Biographies of Eratosthenes

By the Students in the AP Calculus Class
at the HS of Commerce, Springfield, MA

In our study of how a mathematician named Eratosthenes was able to accurately determine the circumference of the Earth over 2200 years ago, the students researched the life of this amazing scientist and produced the following biographies.

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!

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Student Submission #1:

Eratosthenes was born in Cyrene, which is in Libya nowadays. He had a lot of famous teachers, like Lysanias of Cyrene (a scholar), Ariston of Chios (a philosopher who had studied under Zeno, the founder of the Stoic school of philosophy) and Callimachus (poet and scholar who was born in Cyrene). Later Eratosthenes spent some time studying in Athens.

Eratosthenes is known for being a mathematician and the librarian of Alexandria. Alexandria is a coastal town in Egypt, founded by Alexander the great. Around 200 BC it was the most important city of education in the world, that is Europe, Western Asia, and Northern Africa. The library had been founded by Ptolemy I, who was known for borrowing books and not returning them, even if it cost him a lot of money (he copied them however and returned the copies to the owner). It was the greatest library at that time, containing hundreds of thousands of papyrus scrolls. Eratosthenes was only the third librarian.

Eratosthenes is known as a great mathematician for finding the prime number sieve, now called the "Sieve of Eratosthenes". It's a way to find all prime numbers below a given integer. It works this way: let's say you want to find all prime numbers under 100. You write down all numbers from 1 to 100. Then you cancel all numbers devisable by 2 (every second number) except 2 itself. Then you cancel all numbers devisable by 3 (every third number) except the 3 itself. The 4 is already canceled, so you go on to 5 and do the same, and so on. All numbers that are not canceled in the end (besides 1) are the prime numbers. This sieve is what Eratosthenes is known for. But it was also him who figured out the circumference of the earth. Pretty good for about 200 BC, when it was not even clear whether the earth is flat or a sphere!

He was pretty lucky all the way long. It began with a traveler telling him one day that in his hometown, Syene (today Aswan, where the dam tames the Nile) there was no shadow at noon on June 21st. (We wouldn't be surprised about this today, because we can look on a map and see that this place is on Tropic of Cancer, and we know that June 21st is the summer solstice.)

It was lucky for Eratosthenes that Syene is in a straight line south to Alexandria (on the same line of longitude). Eratosthenes also knew the distance. How he figured this out is given differently in different sources: some say he paid a man to pace it out. Others say he simply knew that a camel needed 50 days to get there and that camels walk at 5000 stadia a day (stadia is an old Egyptian measurement unit equal to about 607 feet. Scientists today often quarrel how long it is exactly - depending on this Eratosthenes might not be as accurate as we think). Today we can look on a map and see that the distance is 500 miles. He could also find the angle of elevation of the sun at noon on the 21st of June in Alexandria: he measured the length of an obelisk and the length of its shadow. The angle at Syene he knew (90 degrees - there was no shadow, remember? They knew this for sure, because there was a very deep well, that never reflected the son on any other day, but on the 21st). The difference of those two angels turned out to be 7 degrees and 14 minutes (7.2 degrees). This and the distance and the assumption that the earth is round and the rays from the sun are essentially parallel enabled him to calculate the circumference of the earth. How?

Because given two parallel lines cut by a transversal alternate interior angles are congruent, the arc on the circumference of the earth between Syene and Alexandria is 7.2 degrees, and therefore the distance between the cities is 7.2/360 of the circumference of the earth. Thus the circumference of the earth is 25,000 miles. Confused? See the illustration (we are using an illustration by Prof. Dennis Donovan of Rice U. until we can scan in our own images).

He knows the angle in Syene is 90 degrees, because there is no shadow. He can measure the angle in Alexandria through the length of an obelisk and it's shadow. Trigonometry gives the angle:

tan x = (obelisk)/(shadow)

    x = 82.8

The angle between the two radii to the center of the earth has to be 7.2 degrees (see the triangle in the first picture and consider the sum of the 3 angles has to be 180 degrees). The distance is 500 miles.

7.2/360 = 500 miles/circumference

Thus, circumference = 500 miles * 360 / 7.2 = 25,000 miles

Not really difficult, is it? Eratosthenes himself did not use trigonometry, but charts, but it works the same way.

Eratosthenes also made many other great contributions to science. He worked out a calendar that included leap years, and he laid the foundation for the chronological writing of history by giving dates of literary and political events from the time of the siege of Troy. He also made a star catalogue with 675 stars.

Eratosthenes is also known as a great geographer. He sketched the route of the Nile and suggested that its source is in lakes, and that the floods lower on the river occur when heavy rainfall hits those lakes. He was the first one to get this essentially right answer, even though many scientists of his time had worked on this problem. Eratosthenes also divided the Earth with lines going north, south, east, and west.

Eratosthenes also was a philosopher and writer. He wrote literary works on the theater and on ethics, which were favorite topics for the Greeks.

In his old age, Eratosthenes went blind. He committed suicide in a rather unusual way: he reportedly stopped eating and died of starvation.

Sources:
http://www.terraworld.net/devans/Erast.htm

http://www-history.mcs-and.ac.uk/history/Mathematicians/Eratosthenes.html

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Student Submission #2:

ERATOSTHENES

Eratosthenes lived around 276? to 194? B.C. He was born in Cyrene, Greek town in Northern Africa. His teachers included the scholar Lysanias of Cyrene and the philosopher Ariston of Chios who had studied under Zeno, the founder of the Stoic school of philosophy. Eratosthenes also studied under the poet and scholar Callimachus who had also been born in Cyrene. Eratosthenes then spent some years studying in Athens. He was one of the most scientific geographers of ancient Greece, like many other scientists in Alexandria. Eratosthenes studied at the museum, where he became successful and became the director of the museum with his wide knowledge of literature, mathematics and astronomy. He made a huge contribution to the geography. He wrote a text that was the first to provide a mathematical basis for geography and to describe the world as a globe. He found a way to determine the size of the Earth. He also developed the sieve of Eratosthenes; it is a system for identifying prime numbers.

One of Eratosthenes' most important feats was his attempt to find the circumference of the world, which he determined is about 24661 miles, (which is about 39678 kilometers). Eratosthenes based his measurement of the earth on the assumptions that the earth is round and the sun's rays are parallel. He knew that at noon on the day of summer solstice in Alexandria, Egypt, a vertical post casts a shadow. At the same time in Syene, a town directly to the south, a vertical post casts no shadow. Eratosthenes used Euclidean geometry to determine the angle from the post and an imaginary line from the end of the shadow to the top of the post equals an angle at the earth's center formed by imaginary lines from two towns. He calculated the earth's circumference by measuring the distance between Alexandria and Syene, and multiplying it by the number of times the angle at the earth's center is contained in 360 degrees. From the circumference, he found the diameter to be about 7850 miles (12630 kilometers) from North Pole to South Pole. The correct polar diameter is about 7900 miles (12700 kilometers).

Thank you to Cartographic Images and
Henry Davis Consulting for the permission
to use the above image.

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Student Submission #3:

ERATHOSTHENES was born in Cyrene which is now in Libya in North Africa around 276 BC. His teachers were the scholar Lysanias of Cyrene, the philosopher Ariston of Chios and Zeno the founder of "the stoic of philosophy." Eratosthenes also studied under the poet and scholar Callimachus who had also been born in Cyrene then studied in Athens.

Erathothenes also served as the second librarian of the greatest library in the Western World. He developed a technique for measuring the circumference of the earth which we still use. First: he compared the noon shadow at midsummer between Syene and Alexandria. Second: he measured the distances to the sun as 804,000,000 and the distance to the moon as 780,000 stadia. Third: he computed these distances and also measured the tilt of the earth's axis.

Erathothenes did not only measured the circumference of the earth but also gave other contributions to humanity. He worked out a calendar that included leap years, laid the foundation of a systematic chronography of the world with which he tried to give dates of literary and political events. He sketched the route of the Nile to Khartoum, he thought that the lakes were a source of the river.

Before Eratosthenes ancient Egyptians thought the earth was like an egg guarded at night by the moon (a great white bird). Today he's known as the greatest of ancient geographers.

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