In the mid-19th century, as they studied the hypergeometric functions and their associated differential equations, mathematicians began to notice connections between these equations and various kinds of symmetry groups. Riemann, Fuchs, Klein, and Poincaré are the names that come most immediately to mind when one thinks about this subject, but Gray's book shows that the story is much more complicated — and, of course, much more interesting.
Gray's book surveys these developments and highlights the gradual development of crucial unifying ideas. Mathematically, the book is quite dense and assumes the reader is an able mathematician willing to grapple with ideas that may be unfamiliar. Nevertheless, for those willing to put in the effort, reading this book will deepen one's knowledge both of the theories discussed and of their history.
The second edition is quite a bit larger than the first, the major additions being discussions of the history of the Riemann-Hilbert problem and of the Picard-Vessiot "Galois theory of differential equations". The result is an essential reference for those interested in the history of how group-theoretic ideas were applied to the study of differential equations.
Fernando Q. Gouvêa is the Secret Master of MAA Reviews.