The Librarian Who Measured the Earth With a Shadow

Around 240 BC, in the city of Alexandria, the chief librarian of the Library of Alexandria — a polymath named Eratosthenes of Cyrene, who at the time was about forty-five years old and had been running the library for roughly five years — measured the circumference of the Earth to within approximately two percent of its actual value, using a measurement of two shadows and a piece of straightforward arithmetic. He did this without leaving Alexandria. He did it using a single piece of geographical information (the distance from Alexandria south to the town of Syene, modern Aswan) and a single piece of observational data (the angle of the noon sun in Alexandria on the day of the summer solstice, compared to the noon sun in Syene on the same day).

The measurement, in its details, is one of the most elegantly simple pieces of experimental physics from the ancient world.

What he did

Eratosthenes had read, in one of the geographical or travel-account texts in the Library, that on the day of the summer solstice (around 21 June in the modern calendar) the sun shone directly into a deep well at the Egyptian town of Syene at noon — meaning the sun was at that moment directly overhead at that location, so that a vertical well had no shadow on its inner walls. This is a real astronomical phenomenon: Syene sits very close to the Tropic of Cancer (modern Aswan is at 24.09° N, the Tropic in 240 BC was at approximately 23.74° N), so the sun was almost exactly overhead at noon on the solstice. Eratosthenes apparently did not verify this personally; he took the reported observation on faith from the geographic literature available to him in Alexandria.

Alexandria is about 800 kilometers north of Syene, almost due north (the longitudes differ by less than a degree). If at the same moment in Syene the sun was directly overhead, then at noon in Alexandria the sun must be displaced from overhead by an angle equal to the angular distance between the two cities as seen from the center of the Earth. Eratosthenes measured this angle in Alexandria by placing a vertical pole — a gnomon — in a public square and measuring the length of its noon shadow on the day of the solstice. The ratio of the shadow length to the pole height gives the tangent of the sun’s displacement from overhead, which converts directly to the angular displacement.

Eratosthenes’s measurement gave a sun displacement of approximately 7°12′ — or as he expressed it, one fiftieth of a complete circle.

If 7°12′ of arc corresponds to the Alexandria-to-Syene distance, then the full 360° corresponds to fifty times that distance. The distance from Alexandria to Syene was given in the standard geographical sources of the time as approximately 5,000 stadia. Therefore the circumference of the Earth was approximately 250,000 stadia.

How close he got

The accuracy of the result depends on the value of the stadion — the Greek unit of length Eratosthenes used. The Greek world used several different stadia. The most common values in the third century BC were:

• The Attic stadion of approximately 185 meters
• The Egyptian/Itinerary stadion of approximately 157.5 meters
• The Olympic stadion of approximately 176 meters

Modern scholars are split on which stadion Eratosthenes used. The answer matters a great deal:

• 250,000 × 185 m = 46,250 km (15% too large)
• 250,000 × 176 m = 44,000 km (10% too large)
• 250,000 × 157.5 m = 39,375 km (1.7% too small)

The Earth’s actual mean circumference is 40,008 km. If Eratosthenes used the Egyptian/Itinerary stadion — which is plausible given that he was working in Egypt and using Egyptian travel-distance reports — he was off by less than two percent. If he used the Attic stadion he was off by fifteen percent. Most modern scholarship favors the Egyptian/Itinerary interpretation, partly on grounds of geographic plausibility and partly because Cleomedes (writing about 450 years later) gives a value of the stadion that supports the more accurate reading.

Eratosthenes himself does not appear to have published the specific stadion value he used. His original text — a book titled On the Measurement of the Earth — was lost in late antiquity. What survives is a description of the method in the On the Circular Motions of the Celestial Bodies by Cleomedes, a Greek astronomer of the late Hellenistic or early Imperial period whose dates are uncertain. Cleomedes’s account is the only complete description of Eratosthenes’s procedure that survives.

What this didn’t prove

The result is sometimes described as “the first proof that the Earth is round.” It was not. Greek astronomy and natural philosophy had accepted the spherical shape of the Earth at least since Pythagoras in the sixth century BC, and the proof had been argued from independent grounds (the shape of the Earth’s shadow on the moon during a lunar eclipse, the visibility horizon at sea, the apparent motion of stars as one travelled north or south) since at least Aristotle in the fourth century. Eratosthenes assumed the spherical shape of the Earth in his calculation; the calculation gave the size of the sphere, not its shape.

What Eratosthenes’s result did prove, for the first time, was that the spherical Earth was very large — much larger than the inhabited Greek world. If the planet was 250,000 stadia around and the known inhabited world (from the Atlantic coast of Iberia to the Indus valley) was about 70,000 stadia across, then the inhabited world was about one-fifth of the total circumference. The other four-fifths must therefore be either ocean or unknown land. This conclusion was the basis of the medieval and Renaissance speculation that one could reach the eastern coast of Asia by sailing west from Europe across an ocean of measurable extent. Columbus, who used a substantially smaller estimate of the Earth’s circumference based on the work of the medieval Arab geographer al-Farghani, thought the westward voyage to Asia would be about 4,000 km. Eratosthenes’s larger and more accurate estimate would have put it at over 20,000 km. The intervening continents had not been imagined.

What survives

No physical instrument used by Eratosthenes has been identified. The gnomon-pole he used in Alexandria was a public sundial pole that was very likely reused for ordinary timekeeping for centuries afterward and is not separately documented. The well at Syene — the well in which the noon sun shone with no shadow on the summer solstice — is still in Aswan, in the basement of an active dwelling on the modern street called Sharia el-Tahrir. It is a circular stone-lined shaft about 2 meters across and approximately 9 meters deep. It is no longer used for water. Visitors can request to see it. The current owners are accustomed to occasional foreign tourists asking after it.

The well, on the noon of the summer solstice, still has no shadow on its walls. Egypt’s continental drift over the past 2,300 years is small enough that the geographical observation is approximately as true now as it was in 240 BC.