*Convergence* Articles, 2004–2021

#### 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.

The Life of Sir Charles Scarburgh, by Michael Molinksy

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).

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.

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.

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 mathematical glyphs in Rapanui writing. Four activities involving ellipses help instructors share this example of ethnomathematics with their students.

Misterios Matemáticos de Rapa Nui con Actividades para el Aula de Clases, por Ximena Catepillán, Cynthia Huffman, and 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.

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*

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

#### 2020

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

#### 2019

Correspondence from Mathematicians by Jennifer Horn, Amy Zamierowski and Rita Barger

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

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

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

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

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.

A readers' guide and complete English translation of Servois' 1817 contribution to a debate on numerical integration.

HOM SIGMAA 2019 Student Paper Contest Winner

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

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

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

Advice on promoting excellence in student research and writing in the history of mathematics.

#### 2018

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 Treasure: Japanese Mathematics in the Edo Period, by Frank J. Swetz

Twelve distinct works illustrate the range of mathematics produced in Japan from 1603 to 1867.

Mathematical Treasure: Billingsley's Sources for the First English Euclid's Elements, by Frank J. Swetz

Two Greek sources 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

#### 2017

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.

Exploring Liu Hui's Cube Puzzle: From Paper Folding to 3-D Design, by Lingguo Bu

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

Illustrating *The Nine Chapters on the Mathematical Art*: Their Use in a College Mathematics History Classroom, by Joel K. Haack

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*)

HOM SIGMAA 2017 Student Paper Contest Winners

Download the three winning papers from the 14th annual competition.

Impacts of a Unique Course on the History of Mathematics in the Islamic World, by Nuh Aydin

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

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.

#### 2016

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

President James A. Garfield's Proof of the Pythagorean Theorem, by Sid Kolpas

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

HOM SIGMAA 2016 Student Paper Contest Winners

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.

#### 2015

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.

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

Bridging the Gap Between Theory and Practice: Astronomical Instruments, by Toke Knudsen

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

Pantas’ Cabinet of Mathematical Wonders: Images and the History of Mathematics, by Frank J. Swetz

Engage your students by using images, especially those of historical objects, manuscripts, and texts, in teaching mathematics.

HOMSIGMAA 2015 Student Paper Contest Winners

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

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

#### 2014

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.

David Hilbert's Radio Address, by James T. Smith

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

HOM SIGMAA 2014 Student Paper Contest Winners

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.

#### 2013

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.

HOM SIGMAA 2013 Student Paper Contest Winner

Read "Paradigms and Mathematics: A Creative Perspective," by Hood College graduate Matthew Shives.

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!

External Influences on U.S. Undergraduate Mathematics Curricula: 1950-2000, by Walter Meyer

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)

7. Origins of Boolean Algebra in the Logic of Classes: George Boole, John Venn, and C. S. Peirce, by Janet Barnett

- Project in which discrete mathematics students are introduced to set operations, Venn diagrams, and Boolean algebra by the masters

8. Boolean Algebra as an Abstract Structure: Edward V. Huntington and Axiomatization, by Janet Barnett

- 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

#### 2012

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 15^{th} 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

#### 2011

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.

#### 2010

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 Enigmatic Number *e:* A History in Verse and Its Uses in the Mathematics Classroom, by Sarah Glaz

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 18^{th} century was introduced in a manuscript by the 12^{th} 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.

Combining Strands of Many Colors: Episodes from Medieval Islam for the Mathematics Classroom, by Randy K. Schwartz

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

#### 2009

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.

“In these numbers we use no fractions”: A Classroom Module on Stevin’s Decimal Numbers, by Kathleen Clark

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.

#### 2008

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 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.

Introducing the History of Mathematics: An Italian Experience Using Original Documents, by Adriano Dematte

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.

#### 2007

Euler Squares, by Elaine Young

An elementary introduction to Euler squares.

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 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

Four episodes in the history of geometry are discussed, where dynamic geometry helps in understanding the ideas.

#### 2006

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.

#### 2005

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.

#### 2004

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.