Radius of the Earth

In this assignment we will investigate one of the first known methods to calculate of the radius of the earth. This method was discovered by Erastothenes of Cyrene (third century B.C.). You may have heard his name in connection with a method for finding prime numbers (the Sieve of Erastothenes). The ancient Greeks believed that the earth was round, and, as we will see, Erastothenes was able to estimate the size of the earth quite well. This knowledge was lost to Europe during the dark ages, when people believed that the earth was flat. This belief lasted up to the voyages of Columbus.

Erastothenes observed that at noon on the day of the summer solstice the sun shone directly down a deep well at Syene (present day Aswan). At the same time in Alexandria, the sun was found to cast a shadow corresponding to an angle of 1/50 of a full circle (about 7 degrees in our current measurement of angles). Erastothenes knew that the distance between Alexandria and Syene was approximately 5000 stades (1 stade is about 1/10 mile). He then estimated the radius of the earth from this information. How did he do this, and what value for the radius did he obtain?

To help you figure out how Erastothenes figured this out, consider how the sun's rays shine on the earth.

Because the sun is so large compared to the size of the earth, the sun's rays are practically parallel to each other as they hit the earth. Draw a portion of a large circle to represent the surface of the earth. Then draw a stick placed at Syene and one at Alexandria, and then draw a line representing how the shadow of the stick is formed. Finally compare the angle made by the shadow at Alexandria with the angle formed by connecting the center of the earth with Alexandria and Syene.

There are a couple of geometric facts that you should find useful in this assignment. First, there is a close relationship between an angle and the corresponding arc length on a circle. This relation states that the fraction of a full circle taken up by an angle is equal to the fraction of the circumference taken up by the corresponding arc length. In terms of an equation using proportions, we have

The following picture can help to understand this formula. For example, an angle of 36 degrees is 1/10 of a full 360 degree angle. The corresponding arc length is 1/10 of the full circumference of the circle. The formula says the fraction of the full 360 degree angle made up by an angle is equal to the fraction of the circumference made up by the arc length corresponding to the angle.

Second, because the circumference of a circle is equal to 2 times pi times the radius, if we know the circumference, we can find the radius, and vice-versa. For the third and final fact, if two parallel lines are crossed by a diagonal line, then the angles marked as a in the picture below are all equal.

For some geographical information, Syene (present day Aswan) and Alexandria are marked in the following map of Egypt.