It is a tremendously difficult job to write a book about the life and work of Henri Poincaré, who is often called “the last encyclopedist,” a man who left his imprint in all branches of mathematics known in his time, adding a few himself, and who also made top contributions to physics, astronomy, and the worlds of letters and philosophy. Ferdinand Verhulst, a well-known figure in the field of differential equations, now an emeritus professor at the University of Utrecht in the Netherlands, was well aware of the challenges he would encounter when starting on this project. His volume was more than a decade in the making, and he often doubted that he could bring it to an end. But he succeeded. The result is a gem of a book, which traces in only 260 pages the life of Henri Poincaré and the remarkable results the French mathematician added to the cultural edifice of humankind.
Born in 1854 in Nancy in a physician’s family, Poincaré showed unusual intellectual skills from an early age. He had a good childhood, in spite of suffering with diphtheria, and the atmosphere in which he grew up was prone to studying and learning. His close relatives shared similar values. One of Henri’s first cousins, Raymond Poincaré, would become the 10th President of France, holding his seat during the gruesome years of World War I. Henri was not much interested in politics, but found science attractive. After an aborted attempt at an engineering career, he found his true vocation in mathematics, holding first a lecturer position in Caen and later becoming a professor at the University of Paris. He rose to world fame at the age of 34, when awarded the prize bestowed by King Oscar II of Sweden and Norway for his contributions to the 3-body problem of celestial mechanics. After that Poincaré’s career boomed. For the almost two-and-a-half decades that followed, his work changed the way mathematicians, physicists, astronomers, and philosophers thought. Unfortunately, he died due to complication following surgery, aged 58, at the height of his intellectual powers.
The first part of Verhulst’s book deals with Poincaré’s life from early childhood to his university years. The reader gets well acquainted with his parents, sister, and relatives, and the portrait of the future mathematician comes very much alive. Verhulst succeeds marvelously, in sparse prose, to make the times of Poincaré real.
Then he moves to the period when the French mathematician started to come up with his first results in Caen and the contacts he had with Fuchs and Klein on the theory of automorphic functions. Poincaré’s travels and relationships with colleagues, such as Paul Appell and Mittag-Leffler, are further described in detail. Verhulst skillfully includes sections about Poincaré’s social involvement and his reaction to the Dreyfus affair. Then he dedicates an entire chapter to the important issue of King Oscar’s Prize, describing Poincaré’s initial mistake in the memoir, the way he corrected it, and the discovery of the chaos phenomenon that followed.
The biographical part of the book also deals with the essays and philosophical writing for which Poincaré was elected a member of the French Academy. Verhulst spends here a couple of pages to tell the remarkable story of Poincaré’s last, and less known, volume of essays,
Scientific Opportunism: An Anthology.
I won’t disclose any details, but assure the readers that they will be thrilled to learn how this posthumous work was published. The first part ends with a chapter entitled “At the End, What Kind of a Man?” It is a courageous attempt to capture the image of a giant and put his life into perspective. Even if it stopped here, Verhulst’s book would have been a success. But the author felt that he had to add more.
The second part of the book gets into details about Poincaré’s work, to complete the mathematical references mentioned in connection with his life. This interesting review consists of four chapters. The first deals with automorphic functions and includes comments on the lectures delivered on this topic by Felix Klein. The second, and richest, chapter discusses differential equations and dynamical systems, a subject dear to Verhulst. After analyzing Poincaré’s thesis of 1879, he focuses on the revolutionary memoir on differential equations of 1881/82. These are preludes to a deep and thorough discussion of Poincaré’s masterpiece, “Les Méthodes Nouvelles de la Mécanique Céleste,” to which Verhulst dedicates 55 pages, almost one fifth of his entire book. This is a great choice, and anyone who wants to study the original work (of which the only
English translation is due to D. Goroff in 1993)
can now rely on Verhulst to get over the obstacles Poincaré’s writing of this treatise raises. The chapter ends with the Hopf-Poincaré bifurcation and the Poincaré-Birkhoff theorem.
Verhulst then spends a chapter on analysis situs, the name the French mathematician gave to what is known today as topology. The last chapter dwells on Poincaré’s contributions to mathematical physics, including his work on relativity and cosmogony.
The book ends with Poincaré’s address to the Society for Moral Education and a chapter on historical data and biographical details. These are good choices for bringing Poincaré’s words to life again and guiding the reader into chronological aspects, as well as offering short biographies of the people Poincaré connected with during his career.
A couple of other volumes about Poincaré appeared recently, and each of them is worth reading. But Verhulst’s book stands out through conciseness, completeness, and excellent writing. I highly recommend it to anyone interested first of all in Poincaré, but also in the culture and society of his time.