# What's in Convergence? - Tables of Contents

Annual Volumes

Articles, Ongoing Series, Mathematical Treasures, and Reviews are divided into annual volumes as follows. Browse these tables of contents for interesting and informative articles, images for classroom use, and book and resource reviews. To find Convergence material on specific topics, try these tips for effectively searching the MAA website.

### Special Features and Collections

Explore the following Special Features and Collections, many of which can also be accessed from the Convergence homepage.

# What's in Convergence? - Tables of Contents

### Annual Volumes

Articles, Ongoing Series, Mathematical Treasures, and Reviews are divided into annual volumes as follows. Browse these tables of contents for interesting and informative articles, images for classroom use, and book and resource reviews.

### Special Features and Collections

Explore the following Special Features and Collections, many of which can also be accessed from the Convergence homepage.

# What's in Convergence? - Images for Classroom Use

Use these images of mathematical people, texts, and objects in your classroom!

### Mathematical People

Portrait Gallery: Images of mathematicians from ancient to modern times.

Paul R. Halmos Photograph Collection:  Photos of mathematicians from the 1950s through the 1990s.

### Mathematical Treasures

Mathematical Treasures Collection: Index of hundreds of images of historical objects and texts from libraries, museums, and individuals around the world. Highlights:

# What's in Convergence? - Contents of Volume 1 - 2004

Editors: Victor J. Katz, Frank J. Swetz

### Articles

Using Problems from the History of Mathematics, by Frank Swetz

Why should we use historical problems in today's classroom? This article answers that question and serves as an introduction to the problems on this website.

Can You Really Derive Conic Formulae from a Cone?, by Gary S. Stoudt

As is the case with a great deal of interesting mathematics, the conic sections are believed to have been discovered in an attempt to solve a problem, a problem that on the surface seems to have nothing to do with conic sections.

Benjamin Banneker's Trigonometry Puzzle, by Florence Fasanelli, Graham Jagger, and Bea Lumpkin

Benjamin Banneker solved some trigonometry problems in his extant notebooks. One of them is discussed here. The authors have also discovered the probable source of Banneker's trigonometry table.

Euler's Analysis of the Genoese Lottery, by Robert E. Bradley

In the middle of the 18th century, King Frederick the Great of Prussia became interested in creating a lottery to raise money. As was his custom when mathematical matters were involved, he called upon Leonhard Euler for counsel.

Van Schooten's Ruler Constructions, by C. Edward Sandifer

A translation of and commentary on Frans van Schooten's work on constructions using only a straightedge. Van Schooten's postulates for use of the straightedge, however, allow the copying of one line segment onto another.

Measuring the Globe: An Historical Activity, by Barnabas Hughes

Eratosthenes' measurement of the earth, in a form that's easy for teachers to use.

A Euclidean Approach to the FTC, by Andrew Leahy

The Fundamental Theorem of Calculus is presented in the version of Scottish mathematician James Gregory -- without the use of limits.

The Right and Lawful Rood, by Peter Ransom

The "rood", a linear measure dating from many centuries ago, is calculated anew in today's classrooms.

Using Historical Problems in the Middle School, by Karen Michalowicz and Robert McGee

Historical problems can be used to enliven any mathematics class. Here are some examples from medieval times, from a 19th century American textbook, and from a 19th century Armenian textbook, among other sources.

Alien Encounters, by Gavin Hitchcock

A dramatization, in two Acts, of the struggles of European mathematicians of the seventeenth and eighteenth centuries to come to terms with the newly admitted negative numbers.

Mathematics as the Science of Patterns, by Michael N. Fried

Mathematics is often referred to today as the "science of patterns." But has this always been true historically, or is this something that happened in recent times? The question is discussed here with reference to the work of Euclid and Jacob Steiner.

Counting Boards, by Chris Weeks

The author finds a rare and fine example of a counting table in Strasbourg. Article contains two photos of this table.

Teaching Leonardo: An Integrated Approach, by Rick Faloon

The work of the great Renaissance artist/scientist Leonardo da Vinci can be taught in secondary schools through an integrated approach of several disciplines. This article explores the approach of the Ross School.

HOM SIGMAA 2004 Student Paper Contest

Winning papers from the first year of HOM SIGMAA's annual competition.

### Reviews

Sherlock Holmes in Babylon, edited by Marlow Anderson, Victor Katz and Robin Wilson. Reviewed by Frank J. Swetz

A collection of articles in the history of mathematics that appeared in journals of the Mathematical Association of America over the past 90 years.

Famous Problems and Their Mathematicians, by Art Johnson. Reviewed by Frank J. Swetz

This resource consists of a series of 61 worksheets, each focused on a particular problem and related to a particular historical mathematical personality.

Stamping Through Mathematics, by Robin Wilson. Reviewed by Tim Keenan

The author takes us on a historical tour of mathematics through postage stamps of the world that display mathematicians and mathematics.

St. Andrews History of Mathematics Archive Website  Reviewed by Barnabas Hughes

This website offers a collection of biographies of mathematicians and a variety of resources on the development of various branches of mathematics. It is an extremely rich and extensive site.

Statisticians of the Centuries, edited by C. C. Heyde and E. Seneta. Reviewed by Winston Richards

Thumbnail sketches of statisticians throughout history.

Historical Connections in Mathematics, by Wilbert Reimer and Luetta Reimer. Reviewed by Frank J. Swetz

Brief biographies of mathematicians with mathematical activities based on their work.

Multicultural Classroom Posters Sets 3 & 4  Reviewed by by Vincent Corrado

These posters illustrate aspects of the history of mathematics in countries from Babylonia to Ireland.

Celebrating Women in Mathematics and Science, by Miriam P. Cooney. Reviewed by Erica Voolich

This unique and beautiful book features the biographies of twenty-two notable female mathematicians and scientists, showing how their determination, creativity, and intellectual passion helped them excel in their fields.

Agnesi to Zeno, by Sanderson Smith. Reviewed by Linda Shuey

Short biographies of mathematicians with mathematical activities.

Euler: The Master of Us All, by William Dunham. Reviewed by Clifford Wagner

A small selection of Euler's works, explained by a master expositor.

Mathematics Elsewhere, by Marcia Ascher. Reviewed by Lawrence Shirley

A compilation of ethnomathematical ideas from around the world.

Four Colors Suffice, by Robin Wilson. Reviewed by Frank J. Swetz

The history of the four color problem with sketches of the attempted proofs in the nineteenth century and an outline of the computer proof of the twentieth century.

Historical Topics for the Mathematics Classroom, edited by J.K. Baumgart, D.E. Deal, B.R. Vogeli, and A.E. Hallerberg. Reviewed by Tim Keenan

This 1989 revision of the 1969 NCTM yearbook still provides wonderful suggestions for using the history of mathematics in the classroom.

The Art of the Infinite, by Robert Kaplan and Ellen Kaplan. Reviewed by Albert Briggs

This book is a collection of mathematical ideas organized around the themes of infinity and the illumination of the nature of mathematical thought.

Number from Ahmes to Cantor, by Midhat Gazalé. Reviewed by Frank J. Swetz

A lively history of number systems and number theory from earliest times up to the notion of "infinity".

Milestones of Mathematics Posters  Reviewed by Lynn Godshall

Two posters illustrating the major milestones in the history of mathematics, from the first ideas of "number" to the proof of Fermat's Last Theorem.

The Nothing That Is, by Robert Kaplan. Reviewed by Austin Lobo

A history of the concept of zero from as far back as the Babylonian period, with philosophical excursions into the meaning of "nothing".

The Saga of Mathematics, by Marty Lewinter and William Widulski. Reviewed by Jim Kiernan

A brief history of mathematics aimed at college students with little technical knowledge of mathematics.

Multicultural Classroom Posters Sets 1 & 2  Reviewed by Lynn Godshall

Posters illustrating mathematics concepts in such places as China, Japan, India, and the Americas.

Historic Women of Mathematics Poster  Reviewed by Karen Michalowicz

Poster picturing five famous women, from Hypatia to Grace Hopper.

Benjamin Banneker Poster  Reviewed by Karen Michalowicz

Poster of Banneker, with a brief description of his life and work.

Consortium's Historical Notes  Reviewed by Jim Kiernan

Collection of historical articles that appeared in Consortium, the newsletter of COMAP.

Remarkable Mathematicians, by Ioan James. Reviewed by Jim Kiernan

A collection of biographies of sixty mathematicians from the eighteenth century to the twentieth.

Colorful Characters of Mathematics Posters, by Isaac Asimov. Reviewed by Art Johnson

Whimsical posters of fifteen mathematicians with brief biographies.

The Pioneers of Calculus Posters, by Bruce White. Reviewed by Art Johnson

A collection of sixteen posters of contributors to calculus, with brief biographical sketches.

Great Ideas of Mathematics Posters  Reviewed by Sylvia Lazarnick

A set of four posters dealing with the Pythagorean Theorem, infinity, prime numbers, and the history of pi.

Speaking of Mathematics Posters  Reviewed by Lynn Godshall

A collection of 24 mini-posters, each containing a quotation about mathematics.

Archimedes: What Did He Do Besides Cry Eureka?, by Sherman Stein. Reviewed by Tim Keenan

Archimedes' work, The Method, explained, along with many other important ideas of the great Greek geometer.

A History of Analysis, edited by Hans Niels Jahnke. Reviewed by Jeff Suzuki

A superb collection of articles by experts on various areas of the history of analysis, from the Greeks to modern times.

Readings in the History of Mathematics Education, by James K. Bidwell and Robert G. Clason. Reviewed by Lynn Godshall

This collection of readings gives details on the history of mathematics education in the U.S. from 1828 to 1959.

Mathematical Evolutions, edited by Abe Shenitzer and John Stillwell. Reviewed by Lang Moore

A collection of articles from the American Mathematical Monthly by experts on the evolution of various fields of mathematics.

The Mathematical Century, by Piergiorgio Odifreddi. Reviewed by Gary Stoudt

The development of 30 important mathematics subjects during the twentieth century made understandable to undergraduate mathematics majors.

The Story of Mathematics, by Richard Mankiewicz. Reviewed by Anne Loesch

A wonderful survey of the history of mathematics, emphasizing its relationship with the ambient culture.

Great Moments in Mathematics Before 1650, by Howard Eves. Reviewed by Jon Choate

A collection of short lectures by Howard Eves giving details on 20 important happenings in the history of mathematics before 1650.

Cogwheels of the Mind: The Story of Venn Diagrams, by A. W. F. Edwards. Reviewed by Jim Kiernan

An introduction to the work of Venn as well as the work of the author in extending some of Venn's results.

Abel's Proof, by Peter Pesic. Reviewed by by Lynn Godshall

A discussion of the meaning of mathematical unsolvability in the context of the history of Abel's proof of the unsolvability of the quintic equation in terms of radicals.

Math and the Mona Lisa, by Bűlent Atalay. Reviewed by Frank J. Swetz

The author makes the case for Leonardo da Vinci as the first modern scientist, as he discusses Leonardo's mathematics and science.

Pascal's Arithmetical Triangle, by A. W. F. Edwards. Reviewed by Richard M. Davitt

A history of the development of Pascal's triangle in its various manifestations.

# What's in Convergence? - Contents of Volume 2 - 2005

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

### 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.

# What's in Convergence? - Contents of Volume 3 - 2006

Editors: Victor J. Katz, Frank J. Swetz

### Articles

The Rule of False Position and Geometric Problems, by Vicente Meavilla Segui and Alfinio Flores

This article contains examples of the use of the rule of false position in the solution of geometric problems as found in the work of Simon Stevin. We discuss the benefits for future teachers and their students.

Approximate Construction of Regular Polygons: Two Renaissance Artists, by Raul A. Simon

Leonardo da Vinci and Albrecht Durer both offered approximate constructions of regular pentagons for the use of artists. This article explains these constructions.

John Napier: His Life, His Logs, and His Bones, by Michael J. Caulfield

A brief introduction to the life of John Napier, along with an animation of calculations using Napier's bones.

The Sagacity of Circles: A History of the Isoperimetric Problem, by Jennifer Wiegert

A summary of the history of the problem of finding the region of greatest area bounded by a given perimeter. This essay was a winner of the HOM SIGMAA Student Paper Contest in 2006.

Gerbert d'Aurillac and the March of Spain: A Convergence of Cultures, by Betty Mayfield

The story of Gerbert, who became Pope Sylvester II in 999, and his mathematics.

Dear Professor Greitzer, by Joe Richards and Don Crossfield

A letter to Sam Greitzer, late editor of Arbelos, discussing the derivation of two formulas for calculating pi.

The Quadrature of the Circle and Hippocrates’ Lunes, by Daniel E. Otero

A study of some elements of Greek geometry, as part of a course for liberal arts undergraduates dealing with basic concepts of the calculus.

An Investigation of Historical Geometric Constructions, by Suzanne Harper and Shannon Driskell

Dynamic geometry software is used to demonstrate early Greek attempts at the trisection of an angle and the squaring of a circle.

The Great Calculation According to the Indians of Maximus Planudes, by Peter G. Brown

A translation of part of a thirteenth century work by the Byzantine monk Maximus Planudes on the Hindu-Arabic numerals and the algorithms for calculation.

Fibonacci and Square Numbers, by Patrick Headley

A discussion of aspects of Leonardo of Pisa's Book of Squares.

Leonard Euler’s Solution to the Konigsberg Bridge Problem, by Teo Paoletti

A survey of the famous Konigsberg Bridge Problem and its connection to graph theory by an undergraduate student.

A Plague of Ratios, by Benjamin Wardhaugh

The story of Nicolaus Mercator, music, and logarithms.

Student Reports: A Rewarding Undertaking, by Frank J. Swetz

Some ideas on using student reports when you teach a course in the history of mathematics

How Tartaglia Solved the Cubic Equation, by Friedrich Katscher

The method of Tartaglia for solving cubics, that he eventually explained to Cardano.

Who Was Tartaglia Really?, by Friedrich Katscher

In many sources, we see that Tartaglia has the surname Fontana. According to the author of this article, the co-discoverer of the cubic formula did not ever use that name.

### Announcements

Karen Dee Michalowicz, by Victor J. Katz

We sadly announce the untimely death of one of Convergence's editorial board members.

From the Editors, by Victor J. Katz and Frank Swetz

The editors invite contributions, participation, and feedback from Convergence readers.

### Reviews

From Calculus to Computers: Using the Last 200 Years of Mathematics History in the Classroom, edited by Amy Shell-Gellasch and Dick Jardine. Reviewed by Jim Kiernan.

A collection of articles on using the history of mathematics of the past 200 years in the undergraduate classroom.

The Joy of Pi, by David Blatner. Reviewed by Frank J. Swetz.

A highly recommended new book on the history and applications of pi.

Joy of Pi Website. Reviewed by Jon Choate.

This website is connected to the book, The Joy of Pi. It has numerous interesting facts about pi, with links to additional sites.

Negative Math: How Mathematical Rules Can Be Positively Bent, by Alberto A. Martinez. Reviewed by Karen Michalowicz.

The story of the negative numbers.

The Archimedes Website. Reviewed by Marcus Barnes.

This website devoted to miscellanea about Archimedes contains much interesting material about his life and times.

The Equation that Couldn't Be Solved, by Mario Livio. Reviewed by Doris Schattschneider.

A history of attempts to solve cubic and higher degree polynomial equations, including the notions of group theory and their relationship to the idea of symmetry.

The Square Root of 2: A Dialogue Concerning a Number and a Sequence, by David Flannery. Reviewed by Barnabas Hughes.

A wonderful book about the square root of 2, beginning with the search for the side of a square double a given square.

A Contextual History of Mathematics, by Ronald Calinger. Reviewed by Frank J. Swetz.

An overly ambitious textbook on the history of mathematics.

The Prince of Mathematics: Carl Friedrich Gauss, by M. B. W. Tent. Reviewed by Linda Y. Shuey.

A biography of Gauss designed for high school students.

The History of Mathematics: A Brief Course, by Roger Cooke. Reviewed by Gary Stoudt.

A new edition of a brief history text, arranged topically rather than chronologically.

Awakening of Geometrical Thought in Early Culture, by Paulus Gerdes. Reviewed by Lawrence Shirley.

How does geometry begin? This work explores the origins of geometry in the work of artisans.

The Honors Class: Hilbert’s Problems and Their Solvers, by Ben H. Yandell. Reviewed by Frank J. Swetz.

This work discusses the people who solved some of Hilbert's problems from 1900, as well as the mathematics involved in the solutions.

Math Forum Website. Reviewed by Gail Kaplan.

A description of this well-regarded website.

Math Pages Website. Reviewed by Laura Smith.

A general mathematics website with much information on the history of mathematics.

Ancient Mathematics, by Serafina Cuomo. Reviewed by Barnabas Hughes.

A new history of Greek mathematics, taking into account the latest research.

History of Mathematics Archive Website. Reviewed by Don Crossfield.

A wide-ranging site with links to many sources in the history of mathematics.

ABOUT Website on History of Mathematics. Reviewed by Lawrence Shirley.

A section of a much larger website, dealing with some random topics in the history of mathematics.

Mathematics and The Historian’s Craft: The Kenneth O. May Lectures, edited by G. Van Brummelen and M. Kinyon. Reviewed by Jon Choate.

A collection of the Kenneth May lectures in the history of mathematics given at meetings of the Canadian Society for the History and Philosophy of Mathematics.

Thinking about Mathematics: The Philosophy of Mathematics, by Stewart Shapiro. Reviewed by Frank J. Swetz.

An excellent book surveying the history of the philosophy of mathematics from the time of Plato to the nineteenth and early twentieth centuries.

# What's in Convergence? - Contents of Volume 4 - 2007

Editors: Victor J. Katz, Frank J. Swetz

### Articles

Euler Squares, by Elaine Young

An elementary introduction to Euler squares and how they can be used in teacher training.

Maya Cycles of Time, by Sandra Monteferrante

Explorations of the Mayan calendar.

Abel on Elliptic Integrals: A Translation, by Marcus Emmanuel Barnes

A translation of one of the seminal papers in the field of elliptic integrals, but one that can be read by an undergraduate.

Limit Points and Connected Sets in the Plane, by David R. Hill and David E. Zitarelli

A study of Mullikan's Nautilus, using movies to illustrate the important ideas.

Historical Activities for the Calculus Classroom, by Gabriela R. Sanchis

History and mathematics of curve sketching, tangent lines, and optimization, explored using interactive applets.

Proportionality in Similar Triangles: A Cross-Cultural Comparison, by Jerry Lodder

A student module based on a comparison of the Greek and Chinese approach to the idea of similarity.

The Unique Effects of Including History in College Algebra, by G. W. Hagerty, S. Smith, D. Goodwin

Using the history of mathematics in a college algebra class has had significant positive effects on student learning.

Episodes in the History of Geometry through Models in Dynamic Geometry, by Eduardo Veloso and Rita Bastos

Classroom lessons baseed on four episodes in the history of geometry are discussed, where dynamic geometry helps in understanding the ideas.

HOM SIGMAA 2007 Student Paper Contest Winners

Research earning awards in HOM SIGMAA's annual competition.

### Announcements

Euler Tercentenary Year, by Victor J. Katz

2007 marks the three hundredth anniversary of the birth of Leonhard Euler, the most prolific mathematician of all time. There are numerous events throughout the world celebrating this anniversary, many of which are listed here.

2008 Joint Mathematics Meetings, by Victor J. Katz

There are numerous mathematics history events at the upcoming AMS-MAA Joint Mathematics Meetings in San Diego, CA.

### Reviews

Great Feuds in Mathematics, by Hal Hellman. Reviewed by Jim Kiernan.

A lively description of ten of the greatest feuds in mathematics.

History of Mathematics, by Charles Boyer and Uta Merzbach. Reviewed by Kathleen Acker.

Boyer's classic text, as revised by Uta Merzbach, is still worth having.

A Concise History of Mathematics, by Dirk J. Struik. Reviewed by Barnabas Hughes.

The classic work by Dirk Struik is still worth reading, especially for its attention to the social context of the development of mathematical ideas

God Created the Integers: Mathematical Breakthroughs that Changed History, edited by Stephen Hawking. Reviewed by Eugene Boman.

Original source material from seventeen mathematicians, with commentary by Stephen Hawking

Pioneers in Mathematics, by Michael J. Bradley. Reviewed by Linda Y. Shuey.

A five volume set of biographies of mathematicians from ancient times to the twentieth century, aimed at secondary students.

Arthur Cayley: Mathematician Laureate of the Victorian Age, by Tony Crilly. Reviewed by Kathleen M. Clark.

A biography of Arthur Cayley, the outstanding mathematician of Victorian Britain

King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry, by Siobhan Roberts. Reviewed by Jonathan Choate.

A biography of the geometer Donald Coxeter.

A Radical Approach to Real Analysis, by David Bressoud. Reviewed by James Callahan.

A historically minded textbook designed to teach real analysis by considering some of the problems faced by 19th century mathematicians.

Unknown Quantity; A Real and Imaginary History of Algebra, by John Derbyshire. Reviewed by Don Crossfield.

A history of algebra from its early beginnings to the twentieth century.

Equations from God: Pure Mathematics and Victorian Faith, by Daniel J. Cohen. Reviewed by Barnabas Hughes.

A study of Victorian idealism and its relation to religion, as exemplified in the work of three 19th century British mathematicians.

An Introduction to the History of Mathematics, by Howard Eves. Reviewed by Gary Stoudt.

Howard Eves' sixth edition is still worth considering for a textbook.

The Secret Life of Numbers, by George G. Szpiro. Reviewed by Edith Prentice Mendez.

Short sketches on how mathematicians work and think.

Prime Numbers: The Most Mysterious Figures in Math, by David Wells. Reviewed by Gabriela R. Sanchis.

An introduction to the prime numbers in many of their aspects.

James Joseph Sylvester: Jewish Mathematician in a Victorian World, by Karen Hunger Parshall. Reviewed by Gail Kaplan.

The first detailed biography of James Joseph Sylvester.

Yearning for the Impossible: The Surprising Truths of Mathematics, by John Stillwell. Reviewed by Lynn Godshall.

This book explores the history of mathematics from the perspective of the creative tension between common sense and the "impossible" as the author follows the discovery or invention of new concepts that have marked mathematical progress.

Euler Tercentenary Volumes 1 and 2:

The Early Mathematics of Leonhard Euler, by C. Edward Sandifer.

The Genius of Euler: Reflections on his Life and Work, edited by William Dunham.

Reviewed by Frank J. Swetz.

Two excellent volumes on Euler in honor of his three hundredth birthday.

Unexpected Links between Egyptian and Babylonian Mathematics, by Jöran Friberg. Reviewed by Lawrence Shirley.

This book demonstrates the relationship between the mathematics in some recently discovered Babylonian tablets and some standard problems from Egyptian mathematics.

Measuring America, by Andro Linklater. Reviewed by Frank J. Swetz.

The beginnings of land measurement in the early United States and how this affected American democracy.

The Life of Numbers, written by Alberto Manguel, Antonio Duran and George Ifrah; illustrated by Sean Macksouli, Natalia Pintado, and Javier Pagola. Reviewed by Frank J. Swetz.

A creative expression combining text, design and illustrations, originally designed for the International Congress of Mathematicians in Madrid.

Amazing Traces of a Babylonian Origin in Greek Mathematics, by Jöran Friberg. Reviewed by Barnabas Hughes.

Aspects of classical Greek mathematics are compared with areas of Babylonian mathematics.

The Poincare Conjecture: In Search of the Shape of the Universe, by Donal O’Shea. Reviewed by Eugene Boman.

The history of the Poincare conjecture up to its recent proof by Grigori Perelman.

Euler Tercentenary Volume 3:

How Euler Did It, by C. Edward Sandifer. Reviewed by Frank J. Swetz.

A collection of short pieces each detailing how Euler solved a particular mathematics problem.

A History of Mathematics, by Jeff Suzuki. Reviewed by Gary Stoudt.

A solid history of mathematics text that any instructor of a history course should consider.

The Development of Mathematics in Medieval Europe, by Menso Folkerts. Reviewed by Frank J. Swetz.

A collection of articles on mathematics in Europe from the twelfth to the fifteenth century.

The Pythagorean Theorem: A 4,000 Year History, by Eli Maor. Reviewed by Jim Kiernan.

A survey of this theorem's 4000 year history, with applications to many fields.

Mathematics: Powerful Patterns in Nature and Society, by Harry Henderson. Reviewed by Linda Y. Shuey.

The work of ten scientists who thought deeply about patterns.

The Mathematics of Egypt, Mesopotamia, China, India and Islam, edited by Victor J. Katz. Reviewed by Barnabas Hughes.

A new collection of original source materials in the mathematics of five civilizations.

Calculus Gems: Brief Lives and Memorable Moments, by George F. Simmons. Reviewed by Kathleen M. Clark.

A collection of short biographical sketches of people involved in the development of calculus, as well as brief descriptions of important events in that development.

Euler Tercentenary Volumes 4 and 5:

Euler and Modern Science, edited by N.N. Bogolyubov, G.K. Mikhailov, and A.P. Yushkevich.

Euler at 300: An Appreciation, edited by Robert Bradley, Lawrence D’Antonio, and C. Edward Sandifer.

Reviewed by Frank J. Swetz.

The two final volumes of the MAA tercentenary series on Euler present numerous papers on various aspects of Euler's life and work.

The Mathematician's Brain, by David Ruelle. Reviewed by Kathleen Acker.

A book delving into the working of the mathematical mind.

How Mathematics Happened: The First 50,000 Years, by Peter S. Rudman. Reviewed by Gail Kaplan.

A popular history of ancient mathematics, dealing with the mathematics of ancient Egypt and Babylonia.

Math for Mystics: From the Fibonacci Sequence to Luna's Labyrinth to the Golden Section and Other Secrets of Sacred Geometry, by Renna Shesso. Reviewed by Edith Prentice Mendez.

A book connecting mathematics to mysticism, but not recommended.

Numbers at Work: A Cultural Perspective, by Rudolf Taschner. Reviewed by Jonathan Choate.

Essays on how number has been critical to the work of scientists through the ages.

Hypatia of Alexandria: Mathematician and Martyr, by Michael A. B. Deakin. Reviewed by Eugene Boman.

A biography of Hypatia in her times that carefully distinguishes between the known facts of her life and the many speculations about her.

# What's in Convergence? - Contents of Volume 5 - 2008 (Loci - Volume 1)

Editors: Victor J. Katz, Frank J. Swetz

### Articles

Apollonius's Ellipse and Evolute Revisited, by Frederick Hartmann and Robert Jantzen

Apollonius found how to draw normals to an ellipse from points in the ellipse by using hyperbolas. A modern version is presented here.

What is 0^0?, by Michael Huber and V. Frederick Rickey

The expression 0^0 is usually called an indeterminate form. This article details the history of the meaning of this expression and concludes that, in some cases, we should evaluate it as 1.

Leonardo da Vinci’s Geometric Sketches, by Frank J. Swetz

Leonardo da Vinci illustrated Luca Pacioli’s 1509 De divina proportione. Several of his illustrations are shown here.

Mathematics Education at West Point: The First Hundred Years, by V. Frederick Rickey and Amy Shell-Gellasch

A survey of the mathematics education of cadets in the first century after the founding of the U.S. Military Academy.

HOM SIGMAA 2008 Student Paper Contest Winners, by Victor J. Katz

There are four winners of the HOM SIGMAA Student Paper Contest for 2008. The winning papers may be accessed here.

A discussion of a collaborative effort in Italy to produce materials enabling secondary school teachers to use the history of mathematics in the classroom.

Triangles in the Sky: Trigonometry and Early Theories of Planetary Motion, by Sandra M. Caravella

A survey of early theories of planetary motion, with dynamic figures to help in understanding these motions.

Apportioning Representatives in the United States Congress, by Michael J. Caulfield

The history of apportionment of representatives in the U.S. Congress, from the 1790s until today, along with a discussion of the mathematics involved in the various methods.

The Quipu, by Frank J. Swetz

A collection of illustrations of Inca quipus, with references to their earliest descriptions.

Napier's e, by Amy Shell-Gellasch

A discussion of why we use "e" to represent the base of the natural logarithm system.

### Announcement

History and Pedagogy of Mathematics (HPM) 2008

The quadrennial meeting of the International Study Group on the Relations between History and Pedagogy of Mathematics will be in Mexico City, July 14-18, 2008.

### Reviews

Benjamin Franklin's Numbers: An Unsung Mathematical Odyssey, by Paul C. Pasles. Reviewed by Eugene Boman.

A thorough study of Benjamin Franklin's mathematical accomplishments, in particular his work on magic squares.

Hands on History, A Resource for Teaching Math, edited by Amy Shell-Gellasch. Reviewed by Don Crossfield.

A collection of articles about mathematical models and objects and how they can be used in teaching.

A History of Abstract Algebra, by Israel Kleiner. Reviewed by Ueli Daepp.

A history of the various algebraic structures that came together to give us "abstract algebra" by early in the twentieth century.

The World of Maria Gaetana Agnesi, Mathematician of God, by Massimo Mazzotti. Reviewed by Kathleen Ambruso Acker.

A biography stressing Agnesi's deep commitment to help those in need.

Museum of the History of Science, Oxford. Reviewed by Frank J. Swetz.

There is much to see in this museum related to the history of mathematics.

A Biography of Maria Gaetana Agnesi, by Antonella Cupillari. Reviewed by Edith Prentice Mendez.

A biography of the 18th century author of an early calculus text, with some translations from the text.

Tools of American Mathematics Teaching, 1800-2000, by Peggy Aldrich Kidwell, Amy Ackerberg-Hastings, and David Lindsay Roberts. Reviewed by Don Crossfield.

A survey of the use of technology in American mathematics teaching over the past 200 years.

Sacred Mathematics: Japanese Temple Geometry, by Fukagawa Hidetoshi and Tony Rothman. Reviewed by Frank J. Swetz.

This book describes some of the so-called temple geometry problems that Japanese mathematicians posed and solved beginning in the seventeenth century.

Mathematical Expeditions: Chronicles by the Explorers, by Reinhard Laubenbacher and David Pengelley. Reviewed by Jim Kiernan.

A collection of original texts to help students learn some important areas of mathematics.

Mathematics in Ancient Iraq: A Social History, by Eleanor Robson. Reviewed by Frank J. Swetz.

A new history of mathematics in ancient Mesopotamia, concentrating on its social aspects.

Mathematics Emerging: A Sourcebook, by Jacqueline Stedall. Reviewed by Gary Stoudt.

Textbook that provides sources in both original form and largely literal translation by Stedall.

### Mathematical Treasures

Mathematical Treasures from the Smith and Plimpton Collections at Columbia University, by Frank J. Swetz and Victor J. Katz

Index to Mathematical Treasures Collection: Images of historical texts and objects from libraries, museums, and individuals around the world for use in your classroom!

# What's in Convergence? - Contents of Volume 6 - 2009 (Loci - Volume 1)

Editors:  Victor J. Katz, Frank J. Swetz, Janet Beery, Kathleen Clark

### Articles

James Gregory and the Pappus-Guldin Theorem, by Andrew Leahy

An analysis of James Gregory's proof of the Pappus-Guldin Theorem, along with the original documents in both Latin and English.

A Locally Compact REU in the History of Mathematics: Involving Undergraduates in Research, by Betty Mayfield and Kimberly Tysdal

A description of a Research Experience for Undergraduates conducted in 2007 at Hood College.

Sums of Powers of Positive Integers, by Janet Beery

A history of attempts to develop formulas expressing the sums of powers of the first n positive integers from the Pythagoreans to Jakob Bernoulli.

Investigating Euler's Polyhedral Formula Using Original Sources, by Lee Stemkoski

The works of Leonhard Euler are particularly accessible to readers as his papers usually contain many examples as well as a gradual progression of ideas.  The author shows how teachers can use Euler's original works in the classroom to explore the polyhedral formula and related results.

HOM SIGMAA 2009 Award Winners, by Amy Shell-Gellasch

These are the winning papers from the annual History of Mathematics Special Interest Group of the MAA (HOM SIGMAA) 2009 Student Paper Contest.

A Modern Vision of the Work of Cardano and Ferrari on Quartics, by Harald Helfgott and Michel Helfgott

A study of the solution of quartic equations in Cardano's Ars Magna and in the work of Euler and Descartes.

The Classic Greek Ladder and Newton’s Method, by Robert J. Wisner

Greek ladders for approximating square roots may be more ancient than the ancient Greeks. Students at any level can appreciate their beauty and simplicity. Those who have studied calculus can compare them with Newton’s Method for approximating roots.

After completing this assignment on Simon Stevin's treatment of decimal numbers in his 1585 De Thiende, the author's preservice mathematics teachers understood why our usual procedure for multiplying such numbers works.

### Reviews

The Mayan and Other Ancient Calendars, by Geoff Stray.  Reviewed by Lawrence Shirley.

A detailed study of the cycles of the Mayan calendar, along with some other ancient calendars.

Euler's Gem: The Polyhedron Formula and the Birth of Topology, by David S. Richeson.  Reviewed by Clifford Wagner.

A sketch of the history of topology, beginning with the polyhedron formula and continuing up to the present.

Mathematics in India, by Kim Plofker.  Reviewed by Frank J. Swetz.

A survey of over two thousand years of the history of mathematics on the Indian subcontinent.

The Mathematics of the Heavens and the Earth: the Early History of Trigonometry, by Glen Van Brummelen.  Reviewed by Frank J. Swetz.

A comprehensive history of trigonometry from ancient times to the Renaissance.

Mathematical Works Printed in the Americas, 1554-1700, by Bruce Stanley Burdick.  Reviewed by Frank J. Swetz.

A bibliographical reference to mathematics books printed in the New World before 1700.

Mathematicians: An Outer View of the Inner World, by Mariana Cook.  Reviewed by Frank J. Swetz.

Portraits of 92 living mathematicians, with autobiographical comments.

Pythagoras' Revenge: A Mathematical Mystery, by Arturo Sangalli.  Reviewed by James F. Kiernan.

A fictionalized account of Pythagoras and Pythagorean beliefs.

Euclidean and Non-Euclidean Geometries: Development and History, by Marvin Jay Greenberg.  Reviewed by Eugene Boman.

This textbook seamlessly combines the history of non-Euclidean geometry with the mathematical ideas.

# What's in Convergence? - Contents of Volume 7 - 2010 (Loci - Volume 2)

Editors:  Janet Beery, Kathleen Clark

### Articles

Discovering the Beauty of Science, by Christine Latulippe and Joe Latulippe

The authors' math history class visited the "Beautiful Science" exhibit at the Huntington Library in Southern California. Find actual math history texts and artifacts near you and virtual ones online to share with your students.

Johannes Kepler’s Astronomia Nova, by Frank J. Swetz

Images from Kepler’s 1609 New Astronomy.

The Geometry of Rene Descartes, by Frank J. Swetz

Images from a 1659 Latin edition of Descartes’ Geometria, originally published in 1637 in French as La géométrie.

The author uses her poem, "The Enigmatic Number e," to show how poetry about the history of mathematics can be used to enrich and enliven mathematics instruction.

Servois' 1814 Essay on the Principles of the Differential Calculus, with an English Translation, by Robert E. Bradley and Salvatore J. Petrilli, Jr.

The authors provide an analysis and English translation of the argument by a little known French mathematician that calculus should be based on series rather than on infinitesimals.

HOM SIGMAA 2010 Student Paper Contest Winners, featuring essays by Jennifer Nielsen, Palmer Rampell, and Stefanie Streck

Download the three winning essays from the 2010 HOM SIGMAA Student Paper Contest to learn about medieval Islamic dust boards, Old Babylonian similarity, and the Fermat Problem.

Extracting Square Roots Made Easy: A Little Known Medieval Method, by Friedrich Katscher

A method for extracting square roots used in Italy through the 18th century was introduced in a manuscript by the 12th century mathematician al-Hassar.

Logarithms:  The Early History of a Familiar Function, by Kathleen Clark and Clemency Montelle

The authors recount the ‘great tale’ of Napier’s and Burgi’s parallel development of logarithms and urge you to use it in class.

François-Joseph Servois: Priest, Artillery Officer, and Professor of Mathematics, by Salvatore J. Petrilli, Jr.

This biography reveals that, during his life as a military officer and mathematician, Servois fought for Paris and for the foundations of calculus.

A Disquisition on the Square Root of Three, by Robert J. Wisner

The author compares Greek ladder, continued fraction, and Newton's Method approximations, pointing out that the Greek ladder easily produces both of Archimedes' famous bounds.

The author presents five modules based on mathematics from medieval Islamic cultures for use in a variety of high school and college mathematics courses.

Maya Calendar Conversions, by Ximena Catepillan and Waclaw Szymanski

Students learn about Maya calendar systems, including how to convert Maya Long Count dates to Calendar Round (Tzolkin and Haab calendar) dates, on a trip to the Yucatan.

Servois' 1814 Essay on a New Method of Exposition of the Principles of Differential Calculus, with an English Translation, by Robert E. Bradley and Salvatore J. Petrilli, Jr.

A study and English translation of Servois' attempt to place calculus on a foundation of algebraic analysis without recourse to infinitesimals, continuing the work of Lagrange

### Reviews

The Music of Pythagoras, by Kitty Ferguson.  Reviewed by Gail Kaplan.

Subtitled How an Ancient Brotherhood Cracked the Code of the Universe and Lit the Path from Antiquity to Outer Space, this is a book of entertaining stories more so than scholarly research.

Routes of Learning: Highways, Pathways, and Byways in the History of Mathematics, by Ivor Grattan-Guinness.  Reviewed by Frank J. Swetz.

In this collection of essays on modern trends and issues in the history of mathematics, consideration of mathematics history in the classroom is often more theoretical than practical.

The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics, by Clifford A. Pickover.  Reviewed by Frank J. Swetz.

In this fascinating and accessible book, the author devotes one page of lively and informative text and one striking, full-page illustration to each milestone.

Agora, directed by Alejandro Amenábar, written by Amenábar and Mateo Gil.  Reviewed by Shirley Gray.

Our reviewer reports that the movie, Agora, about the mathematician Hypatia of Alexandria (c. 400 AD), is spectacular and intriguing but that Hypatia could and should have been portrayed as the heroine she truly was.

The Babylonian Theorem: The Mathematical Journey to Pythagoras and Euclid, by Peter S. Rudman.  Reviewed by Frank J. Swetz.

The author constructs a possible and plausible path from the mathematics of the ancient Babylonians of 2000-1600 BCE to that of Pythagoras, Euclid, and the ancient Greeks of 600-300 BCE.

De grands défis mathématiques d'Euclide àCondorcet, edited by Evelyne Barbin. Reviewed by Marc Moyon.

Nine examples of using mathematics history in the mathematics classroom -- for those who read French!

# What's in Convergence? - Contents of Volume 8 - 2011 (Loci - Volume 3)

Editors:  Janet Beery, Kathleen Clark

### Articles

SMURCHOM: Providing Opportunities for Undergraduate Research in the History of Mathematics, by Sloan Evans Despeaux

This article describes an MAA Regional Undergraduate Mathematics Conference (RUMC) featuring history of mathematics, and offers great ideas for getting students in your mathematics history course started on their research papers and projects.

Peano on Wronskians: A Translation, by Susannah M. Engdahl and Adam E. Parker

How a translation of Peano's counterexample to the 'theorem' that a zero Wronskian implies linear dependence can help your differential equations students

Teaching and Research with Original Sources from the Euler Archive, by Dominic Klyve, Lee Stemkoski, and Erik Tou

How faculty and students can use and contribute to the MAA Euler Archive!

HOM SIGMAA 2011 Student Paper Contest Winners, featuring essays by Paul Stahl, Sarah Costrell, and Rick Hill

Download the three winning essays to learn about Kepler's mathematical astronomy, the roots of modern algebra, and the Quadrivium of Isidore of Seville.

Extending al-Karaji’s Work on Sums of Odd Powers of Integers, by Hasan Unal and Hakan Kursat Oral

The authors share their discovery of an 1867 article in a Turkish scientific journal that extends al-Karaji's famous formula for the sum of the cubes to sums of higher odd powers.

Kepler: The Volume of a Wine Barrel, by Roberto Cardil

In his analysis of volumes of wine barrels, Kepler used ideas that would become important in differential and integral calculus.  This article provides you with visual imagery, much of it animated, to help share Kepler's ideas with your students.

When Nine Points Are Worth But Eight: Euler's Resolution of Cramer's Paradox, by Robert E. Bradley and Lee Stemkoski

How Euler resolved the paradox first noted by Maclaurin that nine points should determine a curve of order three, yet two such curves can intersect in nine points. Includes a translation of Euler's 'lost' letter to Cramer on this subject.

### Reviews

Eratosthenes' Geography, by Duane W. Roller.  Reviewed by Frederick Sakon.

Subtitled Fragments collected and translated, with commentary and additional material, this book provides the first English translation of Eratosthenes' Geography and includes "On the Measurement of the Earth."

The Chinese Roots of Linear Algebra, by Roger Hart. Reviewed by Frank Swetz.

An excellent, careful, and penetrating study of matrix methods for solving systems of linear equations in first century China.

The Genesis of Science: The Story of Greek Imagination, by Stephen Bertman.  Reviewed by Angelina Kuleshova.

Review of Bertram's text on the history of Greek science.

L’algèbre au temps de Babylone: Quand les mathématiques s’écrivaient sur de l’argile, by Jens Høyrup.  Reviewed by Marc Moyon.

Revised and expanded French translation of the author's original 1998 book for Danish high school teachers. The title in English is Algebra in the Time of Babylon: When Mathematics Were Written on Clay.

# What's in Convergence? - Contents of Volume 9 - 2012 (Loci - Volume 4)

Editors:  Janet Beery, Kathleen Clark

### Articles

Algebraic Formalism within the Works of Servois and Its Influence on the Development of Linear Operator Theory, by Anthony Del Latto and Salvatore Petrilli

This article describes how Servois’ failed attempt to construct a foundation for the calculus nevertheless may have helped shape modern mathematics.

Teaching the Fundamental Theorem of Calculus: A Historical Reflection, by Jorge López Fernández and Omar Hernández Rodríguez

The authors argue that the teaching of elementary integration should better reflect its historical development.

Georg Cantor at the Dawn of Point Set Topology, by Nicholas Scoville

How the history of analysis, and in particular that of Fourier series, can be used to motivate the study of point-set topology

When a Number System Loses Uniqueness: The Case of the Maya, by Amy Shell-Gellasch and Pedro J. Freitas

Considering non-unique representation of Maya calendar numbers may help your students understand their own number system better.

HOM SIGMAA 2012 Student Paper Contest Winners, featuring essays by Jesse Hamer and Kevin Wininger

Download the two winning essays to learn about the history of using indivisibles to find the area under an arch of the cycloid in the 17th century and of the Radon transform and its use in x-ray tomography in the 20th century.

Servois’ 1813 Perpetual Calendar, with an English Translation, by Salvatore J. Petrilli, Jr.

An image of an early 19th century perpetual calendar, together with a translation and explanation of its creator’s instructions for its use

Maya Cycles of Time, by Sandra Monteferrante

Maya calendars as they were developed over time and the Maya modified base 20 number system used in the calendars

“He Advanced Him 200 Lambs of Gold”: The Pamiers Manuscript, by Randy Schwartz

A discussion of the context and content of the 15th century Pamiers manuscript, with translations of its problems, including one for which negative solutions were acceptable

An Analysis of the First Proofs of the Heine-Borel Theorem, by Nicole Andre, Susannah Engdahl, and Adam Parker

A comparison of five circa-1900 proofs of the famous theorem with a view toward improving student understanding of compactness

Learning Geometry in Georgian England, by Benjamin Wardhaugh

A comparison of the geometry found in two 18th century copybooks written with two very different purposes

### Features

Who's That Mathematician? Images from the Paul R. Halmos Photograph Collection, by Janet Beery and Carol Mead

The well-known mathematician took most of these 343 photos of mathematicians from the 1950s through the 1980s. We welcome you to provide additional information about the photo subjects, including fond memories and interesting stories. This article was an expanding feature throughout 2012 (and through March of 2013), with new photos added every week throughout the year.

Mathematical Treasures, by Frank J. Swetz

Index to Mathematical Treasures Collection: Images of historical texts and objects from libraries, museums, and individuals around the world for use in your classroom!

### Reviews

In Pursuit of the Traveling Salesman, by William J. Cook. Reviewed by Christopher Thompson.

Author William Cook recounts the history of and computational progress on the traveling salesman problem, emphasizing connections within mathematics and with other disciplines.

The Man of Numbers: Fibonacci's Arithmetic Revolution, by Keith Devlin. Reviewed by Frank J. Swetz.

Author Keith Devlin brings to life the impact of the Pisan merchant and his Arabic numbers on medieval Europe.

Mathematics Emerging: A Sourcebook 1540–1900, by Jacqueline Stedall. Reviewed by Frank J. Swetz.

Our reviewer praises the selection of excerpts, the use of facsimiles rather than transcriptions, and the commentary and English translation in this collection.

The Lost Millennium: History's Timeline under Siege, by Florin Diacu. Reviewed by Branden Anglin.

This book suggests that the accepted historical chronology is fundamentally flawed.

A Remarkable Collection of Babylonian Mathematical Texts, by Jöran Friberg. Reviewed by Frank J. Swetz.

Our reviewer finds this collection of translations of Babylonian mathematical texts to be both "remarkable" and accessible.

# What's in Convergence? - Contents of Volume 10 - 2013 (Loci - Volume 4)

Editors:  Janet Beery, Kathleen Clark

### Articles

Who's That Mathematician? Images from the Paul R. Halmos Photograph Collection, by Janet Beery and Carol Mead

The well-known mathematician took most of these 343 photos of mathematicians from the 1950s through the 1980s. We welcome you to provide additional information about the photo subjects, including fond memories and interesting stories. This article was an expanding feature throughout 2012 and through March of 2013, with new photos added every week.

Maya Geometry in the Classroom, by John Diamantopoulos and Cynthia Woodburn

Classic Maya people probably used knotted ropes to form desired geometric shapes in art and architecture: here's how!

To what extent did forces outside of mathematics influence such curricular changes as increased emphasis on applications and modeling, discrete mathematics, and calculus reform?

Robert Murphy: Mathematician and Physicist, by Anthony J. Del Latto and Salvatore J. Petrilli, Jr.

The authors show that Murphy (1806-1843) displayed “true genius” in a very short life and they provide a transcription of Murphy’s first published work in 1824.

Solving the Cubic with Cardano, by William B. Branson

The author shows how, in order to solve the cubic, Cardano relied on both classical Greek geometric and abbaco traditions. He illustrates Cardano's geometric thinking with modern manipulatives.

Primary Historical Sources in the Classroom: Discrete Mathematics and Computer Science, by Janet Barnett, Guram Bezhanishvili, Hing Leung, Jerry Lodder, David Pengelley, Inna Pivkina, Desh Ranjan, and Maria Zack

Sixteen projects designed to help students learn important concepts from discrete math, combinatorics, linear algebra, and computer science by studying original sources

1.  Deduction through the Ages: A History of Truth, by Jerry Lodder

• Project in which discrete mathematics students learn about logic, truth tables, and implication by consulting original sources from ancient to modern times

2.  Sums of Numerical Powers in Discrete Mathematics: Archimedes Sums Squares in the Sand, by David Pengelley

• Project in which discrete math or calculus students learn from Archimedes’ writings how he computed the sum of the squares

3.  Euclid's Algorithm for the Greatest Common Divisor, by Jerry Lodder, David Pengelley, and Desh Ranjan

• Project in which discrete math, computer science, or number theory students learn the Euclidean Algorithm from Euclid’s writings

4.  An Introduction to Symbolic Logic, by Guram Bezhanishvili and Wesley Fussner

• Project in which discrete mathematics students learn the basics of symbolic logic by studying excerpts from Russell’s and Whitehead’s Principia Mathematica

5.  An Introduction to Elementary Set Theory, by Guram Bezhanishvili and Eachan Landreth

• Project in which discrete mathematics students learn the basics of set theory by reading Dedekind’s and Cantor’s original papers on the subject

6.  Computing the Determinant Through the Looking Glass, by Maria Zack

• Project in which linear algebra students learn an easy way to compute determinants from a paper by mathematician Charles Dodgson (whose pen name was Lewis Carroll)
• Project in which discrete mathematics students are introduced to set operations, Venn diagrams, and Boolean algebra by the masters
• Project in which discrete math or abstract algebra students develop the 'algebra of logic' along with E. V. Huntington, who built on the work of Boole

9.  Applications of Boolean Algebra: Claude Shannon and Circuit Design, by Janet Barnett

• Project in which discrete mathematics students apply Boolean algebra to circuit design by studying Claude Shannon’s pioneering paper

10. Figurate Numbers and Sums of Numerical Powers: Fermat, Pascal, Bernoulli, by David Pengelley

• Project in which students in upper-level discrete math or combinatorics courses learn connections between sums of powers and binomial coefficients by “reading the masters”

11. Gabriel Lamé's Counting of Triangulations, by Jerry Lodder

• Project in which upper-level discrete mathematics or combinatorics students count triangulations with Lamé and meet the Catalan numbers along the way

12.  Networks and Spanning Trees, by Jerry Lodder

• Project in which students in an upper-level discrete math or combinatorics course are introduced to labeled graphs and minimal spanning trees by Cayley, Prüfer, and Boruvka

13.  Striving for Efficiency in Algorithms: Sorting, by Inna Pivkina

• Project in which computer science students study quicksort, insertion sort, efficiency of algorithms, and stack data structure via its history

14.  Discovery of Huffman Codes, by Inna Pivkina

• Project in which computer science students learn the fundamentals of information theory by reading the papers of Fano and Huffman

15.  Program Correctness, by Hing Leung

• Project in which computer science students learn the fundamentals of partial correctness proof from Robert W Floyd’s original paper on the subject

16.  Regular Languages and Finite Automata, by Hing Leung

• Project in which computer science students discover the connection between these two topics by studying the original paper of S. C. Kleene

Mathematical Treasures, by Frank J. Swetz

Index to Mathematical Treasures Collection: Images of historical texts and objects from libraries, museums, and individuals around the world for use in your classroom!

Mathematical Treasures added during 2012 and 2013:

### Review

Review of Mathematical Expeditions: Exploring Word Problems across the Ages, by Frank J. Swetz. Reviewed by Kathleen M. Clark.

A collection of problems that should be of interest and use to teachers at all levels

# What's in Convergence? - Contents of Volume 11 - 2014

Editor:  Janet Beery

Associate Editors: Amy Ackerberg-Hastings, Janet Heine Barnett, Kathleen Clark, Lawrence D'Antonio, Douglas Ensley, Victor Katz, Daniel Otero, Randy Schwartz, Lee Stemkoski, Frank Swetz

Founding Editors: Victor Katz, Frank Swetz

### Articles

An Investigation of Subtraction Algorithms from the 18th and 19th Centuries, by Nicole M. Wessman-Enzinger

This survey of four subtraction algorithms used in North America includes as sources both handwritten "cyphering books" and printed arithmetic texts.

Connecting Greek Ladders and Continued Fractions, by Kurt Herzinger and Robert Wisner

An exploration of two historical techniques for estimating irrational numbers and a link between them

Did Euler Know Quadratic Reciprocity?: New Insights from a Forgotten Work, by Paul Bialek and Dominic W. Klyve

The authors use their newly translated paper of Leonhard Euler to answer their title question.

Cubes, Conic Sections, and Crockett Johnson, by Stephanie Cawthorne and Judy Green

Author and illustrator Johnson, author of Harold and the Purple Crayon, posed a question about Euclid, cubes, and conic sections, and painted an answer!

When Nine Points Are Worth But Eight: Euler's Resolution of Cramer's Paradox, by Robert Bradley and Lee Stemkoski

Interactive graphics illustrate the seeming paradox that 9 points should determine a curve of order 3, yet two curves of order 3 can intersect in up to 9 distinct points.

Read and listen to this famous 1930 address, with its dramatic conclusion: "Wir müssen wissen; wir werden wissen." ("We must know; we will know.")

Read winning student papers on the statistics of Florence Nightingale and on Legendre's attempts to prove Euclid's Fifth Postulate.

Celebrating a Mathematical Miracle: Logarithms Turn 400, by Glen Van Brummelen

Why John Napier's invention of logarithms in 1614 was hailed as a miracle by astronomers and mathematicians

Wibold's Ludus Regularis, a 10th Century Board Game, by Richard Pulskamp and Daniel Otero

Players competed for virtues in this dice game for clerics.

How to Improve a Math History Assignment, by Christopher Goff

Moving college students' original source mathematics history projects beyond "reporting" to "engagement"

Historical Activities for the Calculus Classroom, by Gabriela R. Sanchis

History and mathematics of curve sketching, tangent lines, and optimization, explored using interactive applets

A Pair of Articles on the Parallelogram Theorem of Pierre Varignon, by Peter N. Oliver

Mathematical life of Varignon, plus ideas for classroom activities and extensions of his famous theorem

Unreasonable Effectiveness of Knot Theory, by Mario Livio

Knot theory has become surprisingly useful in explaining string theory.

Proofs Without Words and Beyond, by Tim Doyle, Lauren Kutler, Robin Miller, and Albert Schueller

History and philosophy of visual proofs, together with dynamic, interactive "proofs without words 2.0"

Van Schooten's Ruler Constructions, by C. Edward Sandifer

Translation of and commentary on Frans van Schooten's work on constructions using only a straightedge -- and a postulate that allows the copying of one line segment onto another.

Led Astray by a Right Triangle: Misconception, Epiphany, and Redemption, by Frank J. Swetz

A well-known historian initially erred in his study of ancient Chinese mathematics.

Euclid21: Euclid's Elements for the 21st Century, by Eugene Boman, Alexandra Milbrand, Tyler Brown, Siddharth Dahiya, Joseph Roberge, and Mary Boman

A dynamic, interactive Euclid's Elements organized as a directed graph via its logical structure

Online Museum Collections in the Mathematics Classroom, by Amy Ackerberg-Hastings and Amy Shell-Gellasch

The Smithsonian Institution's National Museum of American History website features dozens of object groups, collections of digitized object images and detailed catalog records, related to mathematics. View sample objects and read suggestions for using these resources in your teaching.

Mathematical Treasures from the National Museum of American History, Smithsonian Institution, by Amy Ackerberg-Hastings, Judy Green, Peggy Kidwell, and Amy Shell-Gellasch

Mathematical Treasures, by Frank J. Swetz

Index to Mathematical Treasures Collection: Images of historical texts and objects from libraries, museums, and individuals around the world for use in your classroom!

### Review

Review of Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World, by Amir Alexander. Reviewed by Frank J. Swetz.

How 17th century European proponents of indivisibles and infinitesimals clashed with Thomas Hobbes, Christopher Clavius, and the Catholic Church

# What's in Convergence? - Contents of Volume 12 - 2015

Editor:  Janet Beery

Associate Editors: Amy Ackerberg-Hastings, Janet Heine Barnett, Kathleen Clark, Lawrence D'Antonio, Douglas Ensley, Victor Katz, Daniel Otero, Gabriela Sanchis, Randy Schwartz, Lee Stemkoski, Gary Stoudt

Founding Editors: Victor Katz, Frank Swetz

### Articles

Alan Turing in America, by David E. Zitarelli

Alan Turing visited the United States during 1936–38 and 1942–43.  Two of Turing's greatest accomplishments, in logic and computer design, were influenced by the first of these two visits.

Students in the author's Ancient Mathematical Astronomy course build armillary spheres, astrolabes, quadrants, sextants, and sundials.

Engage your students by using images, especially those of historical objects, manuscripts, and texts, in teaching mathematics. A Spanish translation was prepared by Ximena Catepillán in 2022.

Download the two winning papers from the 12th annual competition, a biography of Bernard Bolzano and a philosophical consideration of mistakes in mathematics.

Can You Really Derive Conic Formulae from a Cone?, by Gary S. Stoudt

Attempts to double the cube led ancient Greek mathematicians to discover and develop the conic sections.

Jan Hudde’s Second Letter: On Maxima and Minima. Translated into English, with a Brief Introduction, by Daniel J. Curtin

Optimization via algebra and arithmetic progressions with an early appearance of the Quotient Rule

Problems for Journey Through Genius: The Great Theorems of Mathematics, by William Dunham

The author shares 162 problems to help you turn his popular book into a textbook.

Euler and the Bernoullis: Learning by Teaching, by Paul Bedard

The author reflects on lessons he has learned about mathematics teaching and learning from these great mathematicians.

A GeoGebra Rendition of One of Omar Khayyam's Solutions for a Cubic Equation, by Deborah Kent and Milan Sherman

How the 11th century Persian mathematician, philosopher, and poet geometrically determined a positive real solution to a cubic equation

Oliver Byrne: The Matisse of Mathematics, by Susan M. Hawes and Sid Kolpas

The most complete biography of Byrne to date, along with tips for teaching with his famous Euclid in Colours

The 'Problem of Points' and Perseverance, by Keith Devlin

How Pascal's and Fermat's unfinished game can help teach today's students both probability and persistence

Geometric Algebra in the Classroom, by Patricia R. Allaire and Robert E. Bradley

Geometric approaches to the quadratic equation from 1700 BCE to the present

D'Alembert, Lagrange, and Reduction of Order, by Sarah Cummings and Adam Parker

Two historical approaches, one familiar and one unfamiliar, to enrich your ODE classroom

Ancient Indian Rope Geometry in the Classroom, by Cynthia J. Huffman and Scott V. Thuong

Activities, applets, and information to help students explore the geometry of altar construction in ancient India

The Cambodian Zero, by Frank J. Swetz with photos by Amir Aczel

Popular mathematics writer Amir Aczel (1950-2015) tracked down the rumored Khmer zero carved into a stone stele in 683 CE.

Some Original Sources for Modern Tales of Thales, by Michael Molinsky

Earliest known sources for stories about Thales and applets illustrating methods attributed to him

Pythagorean Cuts, by Martin Bonsangue and Harris Shultz

Euclid's proof of the Pythagorean Theorem can be adapted to shapes other than squares.

Geometrical Representation of Arithmetic Series, by Gautami Bhowmik

Hints of geometry in medieval Sanskrit arithmetic texts developed for your classroom

Edmund Halley, 1740, by Andrew Wynn Owen

An historical poem by a prize-winning Oxford poet in the form of an autobiographical reflection by Edmund Halley

Mathematical Treasures, by Frank J. Swetz

Index to Mathematical Treasures Collection: Images of historical texts and objects from libraries, museums, and individuals around the world for use in your classroom!

### Review

Review of Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers, by Joseph Mazur. Reviewed by Frank J. Swetz.

The reviewer finds the book to be "lively" and "interesting," but wishes the author would "begin at the beginning."

# What's in Convergence? - Contents of Volume 13 - 2016

Editor:  Janet Beery

Associate Editors: Amy Ackerberg-Hastings, Janet Heine Barnett, Kathleen Clark, Lawrence D'Antonio, Douglas Ensley, Victor Katz, Daniel Otero, Gabriela Sanchis, Randy Schwartz, Lee Stemkoski, Gary Stoudt

Founding Editors: Victor Katz, Frank Swetz

### Articles

A GeoGebra Rendition of One of Omar Khayyam's Solutions for a Cubic Equation, by Deborah Kent and Milan Sherman

How the 11th century Persian mathematician, philosopher, and poet geometrically determined a positive real solution to a cubic equation

Edmund Halley, 1740, by Andrew Wynn Owen

An historical poem by a prize-winning Oxford poet in the form of an autobiographical reflection by Edmund Halley

Garfield's 1876 proof, plus a memorial visiting card featuring a photograph of Garfield (1831-1881)

Descartes' Method for Constructing Roots of Polynomials with 'Simple' Curves, by Gary Rubinstein

Descartes' methods from his 1637 'Geometry' explicated and illustrated using interactive applets

Letter and Visiting Card of Augustus De Morgan, by Sid Kolpas

Visiting card with photograph (circa 1866), brief biography, student sketch (1865), An Essay on Probabilities (1838), and letter to Indologist H. H. Wilson (1843)

When Was Pierre de Fermat Born?, by Friedrich Katscher

An argument that Pierre de Fermat was born in 1607 rather than in 1601

The Duplicators: Eutocius's Collection of Cube Duplications, by Colin B. P. McKinney

Solutions by ancient Greek mathematicians of the classical duplicating the cube problem – with extra tools allowed! – featuring translations from the Greek and interactive applets

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, with classroom-ready examples, exercises, and interactive applets

Download the two winning papers from the 13th annual competition, "A Latent Element of Alice's Agency in Wonderland: Conservative Victorian Mathematics" and "The Evolution of the Circle Method in Additive Prime Number Theory."

Al-Maghribî’s Mecca Problem Meets Sudoku, by Ilhan M. Izmirli

Solutions to an early 17th century puzzle from Istanbul can be generated from solutions to modern day Sudoku puzzles.

Johannes Scheubel's 1551 Algebrae Compendiosa, by Sid Kolpas

Selected examples for classroom use feature early algebraic notation and methods.

Misseri-Calendar: A Calendar Embedded in Icelandic Nature, Society, and Culture, by Kristín Bjarnadóttir

History of this two-season calendar from Viking times to today, with animations and ideas for your classroom.

Mathematical Treasures, by Frank J. Swetz

Index to Mathematical Treasures Collection: Images of historical texts and objects from libraries, museums, and individuals around the world for use in your classroom!

### Reviews

Review of Mathematics in Ancient Egypt: A Contextual History, by Annette Imhausen. Reviewed by Frank J. Swetz.

Our reviewer finds the book to be "well written" and "well researched," and is grateful to the author for summarizing scholarship to date.

Review of Elements of Mathematics: From Euclid to Gödel, by John Stillwell. Reviewed by Frank J. Swetz.

Our reviewer concludes: "If you want to teach mathematics with its history, this is a way to do it!"

# What's in Convergence? - Contents of Volume 14 - 2017

Editor:  Janet Beery

Associate Editors: Amy Ackerberg-Hastings, Janet Heine Barnett, Maureen Carroll, Lawrence D'Antonio, Victor Katz, Michael Molinsky, Elyn Rykken, Gabriela Sanchis, Randy Schwartz, Amy Shell-Gellasch, Jody Sorensen, Gary Stoudt, Erik R. Tou

Founding Editors: Victor Katz, Frank Swetz

### Articles

Crossword Puzzle: Mathematicians from A to Z, by Sid Kolpas and Stu Ockman

Or, what happened when a mathematics professor and a professional crossword puzzle constructor became neighbors!

A Translation of Evangelista Torricelli's Quadratura Parabolae per novam indivisibilium Geometriam pluribus modis absoluta, by Andrew Leahy and Kasandara Sullivan

An English translation and study of a work representative of the seventeenth century infinitesimal methods introduced by Cavalieri.

How your students can explore an ancient cube dissection using paper models, computer animations, and/or 3D printing!

The author's experiences on an MAA Mathematical Study Tour to China can enhance your classroom, too.

The Method of the Scales in ibn al-Hāʾim's Book of Delights, by Randy K. Schwartz and Frank J. Swetz

See the “method of the scales” (double false position) in use in Kitāb al-nuzah (Book of Delights), by ibn al-Hāʾim.

Moses ibn Tibbon’s Hebrew Translation of al-Hassar's Kitāb al Bayān, by Jeremy I. Pfeffer

An exploration of Abu Bakr al-Hassar's influential work about arithmetic of fractions, Kitāb al Bayān wa-l-tadhkār (Book of Proof and Recall)

How this course has affected its instructor and students, and how you, too, can teach such a course!

Analysis and Translation of Raffaele Rubini's 1857 'Application of the Theory of Determinants: Note', by Salvatore J. Petrilli, Jr., and Nicole Smolenski

A compendium of early determinant theory offered in defense of "analytic" mathematics

The Mathematics of Levi ben Gershon in the Classroom, by Shai Simonson

Translation of "word problems" and estimates of square roots and of pi for use with your students

Trisecting an Angle Using Mechanical Means, by Keith Dreiling

Four methods in all by Hippias, Archimedes, and Nicomedes illustrated with interactive applets!

The Mathematical Cultures of Medieval Europe, by Victor J. Katz

Cultural influences on the mathematics of Islamic, Jewish, and Catholic scholars

Recreational Problems in Medieval Mathematics, by Victor J. Katz

Two problems that endured across time, space, and culture

A Series of Mini-projects from TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources, by Janet Barnett, Kathy Clark, Dominic Klyve, Jerry Lodder, Danny Otero, Nick Scoville, and Diana White

Math Origins, by Erik R. Tou

How were concepts, definitions, and theorems familiar to today's students of mathematics developed over time?

Mathematical Treasures at the Linda Hall Library, by Cynthia J. Huffman

Description of the physical and digital history of science collections of the Linda Hall Library in Kansas City. At least 75 of LHL's digitized rare books relate to the history of mathematics and can be used in classrooms.

Mathematical Treasures from the Linda Hall Library, by Cynthia J. Huffman

Mathematical Treasures, by Frank J. Swetz

Index to Mathematical Treasures Collection: Images of historical texts and objects from libraries, museums, and individuals around the world for use in your classroom!

# What's in Convergence? - Contents of Volume 15 - 2018

Editor:  Janet Beery

Associate Editors: Amy Ackerberg-Hastings, Janet Heine Barnett, Maureen Carroll, Lawrence D'Antonio, Victor Katz, Michael Molinsky, Elyn Rykken, Randy Schwartz, Amy Shell-Gellasch, Jody Sorensen, Gary Stoudt, Erik R. Tou

Founding Editors: Victor Katz, Frank Swetz

### Articles

Historical Reflections on Teaching Trigonometry, by David M. Bressoud The functional approach of circle trigonometry is the historical approach!

Crossword Puzzle: Mathematical Potpourri, by Sid Kolpas and Stu Ockman Our second puzzle from a mathematics professor and a NY Times crossword puzzle constructor!

Descriptions of the Integer Number Line in United States School Mathematics in the 19th Century, by Nicole M. Wessman-Enzinger Gradual development of the now ubiquitous number line traced through textbooks of the time.

Russian Multiplication, Microprocessors, and Leibniz, by Sid Kolpas A traditional method of multiplication via binary arithmetic finds a modern use.

A Writing Intensive General Education History of Mathematics Course, by Amy Shell-Gellasch ... for students who think they aren't good at or don't like mathematics!

More Classroom Activities Based on Ancient Indian Rope Geometry, by Cynthia J. Huffman and Scott V. Thuong Activities, applets, and information to help students explore the geometry of altar construction in ancient India.

HOM SIGMAA 2018 Student Paper Contest Winners Read the winning entry, "Race to Refraction: The Repeated Discovery of Snell's Law," along with the two runners-up.

Divisibility Tests: A History and User's Guide, by Eric L. McDowell Discoveries, rediscoveries, and generalizations of these tests to pique students' interest.

Elementary Soroban Arithmetic Techniques in Edo Period Japan, by Rosalie Joan Hosking, Tsukane Ogawa, and Mitsuo Morimoto Learn to solve problems from the Taisei Sankei (c. 1700) on the Japanese abacus.

The Root of the Matter: Approximating Roots with the Greeks, by Matthew Haines and Jody Sorensen The ancient method of Theon's Ladder has both geometric and matrix interpretations.

Cuisenaire Art: Modeling Figurate Number Sequences and Gnomonic Structures in a History of Mathematics Classroom, by Günhan Caglayan Students construct Cuisenaire rod models per instructions from Theon and Nicomachus.

Mathematical Treasures of Japan in the Edo Period, by Frank J. Swetz Twelve distinct works illustrate the range of mathematics produced in Japan from 1603 to 1867.

Billingsley's Sources for the First English Euclid's Elements: Two Annotated Mathematical Treasures, by Frank J. Swetz A Greek source and a Latin source annotated by the translator in the mid-16th century now reside in Princeton, New Jersey.

The Ladies' Diary: A True Mathematical Treasure, by Frank J. Swetz An 18th century almanac for "ladies" became a source for mathematical problems and solutions.

On Squares, Rectangles, and Square Roots, by María Burgos and Pablo Beltrán-Pellicer Sixth-graders extract square roots using manipulatives and a method from ancient China.

An Arabic Finger-reckoning Rule Appropriated for Proofs in Algebra, by Jeffrey A. Oaks In a 1301 work, Ibn al-Bannāʾ based his proofs on a common mental multiplication technique.

John Napier's Binary Chessboard Calculator, by Sidney J. Kolpas and Erwin Tomash Napier's lesser known invention: a 5-function calculator via binary arithmetic on a chessboard.

A Classic from China: The Nine Chapters, by Randy K. Schwartz History of and problems for students from this early and influential Chinese work.

### Ongoing Series

A Series of Mini-projects from TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources, by Janet Barnett, Kathy Clark, Dominic Klyve, Jerry Lodder, Danny Otero, Nick Scoville, and Diana White

Math Origins, by Erik R. Tou How were concepts, definitions, and theorems familiar to today's students of mathematics developed over time?

### Mathematical Treasures

Mathematical Treasures at the Linda Hall Library, by Cynthia J. Huffman Description of the physical and digital history of science collections of the Linda Hall Library in Kansas City. At least 75 of LHL's digitized rare books relate to the history of mathematics and can be used in classrooms.

Mathematical Treasures from the Linda Hall Library added during 2018:

Mathematical Treasures, by Frank J. Swetz

Index to Mathematical Treasures Collection: Images of historical texts and objects from libraries, museums, and individuals around the world for use in your classroom!

# What's in Convergence? - Contents of Volume 16 - 2019

Editors:  Amy Ackerberg-Hastings, Janet Heine Barnett, Janet Beery (through 1/31/19)

Associate Editors:  Paul Bialek, Eugene Boman, Maureen Carroll, Lawrence D'Antonio, Sloan Despeaux, Victor Katz (through 1/31/19), Michael Molinsky, Elyn Rykken, Randy Schwartz, Amy Shell-Gellasch, Jody Sorensen, Gary Stoudt, Erik R. Tou, Laura Turner

Founding Editors: Victor Katz, Frank Swetz

### Articles

Correspondence from Mathematicians by Jennifer Horn, Amy Zamierowski and Rita Barger (posted 12/30/2019)
A project designed by the co-authors to provide their students with a research experience that helped them discover the origins of familiar mathematical concepts.

An Explication of the Antilogism in Christine Ladd-Franklin's "Algebra of Logic" by Julia M. Parker (posted 12/12/2019)
An overview of Ladd-Franklin's contributions to symbolic logic, based on an explication of an excerpt from her doctoral dissertation.

Bringing Historical Methods for Astronomical Measurements into the Classroom by Seán P. Madden, Jocelyne M. Comstock, and James P. Downing (posted 10/14/2019)
Student activities that combine data collection with astronomical measurement methods attributed to Eratosthenes, Ptolemy, and Galileo.

Here's Looking at Euclid by Sid Kolpas and Stu Ockman (posted 09/16/2019)
A mathematical crossword puzzle with historical overtones.

To Simplify, or Not To Simplify? A Lesson from Medieval Iraq by Valerio De Angelis and Jeffrey A. Oaks (posted 09/02/2019)
A case where not simplifying fractions explains a curious rule for computing cube roots from medieval Arabic mathematics, with student exercises.

MAA Convergence is Sweet Sixteen!
As Convergence marks its 16th volume, we recognize its long-term former editors by compiling their contributions to the journal’s content and by presenting a brief history of the journal.

Servois' 1817 "Memoir on Quadratures" translated by Robert E. Bradley and Salvatore J. Petrilli, Jr. (posted 05/20/2019)A readers' guide and complete English translation of Servois' 1817 contribution to a debate on numerical integration.

HOM SIGMAA 2019 Student Paper Contest Winner (posted 04/23/2019)Read the winning entry, "Omar Khayyam's Successful Replacement of Euclid's Parallel Postulate" by Amanda Nethington, from the 16th annual edition of this contest.

Teaching Mathematics with Ephemera: John Playfair's Course Outline for Practical Mathematics by Amy Ackerberg-Hastings (posted 04/22/2019)
Ephemera are a category of primary source that may prove especially engaging for students. The article provides examples of ephemera, a sample analysis of one piece of ephemera, and suggestions for incorporating this form of primary source into mathematics classrooms.

Using the Publimath Database to Bring History into our Teaching by Hombeline Languereau and Anne Michel-Pajus (posted 04/08/2019)
Description, with user instructions, of a French online resource cataloging research articles and projects for using history to teach mathematics.

More Than Just a Grade: The HOM SIGMAA Student Contest Fosters Writing Excellence at UMKC by Richard Delaware (posted 02/10/2019)
Advice on promoting excellence in student research and writing in the history of mathematics.

### Ongoing Series

A Series of Mini-projects from TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources, by Janet Barnett, Kathy Clark, Dominic Klyve, Jerry Lodder, Danny Otero, Nick Scoville, and Diana White

Math Origins, by Erik R. Tou
How were concepts, definitions, and theorems familiar to today's students of mathematics developed over time?

### Mathematical Treasures

Mathematical Treasures at the Linda Hall Library, by Cynthia J. Huffman Description of the physical and digital history of science collections of the Linda Hall Library in Kansas City. At least 75 of LHL's digitized rare books relate to the history of mathematics and can be used in classrooms.

Mathematical Treasures from the Linda Hall Library added during 2019:

Mathematical Treasures, by Frank J. Swetz

Index to Mathematical Treasures Collection: Images of historical texts and objects from libraries, museums, and individuals around the world for use in your classroom!

# What's in Convergence? - Contents of Volume 17 - 2020

Editors:  Amy Ackerberg-Hastings, Janet Heine Barnett

Associate Editors:  Paul Bialek, Eugene Boman, Maureen Carroll, Ximena Catepillan, Lawrence D'Antonio (through 1/31/20), Sloan Despeaux, Toke Knudsen, Michael Molinsky, Adrian Rice, Elyn Rykken, Randy Schwartz (through 1/31/20), Amy Shell-Gellasch, Jody Sorensen (through 1/31/20), Gary Stoudt, Erik R. Tou, Laura Turner

Founding Editors: Victor Katz, Frank Swetz

### Articles

Euler’s Letters to a German Princess: Translation and Betrayal, by Dominic Klyve
An exploration of how the translations of Euler’s Letters to a German Princess came to differ from the original text. (posted 12/07/2020)

The Four Curves of Alexis Clairaut, by Taner Kiral, Jonathan Murdock, and Colin B. P. McKinney
Translation of a paper on families of algebraic curves (along with a transcription of the French original) written when Clairaut was only twelve years old. (posted 11/22/2020)

The ‘Piling Up of Squares’ in Ancient China, by Frank Swetz
Description of manipulative activities that were used in ancient China and could be used in current classrooms to geometrically solve algebraic problems. Includes commentary and a brief bibliography covering 40 years of the history of Chinese mathematics (and its use in teaching), provided by Joel Haack. (posted 11/09/2020)

Converting the Old Babylonian Tablet ‘Plimpton 322’ into the Decimal System as a Classroom Exercise, by Antonella Perucca and Deborah Stranen
A student-ready activity, ideal for pre-service elementary mathematics teachers. (posted 10/26/2020)

The French Connection: Borda, Condorcet and the Mathematics of Voting Theory, by Janet Heine Barnett
An overview of two eighteenth-century texts on voting theory with biographical and historical notes about their authors, Jean-Charles de Borda and Nicolas Condorcet, accompanied by a classroom-ready project based on their original writings suitable for use with Liberal Arts and high school students. (posted 09/22/2020)

Apportionment: What's Your Fair Share – An Activity for Liberal Arts and High School Students, by Jeff Suzuki
A self-contained project suitable for individual or group work, inside or outside the classroom, that uses US Census data from 1790 to guide students through an exploration of what it means for each state to get its fair share of congresspersons, and of how different methods of apportionment might have altered the course of American history. (posted 09/08/2020)

Pathways from the Past: Classroom-Ready Materials for Using History to Teach Mathematics, by Bill Berlinghoff and Fernando Gouvêa
Reproducible student activity sheets developed by the authors of the well-regarded textbook, Math through the Ages, and especially suitable for practicing and pre-service teachers of secondary mathematics and those involved in teacher training. (posted 06/07/2020)

HOM SIGMAA 2020 Student Paper Contest Winner
Read the winning entry, “Did Archimedes Do Calculus?” by Jeffrey Powers, from the 17th annual edition of this contest. (posted 05/11/2020)

Word Histories: Melding Mathematics and Meanings, by Rheta N. Rubenstein and Randy K. Schwartz
Etymologies for common mathematical terms—from subjects such as algebra, geometry, functions and discrete mathematics—can be used by instructors to enrich student learning. (posted 04/20/2020)

Mabel Sykes: A Life Untold and an Architectural Geometry Book Rediscovered, by Maureen T. Carroll and Elyn Rykken
Biography of a little-known high-school mathematics teacher and discussion of her publications, particularly the lavishly-illustrated 1912 A Source Book of Problems for Geometry Based upon Industrial Design and Architectural Ornament. The description of Source Book includes diagrams and animations. (posted 2/24/2020)

Why History of Mathematics? by Glen Van Brummelen
Justifications for using history to teach mathematics that were prepared to help secondary teachers in British Columbia understand how to approach a new 11th-grade course but which are widely applicable. (posted 1/27/2020)

A Mathematical History Tour: Reflections on a Study Abroad Program, by R. Abraham Edwards and Marie Savoie
A unique study-abroad course combining the history of mathematics and travel. (posted 1/13/2020)

### Ongoing Series

Teaching and Learning the Trigonometric Functions through Their Origins, by Daniel E. Otero
A series of curricular units based on primary source texts for use in teaching and learning trigonometry.

A Series of Mini-projects from TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources
A collection of student-ready projects for use in teaching standard topics from across the undergraduate curriculum.

Math Origins, by Erik R. Tou
How were concepts, definitions, and theorems familiar to today's students of mathematics developed over time?

### Mathematical Treasures

Mathematical Treasures at the Linda Hall Library, by Cynthia J. Huffman

Mathematical Treasures from the Linda Hall Library added during 2020:

Mathematical Treasures, by Frank J. Swetz

Index to Mathematical Treasures Collection: Images of historical texts and objects from libraries, museums, and individuals around the world for use in your classroom!

# What's in Convergence? - Contents of Volume 18 - 2021

Editors:  Amy Ackerberg-Hastings, Janet Heine Barnett

Associate Editors:  Paul Bialek, Eugene Boman, Maureen Carroll (through 1/31/21), Ximena Catepillan, Sloan Despeaux, Joel Haack, Toke Knudsen, Stacy Langton, Betty Mayfield, Michael Molinsky, Andrew Perry, Adrian Rice, Elyn Rykken (through 1/31/21), Amy Shell-Gellasch, Gary Stoudt (through 1/31/21), Erik R. Tou, Laura Turner

Founding Editors: Victor Katz, Frank Swetz

### Articles

E. G. Ziegenbalg’s Danish Translation of Euclid’s Elements, by Toke Knudsen
Introduces the first (1744) translation into Danish of the classic geometry text, influences on its content that led to a distinctive pedagogical approach, and the notable family who owned the author’s copy. (posted 10/18/2021)

Algebra Tiles Explorations of al-Khwārizmī’s Equation Types, by Günhan Caglayan
Activities for visualizing al-Khwārizmī's algebraic solution methods using algebra tile manipulatives. (posted 10/04/2021)

Helping Ada Lovelace with her Homework: Classroom Exercises from a Victorian Calculus Course, by Adrian Rice
Highlights from Ada Lovelace's correspondence course on calculus with Augustus De Morgan that shed light on common confusions that still arise today. (posted 09/03/2021)

The Life of Sir Charles Scarburgh, by Michael Molinsky
Biography of Sir Charles Scarburgh (ca 1615–1694) and discussion of his impressive mathematical library and potential role in the production of a rare mathematical treasure: The English Euclide (1705). (posted 06/06/2021)

Mark Kac’s First Publication: A Translation of "O nowym sposobie rozwiązywania równań stopnia trzeciego," by David Derbes
English translation of Mark Kac's first publication on a new derivation of Cardano’s formula, written while he was still in high school, with a typescript of the original Polish article, a biographical synopsis of Kac, the tale of the rediscoveries of the paper, and suggestions for classroom discussions of the cubic. (posted 04/18/2021)

HOM SIGMAA 2021 Student Paper Contest Winner
Read the winning entry, “The Suan shu shu and the Nine Chapters on the Mathematical Art: A Comparison” by Megan Ferguson, from the 18th annual edition of this contest. (posted 04/17/2021)

Mathematical Mysteries of Rapa Nui with Classroom Activities, by Ximena Catepillán, Cynthia Huffman, and Scott Thuong
A trip to Rapa Nui, also known as Easter Island, provided opportunities to explore the elliptical shape of the foundations of dwellings known as hare paenga and to learn about calendrical glyphs in Rapanui writing. Four activities involving ellipses help instructors share this example of ethnomathematics with their students. (posted 04/05/2021)

Misterios Matemáticos de Rapa Nui con Actividades para el Aula de Clases, por Ximena Catepillán, Cynthia Huffman, y Scott Thuong; traducido por Ximena Catepillán con la ayuda de Samuel Navarro
Un viaje a Rapa Nui, también conocida como Isla de Pascua, brindó oportunidades para explorar la forma elíptica de los cimientos de las viviendas conocidas como hare paenga y para aprender sobre glifos calendáricos en la escritura rapanui. Cuatro actividades que involucran elipses ayudarán a los profesores a compartir este ejemplo de etnomatemática con sus estudiantes. (publicado 09/03/2021)

The Educational Times Database: Building an Online Database of Mathematics Questions and Solutions Published in a 19th-Century Journal, by Robert M. Manzo
An introduction to a new tool and its potential uses for researchers and educators, with an overview of the significance of the ET and its contributors in the history of mathematics, as well as the history of efforts to index the run of mathematical problems and solutions in the Educational Times and its sister publication Mathematical Questions. (posted 03/22/2021)

The Evolutionary Character of Mathematics, by Richard M. Davitt and Judy Grabiner
Richard Davitt’s classroom application of Judy Grabiner’s “use-discover-explore/develop-define” model for historical change in mathematics, along with commentary by Grabiner. (posted 02/20/2021)

### Ongoing Series

Keys to Mathematical Treasure Chests
A series that offers examples of how online databases of mathematical objects can be mined to unlock the collections that they preserve for use in research and teaching.

Teaching and Learning the Trigonometric Functions through Their Origins, by Daniel E. Otero
A series of curricular units based on primary source texts for use in teaching and learning trigonometry.

A Series of Mini-projects from TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources
A collection of student-ready projects for use in teaching standard topics from across the undergraduate curriculum.

Math Origins, by Erik R. Tou
How were concepts, definitions, and theorems familiar to today's students of mathematics developed over time?

### Mathematical Treasures

Mathematical Treasures, by Frank J. Swetz

Index to Mathematical Treasures Collection: Images of historical texts and objects from libraries, museums, and individuals around the world for use in your classroom!

# What's in Convergence? - Contents of Volume 19 - 2022

Editors:  Amy Ackerberg-Hastings, Janet Heine Barnett

Associate Editors:  Paul Bialek (through 1/31/22), Eugene Boman, Ximena Catepillan, Sloan Despeaux (through 1/31/22), Joel Haack (through 7/21/22), Toke Knudsen, Stacy Langton, Betty Mayfield, Michael Molinsky, Adam Parker, Andrew Perry, Adrian Rice, Amy Shell-Gellasch, Erik R. Tou, Laura Turner

Founding Editors: Victor Katz, Frank Swetz

### Articles

El Gabinete de Maravillas Matemáticas de Pantas: Imágenes e Historia de las Matemáticas, por Frank J. Swetz; traducido por Ximena Catepillán con la ayuda de Samuel Navarro
Involucre a sus estudiantes mediante el uso de imágenes, especialmente las de objetos históricos, manuscritos y textos, en la enseñanza de las matemáticas. Traducido al español de un artículo de Convergence publicado en 2015, “Pantas’ Cabinet of Mathematical Wonders: Images and the History of Mathematics.” (posted 08/07/2022)

Do Teachers Need to Incorporate the History of Mathematics in Their Teaching? by Po-Hung Liu
The author discusses five reasons for using the history of mathematics in its teaching and provides additional references written since the original publication of the article. (posted 06/06/2022)

HOM SIGMAA 2022 Student Paper Contest Winners
Read the winning papers from the 19th annual edition of this contest: “The Assumptive Attitudes of Western Scholars Regarding the Contributions of Mathematics from India: Assessing yukti-s from the Yuktibhāṣā of Jyeṣṭhadeva” by Rye Ledford (first prize) and “Estimations of $$\pi$$: The Kerala School of Astronomy and Mathematics, the Gregory-Leibniz Series, and the Eurocentrism of Math History” by Sarah Szafranski (second prize). (posted 06/06/2022)

An Ancient Egyptian Mathematical Photo Album – Hieroglyph Numerals and More, by Cynthia J. Huffman
Photographs of ancient Egyptian hieroglyphs in authentic contexts that instructors can use when teaching numeration systems. (posted 4/9/2022)

Kepler and the Rhombic Dodecahedron, by Roberto Cardil
Resources for sharing Kepler's fascinating studies of the rhombic dodecahedron with students. (posted 3/19/2022)

The High School Mathematics Curriculum—What Can We Learn from History? by Robert Reys and Barbara Reys
The authors review several of the major programs for reform in American mathematics education that appeared between 1894 and 2010 and conclude that, while calls for change have been constant, the full implementation of different approaches is much more difficult to achieve. (posted 3/05/2022)

Reflections on Chinese Numeration Systems, by Frank J. Swetz
Recommends ancient Chinese rod numerals to the instructors of preservice elementary teachers as an alternative place-value numeration system for helping students understand the structures and operations of arithmetic. Includes historical descriptions and classroom suggestions. (posted 2/20/2022)

Building a Book: HathiTrust, Ancestry.com, Serendipity, and Lifetime Interests, by David Lindsay Roberts
Reveals how personal knowledge, changes in historical research methods, and unexpected discoveries came together in the preparation of a book on the history of American mathematics, and suggests how the lessons learned could be incorporated into history of mathematics and other courses. (posted 1/22/2022)

### Ongoing Series

Keys to Mathematical Treasure Chests
A series that offers examples of how online databases of mathematical objects can be mined to unlock the collections that they preserve for use in research and teaching.

Teaching and Learning the Trigonometric Functions through Their Origins, by Daniel E. Otero
A series of curricular units based on primary source texts for use in teaching and learning trigonometry.

A Series of Mini-projects from TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources
A collection of student-ready projects for use in teaching standard topics from across the undergraduate curriculum.

### Mathematical Treasures

Mathematical Treasures, by Frank J. Swetz

Index to Mathematical Treasures Collection: Images of historical texts and objects from libraries, museums, and individuals around the world for use in your classroom!

# What's in Convergence?

### Annual Volumes

Articles, Ongoing Series (since 2018), Mathematical Treasures (since 2012) and Reviews (until 2016) are divided into annual volumes as follows. Browse these tables of contents for interesting and informative articles, classroom resources, and reviews of books and other materials on the history of mathematics and its use in teaching. To find Convergence material on specific topics, try these tips for effectively searching the MAA website.

### Special Features and Collections

Explore the following Special Features and Collections, many of which can also be accessed from the Convergence homepage.