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

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

*Founding Editors:* Victor Katz, Frank Swetz

### Articles

Euler’s Letters to a German Princess: Translation and Betrayal, by Dominic Klyve

An exploration of how the translations of Euler’s *Letters to a German Princess* came to differ from the original text. (posted 12/07/2020)

The Four Curves of Alexis Clairaut, by Taner Kiral, Jonathan Murdock, and Colin B. P. McKinney

Translation of a paper on families of algebraic curves (along with a transcription of the French original) written when Clairaut was only twelve years old. (posted 11/22/2020)

The ‘Piling Up of Squares’ in Ancient China, by Frank Swetz

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

Converting the Old Babylonian Tablet ‘Plimpton 322’ into the Decimal System as a Classroom Exercise, by Antonella Perucca and Deborah Stranen

A student-ready activity, ideal for pre-service elementary mathematics teachers. (posted 10/26/2020)

The French Connection: Borda, Condorcet and the Mathematics of Voting Theory, by Janet Heine Barnett

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

Apportionment: What's Your Fair Share – An Activity for Liberal Arts and High School Students, by Jeff Suzuki

A self-contained project suitable for individual or group work, inside or outside the classroom, that uses US Census data from 1790 to guide students through an exploration of what it means for each state to get its fair share of congresspersons, and of how different methods of apportionment might have altered the course of American history. (posted 09/08/2020)

Pathways from the Past: Classroom-Ready Materials for Using History to Teach Mathematics, by Bill Berlinghoff and Fernando Gouvêa

Reproducible student activity sheets developed by the authors of the well-regarded textbook, *Math through the Ages*, and especially suitable for practicing and pre-service teachers of secondary mathematics and those involved in teacher training. (posted 06/07/2020)

HOM SIGMAA 2020 Student Paper Contest Winner

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

Word Histories: Melding Mathematics and Meanings, by Rheta N. Rubenstein and Randy K. Schwartz

Etymologies for common mathematical terms—from subjects such as algebra, geometry, functions and discrete mathematics—can be used by instructors to enrich student learning. (posted 04/20/2020)

Mabel Sykes: A Life Untold and an Architectural Geometry Book Rediscovered, by Maureen T. Carroll and Elyn Rykken

Biography of a little-known high-school mathematics teacher and discussion of her publications, particularly the lavishly-illustrated 1912 *A Source Book of Problems for Geometry Based upon Industrial Design and Architectural Ornament*. The description of *Source Book* includes diagrams and animations. (posted 2/24/2020)

Why History of Mathematics? by Glen Van Brummelen

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

A Mathematical History Tour: Reflections on a Study Abroad Program, by R. Abraham Edwards and Marie Savoie

A unique study-abroad course combining the history of mathematics and travel. (posted 1/13/2020)

### Ongoing Series

**Teaching and Learning the Trigonometric Functions through Their Origins**, by Daniel E. Otero

A series of curricular units based on primary source texts for use in teaching and learning trigonometry.

**A Series of Mini-projects from TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources**

A collection of student-ready projects for use in teaching standard topics from across the undergraduate curriculum.

- 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

**Math Origins**, by Erik R. Tou

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

### Mathematical Treasures

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

Mathematical Treasures from the Linda Hall Library added during 2020:

Mathematical Treasures, by Frank J. Swetz

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

Mathematical Treasures added during 2020:

- Works of Boethius (10th-12th centuries)
- Leaves from Arabic abridgment of Euclid's
*Elements* (1108–1109)
- Bhaskara's
*Lilavati* (undated, original 12th century)
- Mathematics of Gerbert of Aurillac (12th-13th centuries)
- Johannes de Sacrobosco's
*Algorismus* (13th century)
- Manuscript illustrations of tools of measurement (15th century)
- Juan de Ortega's
*Suma de arithmetica: ordinate* (1515, original 1512)
*Algorithmus Demonstratus* (1534)
- Oronce Fine's
*Arithmetica practica* (1544), contributed by Sid Kolpas
- Piero Borghi's
* Libro de Abacho* (1550, original 1484)
*Arithmeticae practicae* of Gemma Frisius (1556, original 1540)
- Johannes Schöner's
*Opera Mathematica* (1561)
- Pierre Forcadel's
*Les six premiers livres des elements d'Euclide* (1573)
- Raphael Sadeler's engraving of Minerva giving instructions to a geometer (1591)
- Alexandre Vallée's engravings of the Muses of Arithmetic, Geometry, and Astronomy (ca 1600)
- Cornelis Schut's engravings of Arithmetic and Geometry (17th century)
- Bartholomaeus Keckermann's
*Systema compendiosum totius mathematices* (1621, original 1617)
- Adriaan Metius's
*Manuale arithmetice et geometrie practice* (1634)
- Bonaventura Cavalieri's
*Trigonometria Plana et Sphaerica, Linearis, & Logarithmica *(1643)
- Jean François's
*Traité de la Quantitée* (1655)
- Jean François's
*L’Arithmétique et la Géométrie Pratique* (1657)
- Giovanni Alfonso Borelli's
*Euclide rinnovato* (1663)
- Pietro Aquila's engraving of Hercules between an astronomer and mathematician (1674)
- Lorenzo Forestani's
*Pratica d'arithmetica e geometria* (1682, original 1603)
- Luigi Maria Cagnacci's
*Libro per brevem apprend[ere] le discipline mattemat[iche]* (after 1689)
- Jean Boulenger's
*La Geometrie Pratique *(1691, original 1624)
- Edmund Pourchot's
*Elementa Geometria* (1692)
- John Newton's
*Cosmographia* (1694)
- Advertisements by British mathematics masters and makers of mathematical instruments (mainly 18th century)
- Ignatius Pardies's
*Short but Plain Elements of* *Geometry and Plain Trigonometry *(1701)
- Jerome [Hieronymus] Ditzel’s
*Geographiae* (1716)
- Valentin Heins’s
*Tyrocinium mercatorio-arithmeticum* (1726, original 1684)
- Bernard Picart's engraving representing geometry and military science (1729)
- Thomas Le Seur's and Francois Jacquier's annotated edition of Newton's
*Principia* (1739–1742)
- Jacques Ozanam's
*La Trigonometrie Rectiligne et Spherique* (1741)
- José Fernandes Pinto Alpoim's
*Exame de Artilheiros* (1744)
- James Dodson's
*The Mathematical Repository*, 3 vol. (1748, 1753, 1755)
- Robert Bénard's engravings of mathematical and surveying instruments (mid-18th century)
- Anonymous engraving of a man using an inclinometer (mid-18th century)
- Daniel Fenning’s
*The Young Algebraist’s Companion* (1751)
- Thomas Simpson’s
*Miscellaneous Tracts* (1757), contributed by Sid Kolpas
- Daniel Fenning’s
*The British Youth’s Instructor *(1765, original 1754)
- J. Leadbeater’s
*The Gentleman and Tradesman’s Compleat Assistant* (1769, original 1768)
- Edward Waring's
*Proprietates algebraicarum curvarum* (1772)
- Colin Maclaurin's
*Treatise of Algebra *(1796, original 1748)
- Étienne Bézout's
*Cours de Mathématiques *(1815, original 1770–1782)
- Condorcet's books on Probability (1785, 1805)
- William Frend's Algebra and Arithmetic texts (1796, 1805 respectively)
- Sylvestre Lacroix’s
*Complément des Élémens d’Algébre* (1804, original 1800)
- French translation of Gauss's
*Disquistiones Arithmeticae* (1807, original 1801)
- Francisco Villela Barbosa's
*Breve Tratado de Geometria Spherica* (1817)
- Hugh Hansford’s Ciphering Book (1818)
- August Leopold Crelle's
*Journal für die reine und angewandte Mathematik* (1826–present)
- George Peacock's
*Treatise on Algebra* (1830)
- Farkas Bolyai's
*Tentamen juventutem studiosam* (1832)
- George Boole's
*A Treatise on **Differential Equations *(1865, original 1859)
- Photograph of Li Shanlan with modern mathematics students (1868–1870)
- Lucien de Puydt’s Vocabulary and Phrases of the Cuna Language, copied and arranged by Carl Hermann Berendt (1869, transcribed 1873)
- Signature of Charles Babbage and
*Babbage's Calculating Machine *museum guide (1821, 1872), contributed by Sid Kolpas
- Jules Hoüel's
*Eléments de la théorie des quaternions* (1874)
- William Clifford's
*Mathematical Fragments* (1881)
- Oliver Heaviside's
*Electromagnetic Theory* (1893–1912)
- “Consul,” the Educated Monkey, invented by William H. Robertson (1915), contributed by Sid Kolpas
- William L. Ames’s and Carl Wischmeyer’s
*Descriptive Geometry* (1918, original 1893)
- Walt Disney's
*Donald in Mathmagic Land *(1959), contributed by Sid Kolpas
- RCA Flowcharting Template (ca 1960), contributed by Amy Shell-Gellasch