This book comprises five parts. The first three contain ten historical essays on important topics: number theory, calculus/analysis, and proof, respectively. Part four deals with several historically oriented courses, and Part five provides biographies of five mathematicians who played major roles in the historical events described in the first four parts of the work.

"Excursions in the History of Mathematics" was written with several goals in mind: to arouse mathematics teachers' interest in the history of their subject; to encourage mathematics teachers with at least some knowledge of the history of mathematics to offer courses with a strong historical component; and to provide an historical perspective on a number of basic topics taught in mathematics courses."

This book explores some of the major turning points in the history of mathematics, ranging from ancient Greece to the present, demonstrating the drama that has often been a part of its evolution. Studying these breakthroughs, transitions, and revolutions, their stumbling-blocks and their triumphs, can help illuminate the importance of the history of mathematics for its teaching, learning, and appreciation. Some of the turning points considered are the rise of the axiomatic method (most famously in Euclid), and the subsequent major changes in it (for example, by David Hilbert); the "wedding," via analytic geometry, of algebra and geometry; the "taming" of the infinitely small and the infinitely large; the passages from algebra to algebras, from geometry to geometries, and from arithmetic to arithmetics; and the revolutions in the late nineteenth and early twentieth centuries that resulted from Georg Cantor's creation of transfinite set theory. The origin of each turning point is discussed, along with the mathematicians involved and some of the mathematics that resulted. Problems and projects are included in each chapter to extend and increase understanding of the material. Substantial reference lists are also provided. Turning Points in the History of Mathematics will be a valuable resource for teachers of, and students in, courses in mathematics or its history. The book should also be of interest to anyone with a background in mathematics who wishes to learn more about the important moments in its development.

Prior to the nineteenth century, algebra meant the study of the solution of polynomial equations. By the twentieth century it came to encompass the study of abstract, axiomatic systems such as groups, rings, and fields. This presentation provides an account of the intellectual lineage behind many of the basic concepts, results, and theories of abstract algebra. The development of abstract algebra was propelled by the need for new tools to address certain classical problems that appeared unsolvable by classical means. A major theme of the approach in this book is to show how abstract algebra has arisen in attempts to solve some of these classical problems, providing a context from which the reader may gain a deeper appreciation of the mathematics involved. Mathematics instructors, algebraists, and historians of science will find the work a valuable reference. The book may also serve as a supplemental text for courses in abstract algebra or the history of mathematics.

This exposition provides a comprehensive historical account of the intellectual lineage behind the basic concepts, results, and theories of abstract algebra. Prior to the 19th century, the concept of "algebra" strictly referred to the study of the solution of polynomial equations. By the 20th century it came to encompass the study of abstract, axiomatic systems such as groups, rings, and fields.

Abstract algebra came into existence largely because mathematicians were unable to solve classical problems by classical means. A major theme of this book is to demonstrate how abstract algebra can illuminate the solution to some of these concrete problems. By focusing on the history of the sources of the subject and considering the context in which the originator of an idea was operating, the reader can better comprehend the "burning problem" that spurred the innovation, while gaining a deeper appreciation of the mathematics involved.

Key features:

* Begins with a substantial discussion of the history of classical algebra

* Contains separate chapters detailing the development of groups, rings, and fields

* Examines the history of linear algebra and relates its evolution to abstract algebra

* Highlights the lives and works of six notables: Cayley, Dedekind, Galois, Gauss, Hamilton, and especially the pioneering work of Noether

* Suggests a detailed outline for instructors on methods for integrating the history of abstract algebra into their teaching

* Each chapter concludes with suggested project assignments and extensive references to the relevant literature

The book may serve as a primary text for a course on the history of modern algebra, or as a supplementaltext for an abstract algebra course or a more general course on the history of mathematics. Additionally, graduate students studying to become mathematics instructors or existing math teachers will find the work particularly useful as a guide for designing a course in abstract algebra inspired by history. Finally, algebraists and historians of science will find the book a valuable reference.