Throughout the history of mathematics, maximum and minimum problems have played an important role in the evolution of the field. Many beautiful and important problems have appeared in a variety of branches of mathematics and physics, as well as in other fields of sciences. The greatest scientists of the past - Euclid, Archimedes, Heron, the Bernoullis, Newton and many others - took part in seeking solutions to these concrete problems. The solutions stimulated the development of the theory, and, as a result, techniques were elaborated that made possible the solution of a tremendous variety of problems by a single method. This book, copublished with the Mathematical Association of America (MAA), presents fifteen "stories" designed to acquaint readers with the central concepts of the theory of maxima and minima, as well as with its illustrious history. Unlike most AMS publications, the book is accessible to high school students and would likely be of interest to a wide variety of readers. In Part One, the author familiarizes readers with many concrete problems that lead to discussion of the work of some of the greatest mathematicians of all time. Part Two introduces a method for solving maximum and minimum problems that originated with Lagrange. While the content of this method has varied constantly, its basic conception has endured for over two centuries. The final story is addressed primarily to those who teach mathematics, for it impinges on the question of how and why to teach. Throughout the book, the author strives to show how the analysis of diverse facts gives rise to a general idea, how this idea is transformed, how it is enriched by new content, and how to remains the same in spite of these changes.

Presents hundreds of extreme value problems, examples, and solutions primarily through Euclidean geometry Unified approach to the subject, with emphasis on geometric, algebraic, analytic, and combinatorial reasoning Applications to physics, engineering, and economics Ideal for use at the junior and senior undergraduate level, with wide appeal to students, teachers, professional mathematicians, and puzzle enthusiasts

Describes techniques for solving problems in maxima and minima other than the methods of calculus.

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This book began with my edition of the anonymous treatise. A translation and notes seemed essential if the material of the treatise was to be understood. It then seemed that Chapter 5 of Heytesbury's Rules for Solving Sophismata, on which the treatise was based, should also be included. My translation of the Heytesbury treatise is based on a fifteenth-century edition, supplemented by readings from a few of the better manuscripts. (A critical edition from all the manuscripts, of which Chapter 5 will be mine, is now in progress under the supervision of Paul Spade, but only a few insignificant changes in the translation should be necessitated by the completed edition. ) An examination of related materials seemed reasonable, and these included Heytesbury's commentator Gaetano, as well as a chapter from a treatise by Johannes Venator (in an edition in progress provided by Francesco del Punta). It seemed unnecessary to publish Gaetano's and Venator's related works in this volume, but all their departures from Heytesbury and the anonymous treatise are noted here. I have not examined other works in the tradition in any detail. I owe a great deal to my teacher, Norman Kretzmann, not only as regards the edition and translations, but also as regards the notes, study and introduction. The referees of the typescript (to me unknown) made unusually thorough criticisms and suggestions to which I have paid close attention. The book is far better for my having done so.

The purpose of this book is to put together in one place the basic elementary techniques for solving problems in maxima minima other than the methods of calculus and linear programming. The emphasis is not on individual problems, but on methods that solve large classes of problems. The many chapters of the book can be read independently, without references to what precedes or follows. Besides the many problems solved in the book, others are left to the reader to solve, with sketches of solutions given in the later pages.

This book provides a unified account of the theory required to establish upper and lower bounds.