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**Poincaré and the three body problem.**
*(English)*
Zbl 0877.01022

History of Mathematics (Providence) 11. Providence, RI: AMS, American Mathematical Society; London: London Mathematical Society (ISBN 0-8218-0367-0). xv, 272 p. (1997).

The book is on Poincaré’s memoir on the three-body problem, published in Acta Mathematica in 1890 as the winning paper in an international competition. Today the memoir is widely applauded for, among others, the first mathematical description of chaotic behaviour in a dynamical system. In fact, however, the memoir is very different from the version which won the prize: after having discovered a critical error in the original paper, Poincaré was forced to rewrite a substantial part of it. “This book is an account of Poincaré’s memoir, both from a mathematical and a historical perspective. At the centre is a detailed study of the mathematics of the memoir in that the versions are compared and the error explained. The memoir is put into a historical context through an examination of the mathematical environment in which it was created as well as of its place within Poincaré’s oeuvre.” And more than that: we are also offered an interesting insight into the late 19th-century mathematical community and an account of the memoir’s reception and impact by following both the progress of the three-body problem and the development of foundations of dynamical systems theory. The author explains in particular, how Poincaré’s methods, characterized by global geometric viewpoint, opened a qualitative approach to general problems of dynamics, thus greatly influencing J. Hadamard and G. Birkhoff. And leaving aside its influence upon the modern chaos theory (as “amply dealt with elsewhere”), she provides “a glance at the work of some later mathematicians whose research into particular aspects of dynamical systems derives ultimately from Poincaré”. These mathematicians include M. Morse, V. K. Melnikov, and founders of KAM-theory, that is, A. Kolmogorov, V. I. Arnold, and J. Moser. In that way we are able to trace the influence of some of Poincaré’s ideas from his memoir deeply into our century. Clearly organized, well written, richly documented, with 6 appendices and a dozen or so illustrations – the book is a highly valuable contribution to the history of modern mathematics, adding value to the series in which it appeared.

Reviewer: R.Duda (Wrocław)