*Editors*: Victor J. Katz, Frank J. Swetz

### Articles

Apollonius's Ellipse and Evolute Revisited, by Frederick Hartmann and Robert Jantzen

Apollonius found how to draw normals to an ellipse from points in the ellipse by using hyperbolas. A modern version is presented here.

What is 0^0?, by Michael Huber and V. Frederick Rickey

The expression 0^0 is usually called an indeterminate form. This article details the history of the meaning of this expression and concludes that, in some cases, we should evaluate it as 1.

Leonardo da Vinci’s Geometric Sketches, by Frank J. Swetz

Leonardo da Vinci illustrated Luca Pacioli’s 1509 *De divina proportione.* Several of his illustrations are shown here.

Mathematics Education at West Point: The First Hundred Years, by V. Frederick Rickey and Amy Shell-Gellasch

A survey of the mathematics education of cadets in the first century after the founding of the U.S. Military Academy.

HOM SIGMAA 2008 Student Paper Contest Winners, by Victor J. Katz

There are four winners of the HOM SIGMAA Student Paper Contest for 2008. The winning papers may be accessed here.

Introducing the History of Mathematics: An Italian Experience Using Original Documents, by Adriano Dematte

A discussion of a collaborative effort in Italy to produce materials enabling secondary school teachers to use the history of mathematics in the classroom.

Triangles in the Sky: Trigonometry and Early Theories of Planetary Motion, by Sandra M. Caravella

A survey of early theories of planetary motion, with dynamic figures to help in understanding these motions.

Apportioning Representatives in the United States Congress, by Michael J. Caulfield

The history of apportionment of representatives in the U.S. Congress, from the 1790s until today, along with a discussion of the mathematics involved in the various methods.

The Quipu, by Frank J. Swetz

A collection of illustrations of Inca quipus, with references to their earliest descriptions.

Napier's *e,* by Amy Shell-Gellasch

A discussion of why we use "*e*" to represent the base of the natural logarithm system.

### Announcement

History and Pedagogy of Mathematics (HPM) 2008

The quadrennial meeting of the International Study Group on the Relations between History and Pedagogy of Mathematics will be in Mexico City, July 14-18, 2008.

### Reviews

*Benjamin Franklin's Numbers: An Unsung Mathematical Odyssey,* by Paul C. Pasles. Reviewed by Eugene Boman.

A thorough study of Benjamin Franklin's mathematical accomplishments, in particular his work on magic squares.

*Hands on History, A Resource for Teaching Math,* edited by Amy Shell-Gellasch. Reviewed by Don Crossfield.

A collection of articles about mathematical models and objects and how they can be used in teaching.

*A History of Abstract Algebra,* by Israel Kleiner. Reviewed by Ueli Daepp.

A history of the various algebraic structures that came together to give us "abstract algebra" by early in the twentieth century.

*The World of Maria Gaetana Agnesi, Mathematician of God,* by Massimo Mazzotti. Reviewed by Kathleen Ambruso Acker.

A biography stressing Agnesi's deep commitment to help those in need.

Museum of the History of Science, Oxford. Reviewed by Frank J. Swetz.

There is much to see in this museum related to the history of mathematics.

*A Biography of Maria Gaetana Agnesi,* by Antonella Cupillari. Reviewed by Edith Prentice Mendez.

A biography of the 18th century author of an early calculus text, with some translations from the text.

*Tools of American Mathematics Teaching, 1800-2000,* by Peggy Aldrich Kidwell, Amy Ackerberg-Hastings, and David Lindsay Roberts. Reviewed by Don Crossfield.

A survey of the use of technology in American mathematics teaching over the past 200 years.

*Sacred Mathematics: Japanese Temple Geometry,* by Fukagawa Hidetoshi and Tony Rothman. Reviewed by Frank J. Swetz.

This book describes some of the so-called temple geometry problems that Japanese mathematicians posed and solved beginning in the seventeenth century.

*Mathematical Expeditions: Chronicles by the Explorers,* by Reinhard Laubenbacher and David Pengelley. Reviewed by Jim Kiernan.

A collection of original texts to help students learn some important areas of mathematics.

*Mathematics in Ancient Iraq: A Social History, *by Eleanor Robson. Reviewed by Frank J. Swetz.

A new history of mathematics in ancient Mesopotamia, concentrating on its social aspects.

*Mathematics Emerging: A Sourcebook*, by Jacqueline Stedall. Reviewed by Gary Stoudt.

Textbook that provides sources in both original form and largely literal translation by Stedall.

###

### Mathematical Treasures

Mathematical Treasures from the Smith and Plimpton Collections at Columbia University, by Frank J. Swetz and Victor J. Katz

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 2008:

- Plimpton 322 (c. 1850 BCE)
- al-Khwārizmī's
*Kitāb al-jabr wa'l-muqābala* (original c. 825)
- Boethius's
*Arithmetic* (c. 1294, original 6th century)
- Manuscript of Euclid's
*Elements* (c. 1294)
- English tally sticks (1296)
- al-Ṭūsī's commentary on Euclid’s
*Elements* (original 13th century)
- de Nemore's
*De Elementis Arismetice Artis* (original 13th century)
- Omar Khayyam's
*Algebra* (13th century, original c. 1100)
*Antichissimo di Algorismo* (14th century)
- Latin edition of Euclid's
*Elements* (14th century)
- Benedetto da Firenze's
*Trattato d'arismetricha* (c. 1460)
- Treviso Arithmetic (1478)
- Pacioli's
*Summa de arithmetica, geometria, proportioni et proportionalita* (1494)
- Italian compass (15th century)
- Boethius's
*Liber Circuli* (1503)
- Reisch's
*Margarita Philosophica* (1503)
- Pacioli's
*De Divina Proportione* (1509)
- Böschenstein's
*Ain neu geordnet Rechenbiechlin* (1514)
- Oronce Fine's Geometry (1530)
- Regiomontanus's
*De Triangulis Omnimodis *(1533)
- Petrus Apianus's
*Instrumentum primi mobilis* (1534)
- Petrus Apianus's
*Introductio geographica* (1534)
- Feliciano's
*Libro di Arithmetica i Geometria* (1536)
- Tartaglia's
*Nova Scientia* (1537)
- Dürer's
*Treatise on Mensuration* (1538, original 1525)
- Cardano's
*Practica Arithmetice* (1539)
- Frisius's
*Arithmeticae Practicae Methodus Facilis* (1540)
- Recorde's
*The Grounde of Artes* (1543)
- Stifel's
*Arithmetica Integra* (1544)
- Cardano's
*Ars Magna* (1545)
- Scheubel's
*De Numeris et Diversis Rationibus* (1545)
- Italian armillary sphere (1550)
- Adam Riese's Rechenbuch (1550)
- Fine's
*Le Sphere du Monde* (1551, original 1549)
- Recorde's
*Pathway to Knowledge* (1551)
- Ghaligai's
*Pratica d'Arithmetica* (1552, original 1521)
- Tartaglia's
*General Trattato di Numeri* (1556)
- Recorde's
*The Whetstone of Witte* (1557)
- Italian astrolabe (1558)
- Barozzi's
*Procli Diadochi *(1560)
- Honoratus's
*Opus Arithmetica* (c. 1560)
- Simon Jacob's
*Rechenbuch* (1565 and 1599)
- Billingsley's edition of Euclid's
*Elements* (1570)
- Thomas Digges'
*Pantometria* (1571)
- Clavius's edition of Euclid's
*Elements* (1574)
- Clavius's
*Epitome Arithmetica Practica* (1584, original 1583)
- Pitiscus's
*Trigonometria* (1600, original 1595)
- al-Rūmī's
*Sharḥ ashkāl al-taʾsīs* (16th century, original 1412)
- Brahe's
*Astronomiae instauratae mechanica* (1602)
*Zhoubi suanjing* (1603, original c. 100 BCE)
*Jiuzhang suanshu* (1603, original c. 250)
- Zubler's
*Nova Instrumentum Geometricum* (1607)
- Jacob Köbel's
*Geometrey* (1608)
- Galileo's
*Siderius Nuncius* (1610)
- Clavius's
*Opera Mathematica* (1612)
- Kepler's
*Ausszag aus der Uralten Messekunst Archimedes* (1616)
- Kepler's
*Chilias Logarithmorum* (1624)
- Kepler's
*Tabulae Rudolphinae* (1627)
- Faulhaber's
*Academia Algebrae* (1631)
- Leonard Digges'
*Tectonicon* (1634, original 1556)
- Stevin's
*Oeuvres Mathematiques* (1634)
- Kepler's
*Epitome Astronomia Copernicanae* (1635, original 1618)
- Ramus's
*The Way to Geometry* (1636)
- Descartes'
*Discourse de la Methode* (1637)
- Galileo's
*La Operazione del Compasso Geometrico, et Militare* (1640)
- Bettini's
*Aerarium Philosophiae Mathematicae* (1648)
*Lilavati* of Bhaskara (1650)
- Eyre's
*The Exact Surveyor* (1654)
- Huygens's
*De Circuli Magnitudine Inventa* (1654)
- van Schooten's
*Exercitationes Mathematicae* (1657)
- Desargues'
*Universal Way of Dyaling* (1659, original 1640)
- van Heuraet's Rectification of Curves (1659)
- Barrow's edition of Euclid's
*Elements* (1660)
- Barrow's Latin translations of Archimedes, Apollonius and Theodosius (1675)
- Jan de Witt's
*Elementa Curvarum Linearum* (1683, original 1661)
- Wallis's
*A Treatise of Algebra* (1685)
- Moore's
*Arithmetick* (1688, original 1650)
- Sault's
*A New Treatise of Algebra* (1694)
- German protractor (c. 1700)
- Seki Kowa's
*Katsuyo sampo* (1712)
- Jacob Bernoulli's
*Ars Conjectandi* (1713)
- Ward's
*A Compendium of Algebra* (1724)
- Austrian measuring rod (1732)
- Barrow's
*Geometrical Lectures* (1735)
- Euler's Calculus of Variations textbook (1744)
- Agnesi's
*Instituzioni analitiche* (1748)
- Euler's
*Introductio in analysin infinitorum* (1748)
- Emerson's
*Treatise of Algebra* (1764)
- Wolff's
*Treatise of Algebra* (1765, original 1739)
- Euler's
*Institutiones calculi integralis* (1768)
- Alberti’s
*Instruzioni pratiche per l’ingenero civile* (1774)
- Bossut's
*Traité élémentaire de géométrie* (1775)
- Works of Isaac Newton (1779)
- Lagrange's
*Méchanique Analitique* (1788)
- Euler's
*Élémens d'Algebre* (1795, original 1767)
- Maseres'
*The Doctrine of Permutations and Combinations* (1795)
- Austrian weights (18th century)
- English gauger's scale (18th century)
- Lacroix's
*Calcul des Probabilités* (1816)
- Bowditch's translation of Laplace's
*Mécanique Céleste* (1829)
- Byrne's edition of Euclid's
*Elements* (1847)
- Korean sangi rods (19th century)