Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse. Eine Abhandlung, zu deren öffentlicher Vertheidigung behuf Erlangung der Doctorwürde der Verfasser bereit sein wird am 16. December 1851.

Göttingen: E. A. Huth, 1851.

First edition, very rare, and an extraordinary association copy from the library of Carl Friedrich Gauss, of Riemann’s Dissertation, “one of the most important achievements of 19th century mathematics” (Laugwitz), “which marked a new era in the development of the theory of analytic functions” (Kolmogorov & Yushkevich, p. 199), introducing geometric and topological methods, notably the idea of a ‘Riemann surface.’ “Riemann’s doctoral thesis is, in short, a masterpiece” (Derbyshire, p. 121). This paper, and its sequel Theorie der Abel’schen Functionen (1857), “have become famous for several reasons. They introduced what are now called Riemann surfaces, in the form of domains spread out over the complex plane. They presented enough tools to classify all compact orientable surfaces, and so gave a great impetus to topology. They provided a topological meaning for an otherwise unexplained constant which entered into Abel's work on Abelian integrals, and more generally gave a geometric framework for all of complex analysis. They are thus the first mature, obscure, papers in the study of the topology of manifolds and are equally decisive for the development of algebraic geometry and the geometric treatment of complex analysis” (Gray, p. 21). Riemann’s Dissertation is of great rarity, for “although [a Dissertation] was a printed booklet, it was not usually published or publicised in the normal way; the candidate had to pay for the print-run, and sales and marketing were executed on an infinitesimal scale. So the first printing of Riemann’s thesis consisted only of the obligatory copies he had to hand in at Göttingen University, and a few copies for his personal use” (Landmark Writings, p. 454). Only three copies of Riemann’s Dissertation have appeared at auction in the last 30 years.

Provenance: Carl Friedrich Gauss (stamp ‘Gauss Bibliothek’ on title).

Riemann begins his Dissertation by offering a new foundation for the theory of analytic functions, based not on analytic expressions but on the assumption that the complex function w = u + iv of the complex variable z = x + iy is ‘differentiable’. Riemann noted that this condition is equivalent to requiring that u and v satisfy the ‘Cauchy-Riemann equations’ (as they are now called), and that, when the derivative is non-zero, it is also equivalent to requiring that the function determines a conformal mapping from the z-plane to the w-plane. In order to deal with multi-valued functions such as algebraic functions and their integrals, Riemann introduced the surfaces now named after him: the Riemann surface associated with a function is composed of as many sheets as there are branches of the function, connected in a particular way so that continuity is preserved and a single-valued function on the surface is obtained. Such a surface can be represented on a plane by a series of ‘cross-cuts’, which divide the surface into simply-connected regions. 

The rest of the Dissertation is devoted to the study of functions on Riemann surfaces. From the Cauchy-Riemann equations it follows that if w = u + iv is an analytic function, then u and v are harmonic functions, i.e. solutions of Laplace’s equation. To construct harmonic functions such as u and v, Riemann began with the case of a simply-connected region and made use of what he called ‘Dirichlet’s principle (he had learned it from Dirichlet’s lectures in Berlin): this asserts that the harmonic functions are exactly those which minimize the value of a certain integral. He then extended this to the non-simply connected case using cross-cuts and other variants. This approach was later to prove controversial, as Weierstrass gave examples of situations in which the minimizing function does not exist, but it was rehabilitated by Hilbert early in the next century. 

The crowning glory of the Dissertation, and the most difficult part of the theory of conformal mappings, is his celebrated mapping theorem. “As an application of his approach he gave a ‘worked-out example’, showing that two simply-connected plane surfaces can always be made to correspond in such a way that each point of one corresponds continuously with its image in the other, and so that corresponding parts are ‘similar in the small’, or conformal ... what is nowadays called the ‘Riemann mapping theorem’.” (Landmark Writings, p. 454). 

According to Richard Dedekind (Bernhard Riemann’s Lebenslauf, p. 7), Riemann probably conceived the main ideas of the Dissertation in autumn 1847. It was submitted on 14 November, 1851 and the Dean of the Faculty asked Gauss for his opinion. Always sparing with his praise, Gauss nevertheless wrote: “The paper submitted by Mr Riemann bears conclusive evidence of the profound and penetrating studies of the author in the area to which the topic dealt with belongs” (quoted from R. Remmert, “From Riemann surfaces to complex spaces”, Bull. Soc. Math. France (1998), p. 207). Following the thesis examination on 16 December, 1851, Riemann was awarded his Doctor Philosophiae and Gauss recommended that he be formally appointed to a post at Göttingen.

DSB XI 449-451; U. Bottazzini & J. Gray, Hidden Harmony – Geometric Fantasies – The Rise of Complex Function Theory, 2013; J. Derbyshire, Prime Obsession, 2003; J. Ferreiros, Labyrinth of Thought: A History of Set Theory and its Role in Modern Mathematics, 2010; I. Grattan-Guiness, Landmark Writings in Western Mathematics, Chapter 34; J. Gray, Linear Differential Equations and Group Theory from Riemann to Poincaré, 1996; A. N. Kolmogorov & A. P. Yushkevich (eds.), Mathematics in the 19th century, Vol. II, 1996; D. Laugwitz, Bernhard Riemann, 1826-1866, 1998; E. Scholz, Geschichte des Mannigfaltigkeitsbegriffs von Riemann bis Poincaré, 1980; J. Stillwell, Mathematics and its History, 3rd edition, 2010.

4to (263 x 212 mm), pp. [ii], [1], 2-32. Green paper spine strip as issued. Custom half leather clam sell box with gil spine lettering.

Item #5632

Price: $75,000.00