What's in Convergence? - Contents of Volume 12 - 2015

Editor: Janet Beery

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

Founding Editors: Victor Katz, Frank Swetz

Articles

Alan Turing in America, by David E. Zitarelli

Alan Turing visited the United States during 1936–38 and 1942–43.  Two of Turing's greatest accomplishments, in logic and computer design, were influenced by the first of these two visits.

Bridging the Gap Between Theory and Practice: Astronomical Instruments, by Toke Knudsen

Students in the author's Ancient Mathematical Astronomy course build armillary spheres, astrolabes, quadrants, sextants, and sundials.

Pantas’ Cabinet of Mathematical Wonders: Images and the History of Mathematics, by Frank J. Swetz

Engage your students by using images, especially those of historical objects, manuscripts, and texts, in teaching mathematics. A Spanish translation was prepared by Ximena Catepillán in 2022.

HOM SIGMAA 2015 Student Paper Contest Winners

Download the two winning papers from the 12th annual competition, a biography of Bernard Bolzano and a philosophical consideration of mistakes in mathematics.

Can You Really Derive Conic Formulae from a Cone?, by Gary S. Stoudt

Attempts to double the cube led ancient Greek mathematicians to discover and develop the conic sections.

Jan Hudde’s Second Letter: On Maxima and Minima – Translated into English, with a Brief Introduction, by Daniel J. Curtin

Optimization via algebra and arithmetic progressions with an early appearance of the Quotient Rule

Problems for Journey Through Genius: The Great Theorems of Mathematics, by William Dunham

The author shares 162 problems to help you turn his popular book into a textbook.

Euler and the Bernoullis: Learning by Teaching, by Paul Bedard

The author reflects on lessons he has learned about mathematics teaching and learning from these great mathematicians.

A GeoGebra Rendition of One of Omar Khayyam's Solutions for a Cubic Equation, by Deborah Kent and Milan Sherman

How the 11th century Persian mathematician, philosopher, and poet geometrically determined a positive real solution to a cubic equation

Oliver Byrne: The Matisse of Mathematics, by Susan M. Hawes and Sid Kolpas

The most complete biography of Byrne to date, along with tips for teaching with his famous Euclid in Colours

The 'Problem of Points' and Perseverance, by Keith Devlin

How Pascal's and Fermat's unfinished game can help teach today's students both probability and persistence

Geometric Algebra in the Classroom, by Patricia R. Allaire and Robert E. Bradley

Geometric approaches to the quadratic equation from 1700 BCE to the present

D'Alembert, Lagrange, and Reduction of Order, by Sarah Cummings and Adam Parker

Two historical approaches, one familiar and one unfamiliar, to enrich your ODE classroom

Ancient Indian Rope Geometry in the Classroom, by Cynthia J. Huffman and Scott V. Thuong

Activities, applets, and information to help students explore the geometry of altar construction in ancient India

The Cambodian Zero, by Frank J. Swetz with photos by Amir Aczel

Popular mathematics writer Amir Aczel (1950-2015) tracked down the rumored Khmer zero carved into a stone stele in 683 CE.

Some Original Sources for Modern Tales of Thales, by Michael Molinsky

Earliest known sources for stories about Thales and applets illustrating methods attributed to him

Pythagorean Cuts, by Martin Bonsangue and Harris Shultz

Euclid's proof of the Pythagorean Theorem can be adapted to shapes other than squares.

Geometrical Representation of Arithmetic Series, by Gautami Bhowmik

Hints of geometry in medieval Sanskrit arithmetic texts developed for your classroom

Edmund Halley, 1740, by Andrew Wynn Owen

An historical poem by a prize-winning Oxford poet in the form of an autobiographical reflection by Edmund Halley

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 2015:

• Early Calculating Machines (1st C. - 19th C.)

• Bakhshali Manuscript (date unknown)

• The Cambodian Zero (683 CE)

• Codex Mexicanus of the Maya (12th C.–1814)

• Aristotelian Logic in Arithmetic (1556)

• Commandino's Archimedes (1558)

• First Euclid's Elements in Both Greek and Latin (1573)

• Works of Christopher Clavius (1586, 1587, 1596, 1608)

• Keckermann's Compendium of Mathematics (1603/1661)

• Euclidis Posteriores of Christopher Clavius (1607/1654)

• Johannes Sem's Practijck des Landmetens (1620)

• Bachet's Arithmetic of Diophantus (1621/1670)

• Mathematicall Recreations in English (1633/1674)

• Schwenter’s Geometriae practicae (1641)

• Tacquet's Euclid and Archimedes (1654/1747)

• Oughtred's Euclid Clarified (1662)

• Jonas Moore's Mathematicks (1681)

• Treasury of Mathematicks (1687)

• Leibniz's Papers on Calculus

  • Differential Calculus (1684)
  • Integral Calculus (1686)
  • Fundamental Theorem (1693)

• Berkeley's "Arithmeticae" (1707)

• Berkeley's "Miscellanea Mathematica" (1707)

• Newton's Principia Mathematica (1714)

• Torricelli's Academic Lectures (1715)

• Chambers' English Cyclopaedia (1728/1738)

• George Berkeley's The Analyst (1734/1871)

• Berkeley: "In Defence of Free-Thinking in Mathematics" (1735)

• Daniel Bernoulli's Hydrodynamica (1738)

• Maclaurin's Treatise of Fluxions (1742)

• Maclaurin's Treatise on Fluxions (1801)

• Robert Simson on Conic Sections (1749/1804)

• Thomas Simpson on Fluxions (1750/1805)

• History of Mathematics for a School (1768)

• Hill's Arithmetick (11th ed., 1772)

• Early American Arithmetic Texts

  • Dilworth's Schoolmaster's Assistant (1773)
  • Nicolas Pike's Arithmetic (1788)
  • American Tutor’s Assistant, or Practical Arithmetic (1794)
  • Mercantile Arithmetic for the U.S. (1801)
  • New Athenian Oracle, or Ladies' Companion (1806)
  • Hutton's Course of Mathematics (1812/1818)
  • United States' Arithmetician (5th ed., 1832)

• Textbooks of Sylvestre Lacroix (1795-1826)

  • Differential and Integral Calculus (1802/1816)
  • Arithmetic and Descriptive Geometry (1795-97/1808-10)
  • Elements of Algebra and Geometry (1799-1800/1810-11)
  • A Text by Lacroix and Bézout in America (1820/1826)

• Davies' Geometry and Trigonometry (1828/1848)

• Boole's Calculus of Finite Differences (1859/1880)

• Cayley's Elliptic Functions (1876)

• Letter from Sylvester to Kronecker (1884), contributed by Sid Kolpas

• Lewis Carroll’s A Tangled Tale (1885)

• Minkowski's Collected Papers (1911)

• Geometric Model by Harry Wheeler (1915), contributed by Peggy Kidwell

Review

Review of Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers, by Joseph Mazur. Reviewed by Frank J. Swetz.

The reviewer finds the book to be "lively" and "interesting," but wishes the author would "begin at the beginning."