CALCULUS Understanding Its Concepts and Methods

Eratosthenes (276--194 BC) --- Historical Sketch

Eratosthenes (276-194 BC) was born in Cyrene, of ancient Egypt that is now part of Libya. He was widely recognized as a polymath, a person of wide learning,* and he was an intellectual descendent of Zeno of Elea (495-430 BC) in that he was devoted to high standards of logical rigor. He devoted his huge mental capacities to mathematics, astronomy, music, and poetry. Eratosthenes studied for a while in Athens, and in his middle thirties became the librarian of the famed library at Alexandria, Egypt (see http://cosmopolis.com/).

One outstanding piece of work by Eratosthenes was his computation of the circumference of the Earth, a calculation of astonishing accuracy. Another was his computation of the distances from the Earth to the Sun and to the Moon. He also computed the tilt of the Earth's axis.

Eratosthenes was also interested in pure mathematics. In fact, mathematicians all seem to like to tell about what is called the sieve of Eratosthenes, a marvelously simple and efficient method of listing the prime numbers. In this method, you first write down the positive integers beginning with , as many as you have the time for:Then you cross out every multiple of the first number not crossed out. So the first step is to cross out, not , but every multiple of . The next table shows which numbers are left after this first step.The next number not crossed out is , so now omit every third number after (lots have already been omitted, of course). The result isThe next number not already omitted is , so now omit every fifth number after , and the result is The next number not already omitted is , so now omit every seventh number after , and the result isAt this stage, what are left are all the primes up to You can be sure that they are primes because their square roots are all less than .

It is generally agreed that Eratosthenes committed suicide by self-imposed starvation, and some historians believe he did such to fulfill a prediction he made as to the age he would be at death.

*T. L. Heath, A History of Greek Mathematics, Oxford, 1921.

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Copyright © 2006 Darel Hardy, Fred Richman, Carol Walker, Robert Wisner. All rights reserved. Except upon the express prior permission in writing, from the authors, no part of this work may be reproduced, transcribed, stored electronically, or transmitted in any form by any method.