wore simple flowing robes called chitons — one big piece of linen pinned at the shoulder. No buttons, no zip, no pockets! Sandals for everyone, rich or poor.
📜There were no printed books — all knowledge was written on papyrus scrolls rolled up like tubes. A library was a room full of thousands of rolled-up tubes!
