Eratosthenes and the Measurement of Earth's Circumference

The /learn/history-of-latitude-and-longitude pillar mentioned Eratosthenes' ~240 BC measurement as the starting point of practical geodesy. This article tells the full story — the methodology, the historical context, the modern reproductions, and what it means for the coordinate work every smartphone does 2,260 years later.

Eratosthenes the man

Eratosthenes of Cyrene (c. 276 BC – c. 194 BC) was a Greek polymath who served as the chief librarian of the Library of Alexandria — the most important scholarly institution of the ancient Mediterranean. He's known for:

• Geographic work: the Geographika (no longer surviving) — a three-volume treatise on Earth and geography. Established the science of geodesy.
• The Sieve of Eratosthenes: an algorithm for finding prime numbers that still bears his name and is taught in introductory computer science.
• Geographical coordinate system: he developed an early lat/lon-like grid for terrestrial mapping (later refined by Hipparchus and Ptolemy).
• The Earth's tilt measurement: he measured the Earth's axial tilt at 11/83 of a circle (~23°51'), remarkably close to the modern value (23°26').
• The Earth's circumference: the topic of this article.

Contemporary Greek scholars nicknamed him “Beta” (B), implying he was second-best at everything — an intellectual jibe rather than a compliment. He was also called “Pentathlos” (the all-rounder), suggesting he was competent across multiple disciplines without being the very best at any one. Modern scholars consider this ungenerous: he was first-rank in several fields.

The measurement methodology

The key observation: at noon on the summer solstice (~21 June by modern calendar), the sun was directly overhead at Syene (modern Aswan, Egypt). Eratosthenes knew this because:

• Travellers reported that a deep well in Syene was illuminated all the way to the bottom at noon on that day.
• Vertical poles in Syene cast no shadow at that moment.
• Syene was reported to be on the Tropic of Cancer (the latitude where the sun is directly overhead at the summer solstice; Cancer is at ~23°26' N).

At Alexandria, ~800 km north of Syene, the same day at the same time:

• A vertical pole (called a “gnomon”) cast a shadow.
• The angle between the pole and the sun's rays — derived from the shadow length and the pole height — was measured as about 7.2° (1/50 of a full circle).

The reasoning:

If the Earth is a sphere and the sun is far enough away that
its rays are parallel everywhere on Earth, then:

  - At Syene, sun directly overhead → angle from vertical = 0°.
  - At Alexandria, sun's angle from vertical = 7.2°.
  - The 7.2° difference equals the angular separation
    between the two cities, measured at Earth's centre.

If 7.2° corresponds to ~800 km of distance along Earth's
surface, then a full 360° corresponds to:

  360° / 7.2° = 50

  50 × 800 km = 40,000 km

Eratosthenes' result was 250,000 stadia (a Greek unit of distance). Multiplying by 50, the implied circumference. Then converting stadia to a modern unit gives the modern-equivalent figure.

The stadion problem

How long was a stadion? Multiple competing answers in modern scholarship:

• Attic stadion: ~185 m (the most common Greek stadion).
• Egyptian / Olympic stadion: ~157.5 m (the stadion most likely used in Egypt).
• Ptolemaic stadion: ~158 m.
• Royal stadion: ~209 m.

At each:

| Stadion length | 250,000 × length | Modern equivalent | Error vs 40,075 km | | -------------- | ---------------- | ----------------- | -------------------- | | 157.5 m | 39,375 km | very close | 1.7 % low | | 158 m | 39,500 km | very close | 1.4 % low | | 185 m | 46,250 km | far too high | 15.4 % high | | 209 m | 52,250 km | much too high | 30.4 % high |

Modern scholars usually argue that Eratosthenes used the Egyptian / Olympic stadion of ~157.5 m, giving a result within ~2 % of the modern value. The other plausible stadia give worse results, but they aren't implausible historically.

The honest position: the exact accuracy of Eratosthenes' measurement is uncertain because the stadion length is uncertain. The methodology was sound; the conversion to modern units is fuzzy.

What Eratosthenes assumed (correctly)

The measurement depended on several assumptions:

1. Earth is a sphere. By 240 BC this was standard among educated Greeks; Eratosthenes didn't need to argue it.
2. Sunlight rays are parallel when they reach Earth. Effectively true at Earth's scale because the sun is ~150 million km away vs Earth's ~12,742 km diameter (the angular variation in incoming sunlight is ~0.005°, negligible at Eratosthenes' precision).
3. Syene is on the Tropic of Cancer. Approximately true in 240 BC; Syene is actually about 5 km north of the Tropic, but for the precision of the measurement this is negligible.
4. Alexandria and Syene are on the same meridian. Slightly false: Alexandria is at ~29°E longitude; Aswan is at ~33°E (about 250 km east). The east-west offset introduces a small error in the latitude-only measurement.
5. Sunlight at Syene actually reaches the bottom of the well at noon on the solstice. Probably true at the time, though no observational record from Syene survives — Eratosthenes relied on traveller reports.

The fact that the assumptions held well enough to produce a ~2 %-accurate result is partially methodological skill, partially compensating errors.

Modern reproductions

Eratosthenes' experiment has been reproduced many times in the modern era — by classrooms, amateur astronomy groups, professional historians of science. The most famous reproduction:

Carl Sagan in Cosmos: A Personal Voyage (1980) — demonstrated the method on television, with a planted-pole shadow comparison. The Cosmos episode helped popularise the historical achievement to millions of viewers.

A typical school reproduction:

1. Choose two cooperating sites on roughly the same meridian, ideally hundreds of km apart (north–south).
2. On a clear day, both sites measure the sun's angle at solar noon (the moment when the sun is highest in the sky).
3. Take the angular difference between the two sites.
4. Multiply by the known distance between the sites, divided by the angular difference, scaled to 360° → get the circumference.

Modern reproductions routinely achieve ~5 % accuracy with careful measurement and a baseline of hundreds of km.

The Eratosthenes Project is an international school program where students at different latitudes cooperatively reproduce the measurement each year. Results published online; thousands of students have participated.

The historical impact

Eratosthenes' measurement was the first quantitative measurement of a planetary parameter. Before Eratosthenes, “how big is the Earth?” was a question with philosophical opinions but no numerical answer. After Eratosthenes, it was a question with a specific value derivable from observation.

The methodology — observe, measure angles, apply geometric reasoning to derive a physical quantity — became the template for all subsequent geodesy. Posidonius (~135 BC) refined the measurement using a different method (observing the elevation of a star at two locations); later Greek and Islamic-world astronomers continued the refinement.

By the time Ptolemy wrote his Geography (~150 AD), Eratosthenes' result was the accepted Earth-size value. Christopher Columbus in 1492 famously used a less accurate (smaller) circumference estimate from a different source — leading him to expect Asia much closer than it actually was. Had Columbus used Eratosthenes' figure, the encounter with the Americas might have been delayed by decades.

What Eratosthenes didn't do

A few clarifications:

• He didn't invent the spherical Earth concept. That was established 200+ years earlier (Pythagoras, c. 500 BC; Aristotle, c. 350 BC).
• He didn't measure Earth's polar radius vs equatorial radius. The Earth's oblate-spheroid flattening wasn't measured until the 1730s expeditions to Lapland and Peru (covered in /learn/why-the-earth-is-not-a-sphere).
• He didn't survey the planet. His measurement involved two cities; subsequent geodesy expanded to triangulation across continents.

His contribution was the methodological breakthrough — show that a planetary parameter could be measured rather than postulated.

Common misconceptions

“Eratosthenes proved the Earth was round.” He didn't — the spherical Earth was already accepted by educated Greeks. He measured the size of an Earth already understood to be spherical.

“Eratosthenes' result was a lucky guess.” It was a deliberate, methodical experiment. The ~2 %-accurate result was likely partly luck (some errors cancelled) and partly skill — the underlying geometry was correct, the observations were carefully made for the era's instruments. Even at 17 % error (worst plausible stadion conversion), the result is remarkable for the 3rd century BC.

“The Greeks knew Earth's exact circumference.” They had a measurement; they didn't consider it exact. Subsequent Greek and Islamic-world refinements continued for centuries. The exact value wasn't known to modern precision until 20th-century satellite geodesy.

“Anyone could have done this experiment.” The experiment is conceptually simple but requires specific conditions (two cities on roughly the same meridian, coordinated solstice observation, accurate ground-distance measurement). Eratosthenes had the intellectual resources of the Library of Alexandria and the data infrastructure of the Ptolemaic Egyptian state. The combination of opportunity, skill, and infrastructure made the measurement possible.

“The Earth is too small to be measured this way.” It's exactly the right size. Earth's ~40,000 km circumference with a 7.2° angular separation corresponds to ~800 km between cities — a distance measurable by foot-survey or camel-train in antiquity. Smaller bodies (the Moon) or bigger objects (the Sun) require different methodology.

“Modern GPS doesn't care about ancient measurements.” GPS computations use Earth's modern measured parameters (WGS 84 ellipsoid, with a = 6,378,137 m), refined continuously since Eratosthenes. The methodology — measure observable quantities, compute geometric parameters — is the same one Eratosthenes established. Modern geodesy is the continuation of the arc he started.

“The well at Syene was a real specific well.” The well-illuminated-at-noon detail is reported in ancient sources but the specific well isn't archaeologically identified. It may have been a real well that Eratosthenes knew about; it may have been a metaphor for “at noon, in this town, the sun is directly overhead.” The methodological point — observing the sun's elevation at two cities — is what matters; the specific instrument (well vs gnomon vs other shadow-caster) is incidental.

“Eratosthenes' original writings survive.” They don't. The Geographika is known only through later authors who cited it (Strabo, Cleomedes, Pliny the Elder). Our knowledge of the methodology comes from these secondary citations. This is typical for Greek antiquity — most original works are lost; we know about them through subsequent references.