*Editors*: Victor J. Katz, Frank J. Swetz

### Articles

The Magic Squares of Manuel Moschopoulos, by P. G. Brown

This is a translation from the original Greek of a manuscript on magic squares by the Byzantine scholar Manuel Moschopoulos, written about 1315.

Benjamin Banneker's Inscribed Equilateral Triangle, by John F. Mahoney

An interesting problem from Banneker's notebook as well as other problems to use with students.

Completing the Square, by Barnabas Hughes

Explain the geometric basis of "completing the square," the original method of solving quadratic equations, to your students.

American Pi, by Larry Lesser

A song for Pi Day.

Thomas Simpson and Maxima and Minima, by Michel Helfgott

Simpson's methods for finding maxima and minima are explored by using examples from his "Doctrine and Application of Fluxions". Many of his techniques could be used in today's classroom.

Leonardo of Pisa: Bunny Rabbits to Bull Markets, by Sandra Monteferrante

The Fibonacci numbers and applications to areas such as plant growth and stock market predictions.

Archimedes' Method for Computing Areas and Volumes, by Gabriela R. Sanchis

Archimedes' use of the Law of the Lever to compute areas and volumes in *The Method* is discussed. Classroom ready examples are presented.

Euler's Investigations on the Roots of Equations, by Todd Doucet

This is a translation of an article of Leonhard Euler in which he attempts to prove the fundamental theorem of algebra. In addition, he discusses in detail his understanding of the nature of complex numbers.

Eratosthenes and the Mystery of the Stades, by Newlyn Walkup

In this article, which won the 2005 HOM SIGMAA Student Paper Contest, the author discusses Eratosthenes' argument to determine the size of the earth as well as possibilities for the size which Eratosthenes found (in modern measures).

Websites to Visit: Plus Magazine and National Curve Bank, by Victor J. Katz and Frank J. Swetz

There are a number of wonderful mathematics websites that readers of *Convergence* should be aware of. We describe two of them here, Plus Magazine and the National Curve Bank.

The Nodding Sphere and the Bird's Beak: D'Alembert's Dispute with Euler, by Robert E. Bradley

An introduction to the priority dispute between Euler and D'Alembert relating to several mathematical ideas that both worked on in the 1740s and 1750s.

HOM SIGMAA 2005 Student Paper Contest Winners

Adds the second-place winner as well as a PDF version of Walkup's paper on Eratosthenes.

### Announcement

Fifth European Summer University

This will be the fifth meeting of this kind, which brings together mathematics educators and researchers to share their teaching ideas and classroom experience in teaching mathematics based on historical, epistemological and cultural approaches.

### Reviews

*A History of Mathematics: Brief Version,* by Victor J. Katz. Reviewed by Robert McGee.

A brief version of the author's well-known history of mathematics text.

*Fibonacci's *Liber Abaci:* Leonardo Pisano’s *Book of Calculation, by Laurence Sigler. Reviewed by Frank J. Swetz.

A translation of one of the earliest European mathematical texts to use the Hindu-Arabic number system.

*Everything and More: A Compact History of ∞,* by David Foster Wallace. Reviewed by Gabriela Sanchis.

A survey of concepts of infinite sets over the centuries.

*Science and Mathematics in Ancient Greek Culture,* edited by C. J. Tuplin and T. E. Rhill. Reviewed by Barnabas Hughes.

Essays on various aspects of Greek science and mathematics, which help give a context for those aspects of Greek culture.

*The Mystery of Numbers,* by Marc-Alain Quaknin. Reviewed by Art Johnson.

A journey through numbers from their earliest beginning in India to the proof of Fermat's Last Theorem by Wiles.

*Companion Encyclopedia of History of Math Sciences,* edited by I. Grattan-Guinness. Reviewed by Frank J. Swetz.

The paperback reprint of this large collection of articles by experts on all aspects of the history and philosophy of mathematics.

*The Calculus Gallery,* by William Dunham. Reviewed by Gary Stoudt.

A gallery of episodes from the history of calculus.

*Pi: A Biography of the World's Most Mysterious Number,* by Alfred S. Posamentier and Ingmar Lehman. Reviewed by Frank J. Swetz.

A discussion not only of the mathematics of pi, but of its applications through the centuries.

*A Discourse Concerning Algebra,* by Jacqueline Stedall. Reviewed by Kathleen Acker.

A study of the rise of English algebra from the Medieval period to the end of the seventeenth century.

*When Least is Best: How Mathematicians Discovered Many Clever Ways to Make Things as Small (or as Large) as Possible,* by Paul J. Nahin. Reviewed by Clifford Wagner.

A survey of techniques of minimizing and maximizing over the centuries.

*Musings of the Masters,* edited by Raymond G. Ayoub. Reviewed by Jeff Suzuki.

Essays on various aspects of mathematical thought by prominent mathematicians of the past century.

*Gauss: Titan of Science,* by G. Waldo Dunnington. Reviewed by Jon Choate.

Reprint of a classic biography of Gauss, with a new foreword by Jeremy Gray.

History of Mathematical Symbolism and Terminology Website, by Jeff Miller. Reviewed by Anthony Piccolino.

Website containing much information on mathematical symbolism and terminology, as well as portraits of mathematicians from postage stamps.

*The Greate Invention of Algebra,* by Jacqueline Stedall. Reviewed by Frank J. Swetz.

Jacqueline Stedall has uncovered the numerous algebraic ideas of Thomas Harriot from the early 17th century and has organized them into a readable treatise.

*Architecture and Mathematics in Ancient Egypt,* by Corinna Rossi. Reviewed by Dorothee Jane Blum.

A study of the nature of architecture in ancient Egypt and its relationship to Egyptian mathematics.

David Joyce's Website. Reviewed by Jim Kiernan.

This website contains a complete version of Euclid's *Elements,* with all the proofs.

*Kepler's Conjecture,* by George Z. Szpiro. Reviewed by Jonathan Choate.

A survey of the attempts to prove Kepler's conjecture over the past 400 years.

*Trigonometric Delights,* by Eli Maor. Reviewed by Dorothee Jane Blum.

A delightful survey of the history of trigonometry, along with discussions of its uses, both ancient and modern.

Math Archives: History of Mathematics Website. Reviewed by Laura Smith.

A website with links to information about numerous topics in the history of mathematics.

*Historical Modules for the Teaching and Learning of Mathematics,* edited by Victor Katz and Karen Dee Michalowicz. Reviewed by Lynn Godshall, David Lutz, and Neil Via.

A CD with eleven modules, each containing numerous activities designed to help secondary teachers use the history of mathematics to teach mathematics.

*Complexities: Women in Mathematics,* edited by Bettye Anne Case and Anne M. Leggett. Reviewed by Erica Voolich.

A collection of articles about historical and contemporary women in mathematics.

*Classics of Mathematics,* edited by Ronald Calinger. Reviewed by Gary Stoudt.

A sourcebook of original materials in the history of mathematics from ancient times to the early twentieth century.

*History of Mathematics - Mesopotamia to Modernity,* by Luke Hodgkin. Reviewed by T. M. Mills.

A new history of mathematics text that asks lots of questions about the history and the mathematics.

*It's About Time,* by N. David Mermin. Reviewed by James Callahan.

A great book from which to learn and teach the subject of relativity.