In about 240 BCE, in the Ptolemaic city of Alexandria, the librarian Eratosthenes worked out the circumference of the spherical Earth to an accuracy of roughly two percent. He used no instrument that would not be available to a contemporary primary school class. He used, in fact, almost nothing.
The story is sometimes told as a curiosity. It is more usefully told as a marker — the moment at which the practice of cartography stops being a literary tradition about distant lands and starts being a quantitative, geometrical discipline.
What he did
Eratosthenes had heard a report — from travelers' notes preserved in the Library of Alexandria, the institution he had been hired to direct — that at noon on the summer solstice, in the city of Syene (modern Aswan, in southern Egypt), a vertical pole cast no shadow. The sun was, at that moment, directly overhead, and a person looking down a deep well in Syene could see the sun reflected from the water at the bottom.
Eratosthenes knew that on the same day, at the same noon, a vertical pole in Alexandria did cast a shadow. He measured the angle of that shadow as approximately one fiftieth of a circle — about 7.2 degrees.
The argument from there is simple but not obvious. If the Earth is spherical, and the sun is so distant that its rays reach the Earth as effectively parallel lines, then the angular difference between the two cities — Syene with no shadow, Alexandria with a 7.2-degree shadow — corresponds to the angular distance between them along the surface of the Earth. One fiftieth of the full 360-degree circumference of the Earth, in arc, equals the linear distance between Syene and Alexandria.
The Greek conventional unit was the stade — a measure derived from athletic stadia, perhaps 157.5 meters in the most plausible reconstruction. The conventionally cited distance from Alexandria to Syene was 5,000 stades, measured by professional pace-counters (bematistai) hired by the Ptolemaic state. Multiply by fifty: 250,000 stades for the Earth's circumference. Convert to modern units: approximately 39,375 km.
The actual polar circumference of the Earth is 40,008 km. Eratosthenes was off by less than two percent, using a stick, a calendar, and a courier's distance estimate.
What he was working with
To put the measurement in context, Eratosthenes was working with a set of facts that had themselves been argued out only over the previous two centuries. That the Earth was a sphere had been a Greek consensus since Pythagoras and was strengthened by Aristotle's observational arguments — the round shadow Earth casts on the moon during a lunar eclipse, the way ships disappear hull-first below the horizon, the changing constellations as one travels north or south. The geometry of similar triangles and proportional reasoning was Eucid's; the Elements had been compiled at Alexandria a few decades before Eratosthenes.
What was new in Eratosthenes' work was the empirical reduction: to extract a single quantitative fact about the Earth from two specific observations and a piece of geometry. The technique is a kind of inverse triangulation; the surveying tradition that culminates two thousand years later in the Cassini family's mapping of France is, in a real sense, a continuous lineage from this measurement.
The other measurement
Eratosthenes also constructed what is conventionally called the first map of the world based on systematic geographical reasoning, with an explicit grid of latitudes and longitudes and the known landmasses placed on it according to the best available distances and bearings. The map itself does not survive; we know its outlines from descriptions in later authors, especially Strabo, writing two centuries afterward. The map covered, in modern terms, the Mediterranean, North Africa, Europe to roughly the Baltic, southwestern Asia, India, and parts of inner Asia. It was wrong in many particulars — the size of the Caspian was off, the shape of the Indian subcontinent was distorted, the African coast trailed off into speculation — but it was systematic. The known world had been reduced to a coordinate system.
Why the achievement faded
The Library of Alexandria's destruction is overstated in most popular accounts, but the institutional collapse was real and gradual. By the second century CE, when Ptolemy (no relation to the dynasty) wrote the Geographia — the cartographic treatise that dominated medieval and Renaissance European mapmaking — Eratosthenes' figure for the Earth's circumference had been displaced by a smaller, less accurate figure, derived by Posidonius and adopted by Ptolemy. The error was in the wrong direction: Ptolemy's Earth was too small, by perhaps fifteen percent.
This mattered. When Christopher Columbus, sixteen centuries later, calculated the westward sailing distance from Iberia to Asia, he was using Ptolemy's smaller Earth and an over-large estimate of Asia's eastward extent. The two errors added: he believed Japan was perhaps 4,500 km west of the Canaries. The actual distance is about 19,000 km. Had Columbus believed Eratosthenes, he would not have sailed.
That the Americas were in the way is, in this sense, a happy collision of two errors with a continent.
What stayed with me
That the measurement is, in retrospect, trivially reproducible. A high school class with a meter stick, two cities a few hundred kilometers apart, and a phone clock can do it on the next solstice and get within a few percent. The ancient achievement was not the equipment. It was the willingness to believe that a quantity as cosmically large as the size of the planet could be inferred from two ordinary local observations and a piece of geometry. That move — the move from the world is too big for us to measure to the world can be measured by us, with what we have here — is the founding move of empirical science.