Mathematical Expeditions: Chronicles by the Explorers, by Reinhard Laubenbacher and David Pengelley, 1999, X, 275 pages, 93 illus., Hardcover, ISBN 978-0-387-98434-6, $79.95. Series: Undergraduate Texts in Mathematics, Springer-Verlag, New York, Inc
Laubenbacher and Pengelley are on a crusade to clarify the development of mathematics for students through the use of original source materials. Mathematical Expeditions was never intended to be a full exposition of the history of mathematics. Instead the authors have picked five essential threads : the concept of parallel lines, set theory, calculating areas and volumes, Fermat’s last theorem, and the search for formulas that will solve polynomial equations. Each of the five sections are organized in similar fashion and can be followed independently. These sections vary from 40-60 pages in length beginning with a 10-20 page introduction followed by 3-6 original sources supplemented by sufficient background information and 30-50 exercises.
As an example, the longest section, on calculating area and volume, has a 14 page introduction followed by selections from Archimedes’ Quadrature of the Parabola, Archimedes’ Method, Cavalieri’s Six Geometrical Exercises, Leibniz’ More on Geometric Measurement, Cauchy’s Lectures on the Infinitesimal Calculus, and Robinson’s Non-Standard Analysis. While the 47 pages which contain these selections are not all strictly original text, the student will be sufficiently exposed to the words of its original creators. This section has 48 exercises sprinkled throughout its eight parts which include an appendix on infinite series.
There is much to be commended in this work. Several of the reading selections are not available in standard sourcebooks. Many are not available in English. To place them in context with the more familiar readings is a great asset. On the whole the exercises are thought provoking but sometimes too broad in scope. Consider 1.1:“Read about world history from 1750 until 1850.” The independent nature of the sections is a cause for some repetition of background material. There are numerous diagrams and photos scattered throughout the book. I am somewhat stymied by the use of a “photo” of Pythagoras ( p.172) after the wiser choice of a picture of a plaque portraying Euclid ( p.18). While the book is not one that I would use as the sole textbook for a standard undergraduate course on the history of mathematics, I do think that it would be very useful for either a second semester or graduate course. These faults are minor compared to the overall worth of the book. I find Mathematical Expeditions to be a resource to which I constantly refer.. I recommend its placement on the bookshelf of anyone planning to teach a course on the history of mathematics.
James F. Kiernan, Adjunct Professor, Brooklyn College, CUNY, New York