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.

• Series Introduction, by Peggy Aldrich Kidwell
• 19th-century String Models, by Peggy Aldrich Kidwell

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
• Episode 5 –  al-Bīrūnī Does Trigonometry in the Shadows
• Episode 6 –  Regiomontanus and the Beginnings of Modern 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.

• 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 1, 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 2 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 2 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 Calculus 2 Students, by Kenneth M Monks
• The Logarithm of -1: A Mini-Primary Source Project for Complex Variables Students, by Dominic Klyve
• Gaussian Guesswork: Three Mini-Primary Source Projects for Calculus 2 Students, by Janet Heine Barnett

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:

• Egyptian limestone slab stele depicting numbers (2613–2494 BCE)
• Old Babylonian tablet depicting heptagon problem (ca 1894–1595 BCE)
• Moscow Mathematical Papyrus (ca 1850 BCE)
• Rhind Mathematical Papyrus (ca 1650 BCE)
• Byrhtferth’s Manuscript (ca 1102–1111)
• Nasīr al-dīn al-Ṭūsī’s Jami’ al-hisab bi-‘l-Takht wa-‘l-turab (The Collection of Arithmetic, 1265 original)
• Nasīr al-dīn al-Ṭūsī’s A Compendium of Treatises on Astronomy and Mathematics (copied in 1279)
• 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)
• Petrus Cracoviensis’ Computus novus ecclesiasticus (1515, 1505 original)
• Gaspar Nicolas’s Tratado da Prática D'arismetyca (1519), contributed by Pedro Jorge Freitas and Jorge Nuno Silva
• 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)
• Gallus Spänlin’s Arithmetica: künstlichen Rechnung, lustiger Exempeln (1546)
• Christoff Rudolff’s Die Coss (1553, original 1525)
• Francesco Maurolico’s Opuscula mathematica : nunc primum in lucem edita (1575)
• Rafael Bombelli’s L’Algebra Opera (1579, original 1572)
• Peter Ramus's Arithmeticae libri duo: Geometriae septem et viginti (1580)
• Simon Stevin’s L’Arithmetique (1585)
• Oberto Cantone's L'vso prattico dell' aritmetica e geometria (1599 first and 1606 second editions)
• Edward Wright’s Certaine Errors in Navigation (1610, original 1599)
• Matthias Bernegger’s Manuale mathematicum, darinn begriffen die tabulae sinuum, tangentiũ, secantium (1619)
• Frans van Schooten’s Latin translation of François Viéte’s works, Opera mathematica (1641)
• Bonaventura Cavalieri’s Geometria indivisibilibus continuorum (1653, original 1635)
• Francesco Maurolico’s Messanensis, Emendatio, et restitutio conicorum Apollonij Pergaei (1654)
• Galileo Galilei's Della Scienza Mecanica (1655), contributed by Sid Kolpas
• Claude Richard’s Apollonii Conicorum libri IV. Cum commentariis R. P. Claudii Richardi (1655)
• Abraham de Graaf’s Principia arithmeticae, theoreticae, & practicae (1662)
• Henry Walrond’s Arithmetical Tables (1663)
• Abraham de Graaf’s De beginselen van de algebra of stelkonst (1672)
• Jonas Moore’s A Mathematical Compendium (1681, original 1674)
• Adriaan de Neeff’s Cijffer-boeck der heylige schrift (1682)
• Peter Perkins’s The Seaman’s Tutor (1682)
• Jacques Ozanam’s Dictionaire mathematique (1691)
• Abraham de Graaf’s De geheele mathesis of wiskunst (1694, original 1676)
• Abraham de Graaf’s Exemplaar-boekje van de arithmetica, zynde een vervolg van de Wiskonstige arithmetica (1702)
• Mathematical playing cards (1702)
• Abraham de Graaf’s Analysis of stelkunstige ontknoping in de meetkunstige werkstukken (1706)
• Abraham de Graaf’s Inleyding tot de wiskunst, of de beginselen van de geometria en algebra (1706)
• Abraham de Graaf’s De vervulling van de geometria en algebra (1708)
• John Collins’s Commercium Epistolicum (1712)
• Abraham de Graaf’s Instructie van het Italiaans boekhouden (1720, original 1688)
• Jean Lepine’s Stylus-Operated Adding Machine (1725)
• 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)
• Maximilian Hell’s Elementa arithmeticae Numericae & Literalis seu Algebrae (1768)
• Alexis Fontaine’s Traité de calcul différentiel et intégral (1770)
• Joseph Fenn’s Instructions Given in the Drawing School (vol. 2, 1772)
• Reinier Vinkele’s The Mathematics Lesson (1776)
• Thaddeus Mason Harris’s cyphering book on Duodecimals (late 18th century)
• Georg Herth’s engraving of the French Academy of Sciences (ca 1790)
• Reuben Burrow’s “A Proof that the Hindoos Had the Binomial Theorem“ (1799)
• Adrien-Marie Legendre’s Éléments de Géométrie (1800, original 1794)
• Peter Barlow’s An Elementary Investigation of the Theory of Numbers (1811)
• Peter Barlow’s A New Mathematical and Philosophical Dictionary (1814)
• Jeremiah Day’s A Treatise of Plane Trigonometry (1815)
• Jeremiah Day’s The Mathematical Principles of Navigation and Surveying (1817)
• John Farrar’s Elements of Geometry, by A.M. Legendre (1819)
• John Playfair’s On the Progress of Mathematical and Physical Science Since the Revival of Letters in Europe (1820, original 1816–1820)
• Louis Marie Pierre Bourdon’s Élémens d'algèbre (1820, original 1817)
• Sophie Germain’s Recherches sur la théorie des surfaces élastiques (1821)
• Augustus De Morgan’s The Elements of Algebra, translated from the first three chapters of the algebra of M. Bourdon (1828)
• Nathan Daboll’s Schoolmaster’s Assistant (1829, original 1799)
• John Farrar’s Elements of Algebra, by Bourdon (1831, translated by George Barrell Emerson and corrected by Nathaniel Bowditch)
• Charles Davies’s Elements of Algebra: Translated from the French of M. Bourdon (1835)
• Jeremiah Day’s The Teacher’s Assistant (1836)
• James B. Thomson’s A Key to the Abridgment of Day’s Algebra (1844)
• 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)
• Charles Davies’s Key to Davies’ Bourdon (1866, original 1856)
• Charles Scott Venable’s Arithmetic, Pure and Commercial (1868)
• Charles Scott Venable’s An Elementary Algebra (1869)
• Sofia Kovalevskaya’s “Theorie der partiellen Differentialgleichungen” (1875)
• Giuseppe Peano’s Arithmetices Principia (1889)
• Model for Soap Film Miminal Surface by Alexander and Ludwig Brill (ca 1892), contributed by Peggy Kidwell
• Paper Model of Elliptic Paraboloid by Alexander and Ludwig Brill (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)
• Model of a Thermodynamic Surface by Richard P. Baker (ca 1906–1935), contributed by Peggy Kidwell
• David Hilbert’s Théorie des Corps de Nombres Algébriques (1913 French translation, original 1897)
• David Hilbert’s “Die logischen Grundlagen der Mathematik” (1922)
• Product integraph by Vannevar Bush and colleagues (ca 1925), contributed by Peggy Kidwell
• Dissected rhombic dodecahedron by A. Harry Wheeler (1930s), contributed by Peggy Kidwell
• A. Harry Wheeler’s Transformable Plane Dissections (1931), contributed by Peggy Kidwell
• David Hilbert’s and Paul Bernays’s Grundlagen der Mathematik (1934)
• Edith Dummer Mintzer’s Boole Senior Blocks (ca 1935), contributed by Peggy Kidwell
• John D. Elder’s Factor Stencil Punch Cards (1939), contributed by Peggy Kidwell
• Trigonometer for Use with Artillery (ca 1942), contributed by Peggy Kidwell
• Model of Spherical Triangle by A. Harry Wheeler (1945), contributed by Peggy Kidwell
• B. F. Skinner’s Arithmetic Teaching Machine (1954), contributed by Peggy Kidwell
• IBM’s Men of Modern Mathematics Timeline (1966), contributed by Sid Kolpas