What's in Convergence? - Contents of Volume 11 - 2014

Editor:  Janet Beery

Associate Editors: Amy Ackerberg-Hastings, Janet Heine Barnett, Kathleen Clark, Lawrence D'Antonio, Douglas Ensley, Victor Katz, Daniel Otero, Randy Schwartz, Lee Stemkoski, Frank Swetz

Founding Editors: Victor Katz, Frank Swetz

Articles

An Investigation of Subtraction Algorithms from the 18th and 19th Centuries, by Nicole M. Wessman-Enzinger

This survey of four subtraction algorithms used in North America includes as sources both handwritten "cyphering books" and printed arithmetic texts.

Connecting Greek Ladders and Continued Fractions, by Kurt Herzinger and Robert Wisner

An exploration of two historical techniques for estimating irrational numbers and a link between them

Did Euler Know Quadratic Reciprocity?: New Insights from a Forgotten Work, by Paul Bialek and Dominic W. Klyve

The authors use their newly translated paper of Leonhard Euler to answer their title question.

Cubes, Conic Sections, and Crockett Johnson, by Stephanie Cawthorne and Judy Green

Author and illustrator Johnson, author of Harold and the Purple Crayon, posed a question about Euclid, cubes, and conic sections, and painted an answer!

When Nine Points Are Worth But Eight: Euler's Resolution of Cramer's Paradox, by Robert Bradley and Lee Stemkoski

Interactive graphics illustrate the seeming paradox that 9 points should determine a curve of order 3, yet two curves of order 3 can intersect in up to 9 distinct points.

David Hilbert's Radio Address, by James T. Smith

Read and listen to this famous 1930 address, with its dramatic conclusion: "Wir müssen wissen; wir werden wissen." ("We must know; we will know.")

HOM SIGMAA 2014 Student Paper Contest Winners

Read winning student papers on the statistics of Florence Nightingale and on Legendre's attempts to prove Euclid's Fifth Postulate.

Celebrating a Mathematical Miracle: Logarithms Turn 400, by Glen Van Brummelen

Why John Napier's invention of logarithms in 1614 was hailed as a miracle by astronomers and mathematicians

Wibold's Ludus Regularis, a 10th Century Board Game, by Richard Pulskamp and Daniel Otero

Players competed for virtues in this dice game for clerics.

How to Improve a Math History Assignment, by Christopher Goff

Moving college students' original source mathematics history projects beyond "reporting" to "engagement"

Historical Activities for the Calculus Classroom, by Gabriela R. Sanchis

History and mathematics of curve sketching, tangent lines, and optimization, explored using interactive applets

A Pair of Articles on the Parallelogram Theorem of Pierre Varignon, by Peter N. Oliver

Mathematical life of Varignon, plus ideas for classroom activities and extensions of his famous theorem

Unreasonable Effectiveness of Knot Theory, by Mario Livio

Knot theory has become surprisingly useful in explaining string theory.

Proofs Without Words and Beyond, by Tim Doyle, Lauren Kutler, Robin Miller, and Albert Schueller

History and philosophy of visual proofs, together with dynamic, interactive "proofs without words 2.0"

Van Schooten's Ruler Constructions, by C. Edward Sandifer

Translation of and commentary on Frans van Schooten's work on constructions using only a straightedge -- and a postulate that allows the copying of one line segment onto another.

Led Astray by a Right Triangle: Misconception, Epiphany, and Redemption, by Frank J. Swetz

A well-known historian initially erred in his study of ancient Chinese mathematics.

Euclid21: Euclid's Elements for the 21st Century, by Eugene Boman, Alexandra Milbrand, Tyler Brown, Siddharth Dahiya, Joseph Roberge, and Mary Boman

A dynamic, interactive Euclid's Elements organized as a directed graph via its logical structure

Online Museum Collections in the Mathematics Classroom, by Amy Ackerberg-Hastings and Amy Shell-Gellasch

The Smithsonian Institution's National Museum of American History website features dozens of object groups, collections of digitized object images and detailed catalog records, related to mathematics. View sample objects and read suggestions for using these resources in your teaching.

Mathematical Treasures from the National Museum of American History, Smithsonian Institution, by Amy Ackerberg-Hastings, Judy Green, Peggy Kidwell, and Amy Shell-Gellasch

• Abaci & Numeral Frames – Modern Chinese Abacus
• Adders – Troncet Arithmographe Owned by Daniel Draper
• Adding Machines – Thomas Hill's Patent Model for a Key-Driven Adding Machine
• Arithmetic Teaching Apparatus – Add-A-Count Scale for Teaching Arithmetic
• Bookkeeping Machines – National Class 3000 Bookkeeping Machine
• Calculating Machines – Frederick A. P. Barnard's Calculating Machine
• Cash & Credit Registers – Model Cash Register Designed by the Ritty Brothers
• Charter Members of MAA – Geometric Model by Harry Wheeler, 1915
• Counters – Moffett Register for Recording Alcohol Sales
• Dividers & Drawing Compasses – Maurice Kidjel's Ratio Cali-Pro
• Electronic Calculators (Desktop) – Wang Loci-2 Programmable Calculator
• Ellipsographs – Keuffel & Esser Trammel Ellipsograph
• Geometric Models – Elongated Pentagonal Pyramid
• Kinematic Models – Hypotrochoid Kinematic Model by M. Schilling
• Mathematical Charts & Tables – Number Line Owned by Andrew Gleason
• Mathematical Paintings – Crockett Johnson's Mathematical Paintings
• Metric System Demonstration Apparatus – Demonstration Metric Grain Measures
• Parallel Rules – Captain Field's Improved Parallel Rule
• Pens & Pencils – Rubik's Cube Museum Promotional Pen
• Planimeters – Polar Planimeter Invented by Jacob Amsler
• Protractors – Japanese Protractor for the 1876 World's Fair
• Scale Rules – Nels Ockerlund's Patent Model for a Combination Ruler
• Sectors – Sector from Jesse Ramsden's Workshop
• Sets of Drawing Instruments – Early Modern Drawing Instruments
• Slide Rules – Raymond Clare Archibald's Slide Rule
• Spherometers – Grunow Spherometer
• Tabulating Equipment – Statistical Tabulator of Benjamin Wood
• Triangles & Squares – Blackboard-Sized Triangles
• Women in Mathematics – Objects Related to Women Mathematicians

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 2014:

• Ishango Bone (20000-25000 years ago, Congo)

• Old Babylonian Area Calculation (circa 1800-1600 BCE)

• Ptolemy's Almagest (1515 edition of 2nd century work)

• The Arithmetic of Boethius (circa 500 CE)

• Zhu Shijie’s New Arithmetic Enlightenment (1299, China)

• Sacrobosco's 13th Century Astronomy (1482 copy)

• Sacrobosco's 13th Century Arithmetic (1490 copy)

• Al-Banna's Lifting of the Veil in the Operations of Calculation (c. 1300, North Africa)

• Islamic Astronomical Chart (15th century)

• Piero della Francesca's illustrated Works of Archimedes (late 1450s, Italy)

• Luca Pacioli's Elements of Euclid (1509)

• Luca Pacioli's De Divina Proportione (1509)

• Cardano's Acceptance of Ptolemy's Astrology (1554)

• Cornelis Cort's Engravings of Arithmetica and Geometria (1565)

• Italian Arithmetica Pratica (1575 manuscript)

• John Blagrave's Mathematical Jewel (1585)

• Kepler’s Mysterium Cosmographicum (1596)

• Al-‘Amili's Quintessence of Calculation (circa 1600, Persia)

• Ricci's and Xu's Theory of Surveying (circa 1610)

• Bachet's Mathematical Recreations (1612)

• Rathborne’s Surveyor in Four Books (1616)

• Robert Fludd's History of the Macrocosm (1618)

• Edmund Gunter's Measuring Instruments (1624/1636)

• Thomas Harriot's Algebra (1631)

• William Oughtred's Clavis Mathematicae (1631/1667)

• Daniel Schwenter’s Delicia (1636 and 1651)

• Descartes' Principia Philosophiae (1644)

• Galileo's Collected Works (1656)

• Archimedes, as Translated by John Wallis (1676)

• John Seller's Mathematical Tables (1677)

• Isaac Barrow on Mathematics Education (1685/1734)

• Alexander's Synopsis Algebraica (1693)

• Sturm's Elements of Mathematics (1695)

• De Moivre's Doctrine of Chances (1711/1756)

• Mathematics for Army and Navy (1730)

• Johann Bernoulli’s Opera Omnia, Vol. I (1690-1713) (1742)

• Jacob Bernoulli's Collected Works (1744)

• Logic for American Independence (1748/1811)

• Kästner’s Foundations of Mathematics (1764)

• Playfair on Teaching Geometry (1795/1832)

• Lagrange's Theorie des Fonctions Analytiques (1797/1813)

• Lazare Carnot's Geometry (1803)

• Monge on Differential Geometry (1807/1809)

• Legendre's Essai sur la Théorie des Nombres (1808)

• Jeremiah Day's Introduction to Algebra (1814/1834)

• Bolzano's Three Problems (1817)

• Cauchy's Cours d'Analyse (1821)

• Poncelet's Projective Geometry (1822/1865)

• Davies' American Textbooks (1826-1840)

• Babbage's Tables of Logarithms (1827/1872)

• Somerville's Translation of Laplace's Celestial Mechanics (1831)

• Julius Plucker on Geometry (1835 and 1839)

• Charles Gill's Mathematical Miscellany (1836)

• Salmon's Treatise on Conic Sections (1848/1869)

• Michel Chasles’ Higher Geometry (1852/1880)

• Dirichlet's Lectures in Number Theory (1863)

• George Boole’s Differential Equations (1859/1877)

• Hamilton's Elements of Quaternions (1866)

• Riemann's Differential Equations (1869)

• Cantor's 'On Trigonometric Series' (1871)

• Schlegel on Grassman’s Extension Theory (1872)

• Klein's Non-Euclidean Geometry (1892-1893)

• Klein's Famous Problems of Elementary Geometry (1897)

• Poincaré on Probability (1907/1912)

• Dedekind on the Nature of Number (1909)

• Whitehead Introduction to Mathematics (1911)

• L. E. Dickson's Algebraic Invariants (1914)

• Mathematischen Wissenschaften: Felix Klein (1924)

• Volterra’s Theory of Functionals (1930)

• Mathematischen Wissenschaften: Hilbert and Bernays (1939)

• Emile Borel’s Elements of Set Theory (1949)

Review

Review of Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World, by Amir Alexander. Reviewed by Frank J. Swetz.

How 17th century European proponents of indivisibles and infinitesimals clashed with Thomas Hobbes, Christopher Clavius, and the Catholic Church