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Mathematical Masterpieces: Further Chronicles by the Explorers

Arthur Knoebel, Reinhard Laubenbacher, Jerry M. Lodder, and David Pengelley
Springer Verlag
Publication Date: 
Number of Pages: 
Undergraduate Texts in Mathematics
[Reviewed by
Mark Bollman
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I teach an honors course on “7 Big Ideas of Science”, where, as far as possible, students read the original works reporting major scientific advances —  The Origin of Species, The Origin of Continents and Oceans, Watson & Crick’s paper on DNA structure, and the like. One of my primary texts is Alan Lightman’s The Discoveries, which collects 24 seminal papers in science and annotates them for easy understanding. In Mathematical Masterpieces, the authors have assembled a collection of annotated mathematical works that does this same thing for four advanced undergraduate mathematical ideas.

The idea presented here are discrete vs. continuous mathematics, algorithms for equation solving, curvature, and the quadratic reciprocity law — this is high-powered mathematics. In a compilation of this nature, the primary source material is at the mercy of its annotation; fortunately, the annotation here serves exactly its purpose: to illuminate what’s being said without overshadowing the original works. While it is certainly possible — indeed, from time to time it’s necessary — to improve on the original words of Euler or Newton, the real value of this book comes from its willingness to step back and let the original words speak for themselves.

To choose one of the four chapters: the study of numerical algorithms starts with the Rhind papyrus and runs through such notables as Girolamo Cardano, Qin Chiu-Shao, Isaac Newton, and Thomas Simpson before closing with Stephen Smale. To find all of these works in one volume would be exciting in itself; to have them in a thoughtfully executed chapter, with clear explanations as a guide and good exercises as an opportunity to follow these mathematicians, is a true delight. My only regret in reading Mathematical Masterpieces is that my honors students don’t typically have enough of a mathematical background to benefit from this book.

The authors deserve praise not only for writing a fine exposition of four major mathematical topics, but also for providing a great service to the mathematical community in bringing the original works of the masters to new audiences.

Mark Bollman ( is an associate professor of mathematics at Albion College in Michigan. His mathematical interests include number theory, probability, and geometry. His claim to be the only Project NExT fellow (Forest dot, 2002) who has taught both English composition and organic chemistry to college students has not, to his knowledge, been successfully contradicted. If it ever is, he is sure that his experience teaching introductory geology will break the deadlock.

 Preface.- The Bridge Between the Continuous and the Discrete.- Solving Equations Numerically: Finding our Roots.- Curvature and the Notion of Space.- Patterns in Prime Numbers: The Quadratic Reciprocity Law.- References.- Credits.