## Measuring the Circumference of the Earth

*By Cynthia Stokes Brown*

*More than 2,000 years ago Eratosthenes compared the position of the Sun’s rays in two locations to calculatethe spherical size of the Earth with reasonable accuracy.*

Eratosthenes was born in the Greek colony Cyrene, now the city of Shahhat, Libya. As a young man, he traveled to Athens to pursue his studies. He returned to Cyrene and made such a name for himself in scholarly endeavors that the Greek ruler of Egypt brought him to Alexandria to tutor his son. When the chief librarian of the famous Library of Alexandria died in 236 BCE, Eratosthenes was appointed to the prominent position around the age of 40.

A man of many talents, Eratosthenes was a librarian, geographer, mathematician, astronomer, historian, and poet. His friends at the library nicknamed him Pentathlos, or athlete who competes in five different events. The name seemed to fit a scholar who excelled in many fields of study. Most of Eratosthenes’s writings have been lost, but other scholars reported his work and findings — which were extensive.

## Studying the earth

Eratosthenes may have been the first to use the word *geography*. He invented a system of longitude and latitude and made a map of the known world. He also designed a system for finding prime numbers — whole numbers that can only be divided by themselves or by the number 1. This method, still in use today, is called the “Sieve of Eratosthenes.”

Eratosthenes was also the first to calculate the tilt of the Earth’s axis, which he figured with remarkable accuracy; the finding was reported by Ptolemy (85-165 CE). Eratosthenes also calculated the distance from the Earth tothe Moon and to the Sun, but with less accuracy. He made a catalog of 675 stars. He made a calendar with leap years and laid the foundation of chronology in the Western world by organizing the dates of literary and political events from the siege of Troy (about 1194–1184 BCE) to his own time.

Yet his most lasting achievement was his remarkably accurate calculation of the Earth’s circumference (the distance around a circle or sphere). He computed this by using simple geometry and trigonometry and by recognizing Earth as a sphere in space. Most Greek scholars by the time of Aristotle (384–322 BCE) agreed that Earth was a sphere, but none knew how big it was.

How did Greek scholars know the Earth was a sphere? They observed that ships disappeared over the horizon while their masts were still visible. They saw the curved shadow of the Earth on the Moon during lunar eclipses. And they noticed the changing positions of the stars in the sky.

## Measuring the earth

Eratosthenes heard about a famous well in the Egyptian city of Swenet (Syene in Greek, and now known as Aswan), on the Nile River. At noon one day each year — the summer solstice (between June 20 and June 22) — the Sun’s rays shone straight down into the deep pit. They illuminated only the water at the bottom, not the sides of the well as on other days, proving that the Sun was directly overhead. (Syene was located very close to what we call the Tropic of Cancer, 23.5 degrees north, the northernmost latitude at which the Sun is ever directly overhead at noon.)

Eratosthenes erected a pole in Alexandria, and on the summer solstice he observed that it cast a shadow, proving that the Sun was not directly overhead but slightly south. Recognizing the curvature of the Earth and knowing the distance between the two cities enabled Eratosthenes to calculate the planet’s circumference.

Eratosthenes could measure the angle of the Sun’s rays off the vertical by dividing the length of the leg opposite the angle (the length of the shadow) by the leg adjacent to the angle (the height of the pole). This gave him an angle of 7.12 degrees. He knew that the circumference of Earth constituted a circle of 360 degrees, so 7.12 (or 7.2, to divide 360 evenly by 50) degrees wouldbe about one-fiftieth of the circumference. He also knew the approximate distance between Alexandria and Syene, so he could set up this equation:

Eratosthenes estimated the distance from Alexandria to Syene as 5,000 stadia, or about 500 miles (800 kilometers). He made this estimation from the time it took walkers, who were trained to measure distances by taking regular strides, to trek between the cities. By solving the equation, he calculateda circumference of 250,000 stadia, or 25,000 miles (40,000 kilometers).

Several sources of error crept into Eratosthenes’s calculations and our interpretation of them. For one thing, he was using as his unit of measure the Greek unit “stadion,” or the length of an athletic stadium. But not all stadiums were built the same length. In Greece a stadion equaled roughly 185 meters (607 feet), while in Egypt the stadion was about 157.5 meters (517 feet). We don’t know which unit Eratosthenes used. If he used the Greek measure, his calculation would have been off by about 16 percent. If he used the Egyptian one, his error would have been less than 2 percent off the actual Earth’s circumference of 24,860 miles (40,008 kilometers).

A century after Eratosthenes, the Greek astronomer Posidonius of Rhodes (c. 135–51 BCE) calculated the Earth’s circumference. Posidonius used the star Canopus as frame of reference: when the star is visible at the horizon in Rhodes, it is 7.5 degrees above the horizon in Alexandria. His first calculations came out almost exactly correct, but he revised the distance between Rhodes and Alexandria, which resulted in a number comparable to about 18,000 miles (about 29,000 kilometers), some 28 percent smaller than the actual circumference. Ptolemy reported the calculations of Posidonius instead of those of Eratosthenes, and it was Ptolemy’s writings that found their way to Christopher Columbus. If Ptolemy had used Eratosthenes’s larger, more accurate figure for Earth’s circumference, Columbus might never have sailed west.

Eratosthenes lived to be about 82 years old, when he starved himself to death because he feared the onset of blindness.

By Cynthia Stokes Brown

## For Further Discussion

Think about the following and share your ideas in the Questions Area below. If you were living in Greece at the time of Eratosthenes, how do you think you would have reacted to his proof? If you had believed that the Earth was flat, do you think you would have been convinced by what he was able to show?

**Sources**

Eratosthenes Project. Accessed June 13, 2011. http://eaae-astronomy.org/eratosthenes/.

Lasky, Kathryn. *The Librarian Who Measured the Earth*. New York and Boston: Little, Brown, 1994.

Nicastro, Nicholas. *Circumference: Eratosthenes and the Ancient Quest to Measure the Globe*. New York: St. Martin’s, 2008.

Teacher’s Guide: The Eratosthenes Project. Accessed June 13, 2011. http://www.physics2005.org/projects/eratosthenes/TeachersGuide.pdf.

**Image credits**

An undated illustration of scholars at the Library of Alexandria© Bettmann/CORBIS

A reconstruction of Eratosthenes’s c. 194 BCE map of the world, from E.H. Bunbury’s 1883 A History of Ancient Geography among the Greeks and Romans from the Earliest Ages till the Fall of the Roman Empire, public domain

A diagram showing how Eratosthenes measured the Earth,accessed from Simon Fraser University Online, http://www.sfu.ca/phys/100/lectures/lecture3/Eratosthenes.html.