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

The contents of the book is great, but the binding on the paperback was terrible. It literally fell apart while I was reading it. Gave up on finishing it after half the pages had fallen out.

Ottimo testo, per gli appassionati.

I own about five of Paul J. Nahin’s more than dozen math & science books. What shouldn’t again be applied, though, is Sir Roger Penrose’s remark, that every equation in a book reduces the number of potential readers by half. If that would be true, I should be the only one left...By contrast, any reader to follow the text - and this means oftentimes using your brain and paper and pencil – will profit from the author’s consummate sabbatical year-long literature search for prime examples of how to minimize or maximize geometrical, physical, and economical quantities.Nahin starts the paperback edition (which is reviewed here) with a discussion of some corrections as well as feedback to the hard cover edition, and also offers a new challenge problem, which is solved in the appendix. Then there is a longer preface not quite on target to the book’s title, but this too is characteristic of the author’s prose, who as an electrical engineer doesn’t like like to be fenced in mathematically.This is why, in Chapter 1, we are presented with a mixed bag of minimization and maximization problems from algebra, physics, and geometry, which are tackled with different methods besides using derivatives, namely basic algebraic reasoning, or the inequality, stating that the arithmetic mean is always less or equal than the geometric mean, as well as computer solutions.The famous problem of finding the maximum area surrounded by a perimeter of given length, which not unexpectedly ends up to be full or half circular, comprises most of Chapter 2, only to be formally solved much later in the book, though.Chapter 3 sets the stage for the dawn of calculus with Regiomontanus’ medieval problem of maximizing the viewing angle to a hanging picture, which is to be found in almost every modern calculus text, but can still be solved without using derivatives. For other problems, even old ones, Nahin does not shy away from computer solutions, when analytic ones are too cumbersome.Chapter 4 exposes the little known fact, that Pierre de Fermat, in solving quite a few optimization problems, was thereby laying the foundations of calculus already, the formalization of which later was achieved by Leibniz and Newton. In doing this Fermat outclassed his rival Descartes, especially by stating the principle of least time, which correctly explains Snell’s law of refraction. Although Descartes’ proof of this law was flawed, he still was correctly applying it to spherical raindrops and thereby showed that the rainbow is a caustic due to a minimum of the deflection angle for sun-rays.Nahin, at the end of Chapter 5, does perform these geometric optics calculations in the modern way by using derivatives for both the primary and secondary rainbow. He then expands this analysis to the tertiary rainbow, which both Newton and Halley had already predicted. At the time of writing his book, Nahin had no other clue than to accept their predictions, that the tertiary rainbow would never be observed in nature. However, this was proved to be wrong in the years 2011-14, when three amateur photographers from Germany and the Netherlands obtained and published images of the third, fourth, and fifth rainbow order produced by sunlight.For the most part, Chapter 5 explains the birth of the derivative, some of its rules and its application to a wide range of problems in algebra, geometry, kinematics, and mechanics, the latter with an example, that equilibrium is attained, when potential energy is at a minimum. Who has read this far, has finished more than half of the book’s 372 pages.Chapter 6, with 79 pages the book’s longest, bears the title “Beyond Calculus”, and leaves the level of what a typical freshman year student will master without additional resources. Galilei’s fastest track problem for a sliding object (the so called brachistochrone), which was first solved to be the cycloid by Johann Bernoulli, asked for new methods of optimizing whole functions instead of just a certain value. This is now handled systematically by using the calculus of variations, invented by the 18th century mathematicians Lagrange and Euler, which demands the minimization of a certain integral or functional. Nahin demonstrates this method for both the brachistochrone and the catenary (hanging chain equilibrium curve), then continues with showing a proof of the isoperimetric problem at last, and finally exposes the quite advanced problem of minimal area surfaces, the solutions of which happen to be analogous to soap bubble films.Chapter 7 heralds “The Modern Age” of optimization by pointing to a small selection of problems from discrete mathematics, like the optimal placement of service points w.r. to output facilities, the shortest path through a network of nodes, composing a cost-optimal diet from a given selection of nutrients, or deriving optimal production plans and schedules. For lack of space to build up the linear algebraic and graph theoretical foundations of this field, Nahin chooses to give his readers an idea of two widely used methods at least, i.e. linear and dynamic programming, which are linked with the 20th century mathematicians Dantzig and Bellman.The book ends with 36 pages of appendices of mathematical proofs and morsels and solutions to challenge problems.In conclusion, “When Least Is Best”, although meticulously type-set and profusely illustrated with figures and diagrams in black and white, is certainly not a coffee table book, but rather a highly recommended addition to any application math lover’s roster, and a valuable resource of ideas for teachers at the senior high school or college levels. Be prepared, however, to having to re-do many of Nahin’s calculations step by step again, even when perusing his book recurrently.

Fast service. Product in mint condition.

Very well written. Never boring.