## Eratosthenes and the Shadows of Egypt

*by BastWorshipper*

Eratosthenes was a Greek polymath born in 276 BCE in Cyrene in what is now Libya. He studied grammar under Callimachus in Alexandria. He then moved to Athens to study philosophy under Arcesilaus and Ariston. He became a mathematician and scholar, since he was Greek, and I believe philosopher was the only other occupation available. He was invited back to Alexandria by Ptolemy III Euergetes to tutor his son, and Ptolemy also made Eratosthenes the third chief librarian at the Library of Alexandria. When threatened with total blindness, he starved himself to death in 194 BCE.

Many of Eratosthenes' works have been lost to history, destroyed with the Library of Alexandria, whether the result of Julius Caesar's clumsiness or the intolerance of Christians or Muslims. Most of what remains is in the form of authors' references to the works of Eratosthenes. Regardless, we have been able to glean something of his accomplishments. He was known primarily as a mathematician working with number theory, geometry and arithmetic. Although exceptional in a wide range of fields, he was never considered to be the best in any one field. This earned him the nickname "Beta," because he was thought to be second best at everything.

He wrote a criticism of the mathematics which formed the basis of Plato's philosophy, which we know about from the writings of Theon of Smyrna. This work studied the basic definitions of arithmetic and geometry and discussed a wide range of topics which even included music, which we know to be the most mathematical of art forms. We also know from the writings of Theon of Smyrna and Eutocius that Eratosthenes solved the problem of duplicating a cube, one of the three major problems in Greek mathematics (squaring a circle, doubling a cube and trisecting an angle). It may have been more impressive had he managed to square the circle, since we still struggle getting that peg into that hole. Nonetheless, his mechanical solution effectively solved the cube problem.

However, in the history of mathematics, Eratosthenes' greatest contribution is probably his work with prime numbers. Once again, although Eratosthenes' writings on prime numbers have not survived, we know of what is called the Sieve of Eratosthenes through the writings of Nicomachus. The Sieve is a simple algorithm which works to identify all of the prime numbers up to a certain integer and is very accurate up to about 10 million. It works by removing all multiples of 2, 3, 5, 7 and so on. Once you have sifted out the multiples, the remaining numbers are the primes; and this sieve was the '*prime*-ary' method of identifying prime numbers up to a certain limit until Leonhard Euler in the 18th century.

Eratosthenes was also an avid astronomer, recording the locations of hundreds of stars, and even writing a history of the Greek myths surrounding the constellations. He also applied himself to calculating distances between the Earth and the Sun and the Moon. Eratosthenes' figure for the distance from the Earth to the Sun is translated as either 4,080,000 or 804 million stadia. If we use the Greek stadium as the unit of measurement and the latter translation of 804 million, this distance works out to about 149 million kilometers. The modern measurement for the Earth's average distance from the Sun is about 149.5 million kilometers, a margin of error of well under 1 percent. Additionally, he accurately calculated the tilt of the Earth's axis. Unfortunately, Eratosthenes' distance to the moon quoted by Eusebius works out to about 144 thousand kilometers, which is a little more than one-third of the actual distance. We like to think that this is the result of an error in translation.

Eratosthenes is probably best known for his calculation of the circumference of the Earth. Eratosthenes was told that in the city of Syene in Egypt on the summer solstice at noon, the sun shone down a well and cast no shadow. This meant that the sun was directly overhead at its zenith. His measurements showed that in Alexandria, at the same time and date, the sun was more than 7 degrees south of the zenith. Assuming that the Sun was far enough away from the Earth that its rays were parallel when they intersected with the Earth, he calculated the arc distance between Syene and Alexandria. Then, knowing the distance between the two cities, he extrapolated the circumference of the Earth. If the Egyptian stadium is used as the unit of measurement, Eratosthenes' calculation is within 2 percent of the current measurement. This is a phenomenal accomplishment of logic and mathematics, especially when considering that a millennium later and for about a millennium the shape of the Earth was the subject of some debate. Strangely enough, some people still believe the Earth is flat.

Eratosthenes also took a great interest in history, evidenced by his writings on the Greek myths related to the constellations. In addition, he made an attempt to document the dates of historical events starting with the Siege of Troy. This is ironic, since his direct contributions to history were destroyed and we only know of his accomplishments through the writings of others. In his eighties and going blind, Eratosthenes felt his vitality was gone and so made sure of it by starving himself to death.

References:

Carl Sagan's Cosmos: The Shores of the Cosmic Ocean

http://space.about.com/cs/astronomerbios/a/Eratosthenesbio.htm

http://en.wikipedia.org/wiki/Eratosthenes

http://www.astro.cornell.edu/academics/courses/astro2201/eratosthenes.htm

http://www-history.mcs.st-and.ac.uk/HistTopics/Doubling_the_cube.html

http://en.wikipedia.org/wiki/Great_Library_of_Alexandria

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