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

This book does so many things well, that I would get bored trying to explain them all. What really impressed me was the explanation of the Euler-Lagrange equation. What is incredible about the treatment is that it is so easy to understand but doesn't leave out any of the math. For anyone trying to teach themselves the calculus of variations I recommend this book as an intro before jumping into a textbook.

I've recently finished the above book and can't tell the reader just how much I enjoyed it, particularly the connections made that often get lost in the usual silo approach to math topics. Plus, Professor Nahin explains the math extremely well and makes it fascinating. I wish there were more similar books at this level. I appreciate the time it must take to put together a book like that but, if Professor Nahin was ever thinking about the next topic(!), how about Linear Algebra and the connections with geometry and calculus -- something along the lines of books by W.W. Sawyer, but more advanced? I know he would do a superb, and valuable, job.

Mr. Nahin states in his preface that 1st year undergraduate math and physics is enough to manage "a lot of mathematics in this book." He is fairly on the mark, discounting my comments about chapter six below. As usual, the reader must keep pencils and scrap paper ready to fully appreciate this book. I hoped to find a book based on applications of math and physics, an engineer's approach. This is one such fascinating book. I was familiar with the AM-GM inequality technique to find extremas. However, Mr. Nahin dispenses of this method early and shows the reader so much more. And in this book, there is a constant exercise of looking at problems a different way. If you like geometric solutions along with the typical lines of algebraic manipulations, you'll love this book. The first five chapters are packed with problems and solutions with excellent graphic representations. Integration requirements increase throughout. In finding extremas in chapter six, the author goes beyond ordinary calculus with the calculus of variations including the Euler-Lagrange differential equation and Beltrami's identity. The focus problem is the minimal decent time curve. It is in section 6.4 that the author truly breaks from his stated reader requirements of "high school algebra, trigonometry, and geometry, as well as the elementary integration techniques." I think most authors of this book's scope typically underestimate reader requirements. As for my part, I did not understand the calculus of variations technique on the first reading. After reading sections 6.4 through 6.8 again, I gained an appreciation of how the method works. After one more reading of these sections, I might know just enough to be dangerous. These challenging sections are well written, but a struggle within the stated reader requirements. Chapter 7 found me in more comfortable ground where great geometric solutions to problems are shown and there is a keen introduction to linear programming. In various cases, Mr. Nahin works through problems with results generated by computer programs. These are not my favorite problems because I lack access to the high end (very expensive) programs that he uses. This book is well written and engaging; and it is easier to manage than An Imaginary Tale. This is my second book by Mr. Nahin, and I view him as a favorite author of technical books. In this review, I intentionally avoided mentioning specific problems covered because I do not want to spoil the surprises. I found them all quite fascinating. The reader will see so many real world physics in a different light. I highly recommend this book.

Finally, a solid book that challenges the lay reader just like the best math teachers do - by showing the elegance and power of mathematical reasoning.This is top shelf material. Nahin is one heck of writer and must be one hell of a teacher! Bravo!Already ordered his book on the history of imaginary numbers.6 stars: ******

Nahin's book is a tour de force about the deep intellectual threads that surround the notion of optimality. In physics, engineering, and mathematics, while touching on a wide range of applications, he asks over and over again: What is the optimal solution and why does it matter? Since I've spent most of my professional career thinking about optimality in one form or another, I was skeptical about how much new I would find in this book. But I was astounded to find something new and interesting on virtually every page. Some examples:--Preface: Torricelli's funnel, which has finite volume and can be filled, but has infinite surface area and cannot be painted; and a slick proof that an irrational number raised to an irrational power can be rational.--Chapter 1: An optimization problem that is not amenable to calculus, but whose solution can be discerned by some clever insight; an optimization problem that is amenable to calculus, but whose solution can be arrived at by algebra; and the use of the arithmetic mean-geometric mean inequality in optimization.--Chapter 2: The ancient isoperimetric problem of Dido on maximal area, how it remained unsolved until modern times; the fact that there exists a figure in the plane whose area is equal to the area of the period at the end of this sentence and which contains a line segment one million light years in length that can be rotated 360 degrees within the figure (the shape of the figure is a little hard to picture); and the fact that there are two consecutive prime numbers the gap between which is greater than a googolplex (don't ask what they are).--Chapter 3: Optimization problems involving the viewing of a painting, the rings of Saturn, folding envelopes, carrying a pipe around a corner in a hallway, the maximum height of mud ejected from a wheel, and other daily concerns.--Chapter 4: Snell's law, the path of light, and the feud between Descartes and Fermat. --Chapter 5: The power of the calculus, the aiming of basketballs and cannon, Kepler's wine barrel, United Parcel Service package size constraints, L'Hospital's pulley problem, and the geometry of rainbows.Chapter 6: Galileo's work on the descent of a particle sliding along the arc of a circle; the discovery of the minimum-time brachistochrone curve by Jacob Bernoulli arrived at by an argument based on the path of light in a variable-density medium, his feud with Newton, and Newton's anonymously published solution to the problem; the isochronous property of both the circle and brachistochrone, which states that the descent time is independent of the starting location along the cure (a point mentioned in chapter 96 of Moby Dick and which left me wondering which paths are isochronous since a straight line is clearly not); the fact that the brachistochrone is about 1.5% faster than the circular arc and that a brachistochrone tunnel dug from New York to Los Angeles would entail a travel time of a mere 28 minutes assuming frictionless slidin