# Eratosthenes and the Mystery of the Stades - Introduction

Author(s):
Newlyn Walkup (University of Missouri – Kansas City)

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.

All winning papers in the HOM SIGMAA Student Contest are published in Convergence; many are also available through the HOM SIGMAA archives at http://homsigmaa.net/writing-prizes/.

#### Other HOM SIGMAA Student Contest Papers in Convergence

2020: Jeffrey Powers (Grand Rapids Community College), “Did Archimedes Do Calculus?

2019: Amanda Nethington (University of Missouri – Kansas City), "Achieving Philosophical Perfection: Omar Khayyam's Successful Replacement of Euclid's Parallel Postulate."

2018: First place – Callie Lane (University of Missouri – Kansas City), "Race to Refraction: The Repeated Discovery of Snell's Law"; Second place – Christen Peters (Lee University), "The Reality of the Complex: The Discovery and Development of Imaginary Numbers," and Rachel Talmadge (University of Missouri – Kansas City), "François Viète Uses Geometry to Solve Three Problems."

2017: Co-winners – Amanda Akin (Lee University), “To Infinity and Beyond: A Historical Journey on Contemplating the Infinite,” Johann Gaebler (Harvard University), “Traditionalism: 1894 to 1925,” and Nathan Otten (University of Missouri – Kansas City), “Huygens and The Value of all Chances in Games of Fortune.”

2016: Co-winners – Brittany Anne Carlson (Salt Lake Community College), “A Latent Element of Alice’s Agency in Wonderland: Conservative Victorian Mathematics,” and William Cole (Lee University), “The Evolution of the Circle Method in Additive Prime Number Theory.”

2015: Co-winners – Samuel Patterson (University of Missouri – Kansas City), “Bernard Bolzano, a Genius Unnoticed in His Time,” and Briana Yankie (Lee University), “Examining Disproved Mathematical Ideas through the Lens of Philosophy.”

2014: First place – Jenna Miller (University of Missouri – Kansas City), "Casting Light on the Statistical Life of Florence Nightingale," and Anna Riffe (University of Missouri – Kansas City), "The Impossible Proof: An Analysis of Adrien-Marie Legendre's Attempts to Prove Euclid's Fifth Postulate"; Second place – Paul Ayers (University of Missouri – Kansas City), “Gabriel Cramer: Over 260 Years of Crushing the Unknowns," and Mary Ruff (Colorado State University – Pueblo), "Probability to 1750."

2013: Matthew Shives (Hood College), "Paradigms and Mathematics: A Creative Perspective."

2012: First place – Jesse Hamer (University of Missouri – Kansas City), “Indivisibles and the Cycloid in the Early 17th Century”; Second place – Kevin L. Wininger (Otterbein University), “On the Foundations of X-Ray Computed Tomography in Medicine: A Fundamental Review of the 'Radon transform' and a Tribute to Johann Radon.”

2011: First place – Paul Stahl (University of Missouri – Kansas City), “Kepler's Development of Mathematical Astronomy”; Second place – Sarah Costrell (Brandeis University), “Mathematics and Mathematical Thought in the Quadrivium of Isidore of Seville,” and Rick Hill (University of Missouri – Kansas City), “Thomas Harriot's Artis Analyticae Praxis and the Roots of Modern Algebra.”

2010: Co-winners – Jennifer Nielsen (University of Missouri – Kansas City), “The Heart is a Dust Board:  Abu’l Wafa Al-Buzjani, Dissection, Construction, and the Dialog Between Art and Mathematics in Medieval Islamic Culture,” Palmer Rampell (Phillips Academy and Harvard University), “The Use of Similarity in Old Babylonian Mathematics,” and Stefanie Streck (Pacific Lutheran University), “The Fermat Problem.”

2009: First place – Nathan McLaughlin (University of Montana), “The Mathematical Optics of Sir William Hamilton: Conical Refraction and Quaternions”; Second place – Tim Chalberg (Pacific Lutheran University), “Regression Analysis: A Powerful Tool and Riveting Drama”; Honorable Mention – Amy Buchmann (Chapman University), “A Brief History of Quaternions and the Theory of Holomorphic Functions of Quaternionic Variables.”

2008: First place – Mame Maloney (University of Chicago), “Constructivism: A Realistic Approach to Math?”; Second place – Woody Burchett (Georgetown College), “Thinking Inside the Box: Geometric Interpretation of Quadratic Problems in BM 13901,” and Cole McGee (Colorado State University – Pueblo), “Jean Le Rond D'Alembert: Biography of a Mathematician, Philosophe, and a Man of Letters”; Honorable mention – Mame Maloney (University of Chicago), “Pathological Functions in the 18th and 19th Centuries.”

2007: Co-winners – Rory Plante, “The Libra Astronomica and its Mathematics,” and Douglas Smith (Miami University, Ohio), “Lucas’s theorem: A Great Theorem.”

2006: Co-winners – Jennifer Wiegert, “The Sagacity of Circles: A History of the Isoperimetric Problem,” and Samantha Reynolds (University of Missouri – Kansas City), “Maria Gaetana Agnesi: Female Mathematician and Brilliant Expositor of the 18th Century.”

2005: First place – Newlyn Walkup (University of Missouri – Kansas City), “Eratosthenes and the Mystery of the Stades”; Second place – James Collingwood (Drake University), “Rigor in Analysis: From Newton to Cauchy.”

2004: Co-winners – Mark Walters, “It Appears That Four Colors Suffice: A Historical Overview of the Four-Color Theorem,” and Heath Yates (University of Missouri – Kansas City), “An Emanji Temple Tablet.”

Newlyn Walkup (University of Missouri – Kansas City), "Eratosthenes and the Mystery of the Stades - Introduction," Convergence (August 2010)