The Whole Earth from a Shadow

In one line: Around 240 BCE Eratosthenes measured the circumference of the whole planet from within a single country — by comparing the length of a noon shadow in two cities and scaling one small arc up to the full circle.

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The world then

By the third century BCE, educated Greeks already accepted that the Earth is a sphere. Aristotle had argued it from evidence: the Earth casts a round shadow on the Moon during a lunar eclipse, and new stars rise into view as you travel south. What no one had done was measure the sphere. Its size was an open question, bounded only by guesswork.

The question

Could you measure something cosmic — the size of the entire world — without ever leaving the ground? Eratosthenes, the chief librarian of the Library of Alexandria, saw that a shadow holds the answer. On a flat Earth, a vertical rod would cast an identically angled noon shadow everywhere at once. Because the Earth curves, that angle changes from place to place, and the change is a direct measure of the curvature.

The design

He combined two observations made at the same moment — local noon on the summer solstice. At Syene (modern Aswan), which sits close to the Tropic of Cancer, the Sun stands almost directly overhead: upright objects cast almost no shadow. At Alexandria, well to the north, a vertical gnomon does cast a shadow — one that measured one-fiftieth of a full circle (what we would now call about 7.2°). The key step is a single assumption: the Sun is so distant that its rays arrive effectively parallel. Given parallel rays, the shadow angle at Alexandria equals the angle subtended between the two cities at the Earth's centre (alternate angles). The local shadow is a direct read-out of the planet's curvature.

The result

If the arc between the cities is one-fiftieth of a full turn, then the arc is one-fiftieth of the Earth's circumference. The overland distance Syene→Alexandria was a known figure of roughly 5,000 stadia, so the whole circumference is 50 × 5,000 = 250,000 stadia (Strabo and Pliny report a refined 252,000). From two shadows and one distance, Eratosthenes had reckoned the size of the whole world.

What we learned, and why it's beautiful

How accurate was he? That depends entirely on the length of his stadion, a unit we can no longer pin down: by the shortest plausible value he was within a percent or two of the modern figure (~40,000 km); by others he is more like 10–15% high, and some of any closeness comes from errors that happen to cancel. But precision is not the point. The beauty is leverage — a planetary quantity pulled down into a schoolroom exercise in proportions, using no instrument but a stick and the Sun and one bold assumption about distant light. It was the first known measurement of the planet's size by pure reason: geometry, applied with nerve.

Sources

Full claim-by-claim evidence is in references.md. Primary/authoritative anchors:

• Cleomedes, Caelestia (On the Circular Motions of the Celestial Bodies) — the principal surviving account of the calculation (Eratosthenes' own On the Measurement of the Earth is lost).
• Strabo, Geography, and Pliny, Natural History — secondary witnesses reporting 252,000 stadia.
• Aristotle, On the Heavens (De Caelo) — the empirical arguments for a spherical Earth.
• MacTutor (Univ. of St Andrews), Eratosthenes; Wikipedia, Earth's circumference; MAA Convergence, Eratosthenes and the Mystery of the Stades.
• D. Shcheglov, "The 'Itinerary Stade' and the Accuracy of Eratosthenes' Measurement"; Peter Gainsford, Kiwi Hellenist (on the well legend and the stadion controversy).

Accuracy notes: The dramatized image of Eratosthenes looking down a well is a modern popularization (Carl Sagan's Cosmos); the well was an ancient marker that the Sun stood overhead at Syene (Pliny, Strabo), but no ancient source ties it to his calculation, which used a shadow instrument (a skaphē / gnomon). Syene is near but not exactly on the Tropic, and Alexandria is not exactly on Syene's meridian — idealizations the ancient account already made. "1/50 of a circle" is the ancient figure; "7.2°" is the modern gloss. The method yields the circumference directly (radius and diameter follow by division). The famous "accurate to within 1%" is partly an artifact of choosing the short stadion; the honest statement is a range of roughly 1–15%.