*Editor:* Janet Beery

*Associate Editors:* Amy Ackerberg-Hastings, Janet 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

The Mathematical Cultures of Medieval Europe, by Victor J. Katz

Cultural influences on the mathematics of Islamic, Jewish, and Catholic scholars

Recreational Problems in Medieval Mathematics, by Victor J. Katz

Two problems that endured across time, space, and culture

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 Treasure: Japanese Mathematics in the Edo Period, by Frank J. Swetz

Twelve distinct works illustrate the range of mathematics produced in Japan from 1603 to 1867.

Mathematical Treasure: Billingsley's Sources for the First English Euclid's Elements, by Frank J. Swetz

Two Greek sources 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 (posted 08/31/18)

An 18th century almanac for "ladies" became a source for mathematical problems and solutions.

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

Math Origins, by Erik R. Tou

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

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) (posted 09/27/18)
- 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) (posted 09/26/18)
- Euclid's
*Elements* from Italy (c. 1510)
- Bradwardine's Geometry and Arithmetic (1511 and 1513)
- Martinez's
*Liber arithmetice practice* (1513) (posted 09/22/18)
- Jacob Köbel's Book on Gauging (1515) (posted 09/17/18)
- 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) (posted 09/05/18)
- Apian's
*Folium populi: Instrumentum* (1533)
- Schoener's
*Algorithmus* (1534)
- Sfortunati's
*Nuovo lume / Libro di arithmetica* (1534) (posted 09/29/18)
- Dürer's
*Institutionum Geometricarum* (1535 and 1606)
- Tagliente's Commercial Arithmetic (1535, original 1515) (posted 10/02/18)
- 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) (posted 10/01/18)
- Gaspar de Texeda's Commerial Arithmetic (1546)
- Ptolemy's
*Almagest* per Regiomontanus (1550)
- Nicholaus Medler's
*Rudimenta arithmeticae* (1550, original 1543) (posted 09/23/18)
- Andre Tacquet's
*Cylindrica et annularia *(1651)
- Oronce Fine's
*On the Practice of Geometry* (1556) (posted 09/10/18)
- Peverone's Arithmetic and Geometry (1558, 1581)
- Pierre de Savonne's Arithmetic (1571)
- Commandino's
*Elements* of Euclid (1572) (posted 10/05/18)
*Libro de Abbaco* (1574)
- 16th Century Tuscan Arithmetic (1577)
- Johann Kandlern's
*Arithmetica* (1578) (posted 09/14/18)
- 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) (posted 09/12/18)
- 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) (posted 08/29/18)
- Trigonometry of Benjamin Bramer (1617) (posted 08/27/18)
- Geometry of Benjamin Bramer (1618) (posted 08/26/18)
- Henrion's
*Collection*: Arithmetic and Algebra (1621)
- Euclid's
*Data* per Claude Hardy (1625) (posted 10/06/18)
- Kepler's
*Rudolphine Tables* and defense of Brahe (1625 and 1627) (posted 10/13/18)
- 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) (posted 09/28/18)
- Oughtred's Defense of His Slide Rule (1632-1633) (posted 09/24/18)
- Frobenius's Logarithms of Rheticus's Trig Values (1634) (posted 10/04/18)
- Mydorge's Recreational Mathematics (1639)
- Pierre Herigone's
*Cursus Mathematicus* (1644)
- Cavalieri's Indivisibles (1647) (posted 09/09/18)
- William Oughtred's
* Key of the mathematics (section on notation) *(1647)* *
- Tycho Brahe's
*New Astronomy* (1648) (posted 10/15/18)
- 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) (posted 10/10/18)
- Galileo's
*Dialogues* in English (1661) (posted 10/18/18)
- Pietro Mengoli's
*Geometriae speciosae* *elementa (1659)*
- Stephano degli Angeli's
*Accessionis ad Stereometriam et Mecanicam *(1662)
- Tycho Brahe's Work and Observatory (1666) (posted 10/14/18)
- Antoine Arnauld's
*Nouveaux elemens de geometrie *(1667)
- Nicolaus Mercator's
*Logarithmotechnia* (1668)
- Astronomy of Hevelius and Hevelius (1668 and 1673) (posted 10/12/18)
- Bachet's Arithmetic of Diophantus (1670)
- Collected Works of Oronce Fine (1670, original 1587) (posted 10/11/18)
- John Kersey's
*Elements of Algebra* (1673)
- Roman Finger Numerals (1673) (posted 09/13/18)
- Isaac Barrow's Euclidean Geometry (1678, original 1655)
- Isaac Barrow's Lecture on Archimedes (1678)
- Blondel's Course of Mathematics (1683) (posted 09/08/18)
- Jacques Ozanam's
*La géométrie pratique* (1684)
- Benjamin Bramer's Applied Geometry (1684) (posted 08/25/18)
- Venturini's
*Libro di Conti di me* (1686)
- First edition of Newton's
*Principia* (1687) (posted 10/16/18)
- Leybourn's Survey of Applied Mathematics (1690) (posted 09/20/18)
- Euclid's
*Elements* of Henrik Coets (1692, original 1691) (posted 09/06/18)
- De Roberval's
*Divers* *Ouvrages* (1693)
- Jean Prestet's
*New Elements of Mathematics *(1694)
- Kegel's Commercial Arithmetic (1696) (posted 09/15/18)
- Peter Lauremberg's
*Institutiones Arithmeticae* (1698, original 1621) (posted 09/19/18)
- 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) (posted 10/03/18)
- 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) (posted 09/18/18)
- Keill's 'Plain' and Spherical Trigonometry (1726, original 1715) (posted 09/16/18)
- Arnauld's
*Art of Thinking* (1727, original 1662)
- Newton's
*Principia* in English (1728 and 1729) (posted 10/17/18)
- Construction and Use of the Sector (1729)
- Mandey's
*Synopsis Mathematica Universalis* (1729, original 1702) (posted 09/21/18)
- Doppelmayr's History of Mathematics in Nuremberg (1730)
- 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)
- Mathematical Lessons or Interviews (Panchaud, 1743)
- Euler on Maxima and Minima (1744)
- Euler and the Two-body Problem (1744) (posted 10/08/18)
- David Gregory's
*Practical Geometry* (1745)
- Collected Works of Giulio di Fagnano (1750)
- Penther on Geometry and Surveying (1752, original 1732) (posted 09/11/18)
- Clairaut's
*Elements of Algebra* (1760, original 1749)
- Müller's
*Elements of Mathematics* (1765, original 1748)
*Problemi di geometria practica *(1767)
- Isaac Newton's Universal Arithmetik (1769, original Latin 1720)
- William Emerson's
*Cyclomathesis* (1770)
- Tables of Johann Lambert (1770) (posted 08/24/18)
- Clairaut's
*Geometry* in Italian Translation (1771, original 1741)
- Slide rule and gauging sticks (18th century)
*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) (posted 10/09/18)
- French commercial arithmetic manuscript (1778) (posted 09/13/18)
- Blaise Pascal's Collected Works (1779) (posted 09/25/18)
- 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) (posted 09/04/18)
- 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)
*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)
*Mathematical Questions Proposed in the Ladies’ Diary *(1817) (posted 09/01/18)
- Calculus and Analysis of Augustin-Louis Cauchy (1821 and 1823) (posted 08/22/18)
- 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) (posted 09/07/18)
*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* (1847)
- Ramachundra'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) (posted 09/03/18)
- 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) (posted 08/28/18)
- Sturm’s
* **Cours d’Analyse* (1877, original 1857)
- Todhunter's 3D Analytic Geometry (1878, original 1858)
- Collected Works of Laplace (1878) (posted 10/07/18)
- Higher Plane Curves of George Salmon (1879, original 1852) (posted 09/01/18)
- 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 Work 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) (posted 08/23/18)
- Hilbert on Algebraic Number Fields (1897) (posted 09/03/18)
- 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) (posted 09/30/18)
- 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) (posted 09/02/18)
- Solid Analytic Geometry of George Salmon (1912, original 1862) (posted 08/30/18)
- 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)