Each of the students has written (or is in the process of writing...) a summary of the method we used to determine the circumference of the Earth. We tried to recreate the method Eratosthenes used over 2200 years ago, with only a small bit of modern, technological "magic."
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Unfortunately, due to directives from the school system we are unable to give credit on the Web to the students who submitted this work. Congratulations on a job well done, class! Your efforts are appreciated!
We went outside to the school parking lot at about 11:45 AM on March 8, 1999 to determine the sun's highest angle of elevation for that day. We used information from the internet, which reported that between 11:45 AM and 12:15 AM is when the sun would shine at the highest degree that it could shine in that day. We used a ruler as our stick to cast a shadow and chair as our holder for the ruler. We taped the ruler to the chair at 90-degree angles so it looks like the right triangle. We measured the distance of the shadow every 3 minutes.
As an extra activity, when we were done with the measuring of the sun's angle of elevation, we placed a flat mirror in the ground to use find the height of the chimney in the rear of the school. We measure from the mirror to the chimney. One of the students in our calculus class who is about 5'5 (the reason we choose that student, because she volunteer to do it. The student who volunteered to stand near the mirror was told to look at the mirror (not to see her beauty) to see if she could see the chimney. If she did that could be good, if she didn't she need to move a little back or forward to see the chimney. We then measured her distance from the mirror and used "similar triangles" to determine the height of the chimney.
When we done with the mirror experiment, we returned to class to calculate the angle in Springfield from the information we collected from the experiment that we did. The angle we got when the sun was shining at its highest angle of elevation to us is 42.8-degrees. We compared this with information we got from the US Naval Observatory on the Internet, and our angle is very close to the angle given by the Observatory, which was 43.0-degree.
When we were finished with the calculation, we went down to the Commerce library to consult a map. The reason we need to see it we want to locate another city that is north or south to us that have the same or close angle as us. We considered many cities but the US Naval Observation didn't have the information we ask for, and finally we chose Port Jefferson, NY which is located to the south of Springfield. Port Jefferson is about 88 miles (141 kilometers) away from us, and at the sun's angle of elevation at the same time as we were measuring the angle in Springfield was a 44.2 angle degree. Using the information from the two cities, we tried to use Eratosthenes' way to find the circumference of the earth. We set up the ratios:
(44.2-43)/360=88/circumfrence and solved for circumference. The distance we estimated turned out to be within 5 percent of the correct circumference of the earth.When we done with the mirror experiment, we returned to class to calculate the angle in Springfield from the information we collected from the experiment that we did. The angle we got when the sun was shining at its highest angle of elevation to us is 42.8-degrees. We compared this with information we got from the US Naval Observatory on the Internet, and our angle is very close to the angle given by the Observatory, which was 43.0-degree.
When we were finished with the calculation, we went down to the Commerce library to consult a map. The reason we need to see it we want to locate another city that is north or south to us that have the same or close angle as us. We considered many cities but the US Naval Observation didn't have the information we ask for, and finally we chose Port Jefferson, NY which is located to the south of Springfield. Port Jefferson is about 88 miles (141 kilometers) away from us, and at the sun's angle of elevation at the same time as we were measuring the angle in Springfield was a 44.2 angle degree. Using the information from the two cities, we tried to use Eratosthenes' way to find the circumference of the earth. We set up the ratios:
(44.2-43)/360=88/circumfrence and solved for circumference. A second way that we calculated the circumference was as follows:
We know the arc of the circumference is the distance from Springfield to Port Jefferson, which was 88 miles. (We determined this using a map from the Internet.) 360 degrees is the measure of any circle. We then calculated the angle measure between Springfield and Port Jefferson to be 1.2 degrees (44.2 degrees - 43 degrees = 1.2 degrees).
We then created the following ratios:
Circumference/88 miles = 360 degree/1.2 degree
Circumference = 88 miles x 300
Circumference = 26400 miles
This is the circumference of the earth from North Pole to South Pole, if we measure the circumference of the earth from east to west it will be different, because the earth is not a perfect sphere. The "correct" circumference that we got from the Internet is 25046 miles, which they measure from the North Pole to South Pole. The percent error we got was 5.4%, we took the circumference we got which was 26400 miles minus the original correct circumference which was 25046 miles and divided by 25056, the answer we got is 0.054 or 5.4%.
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Submission #2:
Objective: To determine the circumference of the Earth by repeating Eratosthenes' experiment, and see how close to his and the actual measurements our results would be.
Procedure: Eratosthenes determined the circumference of the Earth by calculating the sun's angle of elevation at the same time in two different locations (Alexandria and Syene, Egypt). Knowing the distance between these two cities was key for Eratosthenes to conduct his experiment. He then used ratios to determine the Earth's circumference. The formula he used would have probably looked very similar to:
(distance between two cities) / (circumference) = (sun's angle of elevation) / (360 degrees).
In our experiment we used Springfield, MA and Port Jefferson, New York as the two cities for our experiment. My classmates figured out the angle of elevation of the sun and then verified their results with those of the U.S. Naval Observatory (USNO) by way of their web site. [Note by MW: Juan was at a college interview on the day we collected the information on the sun's angle of elevation.]
Submission #3:
THE QUESTION:
How do you measure something that's too big to measure directly (the earth)?THE HISTORY:
Just like Eratosthenes we wanted to find the circumference of the earth using only sticks and their shadows.WE HAD IT EASIER:
Eratosthenes did not have the Internet available to look up the angle of elevation of the sun in another place. He also did not have maps to look up the distance between his place and the other place.WE HAD IT MORE DIFFICULT:
In contrast to Eratosthenes we did not have a place with the sun directly standing over us. This makes the math more difficult.
WHAT ERATOSTHENES DID WHAT WE DID He measured the length of an obelisk and its shadow at noon on the 21st of June. We measured the length of a stick and its shadow at noon on a day in March. He calculated the angle of elevation of the sun through trigonometry. We calculated the angle of elevation of the sun by using trigonometry. He knew that there was no shadow on this day in a place on the same latitude. We looked up the angle of elevation in a place on the same longitude in the Internet. He paid a man to pace out the distance between the two places. (Other sources say he knew how long camels needed and how fast they were.) We looked up the distance between the two places on a map printed from the Internet. OUR ACTIONS AND CALCULATIONS IN DETAIL
MEASURING: On the 8th of March we went outside shortly before noon and tied the stick (actually a long ruler) to a chair so that it would stand vertically without moving. Then we taped a sheet of paper to the ground in the spot where the end of the shadow of the stick fell. About every 3 minutes we made a mark on the paper where the end of the shadow was. In the end we used the shortest of those distances, as this one must be exactly (the natural) noon (in contrast to the artificially defined noon since the creation of the time zones).
INTERNET: We looked at a map printed from the Internet (http://www.mapquest.com) to find a place on the same longitude (in a line directly due south to us). We found Port Jefferson. We could easily measure the distance between Springfield and Port Jefferson on the map.
distance = 88 miles We looked up the highest angle of elevation of the sun in Port Jefferson on the same day (http://aa.usno.navy.mil/AA/)
angle of elevation in Port Jefferson = 44.2 degrees With these three values (distance and both angles of elevation at the same time) we can calculate the circumference of the earth (assuming that the earth is a sphere and the sun is far enough away so that the rays are essentially parallel).
CALCULATION: The difference between the two angles is 1.2 degrees. That is also the angle between the two radii. This can be proven because given two parallel lines cut by a transversal alternate interior angles are congruent and because the sum of the angles in a triangle is 180 degrees.
1.2 degrees thus is the same percentage of 360 degrees (full circle) as 88 miles is of the full circumference of the earth:
1.2 / 360 = 88 miles / circumference of the earth circumference of the earth = 88 miles * 360 / 1.2 = 26,400 miles PERCENTAGE OF ERROR: The true circumference of the earth is 25046 miles. Thus our percentage of error was 5.4%. This percentage is pretty good for the inaccurate means of measurement we had. Eratosthenes' result was better, however, even though this accuracy depends on the definition of stadia (Egyptian unit of measurement).
REASONS FOR ERROR:
- The inaccurate measuring of the length of the shadow. Even errors of some eighths of an inch cause errors of hundreds of miles in the circumference.
- Port Jefferson is not exactly on the same longitude as Springfield.
- The distance between Port Jefferson and Springfield is not exactly 88 miles.
- It was not possible to measure this more accurately on a map, partly because the towns themselves have a certain diameter. One mile more or less in distance would mean
360 / 1.2 = 300 miles difference in the circumference.
Submission #4:
WHAT WE DID
We took information about Eratosthenes out of the Internet. We got some minibiographies about him in whom they explain how he used stick, shadows and reasoning to measured the circumference of the earth.
We went to the parking lot of The High School of Commerce to determine the sun's angle of elevation. Then we looked in a map for a location that had the same longitude as Springfield, which was Port Jefferson, NY. Logged in the US Naval Observatory to find the sun's angle of elevation at Port Jefferson.
THE QUESTION: CAN EVERYBODY MEASURE THE CIRCUMFERENCE OF THE EARTH WITH A FEW TOOLS?
RESULTS:
The sun's angle of elevation at Springfield was 43.0, 44.2 at Port Jefferson, NY. The result of the equation 1.2(the difference between Springfield and Port Jefferson) is divided by 360 = 88mi(the distance between Springfield and Port Jefferson) is divided by x. This equation gave us the circumference of the earth then we looked for the percent of error that is 5.4%.
The project went well and our results were close to the real one.
CONCLUSION
The project was fun to do as a group project with the help of the teacher and a few tools. We also are going to publish the results in the Internet. [And ... obviously you can see that we completed that portion of this project as well.]
REFERENCES
WEB SITES:
US NAVAL OBSERVATORY--- DATE SERVICES
MacTutor: BIOGRAPHY OF ERATOSTHENES
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