Around 240 BCE, Eratosthenes of Cyrene, the chief librarian at Alexandria, produced an estimate for the circumference of the Earth. He used the angle of a shadow cast by an upright rod, the report that the Sun stood directly overhead at the southern city of Syene on the summer solstice, and the estimated distance between the two cities. His answer was 250,000 stadia, later refined to 252,000.

The figure has been repeated for centuries as one of the great early triumphs of measurement, usually with the claim that he landed within a per cent or two of the modern value. That part deserves a closer look, because the accuracy of his result is not actually something we can pin down.

How the method worked

The reasoning is sound, and that is the genuinely impressive part. Eratosthenes knew that at Syene, modern Aswan, the Sun was reported to reach the point directly overhead at noon on the summer solstice. A vertical object there cast no shadow. At Alexandria, to the north, the same upright rod at the same moment did cast a shadow. By measuring the length of the rod and the length of the shadow, he could work out the angle of the Sun’s rays from vertical.

That angle came to about 7.2 degrees, which is one fiftieth of a full circle. If Alexandria sat due north of Syene, then the arc of the Earth’s surface between them had to be one fiftieth of the whole circumference. The distance between the cities was taken to be about 5,000 stadia. Multiply by fifty and the circumference comes out at 250,000 stadia.

The geometry holds. Given a spherical Earth, parallel sunlight, and two cities on the same meridian, the method works.

The problem with the accuracy claim

Here is where the familiar version gets ahead of the evidence. Converting 250,000 stadia into kilometres requires knowing how long a stadion was, and that length is disputed.

If Eratosthenes used a stadion of around 157 metres, often called the Egyptian value, the result lands near 39,000 to 39,700 kilometres, against a true figure of roughly 40,000. That is the famous error of one or two per cent. If instead he used the Attic or Olympic stadion of about 185 metres, the same 250,000 stadia translate to something closer to 46,000 kilometres, an error of ten to fifteen per cent.

We do not know which unit he used. The original treatise is lost. As Britannica puts it, the length of the stadia is doubtful and the accuracy of the result is therefore uncertain, with estimates of the error ranging from about half a per cent to seventeen per cent depending on the assumption.

So the headline number is not wrong, exactly. It is one reading among several, and it happens to be the most flattering one. The honest position is that his result was in the right range, and that how close it came to the right range is something we cannot settle.

The approximations that did not all pull the same way

Even setting the unit aside, several approximations were baked into the measurement, and they did not all push in the same direction.

Syene is not exactly on the Tropic of Cancer, so the Sun was not perfectly overhead there at solstice. Alexandria is not exactly due north of Syene, so the two cities do not lie on a clean meridian. And the 5,000 stadia between them was an estimate, traditionally credited to professional pacers and checked over repeated trips, not a surveyed line.

Some of these errors offset others. That is part of why the final figure can look so good under the favourable unit. A measurement that lands close after several approximations is not always more careful than one that does not. Sometimes it is the approximations cancelling.

What actually survives

The account most people know does not come from Eratosthenes. His own work, titled “On the Measurement of the Earth,” did not survive. What survives are details preserved by later writers, chiefly the Stoic philosopher Cleomedes, who reduced the procedure to a clean teaching example, complete with the often-repeated detail of sunlight reaching the bottom of a well at Syene. The St Andrews MacTutor history of mathematics notes that the treatise is lost and that what we have comes through Cleomedes, Theon of Smyrna, and Strabo. The rounding of 250,000 to 252,000 stadia, which gives a tidy 700 stadia per degree, belongs to this later tradition of tidying the result for the reader.

None of this diminishes what Eratosthenes did. He recognised that a difference in shadow angles between two places was a measurement of the Earth’s curvature, scaled it correctly to the whole sphere, and arrived at a figure of the right order of magnitude at a time when even that was not assured. The method was the achievement. It could be checked, refined, and repeated, which is what separates it from a lucky guess.

The part worth holding onto is the reasoning, not the digits. The digits depend on a unit we have lost. The reasoning has not needed correcting in 2,200 years.