Theorie der Abel’schen Functionen. Offprint from Journal fur die reine und angewandte Mathematik, Bd. 54.

Berlin: G. Reimer, 1857.

First edition of Riemann's paper on Abelian functions, “one of the most notable masterworks of mathematics” (DSB), which develops much further the methods of his dissertation (Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse, 1851), using them to give the general solution of the Jacobian inversion problem (solved in a special case by Karl Weierstrass in 1854) and to establish an early version of the famous Riemann-Roch theorem. Stillwell (p. 310) considers Riemann’s 1857 paper “perhaps his greatest work, which did for algebraic geometry what his Habilitationsvortrag [Ueber die Hypothesen, welche die Geometrie zu Grunde liegen, 1854] did for differential geometry.” Scholz has shown that the concept of a manifold, one of the most important innovations introduced in the 1854 Habilitationsvortrag, grew out of an attempt to find a satisfactory conceptualization of the Riemann surfaces that he had begun to employ in 1851 and developed fully in 1857. “The papers [Grundlagen für eine allgemeine Theorie der Functionen & Theorie der Abel’schen Functionen] 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). No copy of this offprint has appeared at auction since 1974. No copies listed on COPAC.

“The year 1857 was what we should call, in the language of current celebrity biography, Riemann’s “breakout year.” His 1851 doctoral dissertation is nowadays regarded as a classic of nineteenth-century mathematics, but it drew little attention at the time in spite of having been enthused over by Gauss. His other written papers of the early 1850s were not widely known and were published in an accessible form only after his death. To the degree that he had become known at all, it was mainly through the content of his lectures; and much of that content was too far ahead of its time to be appreciated. In 1857, however, Riemann published a paper on analysis that was at once recognized to be a major contribution. Its title was “Theory of Abelian functions.” In it he tackled topical problems by ingenious and innovative methods. Within a year or two his name was known to mathematicians all over Europe. In 1859 he was promoted to full professor at Göttingen ...” (Derbyshire, p. 31).

“Riemann's function theory, known through an 1857 paper on Abelian functions, was the basis for the renown he enjoyed during his lifetime. That work was an outstanding feat, for it attempted to offer a general solution of the Jacobian inversion problem for integrals of arbitrary algebraic functions. The topic emerged from the fascinating competition that Abel and Jacobi sustained in the late 1820s on the subject of elliptic functions (the inverses of elliptic integrals). As for the importance that was attached to it, suffice it to say that Weierstrass became a rising star with his solution of the inversion problem for hyperelliptic integrals in 1854 and 1856. Riemann was tackling a much more general problem and his work, in spite of gaps in the proofs, aroused enormous excitement” (Ferreiros, p. 53). “The depth of originality in Riemann's remarkable paper can be measured by the fact that on reading it no less an authority than Weierstrass withdrew a paper of his own on the same subject, preferring to wait until he had assimilated what Riemann had to say... While Weierstrass had successfully treated integrals on a curve with equation y2 = f(x), the so-called hyperelliptic case, Riemann dealt with integrals on any algebraic curve whatever... To accomplish this feat, Riemann showed how his ideas of 1851 could be extended to provide a remarkable theory of complex functions on almost any surface" (Bottazzini & Gray, p. 286). 

In his 1857 paper “Riemann found it almost indispensable, in order to study Abelian and related functions, to resort to topological considerations. He developed new methods that enabled him to define the “order of connectivity” of a surface - the Euler characteristic - and, later, what Clebsch would call the “genus” of the surface... In the 1851 dissertation he studied connected surfaces with a boundary, and analysed their topological properties by means of dissection into simply-connected components.... In 1857 he analysed closed surfaces, since he was now considering the complex plane completed by a ‘point at infinity’... The most astounding example of the intimate relations between topological notions and properties of functions was the Riemann-Roch theorem, which determines the number of linearly independent meromorphic functions on a Riemann surface, having a given number of poles, as a function of the genus of the surface” (Ferreiros, p. 56).

In the first half of his 1857 paper, Riemann 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 used a version of Dirichlet’s principle to define algebraic functions in terms of their branching behaviour and their poles. The most important result here is ‘Riemann's inequality,’ which states that the number of linearly independent meromorphic functions (i.e. quotients of analytic functions) with poles of order not greater than n1, n2, ..., nm at m distinct points of a surface of genus g is not greater than n1 + n2 + ... + nm - g + 1. In 1864, Riemann's student Gustav Roch was able to strengthen this result, obtaining a formula for the exact number of meromorphic functions − this is the famous Riemann-Roch theorem. In this part of the paper Riemann also discovered that there is a whole class of transformations that do not change the genus of the surface, and hence its topology, but which lead to different functions - these are called ‘birational transformations.’ Riemann showed that there exists a (3g - 3)-dimensional family of birationally inequivalent surfaces of genus g. These ‘moduli spaces’ of Riemann surfaces continue to be an object of intensive study today. 

In the second half of the 1857 paper, Riemann generalized Jacobi’s theory of theta functions to functions of several variables and showed that quotients of products of the new theta functions represented algebraic functions. Riemann reduced the solution of the Jacobian inversion problem to the problem of determining the zeros of the theta functions. The answer depends on the conditions under which a theta function is not identically zero on the Riemann surface. Riemann was able to obtain necessary and sufficient conditions for the vanishing of theta functions that gave a complete solution to the problem of inverting Abelian integrals. As a by-product of this study, Riemann derived a number of important identities involving theta functions. These ‘Riemann theta-relations’ play a vital role in modern research in the theory of Riemann surfaces, for example, in the solution by Shiota in 1986 of the Schottky problem. 

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 (250 x 194 mm), pp. [4], [1] 2-55, stamped out of German lilbray: rubber stamps to recto and verso of title; some minor spotting throughout, otherwise fresh and clean. 20th century marbled boards with gilt pine label.

Item #4285

Price: $9,500.00