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. / Hardcover. First edition, extremely rare, of Riemann’s doctoral dissertation — Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse, printed in Göttingen by Ernst August Huth in 1851 — in a wonderful association copy bearing the stamp of the Göttingen Royal Observatory, the institution of which Gauss had been director from 1807 until his death in 1855, and to which Riemann would himself be appointed on Gauss’s recommendation immediately after the defence of this thesis. The copy was almost certainly among those Riemann was required to deposit in the University Library, and was sold as a duplicate by the Göttingen State and University Library in October 1951. Such is the rarity of this work that, although it is a printed booklet, it was not published in the normal way: a doctoral candidate paid for the print-run himself, and sales and marketing were executed on an infinitesimal scale. 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. RBH records only one rather poor copy (trimmed, with paper repairs, in a modern binding), and we know of only one other copy to have appeared in commerce. KVK and OCLC together list thirteen copies worldwide, of which only three are in the United States: Brown, Johns Hopkins, and the Bullitt collection at Louisville, Kentucky. The thesis is not in Library Hub. This dissertation is, by almost universal consent, the most consequential doctoral thesis in the history of mathematics. Lars Ahlfors observed that very few mathematical papers have exercised an influence on the later development of mathematics comparable to the stimulus received from Riemann’s dissertation: it contains the germ of a major part of the modern theory of analytic functions, initiated the systematic study of topology, revolutionised algebraic geometry, and paved the way for Riemann’s own approach to differential geometry. Detlef Laugwitz ranked it among the most important achievements of nineteenth-century mathematics; John Derbyshire called it, simply, a masterpiece; Kolmogorov and Yushkevich observed that it marked a new era in the theory of analytic functions. The breathtaking generality with which Riemann attacked the problem of conformal mapping, the introduction of Riemann surfaces (for which, as Ahlfors noted, there is no record of anyone having used a similar device before), and the emphasis on Laplace’s equation — putting virtually an equality sign between two-dimensional potential theory and complex function theory — together define a new mode of mathematical thought whose influence runs continuously from 1851 to the present. To understand the thesis one must understand what it displaced. For thirty-five years before 1851 Augustin-Louis Cauchy had been building the theory of functions of a complex variable. He had arrived at the sound definition of such functions — by differentiability in the complex domain rather than by analytic expression — and had characterised them by what are now called the Cauchy–Riemann equations. He had discovered complex integration, the integral theorem, residues, the integral formula, and power-series development; he had even ventured into multivalent functions, following functions and integrals by continuation through the plane, and had come to understand the periods of elliptic and hyperelliptic integrals, although not the reason for their existence. The field of elliptic functions had grown rapidly, though the fundamental property of double periodicity — discovered by Abel and Jacobi — was treated as an algebraic curiosity rather than a topological necessity; hyperelliptic integrals caused much trouble, but no one knew why. In 1851, the year in which Riemann defended his thesis, Cauchy had reached the height of his own understanding. There was one thing he lacked: Riemann surfaces. Riemann was the first to accept Cauchy’s differentiability-based view of complex functions wholeheartedly, and the first to supply the geometric and topological apparatus in which that view could be made fully effective. Georg Friedrich Bernhard Riemann (1826–1866) was the second of six children of a Protestant minister, Friedrich Bernhard Riemann, and the former Charlotte Ebell. His elementary education came from his father, later assisted by a local teacher; remarkable skill in arithmetic was apparent at an early age. From Easter 1840 he attended the Lyceum in Hannover, living with his grandmother; when she died two years later he entered the Johanneum in Lüneburg, where his mathematical interests swiftly outran what the school could provide. In the spring term of 1846 he enrolled at Göttingen University to study theology and philology — the paternal plan — but attended mathematical lectures alongside, and finally received his father’s permission to devote himself wholly to mathematics. At that time Göttingen offered a rather poor mathematical education: even Gauss taught only elementary courses. In the spring term of 1847 Riemann went to Berlin University, where a host of students flocked around Jacobi, Dirichlet, and Steiner; he became acquainted with Jacobi and Dirichlet, the latter exerting the greatest influence upon him. When Riemann returned to Göttingen in the spring term of 1849, the situation had changed profoundly with the return of the physicist Wilhelm Eduard Weber from Leipzig. For three terms Riemann attended courses and seminars in physics, philosophy, and education. He became Weber’s assistant for eighteen months in physical exercises and experiments — an appointment through which he acquired the unorthodox but suggestive conception of contemporary physics that Ahlfors would later identify as a leading influence on the thesis. Through Weber and through Johann Benedict Listing, appointed Göttingen’s professor of physics in 1849, Riemann gained a strong background in theoretical physics and, from Listing specifically, the ideas in topology that would find their first mathematical application in the thesis. According to Dedekind’s Lebenslauf, Riemann had conceived the main ideas of the dissertation already in the autumn of 1847, during his Berlin period. The thesis begins (§1) 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 — meaning that the value of the differential quotient dw/dz is independent of the direction of the differential dz. In §4 Riemann notes that this condition is equivalent to the requirement that u and v satisfy the Cauchy–Riemann equations, ∂u/∂x = ∂v/∂y and ∂v/∂x = −∂u/∂y, and that (when the derivative is non-zero) it is equivalent to the requirement that the function determines a conformal mapping from the z-plane to the w-plane — a mapping that preserves the angles between intersecting curves. This emphasis on the conformal properties of a complex function was original. It cannot be found in the work of Cauchy, but it is very much in line with the views of Gauss. Riemann had read in 1849 Gauss’s memoir on conformal maps, Allgemeine Auflösung der Aufgabe: die Theile einer gegebenen Fläche auf einer andern gegebenen Fläche so abzubilden, dass die Abbildung dem Abgebildeten in den kleinsten Theilen ähnlich wird (Astronomische Abhandlungen 3, 1825, pp. 1–30), and this is in fact the only work Riemann specifically refers to in his entire thesis (footnote 2 at the end of §3). In §5 Riemann introduces the device that bears his name. To present a complex function effectively — in particular when it is what nineteenth-century mathematicians called a many-valued function — it should be thought of as a surface spread out over the z-plane. Several isolated points on the surface may correspond to the same point of the plane, but a line of such points corresponding to a fold is not allowed. The surface may wind around certain points, which Riemann calls branch points (Windungspunkte). The reader is led to believe, as Ahlfors observed, that this is a commonplace convention — but there is no record of anyone having used a similar device before, and from a modern point of view the introduction of Riemann surfaces foreshadows the use of arbitrary topological spaces, among the most consequential mathematical innovations of the following century. In §6 Riemann explains how a surface may be broken down into manageable pieces by means of cuts. A piece of surface is connected when any two points in it can be joined by a curve. He decomposes the surface by means of boundary cuts (Querschnitte) — simple curves joining two points on the boundary. A surface is called simply connected if any such cut divides it into two pieces. He argues that if a surface is cut into m simply connected pieces by n boundary cuts, then the number n − m is a constant, which he proposes to call the order of connectivity of the surface. The surface defined by an algebraic equation F(x, y) = 0, he claims, has order of connectivity 2p − 1 for some integer p; the number p was later called the genus of the surface. This is the inaugural treatment of what is now the central invariant of algebraic curves. The remainder of the thesis is devoted to the study of functions on Riemann surfaces. From the Cauchy–Riemann equations it follows that if w = u + iv is analytic, then u and v are harmonic — solutions of Laplace’s equation ∂2u/∂x2 + ∂2u/∂y2 = 0. To construct such harmonic functions Riemann begins with the case of a simply connected region and invokes (§16) a variational principle that he learned from Dirichlet’s Berlin lectures and therefore called Dirichlet’s principle: the harmonic functions are precisely those which minimise the value of a certain energy integral. The principle itself did not originate with Dirichlet — Gauss, Green, and William Thomson had all used it — but it is Dirichlet’s formulation that Riemann takes. He then extends the construction to non-simply-connected cases by means of cross-cuts and further variants. The dissertation thus offers, in contrast to the conception of a function as something given by an analytic expression, the determination of a function from its singularities, founded on existence and uniqueness theorems proved by a variational principle. It breaks entirely with the idea that complex function theory has essentially to do with formal expressions such as power-series expansions, and asserts instead that the proper way to think of a complex function is in terms of a pair of component harmonic functions. The crowning glory of the thesis, in Laugwitz’s phrase, is the celebrated Riemann mapping theorem (§21): any two simply connected plane surfaces can be mapped onto one another in such a way that each point of one corresponds to a unique point of the other in a continuous way and the correspondence is conformal; moreover, the correspondence between an arbitrary interior point and an arbitrary boundary point of the one and the other may be given arbitrarily, but when this is done the correspondence is determined completely. The consequences for Riemann’s theory were profound: it established that every such domain admits complex functions with prescribed boundary values and with prescribed points at which they become infinite in allowable ways. The mapping theorem, Laugwitz observed, is an instance of Riemann’s novel view of mathematics: it illustrates the fruitfulness of the notion that functions are simply mappings; it is a global proposition (all Gauss had proved was the conformal equivalence of small pieces of surfaces); and it was one of the deeper existence theorems to emerge after Cauchy’s existence theorems about solutions of differential equations. Had Riemann’s dissertation consisted of just the mapping theorem, its influence would still have been considerable. The thesis was submitted on 14 November 1851; the Dean of the Faculty of Philosophy sent it to Gauss for his opinion. Gauss, always sparing with his praise, recorded that Riemann’s dissertation gave conclusive evidence of his profound and penetrating studies in the field, and credited him — in Dedekind’s recollection of Gauss’s words — with “a gloriously fertile originality”. 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. A Habilitation followed in 1854 — famously on the hypotheses which lie at the foundations of geometry, the lecture Gauss himself chose from three the candidate had prepared. Karl Weierstrass observed in 1870 that Riemann’s justification of the existence of a minimiser for the energy integral was inadequate, and this produced a thirty-year crisis in the programme of the thesis. Mathematicians responded in two ways. Some, notably Hermann Amandus Schwarz, tried — successfully — to prove the relevant results without using the Dirichlet principle at all. Others held that since no counter-examples existed and the principle itself was plausible, it must be provable. David Hilbert obtained a rigorous proof after 1900, fully acknowledging Weierstrass’s criticism; the vindication was complete by the early twentieth century, and the Dirichlet principle is again in standard use. The most significant difficulty with the diffusion of Riemann’s message was that the thesis was not properly published. It was printed as a separate and distributed to a number of German universities, as was the custom, but it was not reprinted in a journal such as Crelle’s. This alone goes a long way towards explaining why there are no signs of his approach winning converts in the early 1850s. But for any who took the trouble to consult the paper, its many novelties worked against its immediate acceptance. So too did its vagueness at certain crucial points and its often murky language. Ahlfors wrote a century later that Riemann composed “almost cryptic messages to the future”, and that he stated his mapping theorem in a form that “would defy any attempt at proof, even with modern methods”. Recognition came slowly. The thesis was reprinted separately in 1867 by the Göttingen publisher Adalbert Rente, and in 1876 the Gesammelte mathematische Werke, edited by Heinrich Weber and Richard Dedekind, appeared from Teubner in Leipzig. Dedekind and Weber’s 1882 article on the theory of algebraic functions drew out the full algebraic-geometric consequences of Riemann’s approach. Riemann’s own bibliography is, in number, astonishingly small: only eleven published works in a career of fifteen years, a reflection of the extraordinary care with which he prepared his papers for publication; the present thesis was the first. Today Riemann surfaces are fundamental objects in complex analysis, algebraic geometry, and topology, and related to many other areas of mathematics and of physics. Among the major applications in physics is string theory, in which zero-dimensional particles are replaced by one-dimensional strings: when a closed string (a loop in three-space) propagates, it sweeps out a surface; for consistency of the theory that surface must carry a one-dimensional complex structure, which is to say that it must be a Riemann surface. If the string does not interact, the surface is a cylinder; under interaction the string splits into two other strings and later rejoins, producing a Riemann surface of higher genus. Seen from very large distances, strings look like ordinary particles, having mass and charge, but they can also vibrate — a vibration that leads to the hypothetical quantum-mechanical particle called the graviton, and it is in this sense that string theory is a theory of quantum gravity. More recently, Nathan Seiberg and Edward Witten have discovered that many properties of N = 2 supersymmetric gauge theories admit a geometrical interpretation in terms of an auxiliary Riemann surface, now known as the Seiberg–Witten curve; one of the successes of string theory is the physical embedding of this curve in a ten- or eleven-dimensional geometry, deepening the insight into supersymmetric gauge theories. The Riemann surface, introduced in §5 of the thesis, is now — one hundred and seventy-five years later — as much a tool of physics as of mathematics. The provenance of the present copy is essentially as good as it could be. The title page carries the stamps of the Göttingen Royal Observatory (of which Gauss was director from 1807 until his death in 1855) and of the Göttingen State and University Library, the latter cancelled by the librarian on 15 October 1951; the old Observatory shelf number, ‘N° 5512’, is inked in the upper-left corner. This copy was sold as a duplicate by the Göttingen State and University Library in 1951, and is, in all likelihood, one of the copies that Riemann was required to deposit with the University at the time of his defence. The conjunction of stamps — the Observatory of Gauss and the University Library that received the candidate’s deposit copies — makes this among the most perfectly provenanced copies of the thesis that could survive. 4to (255 × 211 mm), pp. [ii], [1], 2–32. Stitched as issued, with the original green paper spine strip preserved. Light pencil-underlining to the author’s name on the title and another pencil annotation in the upper right corner. A clean copy throughout, the wrapper a little dust-soiled at the edges.
Item #6639
Price: $45,000.00
