02. The Size of the Earth — A Shadow and a Stride

The measurement everyone half-remembers

Around 240 BC, Eratosthenes of Cyrene measured the circumference of the Earth using sunlight, a well, and the distance between two Egyptian cities. At noon on the summer solstice the sun stood directly over Syene and cast no shadow into a deep well there. On the same day at Alexandria, roughly north of Syene, a vertical rod did cast a shadow, and the angle worked out to about one-fiftieth of a circle. If the two cities are separated by one-fiftieth of the way round the Earth, then the whole circumference is fifty times the distance between them. He put the answer at 250,000 stadia, later adjusted to 252,000, and the entire method rests on nothing more exotic than similar triangles. The modern value is about 40,075 kilometres, and the popular claim is that Eratosthenes came within one per cent of it.

Why nobody can say how close he got

The problem is that a stadion is not a fixed length, and no surviving source tells us which one he used. Estimates of the stadion range across roughly 157 to 185 metres depending on the standard assumed, and that range alone moves his answer by more than fifteen per cent. Choose the shorter Egyptian value and he lands almost exactly on the modern figure; choose the Attic value and he is out by a sixth. Dmitry Shcheglov and Jacques Dutka, working decades apart, both conclude that the "within one per cent" claim is an artefact of picking the flattering conversion. His inputs were also generous: Syene is not exactly on the tropic, it is not exactly due south of Alexandria, and the distance between them was a road estimate. The honest summary is that his method was sound, his answer was the right order of magnitude, and his precision is unknowable.

What the argument is really about

Two errors that pull in opposite directions can leave a wrong method looking accurate, which is the oldest trap in measurement. Eratosthenes deserves the credit anyway, because the achievement was showing that the size of the planet is a thing a person can compute from a shadow. The chapter's real lesson is that a result quoted without its unit is not a result, a point the next chapter turns into a seven-year survey and a suicide.

Example: Guessing a stranger's height from their shadow works well until you learn the photograph was taken in the evening.

Verdict. Eratosthenes measured the Earth by a valid method to the correct order of magnitude, and any tighter claim about his accuracy is a claim about the stadion, not about him.

Overturned if: a source is found that states which stadion Eratosthenes used, which would convert the accuracy question from unanswerable to settled.

Sources. 2018 · Dmitry A. Shcheglov, "The so-called 'Itinerary Stade' and the Accuracy of Eratosthenes' Measurement of the Earth," Klio — De Gruyter · 1993 · Jacques Dutka, "Eratosthenes' Measurement of the Earth Reconsidered," Archive for History of Exact Sciences — PhilPapers