Janet Heine Barnett

Enter, Stage Center: The Early Drama of the Hyperbolic Functions

In addition to the standard definitions of the hyperbolic functions (for instance, coshx = (eX + e-X)/2), current calculus textbooks typically share two common features: a comment on the applicability of these functions to certain physical problems (for instance, the shape of a hanging cable knowll as the catenary) and a remark on the analogies that exist between properties of the hyperbolic functions and those of the trigonometric functions (for instance, the identities cosh2x-sinh2x = 1 and cos2 x + sin2 x = 1). Texts that offer historical sidebars are likely to credit development of the hyperbolic functions to the 1 8th-century mathematician Johann Lambert. Implicit in this treatment is the suggestion that Lambert and others were interested in the hyperbolic functions in order to solve problems such as predicting the shape of the catenary. Left hanging is the question of whether hyperbolic functions were developed in a deliberate effort to find functions with trig-like properties that were required by physical problems, or whether these trig-like properties were unintended and unforeseen by-products of the solutions to these physical problems. The drama of the early years of the hyperbolic functions is far ricller than either cf these plot lines would xuggest.

Resources for Teaching Discrete Mathematics: Early Writings on Graph Theory: Euler Circuits and The Königsberg Bridge Problem

The ‘feeble glance’ which Leonhard Euler (1707 1783) directed towards the geometry of position consists of a single paper now considered to be the starting point of modern graph theory. Within the history of mathematics, the eighteenth century itself is commonly known as ‘The Age of Euler’ in recognition of the tremendous contributions that Euler made to mathematics during this period. Born in Basel, Switzerland, Euler studied mathematics under Johann Bernoulli (1667 1748), then one of the leading European mathematicians of the time and among the first — along with his

Learning Mathematics via Primary Historical Sources: Straight from the Source's Mouth.

Abstract Much has been written about the benefits to be derived by incorporating the history of mathematics into its teaching. Of the various ways to do this, the use of original sources is among the most thrilling for the insights and the challenges it offers for students and instructors alike. In this paper, we describe examples of one particular approach to address these challenges that preserves the thrill of the read.

Teaching and Learning Mathematics from Primary Historical Sources.

Abstract Why would anyone think of teaching and learning mathematics directly from primary historical sources? We aim to answer this question while sharing our own experiences, and those of our students across several decades. We will first describe the evolution of our motivation for teaching with primary sources, and our current view of the advantages and challenges of a pedagogy based on teaching with primary sources. We then present three lower-division case studies based on our classroom experience of teaching discrete mathematics courses with student projects based on primary sources, and comment on how these could be adapted for use with other lower-division audiences.

Resources for Teaching Discrete Mathematics: Early Writings on Graph Theory: Topological Connections

The earliest origins of graph theory can be found in puzzles and game, including Euler’s Konigsberg Bridge Problem and Hamilton’s Icosian Game. A second important branch of mathematics that grew out of these same humble beginnings was the study of position (“analysis situs”), known today as topology1. In this project, we examine some important connections between algebra, topology and graph theory that were recognized during the years from 1845 1930. The origin of these connections lie in work done by physicist Gustav Robert Kirchhoff [1824 1887] on the flow of electricity in a network of wires. Kirchhoff showed how the current flow around a network (which may be thought of as a graph) leads to a set of linear equations, one for each circuit in the graph. Because these equations are not necessarily independent, the question of how to determine a complete set of mutually independent equations naturally arose. Following Kirchhoff’s publication of his answer to this question in 1847, mathematicians slowly began to apply his mathematical techniques to problems in topology. The work done by the French mathematician Henri Poincare [1854 1912] was especially important, and laid the foundations of a new subject now known as “algebraic topology.” This project is based on excerpts from a 1922 paper in which the American mathematician Oswald Veblen [1880 1960] shows how Poincare formalized the ideas of Kirchhoff. An American mathematician born in Iowa, Veblen’s father was also a mathematician who taught mathematics and physics at the State University of Iowa. At that time, graduate programs in mathematics were relatively young in the United States. A member of the first generation of American mathematicians to complete their advanced work in the United States rather than Europe, Oswald Veblen completed his Ph.D. at the University of Chicago in 1903. He remained in Chicago for two years before joining the mathematics faculty at Princeton. In 1930, he became the first faculty member of the newly founded Institute for Advanced Study at Princeton.2 A talented fund-raiser and organizer, Veblen also served on the Institute’s Board of Trustees in its early years.

Effect of a random real on\(\kappa \to \left( {\kappa ,{\text{ }}\left( {_{\omega _1 }^\alpha } \right)} \right)^2 \)

AbstractIt is consistent that

An American Postulate Theorist: Edward V. Huntington

\kappa \to (\kappa ,{\text{ }}\left( {_{\omega _1 }^\alpha } \right))^2

The Pedagogy of Primary Historical Sources in Mathematics: Classroom Practice Meets Theoretical Frameworks.

holds in the random extension.