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A History of the Calculus of Variations in the Eighteenth Century
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AMS Chelsea Publishing
1988; 154 pp; hardcover
Volume: 177
Reprint/Revision History:
first AMS printing 2004
ISBN-10: 0-8218-3647-1
ISBN-13: 978-0-8218-3647-7
List Price: US$32 Member Price: US$28.80
Order Code: CHEL/177.H

Shortly after the invention of differential and integral calculus, the calculus of variations was developed. The new calculus looks for functions that minimize or maximize some quantity, such as the brachistochrone problem, which was solved by Johann Bernoulli, Leibniz, Newton, Jacob Bernoulli and l'Hôpital and is sometimes considered as the starting point of the calculus of variations. In Woodhouse's book, first published in 1810, he has interwoven the historical progress with the scientific development of the subject.

The reader will have the opportunity to see how calculus, during its first one hundred years, developed by seemingly tiny increments to become the highly polished subject that we know today. Here, Woodhouse's interweaving of history and science gives his special point of view on the mathematics. As he states in his preface: "Indeed the authors who write near the beginnings of science are, in general, the most instructive; they take the reader more along with them, show him the real difficulties and, which is the main point, teach him the subject, the way they themselves learned it."

Reviews

"Those interested in the beginnings of the Calculus of Variations should read the book by Lagrange and the--unfortunately, very rare--book by Woodhouse. Euler's masterpiece [Methodus ... ] will then offer him very little difficulty. This work of Euler's is distinguished by the abundance of its examples and is one of the most interesting books that the mathematical literature has to offer."

-- C. Carathéodory, in Variationsrechnung und Partielle Differentialgleichungen erster Ordnung

"[Woodhouse's] work details the history of the Calculus of Variations from its origin until the close of the eighteenth century and has obtained a high reputation for accuracy and clearness."

-- I. Todhunter

On Isoperimetrical Problems, and the Calculus of Variations
• Chap. I: The problem of the curve of quickest descent proposed; James Bernoulli's solution of it; Principle of that solution
• Chap. II: Isoperimetrical problems proposed by James Bernoulli; Solved by John Bernoulli; His fundamental and specific equations; Application of them to the curve of quickest descent, and of a given length; Solution of isoperimetrical problems, by Brook Taylor; Imperfections of his and the Bernoullis's methods
• Chap. III: Euler's first memoir on isoperimetrical problems; Table of formulæ; Solution of problems by it; Methods of Thomas Simpson, Emerson, and Mac Laurin
• Chap IV: Euler's second memoir; General formulæ of solution; Characters of distinction, which problems admit of; Exceptions to the general formulæ
• Chap. V: Euler's tract, entitled "Methodus inveniendi Lineas Curvas Proprietate Maximi Minimive gaudentes"; Distribution of cases into absolute and relative maxima and minima; Rules for finding the increment of quantities dependent on their varied state; Formulæ of solution
• Chap. VI: Lagrange's memoir; Use of an appropriate symbol to denote the Variation of a quantity; Rules for finding the variation; New process of deducing Euler's formulæ; Invention of new formulæ
• Chap. VII: Lagrange's general method of treating isoperimetrical problems; Equation of limits; Cases of relative maxima and minima reduced to those of absolute
• Chap. VIII: Particular formulæ deduced from the general one, for the purpose of facilitating the solution of problems; Problems solved