Eratosthenes and the measurement of the Earth

Around 240 BC Eratosthenes of Cyrene, director of the Library of Alexandria, managed to measure the circumference of the Earth with an experiment of astonishing simplicity, resting entirely on the geometry of similar triangles and shadows.

Eratosthenes knew that at Syene (modern Aswan), at noon on the summer solstice, the Sun lit the bottom of a well: its rays therefore fell perfectly vertical. At that same instant, in Alexandria — lying almost on the same meridian, further north — a vertical rod (a gnomon) cast a shadow. By measuring the ratio between the length of the shadow and the height of the rod, Eratosthenes found that the Sun’s rays made an angle of about $7.2°$ with the vertical, that is, one fiftieth of a full turn ($360° / 7.2° = 50$).

Since the Sun is extremely far away, its rays arrive parallel. The shadow angle at Alexandria is therefore equal — by the criterion of angles formed by parallel lines cut by a transversal — to the central angle of the Earth subtended by the arc between the two cities. Hence the proportion between angles and corresponding arcs holds:

$\frac{7.2°}{360°} = \frac{d}{C},$

where $d$ is the Syene–Alexandria distance and $C$ the Earth’s circumference. The caravans estimated that distance at about $5000$ stadia, so

$C = 50 \cdot 5000 = 250\,000 \text{ stadia}.$

Depending on the value assigned to the stadion, the result corresponds to roughly $39\,000$–$46\,000$ km, remarkably close to the true value of about $40\,000$ km. The decisive insight is the very one that governs similar triangles: when two configurations share the same angles, the corresponding quantities — here arcs and central angles — remain proportional, and from a local measurement (a shadow) one recovers a global one (the whole planet) (Boyer; Katz).

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Topics: Similarity
Concepts: Ratio of similarity · Similarity
People: Eratosthenes