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The calculus of variations is a subject whose beginning can be precisely dated. It might be said to begin at the moment that Euler coined the name calculus of variations but this is, of course, not the true moment of inception of the subject. It would not have been unreasonable if I had gone back to the set of isoperimetric problems considered by Greek mathemati cians such as Zenodorus (c. 200 B. C. ) and preserved by Pappus (c. 300 A. D. ). I have not done this since these problems were solved by geometric means. Instead I have arbitrarily chosen to begin with Fermat's elegant principle of least time. He used this principle in 1662 to show how a light ray was refracted at the interface between two optical media of different densities. This analysis of Fermat seems to me especially appropriate as a starting point: He used the methods of the calculus to minimize the time of passage cif a light ray through the two media, and his method was adapted by John Bernoulli to solve the brachystochrone problem. There have been several other histories of the subject, but they are now hopelessly archaic. One by Robert Woodhouse appeared in 1810 and another by Isaac Todhunter in 1861.
Analysis as an independent subject was created as part of the scientific revolution in the seventeenth century. Kepler, Galileo, Descartes, Fermat, Huygens, Newton, and Leibniz, to name but a few, contributed to its genesis. Since the end of the seventeenth century, the historical progress of mathematical analysis has displayed unique vitality and momentum. No other mathematical field has so profoundly influenced the development of modern scientific thinking. Describing this multidimensional historical development requires an in-depth discussion which includes a reconstruction of general trends and an examination of the specific problems. This volume is designed as a collective work of authors who are proven experts in the history of mathematics. It clarifies the conceptual change that analysis underwent during its development while elucidating the influence of specific applications and describing the relevance of biographical and philosophical backgrounds. The first ten chapters of the book outline chronological development and the last three chapters survey the history of differential equations, the calculus of variations, and functional analysis. Special features are a separate chapter on the development of the theory of complex functions in the nineteenth century and two chapters on the influence of physics on analysis. One is about the origins of analytical mechanics, and one treats the development of boundary-value problems of mathematical physics (especially potential theory) in the nineteenth century. The book presents an accurate and very readable account of the history of analysis. Each chapter provides a comprehensive bibliography. Mathematical examples have been carefully chosen so that readers with a modest background in mathematics can follow them. It is suitable for mathematical historians and a general mathematical audience.
Why does nature prefer some shapes and not others? The variety of sizes, shapes, and irregularities in nature is endless. Skillfully integrating striking full-color illustrations, the authors describe the efforts by scientists and mathematicians since the Renaissance to identify and describe the principles underlying the shape of natural forms. But can one set of laws account for both the symmetry and irregularity as well as the infinite variety of nature's designs? A complete answer to this question is likely never to be discovered. Yet, it is fascinating to see how the search for some simple universal laws down through the ages has increased our understanding of nature. The Parsimonious Universe looks at examples from the world around us at a non-mathematical, non-technical level to show that nature achieves efficiency by being stingy with the energy it expends.
* Examines the history and philosophy of the mathematical sciences in a cultural context, tracing their evolution from ancient times up to the twentieth century * 176 articles contributed by authors of 18 nationalities * Chronological table of main events in the development of mathematics * Fully integrated index of people, events and topics * Annotated bibliographies of both classic and contemporary sources * Unique coverage of Ancient and non-Western traditions of mathematics