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We are not very pleased when we are forced to accept a mathematical truth by virtue of a complicated chain of formal conclusions and computations, which we traverse blindly, link by link, feeling our way by touch. We want first an overview of the aim and of the road; we want to understand the idea of the proof, the deeper context.

Unterrichtsblatter fur Mathematik und Naturwissenschaften, 38, 177-188 (1932). Translation by Abe Shenitzer appeared in The American Mathematical Monthly, v. 102, no. 7 (August-September 1995), p. 646.

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# Eratosthenes and the Mystery of the Stades

## Introduction

Editor's Note:  This article was the winning article in the 2005 competition for best history of mathematics article by a student, sponsored by the History of Mathematics SIGMAA of the Mathematical Association of America.

In the third century BCE, the brilliant librarian Eratosthenes of Cyrene (276-195 BCE) devised an ingenious method by which to measure the circumference of the Earth.  Using geometry and the Sun, Eratosthenes accomplished the impossible.  Although his original works have long since been lost, the legendary story has been retold for over two thousand years.  Like all legends it has become difficult to sort the fact from the fiction.  Some scholars claim that Eratosthenes’ approximated the size of the Earth to within 2% of its actual value; while others believe that the accuracy of his measurement is greatly exaggerated.  The key to this ancient riddle is the not-so-standard ancient unit of length – the stade.  There is a great deal of uncertainty as to the actual length of the stade Eratosthenes used.  It is also uncertain whether he made the measurements used in the calculation, or if he relied on the information of others.  Perhaps the most puzzling question is why Eratosthenes inexplicably added 2000 stades to his original figure for the Earth’s circumference.  The mystery is one that drives scholars even today.

Eratosthenes was a man of great distinction among scholars in the ancient world.  He was a good friend of the famous Greek scholar Archimedes of Syracuse (287-212 BCE).  In fact one of Archimedes greatest works, The Method, was dedicated to Eratosthenes [12, p.104 ].

I sent you on a former occasion some of the theorems discovered by me, merely by writing out the enunciations and inviting you to discover the proofs, which at the moment I did not give.  […]  The proofs then of these theorems I have now sent to you.  Seeing moreover in you, as I say, an earnest student, a man of considerable eminence in philosophy, and an admirer […] [13, pp.12-13 ]

Because Eratosthenes was highly knowledgeable in all branches of study, yet he was not the “Alpha” (the greatest) in any one branch, his peers gave him the nickname “Beta” [12, p.104].  Eratosthenes received the equivalent of a college education in Athens and then went to the Egyptian city of Alexandria [1, p.388].  Attracting scholars and students from all over the ancient world, the great library in Alexandria became the center of scholastic achievement.  It is written that the library contained over 500,000 scrolls [15, p.59].  Around 235 BCE, Eratosthenes was appointed head librarian of the library in Alexandria [1, p.388 ].  It was during this period that Eratosthenes would devise his method to approximate the circumference of the Earth.