Abstract

Objective

The goal of this project is to estimate the circumference of the earth by setting up a mathematical proportion from simple measurements.

Introduction

In this project, you will estimate the circumference of the earth, using a method developed about 2,200 years ago, by Eratosthenes, a Greek mathematician and the librarian of the great library at Alexandria, in Egypt.

Eratosthenes knew from his reading that in Syene (a city almost due south of Alexandria) the sun was directly overhead at noon on a particular day of the year. On that day at noon, vertical objects cast no shadow, and the reflection of the sun could be seen in the bottom of a well. He reasoned that the sun was far enough away from the earth so that rays of light from the sun, for all practical purposes, are parallel to each other when they reach earth. If he measured the angle of a shadow in Alexandria at noon on the same day when the sun was directly overhead in Syene, the angle of the shadow would be the same as the central angle of the "wedge" of the earth between Alexandria and Syene.

Figure 1, below, shows Eratosthenes' reasoning. Imagine the earth cut in half along the north-south line between Alexandria and Syene, and then looking at the cut section, face-on (blue circle). Of course, the figure is not meant to be a scale drawing, but it does illustrate the principle of the method. The sun's rays are represented by the yellow lines. In Syene, the sun's rays cast no shadows. In Alexandria, a vertical stick does cast a shadow. Eratosthenes' insight was that the angle of the shadow in Alexandria (the "sun angle", shown by the orange triangle) is also equal to the angle of the wedge of the Earth between the two cities (the "central angle", shown by the other orange triangle).

Figure 1. Eratosthenes' method for measuring the circumference of the Earth (obviously, not to scale).

Since there are 360 degrees in a circle, by dividing the central angle into 360, he could calculate how many similar sectors would be needed to complete a circle. He hired someone to measure the overland distance between the two cities. By multiplying the overland distance by the number of sectors, he arrived at his estimate for the cirumference of the Earth. For example, let's say that the measured sun angle was 7.2°. This means that the central angle between Alexandria and Syene was also 7.2°. Then the number of sectors to make a complete circle would be 360/7.2 = 50. The circumference of the Earth would then be 50 times the overland distance between Alexandria and Syene.

Figure 2, below, shows how a straight line that intersects parallel lines creates equal alternate angles. In Figure 2, angle AGH is equal to angle GHD. The parallel lines (AB and CD) correspond to sun's rays in Figure 1, above. The straight line EF, corresponds to the straight line between the vertical stick in Alexandria and the center of the Earth. If it helps you to see more clearly, you can click and drag the red points in Figure 2 to make the positions of the lines correspond to those in Figure 1. (Figure 2 is from an online version of Euclid's Elements, where you can find a proof that the alternate angles are equal. (Joyce, 1998))

Notes on How to Manipulate the Diagram

The diagram is illustrated using the Geometry Applet (by kind permission of the author, see Bibliography). If you have any questions about the applet, send us an email at: scibuddy@sciencebuddies.org. With the help of the applet, you can manipulate the diagram by dragging points.

In order to take advantage of this applet, be sure that you have enabled Java on your browser. If you disable Java, or if your browser is not Java-capable, then the diagram will still appear, but as a plain, still image.

If you click on a point in the diagram, you can usually move it in some way. The free points, usually colored red, can be freely dragged about, and as they move, the rest of the diagram (except the other free points) will adjust appropriately. Sliding points, usually colored orange, can be dragged about like the free points, except their motion is limited to either a straight line, a circle, a plane, or a sphere, depending on the point. Other points can be dragged to translate the entire diagram. If a pivot point appears, usually colored green, then the diagram will be rotated and scaled around that pivot point. (Note that diagrams will often use only one or two of the above types of points.)

You can't drag a point off the diagram, but frequently parts of the diagram will be moved off as you drag other points around. If you type r or the space key while the cursor is over the diagram, then the diagram will be reset to its original configuration.

You can also lift the diagram off the page into a separate window. When you type u or return the diagram is moved to its own window. Typing d or return while the cursor is over the original window will return the diagram to the page. Note that you can resize the floating window to make the diagram larger.

Making Your Estimate of the Circumference of the Earth

You can use Eratosthenes' method to calculate the circumference of the earth yourself. You can even make the calculation when the sun casts a shadow at noon in both places. The reference from the Stevens Institute of Technology on Eratosthenes method (SIT, 2004a) will show you how to handle this situation. You'll want to study that reference carefully, as part of your background research.

Terms, Concepts, and Questions to Start Background Research

To do this project, you should do research that enables you to understand the following terms and concepts:

• method of Eratosthenes,
• local noon,
• longitude,
• latitude.

Questions

• How do you calculate the central angle between two cities when there are shadows in both of the cities at noon? (see SIT, 2004a)
• How do you calculate the central angle between two cities when they are on opposite sides of the sun's zenith (e.g., cities on opposite sides of the Equator)? (see SIT, 2004a)

Bibliography

• SIT, 2004a. "Estimate the Circumference of the Earth Using Eratosthenes' Method," Stevens Institute of Technology, Center for Innovation in Engineering & Science Education [accessed November 3, 2006] http://www.k12science.org/~jkoen/rwlo/Eratosthenes/Overview/overview.shtml.
• SIT, 2004b. "Measuring the Circumference of the Earth: The Noon Day Project: Project Data," Stevens Institute of Technology, Center for Innovation in Engineering & Science Education [accessed November 3, 2006] http://www.k12science.org/noonday/projectdata.html.
• This webpage has an online calculator for determining the surface distance between two locations on Earth, given their latitude and longitude:
  Byers, J.A., 1997. "Surface Distance Between Two Points of Latitude and Longitude," USDA, Agricultural Research Service [accessed November 3, 2006] http://www.chemical-ecology.net/java/lat-long.htm.
• These webpages have information about Eratosthenes:
  • Haynes, M. and S. Churchman, date unknown. "Eratosthenes," Astronomy 201: Our Home in the Universe, Cornell University [accessed November 2, 2006] http://astrosun2.astro.cornell.edu/academics/courses//astro201/eratosthenes.htm.
  • Wikipedia contributors, 2006. "Eratosthenes," Wikipedia, The Free Encyclopedia [accessed November 2, 2006] http://en.wikipedia.org/w/index.php?title=Eratosthenes&oldid=84842525.
• The Geometry Applet was kindly provided to Science Buddies by its author, Professor David Joyce, who wrote it for illustrating an amazing online edition of Euclid's Elements. Understanding Book I, Proposition 29 will be helpful for this project:
  
  • Joyce, D.E., 1998. "Euclid's Elements, Book I, Proposition 29," Department of Math & Computer Science, Clark University [accessed November 3, 2006] http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI29.html.
  • Joyce, D.E., 1997. "Euclid's Elements, Using the Geometry Applet," Department of Math & Computer Science, Clark University [accessed November 3, 2006] http://aleph0.clarku.edu/~djoyce/java/elements/usingApplet.html.

Materials and Equipment

To do this experiment you will need the following materials and equipment:

• a piece of open, level ground that is in the sun at noon,
• a sunny day,
• meter stick,
• bucket of sand,
• large piece of paper (e.g., newsprint, for measuring shadow length),
• pencil,
• carpenter's T-square or plumb bob (optional, but helpful for making sure your meter stick is vertical),
• measuring tape (metric),
• protractor,
• e-mail pen pal in a distant city who can make the measurement at noon local time on the same day,
• calculator.

Experimental Procedure

1. Do your background research so that you are familiar with the terms, concepts, and questions, above.
2. Figure out what clock time "local noon" will be on the day you make your measurement. Local noon is the time half way between sunrise and sunset, when the sun is highest in the sky. You want to be set up and ready to measure the shadow length about ten minutes before this time so that you don't miss it.
3. Set up your meter stick in the bucket of sand so that it is vertical. Use the T-square to check verticality.
4. Lay out a large piece of paper on which to mark the length of the shadow from the meter stick. The other end of the paper should go under the meter stick (see Figure 3).

   Figure 3. Measuring the length of the meter stick's shadow. (SIT, 2004)

5. You want to measure the length of the meter stick's shadow—from the base of the meter stick to the tip of the shadow—when the sun is at its highest point in the sky.
   1. Mark where the base of the meter stick would meet the paper.
   2. Mark the end of the shadow. It will be a little fuzzy; do your best to be accurate and consistent. Note the time of your measurement next to the mark.
   3. Mark the end of the shadow every two minutes, beginning at least 10 minutes before local noon. The shadow should be at its shortest right at local noon. When the shadow starts increasing in length, you can stop taking measurements.
   4. Remove the paper from under the meter stick, and use the tape measure to measure the length of shadow at local noon (its shortest length).
6. Measure the height of the meter stick, from the ground to the tip (because you are holding it upright in a bucket of sand, it will be slightly taller than one meter).
7. Here's how to use a protractor and your measurements to figure out the sun angle.
   1. Make a right triangle with the base length (BC) equal to the shadow length, and the height (AB) equal to the meter stick height (see Figure 4).
   2. Position the protractor as shown in Figure 4, and read the angle (BAC).

   Figure 4. Measuring the sun angle using a protractor. (SIT, 2002)

8. Send your measurements to your partner in the distant city by email, and have them send their measurements to you. Follow the procedure in step 7 to calculate the sun angle at their location.
9. Find out latitude and longitude of each location.
10. Calculate north-south distance between the two locations.
    • To find the distance, you can use a map scale. Measure how far each city is from the equator (in a straight line). If the cities are on the same side of the equator, subtract the smaller distance from the larger distance. If the two cities are on opposite sides of the equator, add the two distances together.
    • An alternative to using a map scale is to use the latitude and longitude information for each city and an online distance calculator (e.g., Byers, 1997). As with the map scale, you want to calculate the distance of each city from the equator (enter 0° for latitude, and the same longitude as the city of interest), then use that information to figure out the north-south distance between the two cities.
11. Using the north-south distance between the two locations, and the sun angle at each location, you should be able to calculate the circumference of the earth.
12. How close is your estimate to the actual value?

Variations

• Do a comparative study using the sun at different times of day. Which time of day yields the most accurate result?
• What are possible sources of error in your measurements? Can you use error estimates to put upper and lower bounds on your calculation for the circumference of the earth? For example, let's say that you determine that your sun angle measurements are accurate to within 0.5 degrees, what would the upper and lower bounds be for your circumference estimate?
• Use archive data from The Noon Day project to make estimates of the earth's circumference using several different pairs of measurements (SIT, 2004b). Use measurements from locations that are close together, and far apart. Which measurements give you the most accurate estimate of the Earth's circumference?
• If you have good computer and math skills, you could use the Geometry Applet to make an interactive model illustrating the measurements you made in your project, and how they are used to determine the center angle between the two locations.

• For more science project ideas in this area of science, see Astronomy Project Ideas.

Credits

Andrew Olson, Ph.D., Science Buddies

Sources

This project is based on:

• SIT, 2002. "Measuring the Circumference of the Earth: The Noon Day Project," Stevens Institute of Technology, Center for Innovation in Engineering & Science Education [accessed November 3, 2006] http://www.k12science.org/noonday/teacherguide.html.
• SIT, 2004. "Estimate the Circumference of the Earth Using Eratosthenes' Method," Stevens Institute of Technology, Center for Innovation in Engineering & Science Education [accessed November 3, 2006] http://www.k12science.org/~jkoen/rwlo/Eratosthenes/Overview/overview.shtml.

Last edit date: 2010-11-02 09:00:00