Mathematics and Its History (3rd ed.)

Description

cited in the references of the historical introduction of Cantwell's Introduction to Symmetry Analysis (pp. xl-xli, PDF pp. 42-43)

From the reviews of the second edition:"This book covers many interesting topics not usually covered in a present day undergraduate course, as well as certain basic topics such as the development of the calculus and the solution of polynomial equations. The fact that the topics are introduced in their historical contexts will enable students to better appreciate and understand the mathematical ideas involved...If one constructs a list of topics central to a history course, then they would closely resemble those chosen here."(David Parrott, Australian Mathematical Society)"The book...is presented in a lively style without unnecessary detail. It is very stimulating and will be appreciated not only by students. Much attention is paid to problems and to the development of mathematics before the end of the nineteenth century... This book brings to the non-specialist interested in mathematics many interesting results. It can be recommended for seminars and will be enjoyed by the broad mathematical community." (European Mathematical Society)"Since Stillwell treats many topics, most mathematicians will learn a lot from this book as well as they will find pleasant and rather clear expositions of custom materials. The book is accessible to students that have already experienced calculus, algebra and geometry and will give them a good account of how the different branches of mathematics interact."(Denis Bonheure, Bulletin of the Belgian Society)This third edition includes new chapters on simple groups and combinatorics, and new sections on several topics, including the Poincare conjecture. The book has also been enriched by added exercises.

The present book is a very interesting and useful attempt to give a unified view of undergraduate mathematics by approaching the subject through its history. The author motivates this project as follows: "One of the disappointments experienced by most mathematics students is that they never get a course in mathematics. They get courses in calculus, algebra, topology, and so on, but the division of labor in teaching seems to prevent three different topics from being combined into a whole.'' Indeed, there is a gap just of the kind described in most mathematics curricula, and a historical approach seems to be the right way to fill this gap. In the book under review there are 20 chapters dealing with the following topics: Pythagoras' theorem, Greek geometry, Greek number theory, infinity in Greek mathematics, polynomial equations, analytic geometry, projective geometry, calculus, infinite series, number theory, elliptic functions, mechanics, complex numbers in algebra, complex numbers and curves, complex numbers and functions, differential geometry, non-Euclidean geometry, group theory, topology, sets, logic and computation. Each chapter provides a historical survey explaining the origins of the basic ideas, hints at the connections between the subject treated and other fields of mathematics and science, remarks on recent discoveries and results, nice exercises and some biographical notes.
The author presents the mathematics of the past in modern notation and explains it by using modern concepts and results. This has the risk that the students obtain a biased impression of the content and form of classial sources. On the other hand this presentation makes it much easier to grasp the main ideas. This book is highly recommended as the basis for courses, especially for students who want to become teachers at secondary schools. It should be used together with a good source book (those by Struik, Fauvel and Gray, for instance) in order to complete the historical approach to important mathematical fields and their mutual relations by studying selected sources.

Reviewed by Walter Purkert