Eratosthenes: how a shadow measured the Earth

Today we measure the Earth with satellites orbiting in space. But more than 2,200 years ago, in the Egyptian city of Alexandria, the keeper of a library did the same with a straight rod, its shadow and a little geometry. His name was Eratosthenes, and the answer he reached turned out to be remarkably accurate.

The librarian of Alexandria

According to the MacTutor archive of the University of St Andrews, Eratosthenes was born in Cyrene, in present-day Libya, and studied for some years in Athens. Ptolemy III, the king of Egypt, called him to Alexandria as tutor to his son. Around 240 BC, after the death of his teacher Callimachus, he became the third keeper of the famous library of Alexandria, which is said to have held hundreds of thousands of scrolls.

He was an expert in several fields: mathematics, astronomy, geography and history. According to MacTutor, his contemporaries called him “Beta”, meaning second in every field, although many of his works remain the foundation of scientific methods today. In number theory, his method of sorting out prime numbers, known as the “Sieve of Eratosthenes”, is still taught. According to the World History Encyclopedia, he also wrote a three-volume book called “Geography”.

The shadow calculation

Eratosthenes’ own book on the measurement of the Earth is lost. Its details have reached us through later writers, especially Cleomedes. The British historian of mathematics T. L. Heath described the method in his 1913 book as follows.

At noon on the longest day of summer, around 21 June, the sun stood directly overhead at Syene in southern Egypt, today’s Aswan, and cast no shadow. At the same moment in Alexandria, to the north, the pointer of a bowl-shaped sundial cast a shadow. Eratosthenes measured that the shadow made one-fiftieth of a full circle, about 7.2 degrees.

He then made two assumptions. First, the sun is so far away that its rays fall parallel everywhere on the Earth. Second, Alexandria and Syene lie on the same north-south line. If both hold, the angle of the shadow equals the angle at the centre of the Earth between the two cities. This means the distance between the two cities is one-fiftieth of the Earth’s full circumference.

The distance between the two cities was taken to be 5,000 stades. According to Heath, this was probably a round figure. Fifty times 5,000 gives the Earth’s circumference: 250,000 stades.

How accurate was the answer?

There is a complication here. Heath writes that Cleomedes gives the figure of 250,000, but according to the Greek geographer Strabo, Eratosthenes’ figure was 252,000 stades, and this was the one more often quoted in ancient times. Heath thinks Eratosthenes may himself have changed it later to make a round figure of 700 stades for one degree.

The real question is how long a stade was, since the unit varied from place to place in the Greek world. According to MacTutor, if it is taken as about 157 metres, based on the Roman writer Pliny, Eratosthenes’ answer is remarkably accurate. If it is taken as about 167 metres, it is less so. According to NASA, the Earth’s mean radius today is 6,371 kilometres, which gives a circumference of about 40,000 kilometres. With the shorter stade, Eratosthenes’ figure comes close to this.

MacTutor also gives another view. According to the researcher Rawlins, Eratosthenes probably measured only the angle at Alexandria himself and took the rest of the information from other sources. Whatever the debate over details, the method was entirely sound, and that is the real achievement.

Biruni and the fort of Nandana

More than 1,200 years after Eratosthenes, another famous measurement of the Earth was made in present-day Pakistan. According to the astronomical dictionary of the Paris Observatory, Abu Rayhan al-Biruni (973 to 1048) made his measurement at Nandana Fort, in the Salt Range in Jhelum district. Eratosthenes’ method needed two distant cities, but Biruni managed from a single place.

First, he found the height of a hill by taking angles from two points with an astrolabe. Then he climbed to the top and observed how far below eye level the horizon appeared. From these two measurements he worked out the Earth’s radius by trigonometry. According to MacTutor, his answer was 6,339.6 kilometres. The Paris Observatory dictionary adds that such accuracy was partly a coincidence, because the plain was not perfectly level.

The lesson today

Eratosthenes’ experiment is repeated in schools around the world today. All it needs is a straight rod, a tape and a friend in another city who measures the shadow of his rod on the same day at the same time. Two schools in Quetta and Karachi could carry out the experiment together. The lesson is the one the librarian of Alexandria gave: the answer to a big question is sometimes hidden in a shadow, if one knows how to measure it with care.