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.