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Mathematical Expeditions: Chronicles by the Explorers

Reinhard Laubenbacher and David Pengelley
Publication Date: 
Number of Pages: 
Undergraduate Texts in Mathematics
[Reviewed by
Daniel E. Otero
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The 1990s have witnessed a marked increase in interest within the mathematical community in reliable accounts of the history of the subject and its many subdisciplines. Several successful books have appeared telling the stories of mathematicians (such as Constance Reid's biographies of Hilbert, Courant, and Julia Robinson) and mathematical ideas (like Eli Maor's books on e, infinity, and trigonometry). Victor Katz's comprehensive A History of Mathematics won the 1995 Watson Davis Prize from the History of Science Society, and William Dunham's well-received mathematics histories have found wide audiences.

But while researchers have always kept abreast of the genesis of their fields, the greatest surge of interest in this decade has been among mathematics educators. The MAA has come out as a strong champion of supporting these interests by sponsoring the Institute for the History of Mathematics and Its Use in Teaching. This five-year long workshop has provided amateurs and professionals alike with opportunities to discover how to help their students realize that what they learn in their textbooks and classrooms is the result of centuries of (often incremental and hard-fought) discoveries by dozens of the best minds on the planet, people from different eras and cultures, with a variety of motivations, whose commonality is simply doing mathematics.

One of the more interesting trends to emerge from this program to enliven the mathematics classroom with historical topics is the use of original sources. The members of this cult will introduce undergraduates to mathematics in the raw, unadulterated by the typical formalism that appears in most textbooks. They take seriously the sentiment expressed by Abel that "if one wishes to make progress in mathematics, one should study the masters and not the pupils"; moreover, they believe that this exhortation applies to beginners as well as more seasoned practitioners.

The authors of the book under review are among the leaders in this camp. "Nothing captures the excitement of discovery as authentically as a description by the discoverers themselves," they write in their preface. And as the title suggests, Laubenbacher and Pengelley envision their book as something of a travelogue, wherein they bring the reader through five independent Expeditions--one chapter each in geometry, set theory, analysis, number theory and algebra--all by means of a handful of excerpted original sources (in English translations). Each chapter covers about fifty pages of text, and is essentially independent of the others. The format is similar for each: an extended introductory essay charts the basic story line, setting out the mathematical history that is explored in that Expedition. This is followed by three to five primary sources that serve as landmarks within that history. Making sense of these sources is what brings the reader into intimate contact with some of the great minds that shaped the important mathematical developments within that field. In addition, Laubenbacher and Pengelley have prepared extensive bibliographical annotations which guide the reader toward the literature helpful in extending these investigations.

In fact, this is the most striking feature of the book, more important even than the inclusion of the primary sources: the reader who is serious about making headway through this book must be warned that he or she will be at a disadvantage without access to a decent library. Hardly a page passes without two or three references to the 181-item bibliography at the back of the book. And not a small number of the exercises that accompany the Expeditions send us off to the stacks in search of some other treasures, as in this example:

Exercise 3.26: Read and explain Fermat's method of finding the tangent line to a parabola via infinitesimals. Compare the notation of the French and English translations [59, III, pp. 121-23][58, pp. 358-359][166, pp. 223-24] with the Latin original [59, I, pp. 133-35]. Is Fermat's result already in Proposition 2 from Archimedes' "Quadrature of the Parabola" [3]?

This after all is the chief service that Mathematical Expeditions provides. Laubenbacher and Pengelley have furnished us with more than just a glimpse into the heads of great mathematicians. They have armed us with the tools we need to search out the historical literature ourselves and follow the tracks laid by our mathematical forefathers and, in the striking case of Sophie Germain whose work in number theory we read here, a foremother as well!

However, any book that covers such wide-ranging territory cannot be free from some errors. For instance, they claim that "when Euclid makes use of a previously proven proposition, he refers to it in brackets," and that he "identifies rectangles (including squares) by [denoting] opposite corners" when it is impossible for us to expect that the bracketed references or the labeling of figures in the edited text are really Euclid's own. And the statement that the "search for Pythagorean triples goes back at least to the Babylonians" is a common misinterpretation of the Babylonian tablet Plimpton 322. More problematic, however, is the poor quality of the century-old translation of the source by Lobachevsky in the geometry chapter; it is indecipherable in spots and stilted throughout.

Far outweighing any of these difficulties, on the other hand, are the wonderful introductory essays that begin the chapters. The one in Chapter 3, on analysis, is the best short account of the history of the calculus I've ever read. Of course, the sources in this same chapter seemed to me the toughest going in the book. Archimedes' "Quadrature of the Parabola" requires great care on the part of the reader to plow through successfully, and Cavalieri's work on the quadrature of the higher parabolas is nearly impenetrable.

There is a big difference between learning mathematics and its history from a presentation such as we have here and learning it from more traditional works of history (like the ones cited in the first paragraph above). To follow our sojourning metaphor further, it's the difference between getting to know some exotic locale by, on the one hand, hiking off into the countryside, struggling with the native tongue, and surprising oneself with the unfamiliar cuisine, and, on the other, buying a ticket on a bus tour of the same area, where you are whisked from one swanky hotel to another with stops at all the sights that can accommodate parking for large vehicles. The second method of travel provides you with all the comforts of home, but you miss most of the local color; the first method gives a much more satisfying experience, but you've got to work hard for it. Mathematical Expeditions is not an easy book to handle, but it's not meant to be enjoyed in isolation either. Think of it as a key to unlock the mathematical literature that those before us have left behind. Happy travels!

Daniel E. Otero ( is professor of mathematics at Xavier University in Cincinnati, OH. In addition to being active in the Ohio Section of the MAA, he serves on the editorial board for the MAA series of Classroom Resource Materials. His interests are in number theory and the history of mathematics. Prof. Otero is co-founder with Daniel J. Curtin (Northern Kentucky University) of the ORESME (Ohio River Early Sources in Mathematics Exposition) Reading Group, and is designing a calculus course for humanities majors based on original sources.

Preface.- Geometry: The Parallel Postulate.- Set Theory: Taming the Infinite.- Analysis: Calculating Areas and Volumes.- Number Theory: Fermat's Last Theorem.- Algebra: The Search for an Elusive Formula.- References.- Credits.- Index.