# How Eratosthenes Calculated the Circumference of the Earth. Eratosthenes Calculating the Circumference of the EarthGraphic courtesy Oracle Thinkquest In the panel on the left, Eratosthenes is peering down a well in Syene to verify that the sun lights up the water at the bottom and that there are no shadows. In the second panel on the left, Eratosthenes is preparing to measure the length of the shadow of a pillar, so he can calculate the angle of the sun's rays. Portrait of Eratosthenes

Many attempts were made by ancient mathematicians to measure the circumference of the earth. These mathematicians had mixed success. Eratosthenes attempt was, however, quite successful, both from the point of view of accuracy and simplicity. Eratosthenes' estimate was only 245 miles off the accurate value of 24,907, an error of less than 1%. Eratosthenes estimate was based on proportions. The length of an arc of a circle is proportional to the measure of the angle of the rays from the center of the circle to the end points of the arc.

In the example to the left the measure of the angle containing the arc is 90 degrees. Since the measure of the angle of the entire circle is 360 degrees, the proportion of the measure of the arc angle to the measure of the angle of the whole circle is 90/360 = 1/4.

The measure of the arc is given as 400 cm. Since the ratio of the arc angle to the whole circle is 1/4, the ratio of the arc to the whole circle is the same. If we let C represent the circumference of the circle, 400 cm / C = 1/4. Multiplying both sides by 4 gives us 1600 cm / C = 1. Multiplying both sides by C gives us 1600 cm = C. We then know that the circumference of the circle is 1600 cm.