*Editors:* Amy Ackerberg-Hastings, Janet Heine Barnett

*Editor-Elect:* Daniel E. Otero

*Associate Editors:* Eugene Boman, Ximena Catepillan, Abe Edwards, Toke Knudsen, Stacy Langton, Betty Mayfield, Adam Parker, Andrew Perry, Adrian Rice, Laura Turner

*Founding Editors:* Victor Katz, Frank Swetz

### Articles

Pitfalls and Potential Solutions to Your Primary Source Problems, by Adam E. Parker

Suggestions for dealing with difficulties that can arise when an instructor brings primary sources into the mathematics classroom. (posted 12/18/2023)

Primary Source Projects and Reading Apprenticeship in Mathematics History, by Jennifer Clinkenbeard

A classroom testimonial on the effect of combining primary source-based materials designed to teach mathematics with a framework designed to engage students as readers. (posted 11/20/2023)

A Selection of Problems from A.A. Markov’s *Calculus of Probabilities*, by Alan Levine

Translation of excerpts from Markov's *Calculus of Probabilities* (1900), with suggestions for classroom use. (posted 11/06/2023)

The Theorem that Won the War, by Jeff Suzuki

The story of the three Polish mathematicians (Marian Rejewski, Henryk Zygalski, and Jerzy Róẓycki) who first cracked the German encryption system known as Enigma, accompanied by classroom activities that address several key concepts in abstract algebra, including permutation groups, cycle notation, conjugates, and cycle decomposition. (posted 10/09/2023)

Ciclos de Tiempo Maya, por Sandra Monteferrante, traducido por Ximena Catepillán con la ayuda de Samuel Navarro

La autora discute la historia de calendarios Maya, incluyendo el uso del sistema numérico modificado de base 20. Traducido al español de un artículo de *Convergence* publicado en 2007, “Maya Cycles of Time.” (posted 09/11/2023)

Seeking Relevance? Try the History of Mathematics, by Frank Swetz

Reprint of a 1983 NCTM article which argued that a natural integration of historical content into mathematics courses offers multiple benefits to students, with an “epilogue" written by the author in 2023 in which he reflected on developments in the use of history to teach mathematics over the past four decades and assessed the field’s future prospects. (posted 08/21/2023)

Numbers, Infinity, and Reality: An Interdisciplinary Undergraduate Philosophy of Mathematics Course, by Kevin DeLapp and Jessica Sorrells

Description of an undergraduate course at Converse University that brings together advanced topics in the philosophy, history, and sociology of mathematics, providing a fruitful collaboration between mathematics instruction and humanities curricula. (posted 06/19/2023)

HOM SIGMAA 2023 Student Paper Contest Winner

Read the winning paper from the 20th annual edition of this contest: “Nicole Oresme and the Revival of Medieval Mathematics” by Adin Charles Tinsley. (posted 05/23/2023)

Things Certain and Uncertain, by Michael P. Saclolo and Erik R. Tou

The story of a mathematical problem on the mechanics of hot air balloon flight that Euler was working on the very day of his death, presented in its historical context and accompanied by a classroom capsule with suggestions for how the mathematics of balloon flight can be used in a contemporary differential equations or physics course. (posted 5/22/2023)

Who? How? What? A Strategy for Using History to Teach Mathematics, by Patricia Wilson and Jennifer Chauvot

The authors review four benefits of using the history of mathematics in the classroom and suggest a strategy of asking *who* does mathematics, *how* mathematics is done, and *what* mathematics is in order to help students and instructors discover the human story of mathematics by beginning to explore its history. (posted 4/24/2023)

A Mysterious Copy of Lacroix’s *Traité Élémentaire de Calcul Différentiel et de Calcul Intégral*, by Adrian Rice

The author's recounting of an intriguing mystery surrounding his personal copy of the 4th edition of Lacroix's well-known textbook—a tale that involves Augustus De Morgan, James Joseph Sylvester, and the teaching of mathematics at University College London in its early days. (posted 3/27/2023)

Aiding the Teaching of Geometry and Affording Mathematical Recreation: Paper Folding in the Spirit of Sundara Rao of Madras, by Peggy Aldrich Kidwell

A history of paper folding in mathematics education that focuses on the background, publication, and reception of Sundara Rao’s 1893 *Geometrical Exercises in Paper Folding*. The article also describes several potential classroom activities for secondary and undergraduate students of geometry and preservice teachers. (posted 3/13/2023)

### Ongoing Series

**Historical Notes for the Calculus Classroom**, by V. Frederick Rickey

A series of short articles on the history of calculus, developed through the author’s experiences with historical research and teaching and written for the use of instructors.

**Historically Speaking**, by Betty Mayfield

Selections from the short columns on historical mathematics that ran in NCTM’s *Mathematics Teacher* between 1953 and 1969, with new commentary placing the history and mathematics into context.

**Quotations in Context**, by Michael Molinsky

A series of columns that explores the origins and meanings of various quotations about mathematics and mathematicians.

**HoM Toolbox, or Historiography and Methodology for Mathematicians**

A series that guides readers through the basic principles and theoretical approaches for researching and writing the history of mathematics.

**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.

**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
- Fourier’s Heat Equation and the Birth of Modern Climate Science: A Mini-Primary Source Project for Differential Equations and Multivariable Calculus Students, by Kenneth M Monks
- How to Calculate \(\pi\): Buffon's Needle – A Mini-Primary Source Project on Geometric Probability for Calculus 2 Students, Pre-service Teachers and Others, by Dominic Klyve
- Solving Linear Higher Order Differential Equations with Euler and Johann Bernoulli: A Mini-Primary Source Project for Differential Equations Students, by Adam E. Parker
- Fourier’s Infinite Series Proof of the Irrationality of e: A Mini-Primary Source Project for Calculus 2 Students, by Kenneth M Monks
- Fermat’s Method for Finding Maxima and Minima: A Mini-Primary Source Project for Calculus 1 Students, by Kenneth M Monks
- The Closure Operation as the Foundation of Topology: A Mini-Primary Source Project for Topology Students, by Nicholas A. Scoville
- Beyond Riemann Sums: Fermat's Method of Integration – A Mini-Primary Source Project for First-Year Calculus Students, by Dominic Klyve
- Lagrange’s Work on Wilson’s Theorem: Three Mini-Primary Source Projects for Number Theory Students, by Carl Lienert

### 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 2023:

- Old Babylonian Tablet (YBC 07302, ca 1900–1600 BCE)
- De Geometria (English manuscript, ca 900–1100)
- Matthew Paris’s Illuminations of Greek Philosophers for
*Experimentarius* and *The Prognostics* (13th century)
- Medieval French illuminations depicting God as Creator-Geometer (ca 1300–1325)
- John of Erfurt's
*Computus chirometralis* (1443, 14th-century original)
- Benedetto da Firenze’s
*Trattato d’abacho* (ca 1480)
- Belo Moietta’s
*Trattato di geometria* (ca 1486–1501)
- Peter Apian’s
*Cosmographia* (1539, 1524 original)
- Oronce Fine’s
*L'art et manière de trouver certainement la longitude de tous lieux sur la Terre par le cours et mouvement de la Lune et La composition et usage d'un singulier méthéoroscope géographique* (1543)
- Rafael Bombelli’s
*L'algebra parte maggiore dell’aritmetica divisa in tre libri* (1572)
*Geometria Cathoptrica and Aritmetica De Numeri Rotti* (Italian manuscript, ca 1575)
- Bernardino Baldi’s
*Nouae gnomonices libri quinque* (1592)
- Niccolò Tartaglia’s
*Tutte l'opere d'arithmetica *(2 vol., 1592)
- Philippe Danfrie’s
*Declaration de l'usage du Graphometre* (1597)
*Dīwān al-‘adad al-wafq* (ca 16th century, 1123–1124 original)
- Bernardino Baldi’s
*Cronica de Matematici* (16th-century manuscript, 1707 printing)
- Bernardino Baldi’s
*In mechanica Aristotelis problemata exercitationes* (1621)
- Jean Boulenger’s
*La Geometrie pratique* (1634 and 1690 editions)
- Richard White’s
*Hemisphaerium dissectum opus geometricum* (1648)
- Bonaventura Cavalieri’s
*Lo specchio ustorio* (1650, 1632 original)
- Stefano degli Angeli’s
*Problemata geometrica sexaginta. Circa conos, sphaeras, superficies conicas, sphaericasque praecipue versantia* (1658)
- Stefano degli Angeli’s
*De superficie vngulae, et de quartis liliorum parabolicorum & cycloidalium tractatus duo geometrici* (1661)
- Athanasius Kircher’s
*Arithmologia, sive De Abditis Numerorum Mysteriis* (1665), contributed by Jacqueline M Dewar and Sarah J Greenwald
- Christiaan Huygens’s
*Horologium Oscillatorium* (1673)
- Alessandro Marchetti’s
*Problemata sex à Leidensi quodam surveyor Christophoro Sadlerio* (1675)
- John Craig’s
*Methodus figurarum lineis rectis et curvis comprehensarum quadraturas determinandi* (1685)
- Johann Christoph Sturm’s
*Mathesis Enucleata* (1689)
- Mathematics notebook possibly kept by Harvard students (ca 1689–1769)
- Vincenzo Viviani’s
*Formazione, e misvra di tutti i cieli* (1692)
- John Craig’s
*Tractatus mathematicus de figurarum curvilinearum quadraturis et locis geometricis* (1693)
- Claude Irson’s
*Abrégé de l'arithmétique pratique et raisonnée* (1695)
- J. R.’s English translation of Johann Christoph Sturm’s
*Mathesis Enucleata: Or, The Elements of the Mathematicks* (1700, 1689 original)
- Vincenzo Viviani’s manuscript notes on his studies with Galileo (17th century)
- Isaac Newton’s
*Arithmetica universalis: sive de compositione et resolutione arithmetica liber* (1722, 1707 original)
- Manuscript mathematical notes attributed to Isaac Newton (early 18th century)
- Abraham Kästner’s “Cavtionem in qvantitatvm infinite parvarvm neglectv observandam exemplis qvibvsdam illustrat simulque ad audiendam . . . orationem” (1746)
- Christian Wolff’s
*Der Anfangs-Gründe aller mathematischen Wissenschaften* (vol. 3, 1750, 1710 original)
- Thomas Simpson’s
*Select Exercises for Young Proficients in the Mathematicks* (1752)
- Alexis Claude Clairaut’s
*Élémens de géométrie *(1753, 1741 original), contributed by Bruce Burdick and Laura Turner
- Rudolph Ernst Schilling’s
*Die Geometrie und Trigonometrie* (1770)
- Karl Scherffer’s
*Institutionum Geometricarum* (4 vol., 1770–1771)
- John Bonnycastle’s
*An Introduction to Mensuration and Practical Geometry* (1798, 1782 original)
- Gaspard Monge’s
*Applications de L’Analyse a la Géométrie* (1850, 1807 original)
- Edward Strachey’s translation of Bhaskara’s
*Bija Ganita* (1813, 12th-century original)
- Charles Babbage’s “Observations on the analogy which subsists between the calculus of functions and the other branches of analysis” (1817)
- Louis-Benjamin Francœur’s
*Cours complet de mathématiques pures* (2 vol., 1819, 1810 original)
- Daniel Dowling’s
*Key to the course of mathematics, composed for the use of the Royal Military Academy, by Charles Hutton* (1818)
- Charles Babbage's
*Examples of the Solutions of Functional Equations* (1820)
- Charles Babbage’s “Observations on the notation employed in the calculus of functions” (1820)
- John Quincy Adams’s
*Report upon Weights and Measures* (1821)
- Charles Babbage’s “On the application of analysis to the discovery of local theorems and porisms” (1823)
- Charles Babbage’s “On the influence of signs in mathematical reasoning” (1826, read in 1821)
- August Leopold Crelle’s
*Lehrbuch der Elemente der Geometrie und der ebenen und sphärischen Trigonometrie* (vol. 2, 1827)
- Johann Heinrich Moritz von Poppe’s
*Geschichte der Mathematik seit der ältesten bis auf die neueste Zeit* (1828)
- Joab Goldsmith Cooper’s
*The Scholar’s Assistant* (1830)
- Aḥmad ibn Ibrāhīm al-Sharbatlī’s Astrolabe Quadrant (1840–1841)
- J. C. Loudon’s
*Self-Instruction for Young Gardeners, Foresters, Bailiffs, Land-stewards, and Farmers* (1845)
- Florence Nightingale’s
*Notes on Matters Affecting the Health, Efficiency, and Hospital Administration of the British Army* and contributions to *Mortality of the British Army* (both in 1858)
- Henry Rutter’s
*The Metric System of Weights and Measures* (1866)
- Benjamin Williamson’s
*An Elementary Treatise on the Differential Calculus* (1872)
- James Clerk Maxwell’s
*Treatise on Electricity and Magnetism* (1873)
- Francis Galton’s “Typical Laws of Heredity” (1877)
- Gaspard Monge’s
*Darstellende Geometrie* (1900, 1798 original)
- Srinivasa Ramanujan’s “Lost” Notebooks (1910s)
- David Eugene Smith and Louis Charles Karpinski’s
*The Hindu-Arabic Numerals* (1911)
- M. Raṅgācārya’s
*The Ganita-sāra-sangraha of Mahāvīrācārya*, with introduction by David Eugene Smith (1912, 9th-century original)
- David Eugene Smith’s second edition of Augustus De Morgan’s
*Budget of Paradoxes* (1915, 1872 original)
- David Eugene Smith’s
*History of Mathematics* (1923 and 1925)
- Florian Cajori’s
*A History of Mathematical Notations* (2 vol., 1928–1929)