The author set himself the goal of discussing, and putting into proper perspective, all significant contributions to the development of the calculus of variations, beginning with Fermat's principle of least time and continuing through the entire 19th century. He confines his treatment to single integral problems. This book should be of value not only to the many aficionados of the calculus of variations but to all mathematical analysts, considering how fertile a ground the calculus of variations has been for the application of all important results of analysis, and how many problems in analysis and functional analysis were inspired by the calculus of variations.The book is subdivided into 7 chapters: 1. Fermat, Newton, Leibnitz, and the Bernoullis; 2. Euler; 3. Lagrange and Legendre; 4. Jacobi and his School; 5. Weierstrass; 6. Clebsch, Meyer, and Others; 7. Hilbert, Kneser, and Others.Each chapter begins with a brief summary and then proceeds with a somewhat detailed discussion of those contributions that fall into the epoch that is identified by the chapter heading and which the author considers important for the subsequent development of the subject. To make the early contributions readable, the author renders them in his own words and in 20th century terminology and notation, interspersed with direct quotes. One even finds a (very poor) facsimile of one of Newton's computations. As the author proceeds through the 18th and 19th century, the need for such reformulation diminishes and the renditions, to an ever increasing degree, come closer to the originals. There appears, however, to be no compelling reason for having retained the archaic notation h for h under the lim symbol in a discussion of a 1904 paper by Max Mason (pp. 363-365). As far as this reviewer is aware, this is the first time that extensive summaries of some of the classical contributions to the calculus of variations become accessible in the English language. (This cannot be asserted with great conviction, however, as this reviewer has never had the need for seeking English translations of Latin, French or German contributions.)In order to keep this book within reasonable bounds, the author had to make many hard decisions when selecting or rejecting material for inclusion. We believe that he has done this fairly and with great insight. Still, it is somewhat of a disappointment to find mention of von Escherich's contribution to the theory of conjugate points in the Meyer problem, but not Johann Radon's. It was the latter who brought this theory to a successful conclusion by establishing that the determinant of a conjugate solution of the variational equations does not vanish identically in any subinterval of [a, b] (Zum Problem von Lagrange, Abh. aus dem Math. Sem. der Hamburgischen Universitaet, vol. 6, pp. 273-299), a result that had eluded von Escherich.Downloaded 11/25/14 to 129.120.242.61. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php