The Significance of this Discovery

What this means is that if you can measure the sun's angles at two different positions on the earth at the same time, you can figure out the central angle! Let's look at an example to see how this information leads to determining an empirical value for the circumference of the earth.

Our two sites will be at Manasquan, New Jersey and San Juan, Puerto Rico.

Method 1 - Using Scale Drawing

In Manasquan this angle would be a little less than 39 degrees. 

In San Juan the angle was about 18.5 degrees. 

 At local noon on the same day, the experimenters measured the shadow cast by a meter stick. 

In Manasquan, the shadow length equaled 80.5 cm. In San Juan the shadow length was 35.3.  The next step is to figure out the sun angles at Manasquan and Puerto Rico. A protractor can be used to measure this angle.

The central angle equals the difference between these angles. The Geometer's Sketchpad can be used to get more accurate measurement for the angles.
 

Using these measurements, the central angle is 38.85 - 18.59 or 20.26 degrees.

Now that we know the central angle, we can determine how many such angles would make up the full circle. That is, how many "slices" can we make where the angle is 20.26 degrees? Since the total is 360 degrees, then you would have a little less than 18 equal "slices" (or 17.77 to be more precise.)

Each slice's edge represents the distance between Manasquan and Puerto Rico, or more accurately, the lateral distance between the two sites.

But why the lateral distance and not just the direct distance? (more about this later.)
 

Manasquan, New Jersey
Location: 40:07:34 (40.13) 
N 74:02:59 (74.05) W

San Juan, Puerto
Location: 18:28:00 (18.47) 
N 66:07:00 (66.12) W
The distance between these two places along a north-south line is approximately 21.7 degrees or 2404 km.* 

So if the length of the "slice" is about 2400 km and there are about 18 slices then the projected circumference is 2400 x 18 = 43,200 km which is about an 8% error since the average circumference is 40,008 kms. (More precisely, it is 2404 km times 17.7 "slices".)

*This can be found by determining the scale of a world map and measuring. Of course, Eratosthenes did not have such luxuries as accurate maps. And he certainly didn't know that each angle of latitude was approximately 111 kilometers long. (He used stadia as a unit of measurement.) But for our purposes which is to understand this experiment we will take some 20th century liberties.

Method 2: Using Trigonometry
Skip the trig. Take me to the "How to do the experiment" pages