Part A: Estimate the circumference of the Earth using Eratosthenes' calculations

Eratosthenes calculated a remarkably accurate measurement of the circumference of earth by doing an experiment. His idea was to think of the earth as an orange cut in half, and if you walked along the edge of the orange, your path would be circular. Since he couldn't walk around the world, he determined that if he could divide the cross-section of the earth into equal-size wedges, like pizza slices, he would only need to measure one of the arcs from one of the wedges (the length of the pizza crust of one slice) and multiply that by the number of arcs (number of pizza slices).

• For example, if a pizza is cut into 8 equal slices, and the crust from one of the slices measures 2", the circumference of the pizza would be 16":
  • 8 (slices) x 2" = 16"
• Now to apply this to the earth, if a cross-section of the earth at its widest point could be divided into 10 equal wedges, and the arc from one of the wedges measured 4,000 km, then the circumference of the earth would be 40,000 km:
  • 10 (wedges) x 4,000 km = 40,000 km

In order to make an accurate calculation of the circumference of the earth using this method, you would therefore need to divide the earth's cross-section into a number of equal-size wedges and then, measure the arc of one of the wedges. However, how did Eratosthenes determine how many equal-size wedges to divide the cross-section of the earth, and how did he make measure the arc of one of the sections?

In order to answer the first question, Eratosthenes used Geometry. If you could determine the angle that one of the wedges makes at the center of the cross-section, you could figure out out how many wedges make up the circle. Let's take a look at the following example:

In the circle to the left, angle ABC, also known as the central angle, is 60 degrees. Since there are 360 degrees in a circle, then there are 6 slices in this "pie" (360/60 = 6). That means that 6 arcs (AC) would fit around the globe. Now, all you need to know is the distance of AC and you would be in business!

However, finding this central angle in a cross-section of the earth seems just as impossible as finding the diameter; but, as it turns out, this was Eratosthenes' brilliant insight. He discovered it by studying shadows!

One day while in his library in Alexandria, Eratosthenes read that in the frontier outpost town of Syene, now Aswan, at noon on June 21st, vertical sticks cast no shadow and a reflection of the sun could be seen at the bottom of the well. This meant that the sun was directly overhead. Being the scientist that he was, he wanted to know if the same thing happened in Alexandria, almost directly north of Syene. He discovered that in fact there were shadows there. He then measured the length of the shadow relative to the length of the pole he used to make the shadow and calculated the "Sun" angle to be approximately 7.2°.

In the diagram to the right, note that there is no shadow at Syene while there is a shadow in Alexandria. Based on this observation, Eratosthenes realized that the sun angle had to equal the central angle since the sun was far enough away so that its rays were parallel at all locations of the earth. Using this angle, 7.2°, he could now determine how many equal-size wedges to divide the cross-section of the earth, the first part of the equation
(EXTRA: Open the Java sketchpad sketch and move the sun angle SA around to change the angle. What do you notice about the sun angle SA and the central angle C? The two angles are equal).

If you remember from above, he also needed to measure one of the arcs, which in this case was the distance from Alexandria to Syene to determine the circumference of the earth. During this time, the common unit of measurement was the stadia, which is equal approximately to 185.4 meters today. Eratosthenes hired a man to pace out the distance between these two cities and determined it to be approximately 5,000 stadia.

1. Using Eratosthenes method and calculations, determine the north-south circumference* of the earth in kilometers: ____________km
   (Use the space below or a notebook to write out your calculations)
   
   
   
   
   
   
   
   
   
   
   
   
    
2. In your own words, describe how Eratosthenes was able to estimate the circumference of the earth and how you could repeat his experiment using his methodology today.