Über die Hypothesen, welche der Geometrie zu Grunde liegen.
Göttingen: Dieterichschen Buchhandlung, 1867. First edition, very rare separately-paginated offprint, and a fine copy without institutional stamps, of the lecture that remade geometry. Riemann’s Über die Hypothesen welche der Geometrie zu Grunde liegen, delivered before the philosophical faculty at Göttingen on 10 June 1854, generalised Gauss’s differential geometry of curved surfaces to spaces of arbitrary dimension, introduced the concepts of the Riemannian manifold and the Riemannian metric, and demonstrated that the geometry of physical space is not a truth of reason but a question to be settled by measurement. The lecture was not published in Riemann’s lifetime; his colleague and friend Richard Dedekind brought it to print in the Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen in 1867, the year after Riemann’s death at the age of thirty-nine. Sixty years later, the mathematical apparatus Riemann had outlined in twenty pages supplied the framework within which Einstein formulated the general theory of relativity. ABPC/RBH lists only two other copies at auction: a defective copy with erased inscriptions (Reiss, 2007, €23,200) and a copy with institutional stamps (Christie’s, 2022, £75,600). OCLC records six copies in the United States. The occasion was itself remarkable. To qualify as a lecturer (Privatdozent) in a German university, a candidate was required to submit a dissertation and then to deliver a public lecture on one of three topics proposed by the candidate and chosen by the faculty. Riemann submitted three titles: the first on the representability of functions by trigonometric series (the subject of his Habilitationsschrift, already substantially complete), the second on the solution of pairs of quadratic equations (a technical algebraic problem), and the third on the hypotheses underlying geometry. Convention dictated that the faculty would choose the first topic; Gauss, who as the senior mathematician controlled the choice, selected the third. Dedekind, who witnessed the event, later wrote that the choice was unexpected and that Gauss was curious to hear how so young a man would handle so difficult a subject. The curiosity was well placed. Riemann was twenty-seven; Gauss was seventy-six and would not live past the following spring. Jeremy Gray has described the resulting lecture as the most consequential work on geometry and physical space since Euclid’s Elements. Riemann opened with an observation that would have been striking in any mathematical lecture and was revolutionary in this one: that a darkness lay at the foundations of geometry which the usual axiomatic treatment did nothing to illuminate. The problem was not with the theorems but with the assumptions. Euclidean geometry assumed a particular structure for space — three dimensions, infinite extent, a specific relationship between distances — without examining whether those assumptions were necessary or even true. Riemann proposed to replace the unexamined assumptions with a general framework within which many different geometries could be formulated, and to let observation determine which of them described the physical world. The framework begins with what Riemann called a Mannigfaltigkeit — a manifold, or multiply-extended magnitude — of n dimensions: a space in which the position of each point is specified by n coordinates. A curve is a one-dimensional manifold; a surface is a two-dimensional manifold; ordinary space is three-dimensional; but there is no reason in principle to stop at three. The concept is general enough to include the space of all possible colours (which Riemann mentions as an example) or any other continuously variable quantity that requires several independent measurements to specify. Once the manifold is given, Riemann asks what it means to measure distances within it. His answer is an infinitesimal metric — a formula, expressed in terms of the coordinates and their differentials, that gives the distance between two infinitesimally close points. If the manifold is a surface embedded in ordinary space, the metric is the formula Gauss had derived in his Disquisitiones generales circa superficies curvas of 1827; Riemann’s innovation was to define the metric without reference to any surrounding space, treating it as an intrinsic property of the manifold itself. A geometry, in Riemann’s conception, is a manifold equipped with a metric — a space of points together with an infinitesimal ruler. From this apparatus Riemann derives a curvature that generalises Gauss’s curvature of surfaces to n dimensions. In two dimensions the curvature at a point is a single number; in higher dimensions it is a more complex object — the Riemann curvature tensor, as later mathematics would call it — that encodes the degree to which the geometry of the manifold deviates from flatness at each point and in each direction. A manifold of zero curvature is flat: its geometry is Euclidean. A manifold of constant positive curvature is the n-dimensional analogue of a sphere; one of constant negative curvature is the n-dimensional analogue of the pseudosphere that Beltrami would identify in 1868 as the natural model for the non-Euclidean geometry of Lobachevsky and Bolyai. A manifold of variable curvature — positive in some regions, negative in others, changing from point to point — admits no simple global description; it is exactly such a manifold that describes the spacetime of general relativity, in which the curvature at each point is determined by the distribution of matter and energy. The intellectual antecedents of the lecture were narrower than its consequences would suggest. Gauss’s 1827 Disquisitiones generales, with its theorema egregium establishing that the curvature of a surface is intrinsic to it and independent of any embedding, was the indispensable mathematical precursor; Gauss had also, in private speculations he never published, considered the possibility of geometries in which the parallel postulate failed, and had even, around 1820, conducted a survey of the great triangle Hohenhagen–Brocken–Inselberg in the Harz mountains, ostensibly for cartographic reasons but in fact to test whether the angles of a sufficiently large terrestrial triangle would sum to less than two right angles. Lobachevsky, in publications of 1829 and 1837, and János Bolyai, in an appendix to his father’s textbook of 1832, had each independently constructed a consistent geometry in which the parallel postulate was replaced by an alternative; their work was little known outside specialised circles, and Riemann does not cite either, though his manifolds of constant negative curvature are the same objects. What Riemann added to this prehistory was a framework general enough to contain all of these as instances and to admit indefinitely many further geometries beside. The most radical passage of the lecture is its conclusion. Riemann draws a distinction between the boundedness of space and its infinitude: a sphere, for example, is unbounded (it has no edges, no barriers) but finite (it has a definite area). All the evidence available to astronomers, he observes, is that space is unbounded; but this does not mean that it is infinite. The assertion that space is infinite — which every cosmology since antiquity had taken for granted — is an empirical claim, and one that might be false. He then observes that the metric properties of space might break down in very small regions, and that if this were to lead to simpler explanations of physical phenomena one should suppose that they do. Space at the smallest scales might not be continuous at all but discrete — a lattice rather than a continuum. The passage anticipates, by a century and a half, the speculations of quantum gravity. Finally, Riemann proposes that the ground of the metric relations of space must be sought in the binding forces which act upon it — a sentence that Einstein, in 1915, would vindicate with an exactness Riemann could not have foreseen: in general relativity, the curvature of spacetime is determined by the stress-energy tensor, and the geometry of the universe is a consequence of the matter it contains. Bernhard Riemann (1826–1866), the son of a Lutheran pastor at Breselenz in the Kingdom of Hanover, originally intended to follow his father into the church. He entered Göttingen in 1846, moved to Berlin for two years to study under Jacobi, Dirichlet, and Steiner, and returned to Göttingen in 1849 to complete his doctoral thesis under Gauss — the thesis on complex-variable theory that is itself a founding text in its field. His Habilitationsschrift on trigonometric series (1854) and the present Habilitationsvortrag followed; he became Privatdozent that year, associate professor in 1857, and, on Dirichlet’s death in 1859, full professor and holder of Gauss’s former chair. His research output was small in volume — fewer than twenty published papers — but almost every one of them opened a new field or transformed an existing one. His health, always fragile, deteriorated from tuberculosis in the early 1860s; he died at Selasca on Lake Maggiore on 20 July 1866, at thirty-nine, with most of his work unpublished. Dedekind’s posthumous edition of the present lecture, the Habilitationsschrift on trigonometric series, and several other papers in the 1867 Abhandlungen brought the scope of Riemann’s achievement before the mathematical public for the first time. The compass of Riemann’s achievement, given the brevity of his career, is not easily exhausted. The 1851 doctoral thesis founded the geometric theory of complex functions and introduced the concept of the Riemann surface, the multi-sheeted complex domain on which a multi-valued function becomes single-valued; this apparatus would underlie the algebraic geometry of Brill, Noether, Klein, and Hilbert through the rest of the century. The 1859 paper on the distribution of prime numbers stated the conjecture, still unresolved, that all the non-trivial zeros of the zeta function lie on a single vertical line in the complex plane — the Riemann hypothesis, the most celebrated open problem in mathematics. The Habilitationsschrift on trigonometric series rebuilt the foundations of the theory of integration and introduced what is now called the Riemann integral. A 1857 paper on Abelian functions extended the Cauchy–Jacobi–Weierstrass theory in directions that would not be exhausted for a generation. Each of these works, like the present lecture, transformed its field; collectively they place Riemann beside Gauss as the deepest mathematical thinker of the nineteenth century, and Felix Klein later observed that the most fertile lines of research in the second half of the century all passed through Riemann at one point or another. The lecture’s reception was slow but deep. The technical apparatus it required — the tensor calculus, the theory of connections, the global analysis of manifolds — was not developed until the work of Christoffel (1869), Ricci-Curbastro (1884–1900), and Levi-Civita (1917). Christoffel’s 1869 paper introduced the symbols that bear his name and gave the transformation laws under which Riemann’s metric retained its meaning under change of coordinates; Ricci-Curbastro and Levi-Civita developed those symbols into a systematic calcolo differenziale assoluto, presented in their 1900 joint paper as a comprehensive language for differential geometry; Levi-Civita’s 1917 introduction of the parallel transport of vectors gave the apparatus its modern geometric foundation. The first profound application outside pure mathematics came in 1915–16, when Einstein and Marcel Grossmann, searching for a mathematical framework adequate to the idea that gravity is not a force but a property of the geometry of spacetime, found in the Riemannian apparatus exactly the language they needed; PMM 293b notes that without Riemann’s framework Einstein could not have developed general relativity. The influence has continued to widen through the twentieth century: the Riemann curvature tensor is fundamental to modern cosmology, to the gauge field theories of particle physics, and to the mathematical programme of classifying manifolds that runs through topology from Poincaré to the resolution of the geometrisation conjecture by Perelman in 2003. The direct influence of Riemann’s lecture on the development of modern geometry was channelled through a small number of canonical transmissions. William Kingdon Clifford’s English translation in Nature in 1873, completed shortly after he first read the German text, brought Riemann’s ideas before the English-speaking mathematical public and was instrumental in shaping the early Anglo-American interest in non-Euclidean geometry; Clifford’s own short paper of the same year on the space-theory of matter applied Riemann’s framework to a hypothesis that what we perceive as the motion of matter through space is in fact the propagation of regions of variable curvature, an anticipation of general relativity that Clifford did not live to develop. Felix Klein’s 1872 Erlangen Programme, which classified geometries by the groups of transformations under which their properties remain invariant, was in part a response to Riemann’s framework, and Klein’s 1876 edition of Riemann’s collected works, prepared with Heinrich Weber, made the full corpus accessible to a generation of working mathematicians. Hermann Weyl’s 1919 commentary on the lecture, included as an afterword in successive editions of his Raum, Zeit, Materie, joined the apparatus of the present lecture to the freshly-published apparatus of general relativity and provided the standard exposition for the inter-war generation. Through these transmissions the lecture has remained a working text rather than a historical curiosity, in continuous use as the canonical statement of the foundational ideas of differential geometry for more than a century and a half after its delivery. Provenance: Henry Taber (1860–1936), American mathematician — signature dated August 1883 on the title page, with a second inscription ‘H. Taber’ on the front wrapper in a different hand. Taber studied at Yale before earning his PhD in mathematics at Johns Hopkins under Charles Sanders Peirce and William Edward Story. He spent most of his career as professor of mathematics at Clark University, Worcester, Massachusetts, which under Story’s direction was the leading centre for mathematics in the United States until 1892. The inscription on the title page records that Taber acquired the offprint in Berlin — Story had studied there in the early 1870s and retained contacts in the German mathematical community, and Berlin was at the time the home of Leopold Kronecker, one of the world’s leading algebraists and a central figure in the reception of Riemann’s ideas. This is a separately-paginated offprint from Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen, mathematische Klasse, vol. 13 (1867), pp. 133–152. The 1867 Abhandlungen, edited posthumously by Dedekind, contains in the same volume Riemann’s Habilitationsschrift on trigonometric series and three further pieces from the unpublished papers; together with the present lecture they form the first comprehensive presentation of his work outside the journals he had himself seen through the press. The offprints of the geometry lecture were issued separately in only a small run for the limited circle of mathematicians who could be expected to make use of them; copies have always been scarce, and copies in good condition without institutional markings are exceptional. The Dieterichsche Buchhandlung, founded at Göttingen in 1760 and the standard publisher of the Royal Society of Sciences’ Abhandlungen through the long nineteenth century, applied to the offprint the same plain typographic standards it had set for the parent volume, and the present copy preserves the offprint exactly as it left the printing house in 1867. References: Printing and the Mind of Man 293(b) — Parkinson, Breakthroughs 1854 & 1867 — Not in Honeyman, Horblit, or Norman — Gray, ch. 39 in Grattan-Guinness (ed.), Landmark Writings in Western Mathematics 1640–1940, 2005 — Laugwitz, Bernhard Riemann 1826–1866: Turning Points in the Conception of Mathematics, 1999 — Monastyrsky, Riemann, Topology, and Physics, 2nd ed., 2008 — Spivak, A Comprehensive Introduction to Differential Geometry, vol. 2 (3rd ed., 1999), ch. 4, for an annotated English translation of the lecture. 4to (245 × 203 mm), pp. [ii], 20. Nineteenth-century plain wrappers.
Item #6438
Price: $150,000.00
