Eratosthenes Measures the Earth (c. 240 BC): Syene and Alexandria

Around 240 BC Eratosthenes of Cyrene, the chief librarian of Alexandria, measured the size of the Earth using nothing more than shadows, geometry and a known distance. By comparing the Sun’s angle at noon on the summer solstice at Syene (modern Aswan) and Alexandria, he estimated the planet’s circumference at 250,000 stadia. Depending on the unit he used, his result was within a few percent to about 16 percent of the true value, a remarkable achievement for the ancient world.

Background: a round Earth

By the 3rd century BC educated Greeks agreed that the Earth is a sphere. Aristotle had pointed to the curved shadow of the Earth on the Moon during lunar eclipses and to the way different stars become visible as a traveller moves north or south. Aristotle also mentioned an estimate of 400,000 stadia by earlier mathematicians, and Archimedes later cited 300,000 stadia, but how these figures were found is not known.

The open question was not the Earth’s shape but its size. Measuring it also had practical value. Hellenistic kings needed maps for administration, trade and war, and Alexandria’s scholars collected reports from travellers, soldiers and sailors across the known world. Eratosthenes was the first known person to measure it by a method that can be checked.

Eratosthenes of Cyrene

Eratosthenes was born around 276 BC in Cyrene, a Greek city in modern Libya, and studied in Athens. Around 245 BC Ptolemy III Euergetes invited him to Alexandria, and he became head of the Library of Alexandria, succeeding Apollonius of Rhodes. He also tutored the future Ptolemy IV.

Alexandria under the early Ptolemies was the best place in the world for such work. The Library collected books from across the Greek world, and scholars at the Museum received royal salaries to research and teach. Eratosthenes could draw on travellers’ reports, earlier maps and astronomical records gathered there.

He worked in so many fields that some contemporaries nicknamed him “Beta”, meaning second best at everything. In fact he was a leading scholar in geography, mathematics, astronomy, chronology and literary criticism, and a correspondent of Archimedes.

The method

Eratosthenes’ own book, On the Measurement of the Earth, is lost. The clearest account is by Cleomedes, a later Greek writer on astronomy whose date is uncertain, probably between the 1st century BC and the 2nd century AD. His version is simplified, but the main steps are clear.

1. At Syene, on the summer solstice, the noon Sun stood directly overhead. Vertical sundial pointers cast no shadow, and, according to later tradition, sunlight reached the bottom of deep wells.
2. At the same moment in Alexandria, which Eratosthenes assumed lay due north of Syene, the pointer of a bowl-shaped sundial (a skaphe) cast a short shadow. The angle was one-fiftieth of a full circle, or 7.2 degrees.
3. Because the Sun is so far away, its rays reach both cities as parallel lines. The angle at Alexandria therefore equals the angle between the two cities measured at the centre of the Earth.
4. The distance between Syene and Alexandria was taken as 5,000 stadia. If 5,000 stadia correspond to one-fiftieth of the circle, the whole circumference is 50 x 5,000 = 250,000 stadia.

Later sources, including Strabo and Pliny the Elder, give 252,000 stadia, perhaps rounded so that the figure divides evenly by 60 into 4,200 stadia per sixtieth of a circle.

How accurate was it?

The answer depends on the length of the stadion, which varied in the ancient world, and scholars disagree about which one Eratosthenes used.

Several of his assumptions were slightly wrong. Syene lies a little north of the Tropic of Cancer, and Aswan is about 3 degrees of longitude east of Alexandria, not due south. The true distance between the cities along the meridian is about 800 to 850 km. Some of these errors cancelled each other out, which helps explain why his final figure was better than his individual measurements. How the 5,000-stadia distance was measured is unclear: it may have come from royal surveys or estimates of travel times along the Nile.

Whatever the exact figure, Eratosthenes’ method was correct, and his result was the right order of magnitude, a huge advance on earlier guesses. Later geographers, including Hipparchus, generally accepted it with only minor adjustments.

Later measurements

Other ancient scholars tried to improve on Eratosthenes. Posidonius of Apamea, in the early 1st century BC, compared the height of the bright star Canopus above the horizon at Rhodes and at Alexandria and reached a figure of 240,000 stadia. A smaller figure of 180,000 stadia, also linked with Posidonius by Strabo, was adopted by Ptolemy in the 2nd century AD. Because Ptolemy’s Geography became a standard work in the Renaissance, this underestimate influenced European maps for centuries.

Scholars in the Islamic world returned to direct measurement. Under the Abbasid caliph al-Ma’mun in the 9th century, teams of astronomers measured the length of one degree of latitude on the plain of Sinjar in Mesopotamia, obtaining a result close to the modern value. In the 11th century al-Biruni calculated the Earth’s radius from the dip of the horizon seen from a mountain.

Modern geodesy began with French expeditions of the 1730s to Lapland and Peru, which confirmed that the Earth is slightly flattened at the poles. Satellite measurements now give the circumference as about 40,075 km around the equator and about 40,008 km through the poles.

Other achievements

Eratosthenes wrote a three-book Geography and drew a world map using lines of latitude and longitude, the first known systematic attempt of its kind. He is often credited with coining the word “geography”.

In mathematics he devised the “sieve of Eratosthenes”, a simple method of finding prime numbers that is still taught. He also produced a chronology of Greek history, dating events from the fall of Troy, and measured the tilt of the Earth’s axis with reasonable accuracy. According to the Byzantine Suda, he starved himself to death around 194 BC after losing his sight, though this tradition cannot be confirmed.

Myths and facts

Myth: he proved the Earth is round. That was already accepted. Eratosthenes measured how big it is.

Myth: medieval Europeans thought the Earth was flat until Columbus. Medieval scholars knew the Earth was a sphere. The debate over Columbus’s voyage in 1492 was about its size: Columbus relied on smaller estimates than Eratosthenes’, which made Asia seem much closer than it was.

Timeline

• c. 276 BC: Eratosthenes is born at Cyrene.
• c. 245 BC: Ptolemy III invites him to Alexandria.
• c. 240 BC: Eratosthenes measures the Earth’s circumference.
• c. 194 BC: Eratosthenes dies in Alexandria.
• 1st century AD: Strabo and Pliny the Elder report his result.
• 1492: Columbus sails west using a much smaller estimate of the Earth’s size.

Why it matters

Eratosthenes showed that the size of the whole planet could be found from a few careful observations and simple geometry. His experiment remains one of the most famous examples of scientific reasoning and is still repeated by schoolchildren around the world on the equinoxes and solstices.