How Eratosthenes Measured Earth’s Circumference With Shadows

Around 240 BC, the Greek scholar Eratosthenes of Cyrene worked out the size of the whole Earth without leaving Egypt. He noticed that at noon on the summer solstice the Sun shone straight down into a well at Syene (modern Aswan), casting no shadow, while at the same moment an upright stick in Alexandria, far to the north, still cast a short shadow. The angle of that shadow was about 7.2 degrees, roughly one fiftieth of a full circle. If the Sun’s rays are parallel, the distance between the two cities must also be one fiftieth of the Earth’s circumference.

The distance was reckoned at about 5,000 stadia, so the full circumference came out at about 250,000 stadia. Depending on which length of stadion he used, that equals somewhere around 40,000 to 46,000 kilometres; the real figure is about 40,000 km around the poles. Either way, it was an astonishing result for a man armed with a shadow, some walking measurements and geometry.

Who was Eratosthenes?

Eratosthenes was born around 276 BC in Cyrene, a Greek city in what is now Libya, and studied in Athens. After Ptolemy III became king of Egypt in 246 BC, he invited Eratosthenes to Alexandria to run its famous Library, the greatest collection of knowledge in the ancient world. He was then around thirty, not in his forties as is sometimes said, and he stayed in Alexandria for the rest of his life.

He was a true all-rounder: mathematician, geographer, astronomer, historian and poet. Admirers called him Pentathlos, after the athletes who competed in five events. Rivals called him Beta, the second letter of the Greek alphabet, implying he was second-best in every field. History has been kinder: few people in any age have contributed to so many subjects. According to the Byzantine encyclopaedia known as the Suda, he lost his sight in old age and died around 194 BC.

The experiment, step by step

The method rests on two assumptions that Greek scholars already accepted: the Earth is a sphere, and the Sun is so far away that its rays arrive essentially parallel.

The overhead Sun at Syene. Eratosthenes knew that at noon on the longest day of the year, sunlight reached the bottom of a deep well at Syene. That meant the Sun stood directly overhead there.
The shadow at Alexandria. At the same time on the same day, a vertical pointer called a gnomon in Alexandria cast a shadow. Measuring it showed the Sun was about 7.2 degrees from straight up.
The geometry. Imagine lines from both cities running to the centre of the Earth. Because the sunbeams are parallel, the angle between those two lines at the Earth’s centre equals the 7.2-degree shadow angle at Alexandria. That is 1/50 of 360 degrees.
The distance. Surveyors known as bematists, trained to measure distances by counting regular steps, put the distance from Syene to Alexandria at about 5,000 stadia.
The answer. Fifty times 5,000 gives 250,000 stadia for the full circumference.

The diagram shows the idea: parallel rays of sunlight strike Syene vertically while arriving at an angle in Alexandria, and that difference reveals how much of the Earth’s curve lies between the two cities.

How accurate was he?

That depends on a question nobody can fully settle: how long was his stadion? Greek stadia varied from place to place, from roughly 160 to 210 metres. Using a stadion of about 185 metres, a common estimate, the result is about 46,000 km, some 15% too large. Using a shorter stadion of about 160 metres, it comes to 40,000 km, almost exactly right. Some historians argue his original figure was a little different and, with a 180-metre stadion, came within about 1%.

He also made small mistakes. Syene is not exactly on the Tropic of Cancer, and it is not due south of Alexandria; it lies a few degrees to the east. As it happens, his errors partly cancelled each other out. His own book, On the Measure of the Earth, is lost. We know his method mainly from a simplified summary by the later Greek writer Cleomedes, and later authors quote the result as 252,000 stadia, a rounded figure that divides neatly by 60.

Whatever the exact accuracy, the method was correct in principle, and it was the first known calculation of the Earth’s size based on observation and reasoning rather than guesswork.

This short animated video walks through the experiment visually. The narration is not in English, so switch on the English subtitles in the player settings.

More than one great idea

The Earth measurement is his most famous work, but far from his only one:

The sieve of Eratosthenes: a simple way of finding prime numbers by crossing out the multiples of each number in turn. It is still taught in schools and used in computing.
Geography: he is credited with coining the word itself. His Geographica mapped the known world using lines of latitude and longitude, listed more than 400 cities and divided the Earth into five climate zones.
The tilt of the Earth: he measured the angle between the Earth’s equator and the Sun’s apparent path at about 23° 51′, close to the modern value.
Chronology: he tried to date historical events scientifically, placing the fall of Troy in 1184/1183 BC, and compiled a list of Olympic winners.
The armillary sphere: the later astronomer Hipparchus credited him with inventing this model of the heavens made of rings.

Myths about ancient views of the Earth

“People thought the Earth was flat until Columbus.” Not true. Educated Greeks had known the Earth was round since at least the time of Pythagoras and Aristotle, and medieval scholars knew it too. The argument over Columbus’s voyage was about the size of the Earth: his critics, relying on figures like Eratosthenes’, correctly said Asia was much too far away to reach by sailing west.

“He did it with just a stick.” The stick, or gnomon, was the elegant part, but the calculation also depended on knowing the distance between the cities and on careful astronomical knowledge of the solstice.

“His answer was exact.” It was impressively close, but because of the uncertainty about the stadion, any claim of perfect accuracy goes beyond the evidence.

Try the experiment yourself

Schools around the world repeat Eratosthenes’ experiment every year, usually at an equinox in March or September, when the Sun is directly over the equator at noon. You need a vertical stick, a level surface, a ruler and a partner school or friend several hundred kilometres to the north or south.

At local solar noon, when the shadow is shortest, measure the height of the stick and the length of its shadow.
Find the Sun’s angle from vertical with a calculator: the inverse tangent of shadow length divided by stick height.
Compare angles with your partner. The difference, as a fraction of 360 degrees, equals the north–south distance between you as a fraction of the Earth’s circumference.

Even with simple tools, classes can get surprisingly close to the true value, which shows how powerful a good idea can be. More than 2,200 years later, Eratosthenes’ shadow still teaches the same lesson: careful observation and clear reasoning can reveal the shape of the world.

Sources: Wikipedia: Eratosthenes; video: YouTube.

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