*Mathematicians: An Outer View of the Inner World*, Mariana Cook, 2009, 199 pp., $35 hardbound, ISE: 978 -- 06911395. Princeton University Press, 41 William St., Princeton, NJ 08540.

This book is a collection of black-and-white portraits of 92 eminent mathematicians. Each portrait is accompanied by autobiographical comments. In my usual review of books, I peruse the contents, note special features and read a few selected sections. However, with this book, my involvement was more prolonged. Each time I picked it up, I became engrossed with the individuals and their stories. The mathematicians chosen for this survey are all well respected for their work and accomplishments. They were picked randomly from a select but limited population. Many are instantly recognizable, some are known for the theorems or theories they have derived, a few are Fields medals winners, while others are just workers in the field. Mariana Cook's portraits have captured something of the spirit and personality of each of their subjects. Robert McPherson's photo has the Princeton mathematician leaning against a tree accompanied by his campus bicycle. Mikhael Gromov of New York University sits amongst garden foliage like a bearded elf. Marie-France Vigneras of Institut de Mathématique de Jussieu, Paris poses barefoot casually holding a glass of wine. William Thurston of Cornell stands embraced by the curve of a gnarled old oak tree. But most appear against a plain black background, staring out at the reader, their knowing gaze conveying the sense that we together have shared mathematical experiences.

Brief autobiographical comments offer advice and personal experiences. Almost all of these testaments are modest in their tone--there is no real bragging about accomplishments. Benoit Mandelbrot notes that as a child he hoped to emulate Johann Kepler in taming the ellipse. Stephen Smale recounts how his education began in a one-room schoolhouse and how eventually he became inspired by the work of Newton and Weyl to explore dynamical systems. Henri Cartan comments on his youthful disillusionment with elementary Euclidean geometry; to him, its foundations lacked necessary formalization. Cartan went on to help establish the Bourbaki group, whose task was to formalize all of mathematics. Joan Birman of Barnard College, Columbia University, remembers how the prevalence of knot theory in nature drew her to the subject. I was particularly pleased to learn that Adebisi Agboola, number theorist at the University of California, developed his interest in mathematics through the history of the subject. As a young boy of twelve, he read David Bergamini’s Life Series volume, *Mathematics *(1963). This historical approach to explaining mathematics changed his attitude toward the subject from negative to favorable. The comments of all participants are thought provoking and inspirational.

When does history begin? Is it when we turn away from an event? Or when a person dies, and we reconsider the accomplishments he or she has left? No, I think history begins when an action is initiated. Cook’s collection of portraits and comments stands as a testament to the spirit and accomplishments of mathematics today. It is a history of mathematics in action. This is a book that will appeal to a limited audience. Certainly, it will be an excellent addition to any university library. Those curious as to the mysterious nature of mathematics would do well to read this book. It would be an excellent gift for a young person inclined toward science and mathematics, and whose career plans may still be in a formative stage.

Frank J. Swetz, Professor Emeritus, the Pennsylvania State University