This is the English translation of a volume originally published only in Russian and now out of print. The book was written by Jacques Hadamard on the work of Poincaré. Poincaré's creation of a theory of automorphic functions in the early 1880s was one of the most significant mathematical achievements of the nineteenth century. It directly inspired the uniformization theorem, led to a class of functions adequate to solve all linear ordinary differential equations, and focused attention on a large new class of discrete groups. It was the first significant application of nonEuclidean geometry. The implications of these discoveries continue to be important to this day in numerous different areas of mathematics. Hadamard begins with hyperbolic geometry, which he compares with plane and spherical geometry. He discusses the corresponding isometry groups, introduces the idea of discrete subgroups, and shows that the corresponding quotient spaces are manifolds. In Chapter 2 he presents the appropriate automorphic functions, in particular, Fuchsian functions. He shows how to represent Fuchsian functions as quotients, and how Fuchsian functions invariant under the same group are related, and indicates how these functions can be used to solve differential equations. Chapter 4 is devoted to the outlines of the more complicated Kleinian case. Chapter 5 discusses algebraic functions and linear algebraic differential equations, and the last chapter sketches the theory of Fuchsian groups and geodesics. This unique exposition by Hadamard offers a fascinating and intuitive introduction to the subject of automorphic functions and illuminates its connection to differential equations, a connection not often found in other texts. This volume is one of an informal sequence of works within the History of Mathematics series. Volumes in this subset, "Sources", are classical mathematical works that served as cornerstones for modern mathematical thought. Readership Graduate students and research mathematicians; mathematical historians. Reviews "Gives a fascinating and highly instructive brief exposition on Poincaré's creation of the theory of automorphic functions ... The historical circumstances, sources and perspectives of Poincaré's discovery are very instructionally and vividly depicted ... this beautiful thin book will please analysts as well as geometers and also all fans of the history of mathematics."  Mathematica Bohemica "The book is substantially enhanced by editor Gray's introduction, `Brief History of Automorphic Function Theory, 18801930', which deftly treats both mathematics and related nontechnical matters ... "Jeremy Gray and Abe Shenitzer deserve the gratitude of the entire mathematical community for bringing out this Hadamard volume. It would be remiss to fail to praise the quality of the writing found here. Gray's insightful introductory remarks and Shenitzer's translation of the Russian text are recognizably English, clearly, carefully and intelligently wrought. This is an important addition to the literature, of great interest both to mathematical historians and to anyone with even a passing acquaintance with nonEuclidian geometry and automorphic functions. For mathematicians actively engaged in research in these areas, this book is essential reading."  Mathematical Reviews "It is cause for joy and celebration that the "Sources" subseries of the AMS/LMS "History of Mathematics" series continues to grow. This little book should be of interest both to historians seeking to understand the evolution of the theory of automorphic functions and to mathematicians working in the area, and thus it is a valuable addition to the (rather short) list of original source material available in English translation. Keep them coming, AMS!"  MAA Online Table of Contents  Historical introduction
 A brief history of automorphic function theory, 18801930
 The group of motions of the hyperbolic plane and its properly discontinuous subgroups
 Discontinuous groups in three geometries. Fuchsian functions
 Fuchsian functions
 Kleinian groups and functions
 Algebraic functions and linear algebraic differential equations
 Fuchsian groups and geodesics
 References
 Additional references
