Eratosthenes' Calculation of Earth's Circumference



Click the green "forward" button to step through the animation. (Note: If you cannot see the animation above, or it is not working properly, you may need to download the latest Flash player.)

In 240 B.C., the Greek astronomer Eratosthenes made the first good measurement of the size of Earth. By noting the angles of shadows in two cities on the Summer Solstice, and by performing the right calculations using his knowledge of geometry and the distance between the cities, Eratosthenes was able to make a remarkably accurate calculation of the circumference of Earth. Let's take a closer look at how he did it!

Eratosthenes lived in the city of Alexandria, near the mouth of the Nile River by the Mediterranean coast, in northern Egypt. He knew that on a certain day each year, the Summer Solstice, in the town of Syene in southern Egypt, there was no shadow at the bottom of a well. He realized that this meant the Sun was directly overhead in Syene at noon on that day each year.

Eratosthenes knew that the Sun was never directly overhead, even on the Summer Solstice, in his home city of Alexandria, which is further north than Syene. He realized that he could determine how far away from directly overhead the Sun was in Alexandria by measuring the angle formed by a shadow from a vertical object. He measured the length of the shadow of a tall tower in Alexandria, and used simple geometry to calculate the angle between the shadow and the vertical tower. This angle turned out to be about 7.2 degrees.

Next, Eratosthenes used a bit more geometry to reason that the shadow's angle would be the same as the angle between Alexandria and Syene as measured from the Earth's center. Conveniently, 7.2 degrees is 1/50th of a full circle ( 50 x 7.2 = 360 ). Eratosthenes understood that if he could determine the distance between Alexandria and Syene, he would merely have to multiply that distance by 50 to find the circumference of Earth!

Here's where things get a bit tricky. Eratosthenes had the distance between the two cities measured. His records show that the distance was found to be 5,000 stadia. The stadion (plural = stadia) was a common distance unit of the time. Unfortunately, there was not a universal, standard length for the stadion; so we don't know exactly which version of the stadion Eratosthenes used, and therefore are not exactly sure how accurate his solution was. He may have been correct to within less than 1%, a remarkable accomplishment! Or, if it was actually a different stadion that he used, he may have been off by about 16%. That is still pretty good! The actual polar circumference of Earth is just a bit over 40 thousand km (about 24,860 miles).

Eratosthenes was a talented mathematician and geographer as well as an astronomer. He made several other important contributions to science. Eratosthenes devised a system of latitude and longitude, and a calendar that included leap years. He invented the armillary sphere, a mechanical device used by early astronomers to demonstrate and predict the apparent motions of the stars in the sky. He also compiled a star catalog that included 675 stars. His measurement of the circumference of Earth was highly respected in his day, and set the standard for many years thereafter. He may have also measured the distances from Earth to both the Moon and to the Sun, but the historical accounts of both deeds are, unfortunately, rather cryptic.

Eratosthenes' Calculation of Earth's Circumference



Click the green "forward" button to step through the animation. (Note: If you cannot see the animation above, or it is not working properly, you may need to download the latest Flash player.)

Eratosthenes was a Greek astronomer in ancient times. Around 240 B.C. he made the first good measurement of the size of the Earth. How did he do that?

Eratosthenes used the lengths of shadows to figure out how high in the sky the Sun was in a certain place on a certain day. He knew of another place where there was no shadow at all on the same day. That meant the Sun was straight overhead. He found out the distance between the two places, then used some geometry to figure out the rest. Let's take a closer look!

Eratosthenes lived in the city of Alexandria. Alexandria is in northern Egypt. It is by the Nile River and the Mediterranean Sea. There was a tall tower in Alexandria. Eratosthenes measured the length of the tower's shadow on the Summer Solstice. He used that information plus some geometry to figure out the angle between the Sun and straight up.

There was a town in southern Egypt called Syene. There was a well in Syene. On the Summer Solstice, the Sun shone straight down the well to the very bottom. That meant the Sun must be straight overhead.

Eratosthenes had someone measure the distance between Alexandria and Syene. He used that distance, what he knew about the Sun's angles, and a bit of geometry to figure out the size of the Earth.

We aren't quite sure what answer Eratosthenes came up with, though. The distance between Alexandria and Syene was measured in stadia. The stadion was a distance unit that was often used in ancient times. However, not everybody used a stadion of the same length. If Eratosthenes used one length for the stadion, his answer was really, really good. The Earth is about 40 thousand kilometers (about 24,860 miles) around. The measurement that Eratosthenes made might have been within about 1% of this. That would be amazing! However, he might have used a different length stadion. If that is true, his answer was off by about 16%. That is still pretty good!

Eratosthenes may have also done some measurements of the distance from Earth to the Moon and from Earth to the Sun. Unfortunately, the records of that are not very clear. So we aren't quite sure whether he ever did make those measurements, or what his answers really were if he did.

Eratosthenes' Calculation of Earth's Circumference



Click the green "forward" button to step through the animation. (Note: If you cannot see the animation above, or it is not working properly, you may need to download the latest Flash player.)

Eratosthenes was a Greek astronomer in ancient times. Around 240 B.C. he made the first good measurement of the size of the Earth. How did he do that?

Eratosthenes used the lengths of shadows to figure out how high in the sky the Sun was in a certain place on a certain day. He knew of another place where there was no shadow at all on the same day. That meant the Sun was straight overhead. He found out the distance between the two places, then used some geometry to figure out the rest. Let's take a closer look!

Eratosthenes lived in the city of Alexandria. Alexandria is in northern Egypt. It is by the Nile River and the Mediterranean Sea. There was a tall tower in Alexandria. Eratosthenes measured the length of the tower's shadow on the Summer Solstice. He used that information plus some geometry to figure out the angle between the Sun and straight up.

There was a town in southern Egypt called Syene. There was a well in Syene. On the Summer Solstice, the Sun shone straight down the well to the very bottom. That meant the Sun must be straight overhead.

Eratosthenes had someone measure the distance between Alexandria and Syene. He used that distance, what he knew about the Sun's angles, and a bit of geometry to figure out the size of the Earth.

So what answer did he get? We know now that Earth is about 40 thousand kilometers (about 24,860 miles) around. We don't know for sure what answer Eratosthenes got. His measurement between Alexandria and Syene was in stadia. The stadion was used to measure distance in ancient times. It was used like a kilometer or a mile is used today. We don't know for sure how long the stadion that Eratosthenes used was. It might have been one length, or it might have been another. So maybe his answer was really, really good. It might have been within 1% of the right answer! Or it might not have been quite so good. It may have been off by about 16%. Either way, his answer was at least pretty good.


Page created July 20, 2007 by Randy Russell. Last modified July 31, 2007 by Randy Russell.
The source of this material is Windows to the Universe, at http://www.windows.ucar.edu/ at the University Corporation for Atmospheric Research (UCAR). © The Regents of the University of Michigan. All Rights Reserved. Site policies and disclaimer