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

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). (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 (posted 4/5/2021)
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.

The Educational Times Database: Building an Online Database of Mathematics Questions and Solutions Published in a 19th-Century Journal, by Robert M. Manzo (posted 3/22/2021)
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 (posted 2/20/2021)
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.

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.

• Series Introduction
• Episode 1 – Babylonian Astronomy and Sexagesimal Numeration
• Episode 2 – Hipparchus’ Table of Chords
• Episode 3 – Ptolemy Finds High Noon in Chords of Circles
• Episode 4 – Varāhamihira and the Poetry of Sines

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.

• Series Introduction, by Janet Barnett, Kathy Clark, Dominic Klyve, Jerry Lodder, Daniel E. Otero, Nick Scoville, and Diana White
• The Derivatives of the Sine and Cosine Functions: A Mini-Primary Source Project for Calculus I, by Dominic Klyve
• Why be so Critical? Nineteenth Century Mathematics and the Origins of Analysis: A Mini-Primary Source Project for Introductory Analysis Students, by Janet Heine Barnett
• Connecting Connectedness: A Mini-Primary Source Project for Topology Students, by Nicholas A. Scoville
• Generating Pythagorean Triples: A Mini-Primary Source Project for Mathematics Majors, Elementary Teachers and Others, by Janet Heine Barnett
• Euler's Rediscovery of e: A Mini-Primary Source Project for Introductory Analysis Students, by Dave Ruch
• How to Calculate \(\pi\): Machin's Inverse Tangents, A Mini-Primary Source Project for Calculus II Students, by Dominic Klyve
• Henri Lebesgue and the Development of the Integral Concept: A Mini-Primary Source Project for Undergraduate Analysis Students, by Janet Heine Barnett
• Seeing and Understanding Data: A Mini-Primary Source Project for Students of Statistics, by Charlotte Bolch and Beverly Woods
• The Origin of the Prime Number Theorem: A Primary Source Project for Number Theory Students, by Dominic Klyve
• The Cantor Set Before Cantor: A Mini-Primary Source Project for Analysis and Topology Students, by Nicholas A. Scoville
• Euler’s Calculation of the Sum of the Reciprocals of the Squares: A Mini-Primary Source Project for Calculus II Students, by Kenneth M Monks
• Completing the Square: From the Roots of Algebra, A Mini-Primary Source Project for Students of Algebra and Their Teachers, by Daniel E. Otero
• Regression to the Mean: A Mini-Primary Source Project for Statistics Students, by Dominic Klyve
• Investigations Into d'Alembert's Definition of Limit: A Mini-Primary Source Project for Students of Real Analysis and Calculus 2, by David Ruch
• Braess’ Paradox in City Planning: A Mini-Primary Source Project for Multivariable Calculus Students, by Kenneth M Monks
• Topology from Analysis: A Mini-Primary Source Project for Topology Students, by Nick Scoville
• Babylonian Numeration: A Mini-Primary Source Project for Pre-service Teachers and Other Students, by Dominic Klyve
• Wronskians and Linear Independence: A Theorem Misunderstood by Many – A Mini-Primary Source Project for Students of Differential Equations, Linear Algebra and Others, by Adam E. Parker
• Bhāskara’s Approximation to and Mādhava’s Series for Sine: A Mini-Primary Source Project for Second-Semester Calculus Students, by Kenneth M Monks

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

• Math Origins: The Totient Function
• Math Origins: Orders of Growth
• Math Origins: Eigenvectors and Eigenvalues
• Math Origins: The Logical Ideas
• Math Origins: The Logical Symbols
• Math Origins: The Language of Change

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!

Mathematical Treasures added during 2021:

• Byrhtferth’s Manuscript (ca 1102–1111)
• German manuscript of Sacrobosco’s Algorismus (mid-15th century)
• German manuscript of Euclid’s Elements (1460)
• Filippo Calandri's Trattato di arithmetica (1491)
• Francisco Pellos’s Compendio de lo abaco (1492)
• Kitab al-Mutawassitat, Arabic “The Book of ‘The Middle Books’” (15th–18th centuries)
• Johannes Widman’s Behend und hüpsch Rechnung uff [auf] allen Kauffmanschafften (1508, original 1489)
• Stefano di Battista’s Summa arismetice (ca 1522)
• Hans Sebold Beham’s engraving of Geometria (16th century)
• Georg Pencz’s engraving of Arithmetica (16th century)
• Eukleidou Stoicheion (printed Greek-language Euclid’s Elements, 1533)
• Italian manuscript of Euclid’s Elements (1541)
• Christoff Rudolff’s Die Coss (1553, original 1525)
• Simon Stevin’s L’Arithmetique (1558)
• Rafael Bombelli’s L’Algebra Opera (1579, original 1572)
• Peter Ramus's Arithmeticae libri duo: Geometriae septem et viginti (1580)
• Frans van Schooten’s Latin translation of François Viéte’s works, Opera mathematica (1641)
• Bonaventura Cavalieri’s Geometria indivisibilibus continuorum (1653, original 1635)
• Galileo Galilei's Della Scienza Mecanica (1655), contributed by Sid Kolpas
• Mathematical playing cards (1702)
• Ḥaydar ibn 'Abd al-Raḥmān Jazarī’s Risālah fī al-ʻamal bi'l-asṭurlāb (1726)
• John Keill’s Euclid’s Elements of Geometry (1728, original 1723)
• Pierre Varignon’s Élémens de Mathématiques (1731)
• Della Mechanica collection of notebooks (1763–1784)
• Joseph Fenn’s Instructions Given in the Drawing School (vol. 2, 1772)
• Sophie Germain’s Recherches sur la théorie des surfaces élastiques (1821)
• Nathan Daboll’s Schoolmaster’s Assistant (1829, original 1799)
• Augustus De Morgan’s Arithmetical Books from the Invention of Printing to the Present Time (1847), contributed by Sid Kolpas
• Benjamin Greenleaf’s Common School Arithmetic (1855)
• Warren Colburn’s First Lessons in Intellectual Arithmetic (1863, original 1821)
• Sofia Kovalevskaya’s “Theorie der partiellen Differentialgleichungen” (1875)
• Model for Soap Film Miminal Surface by Alexander and Ludwig Brill (ca 1892), contributed by Peggy Kidwell
• Charlotte Angas Scott’s An Introductory Account of Certain Modern Ideas and Methods in Plane Analytical Geometry (1894)
• Lejeune Dirichlet’s Untersuchungen über verschiedene Anwendungen Infinitesimalanalysis auf die Zahlentheorie (1897, original 1839–1840)
• David Hilbert’s Théorie des Corps de Nombres Algébriques (1913 French translation, original 1897)
• David Hilbert’s “Die logischen Grundlagen der Mathematik” (1922)
• David Hilbert’s and Paul Bernays’s Grundlagen der Mathematik (1934)