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When Least is Best: How Mathematicians Discovered Many Clever Ways to Make Things as Small (or as Large) as Possible

Princeton University Press
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This book was terrific fun to read! I thought I would skim the chapters to write my review, but I was hooked by the preface, and read through the first 100 pages in one sitting (dinner was late that night). Nahin covers many of the same topics as V.M. Tikhomirov does in Stories About Maxima and Minima (volume 1 in the AMS's Mathematical World Series). In his last story, Tikhomirov says "I think that extremum problems provide wonderful material for teaching thinking, inventiveness, scientific flexibility, and the overcoming of intellectual difficulties (p. 183)," and the book under review overflows with just such material. While the two books are similar in many ways — both embed the mathematics within a historical narrative that motivates the problems and both authors employ a conversational style that talks directly to the reader — Nahin appeals to wider audience and includes discrete optimization problems and methods (probability, linear programming, graph theory) not discussed by Tikhomirov. And the emphasis is different: When Least Is Best is a history of mathematics book with lots of interesting mathematics in it. Stories About Maxima and Minima is a mathematics book with lots of interesting history in it.

So, whom is this book pitched at? Anyone who has taken calculus and a bit of physics should find it accessible. The author is an electrical engineer and makes it clear in the introduction that his themes are "plausibility and/or direct computation (p. 1),"...constructive demonstrations rather than theoretical proofs. I have read quite a few popular science and mathematics books that claim that "the first year of undergraduate calculus and physics is pretty much enough (p. xiii)" as far as prerequisite intellectual background necessary to understand the text. But I often find myself wondering who these gifted undergraduates may be! Nahin does, in fact, adhere to this promise, and although many of the calculations are lengthy and involved, comprehension does require nothing more than patience, perseverance, and first year calculus and physics.

Here are the features of this book that make it so engaging. First is the author's obvious delight and enjoyment — he is having fun and it is contagious. He is particularly enamored of paradoxical results. For example if we toss four fair dice, the most likely number of 3's to show is zero. But an even more likely event is that at least one 3 shows. "This strikes many as a paradoxical result, but that is part of the inexhaustible charm of probability! (p. 9)." Within each chapter there are teasers that egg the reader on by stating problems, facts or flaws that he will justify, prove or explain later. Within the flow of a demonstration he will state a result or generalization and give a tantalizing clue, then invite the reader to see if they can figure it out before he does so later in the chapter. Certain problems are woven throughout the book (Tikhomirov also does this) solved in different ways, connecting the mathematical ideas and encouraging the reader to look at a situation from various angles. Finally, there is no bibliography and ordinarily this would not be listed as a positive feature. However, relevant sources are included as part of the text. I appreciated not having to flip to a list of references at the back of the book, or even move my eyes to the bottom of the page to check a footnote. The citation became part of the story, not interrupting the flow. And the references are eclectic and wide ranging, including Thomas Heath's A History of Greek Mathematics, William Dunham's Journey Through Genius, Herman Melville's Moby Dick, Rebecca Goldstein's Strange Attractors, articles from Isis, Mathematics Teacher, The College Mathematics Journal, Mathematics Magazine, The American Mathematical Monthly, American Journal of Physics, Scientific American, and many more.

So exactly who is the audience for this book? It would make an excellent gift for a bright high school student who has taken or is taking AP Calculus, or her college counterpart. I've already decided to buy a few copies to offer as prizes for a student poster session, and give to my favorite graduating seniors. Mathematics professors could assign individual chapters or parts of chapters as projects in calculus, differential equations, modeling, and perhaps some other classes. I can imagine using the book as a text for a seminar, independent study, or thesis. The references could lead to advanced explorations. I can also imagine generating a series of math club talks (again, following through on some of the references as well) led by students. And, in case I haven't made it perfectly clear, every mathematically inclined colleague, relative or friend will thoroughly enjoy this delicious book.

Bonnie Shulman ( is associate professor of mathematics at Bates College in Lewiston, ME and is looking for clever ways to make the time it takes to grade large stacks of papers as small as possible. Her current research is in history and philosophy of mathematics and she is having a wonderful time teaching a senior capstone seminar on History of the Proof, tracing the development of Abel's proof of the insolvability of the quintic by radicals.
Date Received: 
Sunday, February 2, 2003
Include In BLL Rating: 
Paul J. Nahin
Publication Date: 
Bonnie Shulman

 reface xiii
1. Minimums, Maximums, Derivatives, and Computers 1
1.1 Introduction 1
1.2 When Derivatives Don't Work 4
1.3 Using Algebra to Find Minimums 5
1.4 A Civil Engineering Problem 9
1.5 The AM-GM Inequality 13
1.6 Derivatives from Physics 20
1.7 Minimizing with a Computer 24
2. The First Extremal Problems 37
2.1 The Ancient Confusion of Length and Area 37
2.2 Dido' Problem and the Isoperimetric Quotient 45
2.3 Steiner '"Solution" to Dido' Problem 56
2.4 How Steiner Stumbled 59
2.5 A "Hard "Problem with an Easy Solution 62
2.6 Fagnano' Problem 65
3. Medieval Maximization and Some Modern Twists 71
3.1 The Regiomontanus Problem 71
3.2 The Saturn Problem 77
3.3 The Envelope-Folding Problem 79
3.4 The Pipe-and-Corner Problem 85
3.5 Regiomontanus Redux 89
3.6 The Muddy Wheel Problem 94
4. The Forgotten War of Descartes and Fermat 99
4.1 Two Very Different Men 99
4.2 Snell' Law 101
4.3 Fermat, Tangent Lines, and Extrema 109
4.4 The Birth of the Derivative 114
4.5 Derivatives and Tangents 120
4.6 Snell' Law and the Principle of Least Time 127
4.7 A Popular Textbook Problem 134
4.8 Snell' Law and the Rainbow 137
5. Calculus Steps Forward, Center Stage 140
5.1 The Derivative:Controversy and Triumph 140
5.2 Paintings Again, and Kepler' Wine Barrel 147
5.3 The Mailable Package Paradox 149
5.4 Projectile Motion in a Gravitational Field 152
5.5 The Perfect Basketball Shot 158
5.6 Halley Gunnery Problem 165
5.7 De L' Hospital and His Pulley Problem, and a New Minimum Principle 171
5.8 Derivatives and the Rainbow 179
6. Beyond Calculus 200
6.1 Galileo'Problem 200
6.2 The Brachistochrone Problem 210
6.3 Comparing Galileo and Bernoulli 221
6.4 The Euler-Lagrange Equation 231
6.5 The Straight Line and the Brachistochrone 238
6.6 Galileo' Hanging Chain 240
6.7 The Catenary Again 247
6.8 The Isoperimetric Problem, Solved (at last!) 251
6.9 Minimal Area Surfaces, Plateau' Problem, and Soap Bubbles 259
6.10 The Human Side of Minimal Area Surfaces 271
7. The Modern Age Begins 279
7.1 The Fermat/Steiner Problem 279
7.2 Digging the Optimal Trench, Paving the Shortest Mail Route, and Least-Cost Paths through Directed Graphs 286
7.3 The Traveling Salesman Problem 293
7.4 Minimizing with Inequalities (Linear Programming) 295
7.5 Minimizing by Working Backwards (Dynamic Programming) 312
Appendix A. The AM-GM Inequality 331
Appendix B. The AM-QM Inequality, and Jensen' Inequality 334
Appendix C. "The Sagacity of the Bees" 342
Appendix D. Every Convex Figure Has a Perimeter Bisector 345
Appendix E. The Gravitational Free-Fall Descent Time along a Circle 347
Appendix F. The Area Enclosed by a Closed Curve 352
Appendix G. Beltrami 'Identity 359
Appendix H. The Last Word on the Lost Fisherman Problem 361
Acknowledgments 365
Index 367

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Monday, July 7, 2008