Eratosthenes Measures Earth: The Shadow Experiment

Around 240 BCE, a Greek scholar in Alexandria read a report about a well in Syene — a city roughly 800 kilometers to the south — where, at noon on the summer solstice, vertical pillars cast no shadow and sunlight reached all the way down to the water at the bottom. He put that observation together with the shadow angle he could measure in his own city, applied the geometry of a sphere, and produced a figure for Earth's circumference that was accurate to within a few percent of the modern value. No satellite. No laser. Two sticks and careful reasoning.

Who Was Eratosthenes and Why Was He Thinking About This

Eratosthenes of Cyrene (c. 276–194 BCE) — Greek polymath, chief librarian of the Library of Alexandria, mathematician, geographer, astronomer, and poet. He was the first person to calculate Earth's circumference using geometric methods and the first to use the word "geography" (geographia) in its modern sense.

Eratosthenes was born in Cyrene (in modern-day Libya) and educated in Athens before being invited to Alexandria by Ptolemy III Euergetes to serve as the head of the Library of Alexandria, then the largest repository of written knowledge in the ancient world. He held this post for most of his adult life, giving him access to an extraordinary collection of traveler accounts, astronomical records, and geographical reports.

His approach to problems was characteristically synthetic: he combined data from different sources and applied mathematical reasoning to produce numerical results. He calculated the tilt of Earth's axis (measuring it at 23° 51' 15", which is remarkably close to the actual 23° 26'). He produced the first systematic map of the known world to use a grid of lines of latitude and longitude. He devised the Sieve of Eratosthenes, still used today as an algorithm for finding prime numbers.

The circumference measurement was not a stray lucky observation. It was the result of deliberate synthesis: a travel report about Syene, geometric knowledge about spheres, and a careful measurement made at a known time and place in Alexandria.

Source: American Physical Society — "Eratosthenes Measures Earth" (aps.org/apsnews/2006/06/eratosthenes-measures-earth)

The Report That Started It: A Well With No Shadow

The key piece of data that Eratosthenes worked from was a report — probably from travelers or merchants — about the city of Syene, which is modern-day Aswan in southern Egypt, located at approximately 24.1° N latitude. According to the report, at noon on the summer solstice (around June 21), vertical columns cast no shadow, and the sun shone directly down into a deep well, illuminating its water.

This observation has a precise geometric meaning. If a vertical column casts no shadow at noon on the summer solstice, the sun is directly overhead — that is, at the zenith, exactly 90 degrees above the horizon, exactly in line with a vertical pole. The technical term for this alignment is that Syene is located very close to the Tropic of Cancer, the line of latitude where the sun reaches the zenith on the summer solstice.

The sun is not overhead everywhere on Earth at the same moment. It is overhead on the summer solstice only at the Tropic of Cancer. North of that line, the sun is always somewhat south of the zenith at noon. South of it, the sun is always somewhat north of the zenith at noon. An observer at any latitude other than the Tropic of Cancer will see a vertical post cast a shadow at noon on the summer solstice.

Eratosthenes was in Alexandria, at about 31.2° N — well north of Syene and well north of the Tropic. At noon on the summer solstice in Alexandria, a vertical stick would cast a shadow. The question was: how long?

Source: Bucknell University Astronomy — "Eratosthenes and the Size of the Earth" (eg.bucknell.edu/physics/astronomy/astr101/specials/eratosthenes.html)

The Measurement: Stick, Shadow, Angle

On the summer solstice, at noon, Eratosthenes measured the angle of the shadow cast by a vertical stick (called a gnomon — the same kind of pointer used in a sundial) in Alexandria. The shadow length, combined with the known height of the gnomon, gives a triangle from which the angle between the sun's rays and the vertical can be calculated.

Gnomon — a vertical rod or post used as a sundial pointer, whose shadow angle at a given time encodes the sun's elevation above the horizon.

The angle he measured was approximately 7.2 degrees — that is, the sun's rays were 7.2 degrees away from the vertical in Alexandria. Since the sun was directly vertical (0 degrees from vertical) in Syene at the same moment, the difference in sun angle between the two cities was 7.2 degrees.

Here is the critical geometric step. Because the sun is so far away that its rays arrive at Earth essentially in parallel — like a beam of light from an infinitely distant source — the difference in sun angle between two points on Earth's surface is the same as the angle between those two points as seen from Earth's center. Alexandria and Syene define an arc on Earth's surface, and that arc subtends an angle of 7.2 degrees at Earth's center.

The rest is proportion. A full circle is 360 degrees. The arc from Alexandria to Syene subtends 7.2 degrees, which is 7.2/360 = 1/50 of a full circle. If the distance from Alexandria to Syene is D, then Earth's circumference is 50 × D.

This is one of the most elegant proportions in the history of science. The same ratio that describes how much the sun angle shifts between the two cities also describes what fraction of Earth's total circumference lies between them. The geometry is not an approximation — it is exact, provided Earth is a sphere and the sun's rays are parallel.

Source: University of Texas Astronomy — "Eratosthenes Measures Earth's Circumference" (outreach.as.utexas.edu/marykay/assignments/eratos1.html)

Measuring the Distance: Bematists and the Stade

If Alexandria and Syene are 1/50 of Earth's circumference apart, then multiplying their separation by 50 gives the circumference. Eratosthenes used a distance of 5,000 stadia between the two cities. Multiply by 50: Earth's circumference = 250,000 stadia.

Stade (or stadium) — a unit of distance used in ancient Greece, approximately equal to the length of a standard footrace track. The precise conversion to modern meters is disputed: estimates range from roughly 157 m (the Attic stade) to 185 m (the Attic stadium used for measurements) to 210 m (the Ptolemaic stade used in Egypt).

The 5,000-stadia figure for the Alexandria–Syene distance was likely measured by professional bematists — trained distance estimators whose job was to count their steps and convert step counts to standardized stades while marching between cities. The Ptolemaic administration used bematists extensively for surveying Egypt and surrounding territories.

Converting 250,000 stadia to modern units depends critically on which stade Eratosthenes used. Using a stade of 157.5 m (one estimate for the Attic stade): 250,000 × 157.5 m = 39,375 km. Using 180 m (the so-called Pan-Hellenic stade): 250,000 × 180 m = 45,000 km. The modern circumference of Earth at the poles is about 40,008 km; around the equator it is 40,075 km.

The 39,375 km value derived from the 157.5 m stade falls within about 2% of the modern figure — an extraordinary accuracy. Even with the 180 m stade, the result of 45,000 km is off by only about 12%. Given that he was working with rounded distance estimates and an approximate measurement of the shadow angle, the result is remarkable in either case.

Source: Wikipedia — "Eratosthenes" (en.wikipedia.org/wiki/Eratosthenes)

Source: Encyclopedia.com — "Eratosthenes Calculates the Circumference of the Earth" (encyclopedia.com/science/encyclopedias-almanacs-transcripts-and-maps/eratosthenes-calculates-circumference-earth)

Why the Assumptions Were Valid

Eratosthenes' calculation rests on three geometric assumptions. Each was reasonable, and each can be checked.

First: Earth is a sphere. This was not a novel assumption in Eratosthenes' time. Pythagoras had argued for a spherical Earth in the 6th century BCE on aesthetic grounds. Aristotle (384–322 BCE) had provided empirical arguments: the circular shadow Earth casts on the moon during lunar eclipses, the fact that the horizon drops as you climb higher, and the observation that different stars are visible from different latitudes. By Eratosthenes' time, the spherical Earth was the standard working model among educated Greeks. The measurement itself requires it — on a flat Earth, parallel sun rays would cast identical shadows everywhere, and there would be no shadow difference to measure.

Second: Alexandria and Syene are on the same meridian (the same line of longitude). This is approximately true but not exactly so. Alexandria is at about 29.9° E longitude; Syene (Aswan) is at about 32.9° E. The north-south offset between them, which is the relevant one for Eratosthenes' calculation, is real. The east-west offset introduces a small error, but because the cities are close to the same meridian, the error is modest. If Eratosthenes knew the cities were not exactly on the same meridian, he may have already accounted for the slight offset.

Third: The sun's rays are parallel. This is valid because the sun is so far from Earth that the angular diameter of the sun — about 0.5 degrees — represents the slight non-parallelism of its rays. This introduces an error smaller than 0.5 degrees in a measurement already limited to one-degree precision by the shadow angle observation. It is negligible.

The chain of reasoning — from a travel report about a shadowless well, through a geometric argument about parallel rays and spherical surfaces, to a multiplied distance — is logically complete and essentially free of conceptual errors. The main uncertainty was in the distance measurement, not in the reasoning.

Source: Nature (1943) — "Eratosthenes and the Circumference of the Earth" (nature.com/articles/152473a0)

A Timeline of the Measurement and Its Legacy

Year (approx.)	Event
c. 276 BCE	Eratosthenes born in Cyrene
c. 245 BCE	Appointed chief librarian of the Library of Alexandria
c. 240 BCE	Conducts the shadow angle measurement; computes Earth's circumference as 250,000 stadia
c. 225 BCE	Writes Geographica, the work in which the circumference calculation is described
c. 194 BCE	Eratosthenes dies in Alexandria
c. 150 CE	Claudius Ptolemy adopts a smaller, incorrect estimate for Earth's circumference, setting back geography
1492 CE	Columbus, using Ptolemy's underestimate, believes Asia is reachable by sailing west from Spain
1522 CE	Magellan's crew completes the first circumnavigation, providing direct confirmation of Earth's size
2012 CE	Anthony Abreu Mora repeats Eratosthenes' calculation with precise modern data; gets 40,074 km (0.16% error)

Source: Medium / Roaming Physicist — "Looking Down a Well in Egypt to Measure the Size of the Earth" (medium.com/roaming-physicist/looking-down-a-well-in-egypt-to-measure-the-size-of-the-earth-5e12f2df0b2)

What Eratosthenes Got Wrong — and Why It Didn't Matter

Eratosthenes' result was not perfect. Several sources of error were baked into the method.

The shadow angle measurement in Alexandria was performed with care but could not achieve precision below about half a degree with ancient instruments. At 7.2 degrees, a half-degree error represents roughly 7% of the measured quantity. This alone introduces meaningful uncertainty in the final answer.

The distance between Alexandria and Syene — 5,000 stadia — was a round number. Modern GPS gives the straight-line distance as about 843 km. If we take Eratosthenes' 5,000 stadia as equal to 843 km, then his stade was roughly 169 m, giving 250,000 × 169 m = 42,250 km — about 5.5% above the modern polar circumference. If he was using a stade of 157.5 m, his distance figure was slightly too large, but the errors partially canceled.

Syene is not exactly on the Tropic of Cancer (which in Eratosthenes' time was at approximately 23° 44' N; Aswan is at 24° 5' N). The sun at noon on the summer solstice in Aswan still casts a small shadow — about 0.4 degrees — rather than none. This introduces a small underestimate of the sun's angle difference between the two cities.

None of these errors are large enough to substantially change the conclusion. The method works. The precision was adequate given the technology. The conceptual framework was entirely correct. What is most striking is not that Eratosthenes got within a few percent of the right answer — it is that the method he used is logically sound enough that a modern version of the same calculation, done with GPS distances and a protractor, gives the right answer to within 0.16%.

Source: Study Library — "Eratosthenes' Earth Circumference: Solution and Error Analysis" (studylib.net/doc/8857948)

The Deeper Significance: Measurement Without Travel

Eratosthenes never visited Syene. He measured a global property of Earth — its size — from a single location, using data collected elsewhere. The entire calculation happened at his desk in Alexandria, with a gnomon, a clock (defined by the solstice noon), and a travel report.

This is what makes the experiment philosophically important beyond its numerical result. It demonstrates that local measurements, combined with geometric reasoning, can reveal properties of the whole. You do not need to see Earth from space to know it is a sphere of a certain size. You need two measurements at different latitudes, a known distance between those latitudes, and the insight that the sun's rays are parallel.

The same logical structure underlies much of modern science: astronomers determine the distance to stars using parallax shifts that span only the diameter of Earth's orbit; physicists measure the mass of the entire Milky Way by observing the orbital velocities of stars at its edge; geologists infer the temperature of Earth's core from seismic wave travel times. In each case, local data plus known physics produces a global or remote conclusion. Eratosthenes was doing this 2,200 years ago.

The measurement also had immediate practical consequences. A geographer who knows Earth's circumference can scale any map accurately. Eratosthenes went on to produce the first systematic world map using a grid, and his circumference measurement set the scale for it.

Source: 101 Computing — "Eratosthenes: The Ancient Scholar Who Measured the Earth" (101computing.net/eratosthenes-the-ancient-scholar-who-measured-the-earth/)

What People Get Wrong About This Experiment

The most common misconception is that medieval Europeans thought Earth was flat and that Columbus was a brave pioneer fighting this belief. This is a modern myth. Every educated European in the medieval period — clergy and scholars alike — knew Earth was spherical. The question Columbus was debating with the Spanish crown was not Earth's shape but its size. His critics were right: they used estimates derived from Ptolemy's (incorrect, smaller) circumference and argued that the western ocean route to Asia was too long to be navigable. Columbus was wrong about the size and only survived because a continent he did not know existed happened to be in the way.

A second error is imagining that Eratosthenes looked straight down a well to make his measurement. He measured the shadow of a gnomon in Alexandria — a straightforward angle measurement from shadow length and post height. The well in Syene was the source of the other data point (zero shadow angle), not a device he used personally.

A third confusion is about whether his answer was correct. It was very close, but the uncertainty in the stadia conversion means we cannot say with certainty whether the 250,000 stadia result was within 2% or 12% of the modern value. Both are impressive for 240 BCE, but one is spectacular and the other is merely excellent.

For the broader history of how ancient scientists reasoned about the cosmos, see Aristarchus and the distances to the Sun and Moon. For how modern geodesy measures Earth's shape with satellite data, see how GPS works.

FAQs

Q: Did Eratosthenes actually go to Syene to make measurements?
A: No. Eratosthenes worked entirely from Alexandria. The data about Syene — that vertical pillars cast no shadow at noon on the summer solstice — came from travelers' reports, merchants, or administrative records held in the Library of Alexandria. He measured only the shadow angle in Alexandria personally, then combined that with the reported distance to Syene to complete the calculation.

Q: Why did he choose the summer solstice?
A: The solstice is the one moment each year when the sun reaches its maximum northward position and shines directly overhead at the Tropic of Cancer. Using the solstice made the Syene data point (zero shadow angle) geometrically clean: Syene sat close enough to the Tropic that the effect was exact or nearly so. On any other day, neither city would produce the clean reference condition he needed.

Q: How did Eratosthenes measure the shadow angle?
A: He used a gnomon — a vertical rod of known height — and measured the length of its shadow at noon on the solstice. The angle of the sun from the vertical equals the arctangent of (shadow length / rod height). Ancient astronomers could make this calculation using tables of ratios that served the same function as trigonometric tables. An angle of 7.2 degrees corresponds to a shadow about 1/8 as long as the rod is tall.

Q: Is Syene exactly on the Tropic of Cancer?
A: Not quite. Modern Aswan (ancient Syene) sits at 24° 5' N, while the Tropic of Cancer in Eratosthenes' time was at approximately 23° 44' N — about 21 km north of where it is now (the Tropic drifts slowly due to the precession of Earth's axial tilt). The sun at noon on the solstice in ancient Syene fell very slightly south of the zenith, but close enough that the "no shadow" description was essentially accurate for practical purposes.

Q: How accurate was the final result in modern units?
A: It depends on the stade conversion, which historians still debate. Using a stade of 157.5 m gives a circumference of about 39,375 km — about 1.7% less than the polar circumference of 40,008 km. Using a stade of 180 m gives about 45,000 km, which is about 12% too large. The true value of Eratosthenes' stade is unknown, so the honest answer is that his result was somewhere between about 2% and 12% from the modern value. Either way, it was the most accurate measurement of Earth's size for well over a thousand years.

Q: Was Eratosthenes the first person to attempt this measurement?
A: He was the first to do it geometrically with a recorded method. Earlier Greek thinkers — Aristotle mentions a figure of 400,000 stadia from unnamed predecessors — had produced estimates, but without a clear methodological account that survives. Eratosthenes' calculation in his Geographica (now lost but summarized by later writers including Strabo and Cleomedes) is the first whose method is recoverable in enough detail to evaluate and replicate.

The Takeaway

Eratosthenes measured the size of a planet with two sticks, a distance report, and the geometry of parallel lines on a sphere. The conceptual leap was not large — once you know Earth is round and the sun is far away, the calculation is a single proportion — but making the leap required recognizing that a travel curiosity (a well that mirrors the sun once a year) was a precisely defined geometric fact that could be combined with a local measurement to yield a global result. That recognition is the real achievement. The number is impressive; the method is the thing that endured. Every GPS satellite, every geodetic survey, every planetary science mission that measures a distant world from Earth uses the same logic: take what you can measure here, apply what you know about geometry and physics, and recover what you could never reach directly.

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