*Editor:* Janet Beery

*Associate Editors:* Amy Ackerberg-Hastings, Janet Heine Barnett, Maureen Carroll, Lawrence D'Antonio, Victor Katz, Michael Molinsky, Elyn Rykken, Randy Schwartz, Amy Shell-Gellasch, Jody Sorensen, Gary Stoudt, Erik R. Tou

*Founding Editors:* Victor Katz, Frank Swetz

### Articles

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 Treasures of Japan in the Edo Period, by Frank J. Swetz Twelve distinct works illustrate the range of mathematics produced in Japan from 1603 to 1867.

Billingsley's Sources for the First English Euclid's Elements: Two Annotated Mathematical Treasures, by Frank J. Swetz A Greek source and a Latin source 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.

Ongoing Series

A Series of Mini-projects from **TR**ansforming **I**nstruction in **U**ndergraduate **M**athematics via **P**rimary **H**istorical **S**ources, by Janet Barnett, Kathy Clark, Dominic Klyve, Jerry Lodder, Danny Otero, Nick Scoville, and Diana White

- 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

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 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.

Mathematical Treasures from the Linda Hall Library added during 2018:

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 2018:

- Two mathematical tablets from ancient Sumer (27th-20th cent. BCE)
- The Venerable Bede's Computus (c. 1180, original 725)
- Boethius's
*Arithmetic* (1294 ms. copy of 6th-cent. work)
- Alfonsine Tables (13th century)
- Lilavati of Bhaskara (18th cent. ms. of 11th-cent. work)
- Zhu Shijie's
*Precious Mirror of the Four Elements *(19th-cent. ed. of 1303 work)
- Campanus commentary on Euclid's
*Elements* (c. 1300-1315)
- Paolo Dagomari's 14th Century Arithmetic (1339)
- Euclid's
*Elements *from England (c. 1385-1399)
- Giovanni's 15th-century Commercial Arithmetic (1422)
- 15th-century Compendium of Arithmetic (c. 1424)
- Arithmetic of Joseph ben Moses Sarfati (early 15th cent.)
- Arithmetic of Abu Abd al-Rahman Ya’qub ibn Ayyub (1445, original 13th cent.)
- Al-Khwarizmi's
*Algebra* (1456 ms. of 12th cent. Latin trans. of 9th cent. work)
- Benedetto da Firenze's
*Trattato d'arismetriche *(c. 1460 ms.)
- Early Translation of Euclid's
*Elements* into Arabic (1466, original 9th century)
- Italian arithmetic manuscript (1473)
*Ars numerandi* (*The Art of Numbering*) (anonymous) (1482)
*Computus* of Anianus (1488, original 13th century)
*Kalendarium* of Regiomontanus (1489, original 1474)
- Venetian merchant's mathematical notebook (15th-16th cent.)
- Counting board
*jetons* from Nuremberg (c. 1490-1635)
- Bradwardine on numeration (1503)
- Georg von Peurbach's Arithmetic (1503)
- Euclid's
*Elements* from Italy (c. 1510)
- Bradwardine's Geometry and Arithmetic (1511 and 1513)
- Martinez's
*Liber arithmetice practice* (1513)
- Jacob Köbel's Book on Gauging (1515)
- La Roche's Arithmetic (1520)
*Grahalāghava* of Gaṇeśa Daivajne (original c. 1520)
- Girolamo Tagliente's Commercial Arithmetic (1525)
- Fernel's
*De proportionibus* (1528, 4 new images)
- Arithmetic Books of Johann Neudörffer (c. 1530)
- Koebel's Book on the Astrolabe (1532)
- Apian's
*Quadrans Apiani astronomicus* (1532)
- Apian's
*Folium populi: Instrumentum* (1533)
- Schoener's
*Algorithmus* (1534)
- Sfortunati's
*Nuovo lume / Libro di arithmetica* (1534)
- Dürer's
*Institutionum Geometricarum* (1535 and 1606)
- Tagliente's Commercial Arithmetic (1535, original 1515)
- Jacob Kobel's Rechenbuch (1537, original 1514)
- Commercial Arithmetic by Petrus Apianus (1537, original 1527)
- Georg von Peurbach's
*Arithmetices elementa *(1544)
- Michael Stifel's
*Deutsche Arithmetica* (1545)
- Gaspar de Texeda's Commerial Arithmetic (1546)
- Ptolemy's
*Almagest* per Regiomontanus (1550)
- Nicholaus Medler's
*Rudimenta arithmeticae* (1550, original 1543)
- Andre Tacquet's
*Cylindrica et annularia *(1651)
- Oronce Fine's
*On the Practice of Geometry* (1556)
- Peverone's Arithmetic and Geometry (1558, 1581)
- Pierre de Savonne's Arithmetic (1571)
- Commandino's
*Elements* of Euclid (1572)
*Libro de Abbaco* (1574)
- 16th Century Tuscan Arithmetic (1577)
- Johann Kandlern's
*Arithmetica* (1578)
- Humfrey Baker's Arithmetic (1580, original 1562)
- Ramus's
*Arithmetices *(1586)
- Viète's
*Supplementum Geometriae* (1593)
- Heron's Commentary on Euclid's
*Elements* (16th century, original 1st century)
- Anonymous German Arithmetic (c. 1600)
- Flemish Commercial Arithmetic (c. 1600)
- Early French Commercial Arithmetic (1602, original 1557)
- Set of Napier rods or bones (17th cent.)
- English university student notes (17th cent.)
- Hulsius on Mechanical Instruments (1603-1604)
- The
*Trevelyon Miscellany* (1608)
- Galileo's
* *letter about Jupiter's moons (1609)
- Galileo's
* De proportionum instrumento *(1612)
- Henrion's Military Mathematics (1613)
- Henrion's Translation of
*Sphaerics* of Theodosius (1615)
- Trigonometry of Benjamin Bramer (1617)
- Geometry of Benjamin Bramer (1618)
- Henrion's
*Collection*: Arithmetic and Algebra (1621)
- Euclid's
*Data* per Claude Hardy (1625)
- Kepler's
*Rudolphine Tables* and defense of Brahe (1625 and 1627)
- Henrion's Use of the Mecometre (1630)
- Oughtred's Key to Mathematics (1631)
- Denis Henrion's
*Use of the Proportional Compass* (1631, 1st ed. 1618)
- Van Schooten's Sine Table (1632, original 1627)
- Oughtred's Defense of His Slide Rule (1632–1633)
- Frobenius's Logarithms of Rheticus's Trig Values (1634)
- Mydorge's Recreational Mathematics (1639)
- Pierre Herigone's
*Cursus Mathematicus* (1644)
- Cavalieri's Indivisibles (1647)
- William Oughtred's
* Key of the mathematics *(section on notation)* *(1647)* *
- Tycho Brahe's
*New Astronomy* (1648)
- Wingate's Arithmetic Using Logarithms (1648/1652)
- Andre Tacquet's
*Cylindrica et annularia *(1651)
- Leybourn's
*Compleat Surveyor* (1653)
- Schott's Mathematics and Physics (1657)
- Galileo's
*Dialogues* in English (1661)
- Pietro Mengoli's
*Geometriae speciosae* *elementa *(1659)
- Stephano degli Angeli's
*Accessionis ad Stereometriam et Mecanicam *(1662)
- Tycho Brahe's Work and Observatory (1666)
- Antoine Arnauld's
*Nouveaux elemens de geometrie *(1667)
- Nicolaus Mercator's
*Logarithmotechnia* (1668)
- Astronomy of Hevelius and Hevelius (1668 and 1673)
- Bachet's Arithmetic of Diophantus (1670)
- Collected Works of Oronce Fine (1670, original 1587)
- John Kersey's
*Elements of Algebra* (1673)
- Roman Finger Numerals (1673)
- Isaac Barrow's Euclidean Geometry (1678, original 1655)
- Isaac Barrow's Lecture on Archimedes (1678)
- Blondel's Course of Mathematics (1683)
- Jacques Ozanam's
*La géométrie pratique* (1684)
- Benjamin Bramer's Applied Geometry (1684)
- Venturini's
*Libro di Conti di me* (1686)
- First edition of Newton's
*Principia* (1687)
- Leybourn's Survey of Applied Mathematics (1690)
- Euclid's
*Elements* of Henrik Coets (1692, original 1691)
- De Roberval's
*Divers* *Ouvrages* (1693)
- Jean Prestet's
*New Elements of Mathematics *(1694)
- Kegel's Commercial Arithmetic (1696)
- Peter Lauremberg's
*Institutiones Arithmeticae* (1698, original 1621)
- Malconet's Geometry for Self-Study (1700)
- Leonty Magnitsky's
*Arithmetika *(1703)
- Joseph Raphson's
*Analysis equationum universalis *(1702, original 1690)
- William Jones's
*Synopsis Palmariorum Matheseos *(1706)
- Lydal's
*Vulgar and Decimal Arithmetic* (1710)
- Isaac Newton's
*Analysis per quantitatum series, fluxiones ac differentias *(1711)
- Christian Wolff's
*Trigonometrie* (1711)
- Everard's
*Stereometry or Art of Gauging* (1712)
- Halcken's
*Deliciae Mathematicae* (1719)
- Edward Hatton's
*System of Arithmetic* (1721)
- Kraus's Lengthy Commercial Arithmetic (1722)
- Keill's 'Plain' and Spherical Trigonometry (1726, original 1715)
- Arnauld's
*Art of Thinking* (1727, original 1662)
- Newton's
*Principia* in English (1728 and 1729)
- Cunn's Construction and Use of the Sector (1729)
- Mandey's
*Synopsis Mathematica Universalis* (1729, original 1702)
- Doppelmayr's History of Mathematics in Nuremberg (1730)
- Leybourn's Description and Use of Gunter's Quadrant (1731)
- Lamy's
*Mathematical Works* (1734)
- Leonhard Euler's
*Mechanica* (1736)
- Heilbronner's History of Mathematics (1742)
- Emerson's
*Doctrine of Fluxions* (1743 and 1768)
- Panchaud's
*Mathematical Lessons or Interviews* (1743)
- Euler on Maxima and Minima (1744)
- Euler and the Two-body Problem (1744)
- David Gregory's
*Practical Geometry* (1745)
- Collected Works of Giulio di Fagnano (1750)
- Penther on Geometry and Surveying (1752, original 1732)
- Clairaut's
*Elements of Algebra* (1760, original 1749)
- Müller's
*Elements of Mathematics* (1765, original 1748)
- Francesco Vecelli's
*Problemi di geometria practica *(1767)
- Isaac Newton's
*Universal Arithmetik* (1769, original Latin 1720)
- William Emerson's
*Cyclomathesis* (1770)
- Tables of Johann Lambert (1770)
- Clairaut's
*Geometry* in Italian Translation (1771, original 1741)
- Slide rule and gauging sticks (18th century)
- Samuel Davis's
*Mensuration Made Perfectly Easy* (c. 1770s)
- Sénebier's
*Traite d’Arithmetique* (1774)
- Leupold's Practical Arithmetic and Geometry (1774, original 1727)
- Hawney's
*Complete Measurer* (1775/1813)
- Langsdorf on
*Kaestner's Analysis of Finite Quantities* (1776-1777)
- Euler's Expansion of Robins's
*Gunnery* (1777, original 1745)
- French commercial arithmetic manuscript (1778)
- Blaise Pascal's Collected Works (1779)
- Davison's
*New System of Arithmetic* (1780)
- French Mathematical Encyclopedias (1784 and 1798)
- Guyot's
*Nouvelles récréations physiques et mathématiques* (1786, original 1769)
- Three Books by Simon L'Huilier (1786, 1804 and 1809)
- Euler's
*Letters to a German Princess* (1787, original 1769-73)
- Fujita Kagen's
* Sacred Mathematics* (1789)
- Étienne Bézout's Military Mathematics (1793, original 1770-82)
- Lorenzo Mascheroni's
* La Geometria del Compasso *(1797)
- De Condillac's
*La Langue des Calculs* (1877, original 1798)
- Lacroix on Definite Integrals (1800)
- Maria Agnesi’s
*Analytical Institutions* (1801)
- Sylvestre Lacroix's
*Trigonometry* (1803, original 1798)
- Charles Hutton's
*Mathematical Tables* (4th ed., 1804)
- Michael Walsh's
*Mercantile Arithmetic* for the U.S. (1807)
- Ludlam's
*Rudiments of Mathematics* (1809, original 1785)
- Paolo Ruffini's
*Riflessioni intorno alla soluzione delle equazioni algebraiche* (1813)
- Legendre's Integral Calculus (1816)
- Sylvestre Lacroix on Teaching (1816, original 1805)
- Leslie's Geometry and Trigonometry (1817, original 1809)
- Leybourn's
*Mathematical Questions Proposed in the Ladies’ Diary *(1817)
- Calculus and Analysis of Augustin-Louis Cauchy (1821 and 1823)
- Duncan's
*Supplement to Playfair's Geometry and Wood's Algebra* (1824, original 1822)
- Cauchy on Definite Integrals (1825)
- Cauchy on Applications of Calculus to Geometry (1826)
- Cauchy's
*Exercices de Mathematiques *(1826)
- Simpson's
*A Treatise of Algebra* (1826, original 1745)
- Cauchy on Solving Equations (1829)
*Pure Mathematics* by Francoeur (1829-1830, original 1809)
- Robert Simson's
*Elements of Euclid* (1834, original 1756)
*Hoppus's Tables for Measuring* (1837, original 1736)
- De Morgan's
*Elements of Trigonometry and Trigonometrical Analysis* (1837)
- Halliwell's
*Rara Mathematica* (1839)
- Oliver Byrne's
*Practical Gager* (1840)
- Boole's
*Mathematical Logic* (1847)
*Encyclopedia of Pure Mathematics*, edited by Peter Barlow et al. (1847)
- Ramchundra's
*Treatise on Problems of Maxima and Minima Solved by Algebra *(1850)
- Hamilton's
*Lectures on Quaternions* (1853)
- Albert Church's Geometry for West Point (1867, original 1864)
- Cherokee-English Arithmetic (1870)
- Jordan on Group Theory (1870)
- Potts's
*Elements of Euclid* (1871, original 1845)
- Todhunter's
*Elements of Euclid* (1872, original 1862)
- Todhunter on Laplace, Lame, and Bessel Functions (1875)
- Todhunter's
*Algebra for Beginners* (1876, original 1858)
- Higher Algebra of George Salmon (1876, original 1859)
- Sturm’s
* **Cours d’Analyse* (1877, original 1857)
- Todhunter's 3D Analytic Geometry (1878, original 1858)
- Collected Works of Laplace (1878)
- Higher Plane Curves of George Salmon (1879, original 1852)
- Chinese Encyclopedia of Mathematics (1882, original 1723)
- Peirce's
* **Linear Associative Algebra* (1882)
- Dodgson's Euclid, Books I and II (1883, original 1875)
- Clifford on Riemann Surfaces (1882)
- Weierstrass on Elliptic Functions (1887)
- Todhunter's
*Integral Calculus* (1889, original 1878)
- Two Works of Heinrich Weber (1891 and 1895)
- Forsyth's
* Complex Variables *Text (1893)
- Harkness's and Morley's
*Theory of Functions* (1893)
- Appell and Goursat's
* Analysis *(1895)
- Collected Works of Leopold Kronecker (1895)
- Burkhardt on Analytic Functions (1897)
- Dirichlet's Collected Works (1897)
- Mary Boole on the Psychology of Mathematics (1897)
- Complete Works of Évariste Galois (1897)
- Hilbert on Algebraic Number Fields (1897)
- Peano's
*Formulaire de* *Mathématiques* (1901)
- Snyder and Hutchinson's
* **Differential and Integral Calculus* (1902)
- Mary Boole on the Teaching of Arithmetic (1903)
- Goursat's
* Calculus Classic *(1904, original 1902)
- Eisenhart's
*Differential Geometry* (1906)
- David Eugene Smith's
*Rara Arithmetica* (1908)
- Wilson's
* **Solid Geometry and Conic Sections* (1908, original 1872)
- Boole's
*Philosophy and Fun of Algebra* (1909)
- Hadamard's
* Calculus of Variations *(1910)* *
- Minkowski's
*Geometry of Numbers *(1910)
- Solid Analytic Geometry of George Salmon (1912, original 1862)
- Fubini's
* Analysis *(1920, original 1913)
- Hobson's
* **Theory of Functions of a Real Variable* (1921, original 1907)
- Hardy's
* **A Course in Pure Mathematics* (1921, original 1908)
- Oswald Veblen’s
* **Analysis Situs* (1922)
- Lamb's
* **An Elementary Course of Infinitesimal Calculus* (1934, original 1897)
- Sophus Lie’s
* Collected Papers *(1934)