The first part of this book, describes the life and work of Henri Poincaré (Nancy 1854 - Paris 1912). At the age of five he got diphtheria followed by a paralysis from which he recovered. His teachers recognized his great mathematical talent at young age, which helped him later when Poincaré by some misunderstanding didn't perform well at an exam or when by his impatient mind took nontrivial leaps in his solution method, that were not appreciated by his examiners. He entered the École Polytechnique, a military school in Paris at the age of 19 and 2 years later the École des Mines to become a mine engineer. He presented a dissertation on partial differential equations under supervision of Darboux, Laguerre and Bonnet. He worked briefly as a mine engineer in Vesoul (France) but was soon appointed lecturer at the University of Caen. There we produced his work on automorphic functions, which Poincaré called Fuchsian, a terminology that resulted in a dispute with Klein. His career took off when he was appointed a professorship at the Sorbonne. His way of lecturing, and his scientific contacts (Darboux, Appell, Mittag-Leffer) are described, his marriage and children, and his involvement in the Dreyfus Affair. In 1889 Poincaré was awarded the prize of Oscar II, king of Sweden and Norway. During the editorial process of publishing his paper in the Acta Mathematica, some clarifications were needed, which actually contained the elements of chaos theory, which was only recognized as the KAM (Kolmogorov-Arnold-Moser) theorem in 1960. The first part ends with an analysis of Poincaré as a philosopher, (his views on mathematics, physics and science in general) and how he was as a person.
Poincaré's interests were very diverse and he made contributions about topics that became later whole new research fields like automorphic functions, qualitative theory of differential equations, bifurcation theory, asymptotic expansions, dynamical systems, mathematical physics, and topology. It is impossible to cover all of these topics in one monograph, but nevertheless, the author did a great job in covering many aspects related to differential equations and dynamical systems in part 2 of this book, leaving out almost all of Poincaré's influence on group theory, algebra, and probability. The chapters in this second part have these titles: Automorphic functions (just an introductory chapter), Differential equations and dynamical systems (this is the most extensive one and discusses Poincaré's 1879 thesis, the chapters of his Mécanique Céleste, bifurcation, and the Poincaré-Birkhoff theorem), Analysis Situs (this is a short ne about topology), and Mathematical physics (about several applications of differential equations). In each of these chapters the author summarizes the yeast of Poincaré's texts and gives some background, leaving out most of the unnecessary technicalities, and he adds his own comments and remarks. The remaining two chapters contain Poincaré's address to the Society for Moral Education three weeks before his death and short biographical summaries of mathematicians from late 19th - early 20th century.
With this book, Verhulst did a marvelous job in sketching not only the person of Henri Poincaré, but also by restricting to, or rather emphasizing on, the differential equations and dynamical systems, among the diverse subjects that Poincaré worked on, he succeeds very well in communicating the essence of what the theory is about. Admitted, this is not for a mathematical illiterate since there are indeed quite some formulas floating around, (even the first part discusses his contributions although without formulas) but any mathematician or mathematical physicist should be able to understand what is going on, even if this is not his or her core business. Verhulst is clearly very well familiar with the work of Poincaré but he is also well documented about the historical and human facts behind the writings. The many illustrations (mostly portraits of mathematicians) make it so much more a pleasure to assimilate. This is a book obviously interesting for historians of mathematics, but also for any mathematician with some interest in the origin of his or her research field or who want to catch a glimpse of the person and the mind of a genius.