This book is a study of how a particular vision of the unity of mathematics, often called geometric function theory, was created in the 19th century. The central focus is on the convergence of three mathematical topics: the hypergeometric and related linear differential equations, group theory, and non-Euclidean geometry.

The text for this new edition has been greatly expanded and revised, and the existing appendices enriched with historical accounts of the Riemann--Hilbert problem, the uniformization theorem, Picard-Vessiot theory, and the hypergeometric equation in higher dimensions. The exercises have been retained, making it possible to use the book as a companion to mathematics courses at the graduate level.

This work continues to be the only up-to-date scholarly account of the history of a branch of mathematics that continues to generate important research, for which the mathematics has been the occasion for some of the most profound work by numerous 19th century figures: Riemann, Fuchs, Dedekind, Klein, and Poincar.

Reviews of the first edition:

"The volume...is one among the most interesting books on the history of mathematics published in recent times.... very stimulating reading for both historians of modern mathematics and mathematicians as well." ---Mathematical Reviews

"The book contains an amazing wealth of material relating to the algebra, geometry, and analysis of the nineteenth century.... Written with accurate historical perspective and clear exposition, this book is truly hard to put down." ---Zentralblatt fur Mathematik

"...must reading for every serious student of nineteenth century mathematics...represents a substantial contribution toward filling what is generally acknowledged to be an immense gap in the historical literature."---ISIS

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