## Ueber die Hypothesen, welche der Geometrie zu Grunde liegen. [With:] Über die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe.

Göttingen: Dieterich, 1867.

First edition, very rare separately-paginated offprints, of these two landmark papers: Riemann’s celebrated Habilitationsvortrag on the foundations of geometry which “did more to change our ideas about geometry and physical space than any work on the subject since Euclid’s *Elements*” (Gray, p. 507); bound with his famous Habilitationsschrift on Fourier series which introduced the Riemann integral and “led to the creation of set theory” (Mascré, p. 491). “On Gauss’s recommendation Riemann was appointed to a post in Göttingen and he worked for his Habilitation, the degree which would allow him to become a lecturer. He spent thirty months working on his Habilitation dissertation which was on the representability of functions by trigonometric series [the second of the offered papers]. He gave the conditions for a function to have an integral, what we now call the condition of Riemann integrability … To complete his Habilitation Riemann had to give a lecture. He prepared three lectures … Gauss chose the lecture on geometry [the first of the offered papers] … Among Riemann’s audience, only Gauss was able to appreciate the depth of Riemann’s thought … The lecture exceeded all his expectations and greatly surprised him. Returning to the faculty meeting, he spoke with the greatest praise and rare enthusiasm to Wilhelm Weber about the depth of the thoughts that Riemann had presented” (MacTutor). “Its reading on 10 June 1854 was one of the highlights in the history of mathematics” (Freudenthal in DSB). “The importance of this treatise is not confined to pure mathematics. Without it, Einstein would not have been able to develop his general theory of relativity” (PMM 293b). “In this work [*Über die Darstellbarkeit*], prepared for a doctoral defence in 1854 but published only after his death, Riemann both refined the understanding of the integral but especially opened a new era in the handling of Fourier series. His explorations led to new insights into functions and infinite series … ‘On the developability of a function by a trigonometric series’ lays the foundations of what will open a new epoch in real-variable analysis” (Mascré, pp. 491-2). Riemann had neither of these papers published himself, but when his colleague Richard Dedekind did have them printed, shortly after Riemann’s early death, in 1867, they excited great interest among mathematicians, opened new areas of research, and have had a profound influence on the development mathematics and theoretical physics from the 19^{th} century through to the present day. These two papers are bound with an offprint of Riemann’s posthumously published paper ‘Ueber die Fläche vom kleinsten Inhalt bei gegebener Begrenzung’. “Another question, which Riemann dealt with at about the same time as Weierstrass, refers to minimal surfaces, i.e., surfaces of least area for a given boundary. The relatively long elaboration by Hattendorff, published 1867, is based on Riemann’s formulas” (Laugwitz, *Berhard Riemann*, pp. 142-143). ABPC/RBH lists only one copy of *Ueber die Hypothesen *(Reiss, October 23, 2007, lot 774, €23,200), no copy of *Über die Darstellbarkeit*, and one of *Ueber die Fläche vom kleinsten Inhalt*. Reiss also sold the Hans Merkle copy of *Ueber die Hypothesen* for €50,400 in 2002. OCLC records, in the US, six copies of *Ueber die Hypothesen* and only one of *Über die Darstellbarkeit* (Smithsonian).

*Provenance*: old stamps from the ‘Kabinet I. stolice mathematicky’ of the Czech Technical University in Prague on each title page. The books from this library were moved to an elementary school in 1938 and then acquired by a Czech antiquarian dealer, from whom we purchased this volume.

*Ueber die Hypothesen* “was originally given as a lecture to the Philosophy Faculty of the University of Göttingen in 1854, in partial fulfilment of the requirements for the award of a *Habilitation*, the German qualification needed before one could teach at a German university. Candidates had also to submit a written thesis, and to offer three topics for a lecture. Riemann offered this title as the third of his list of three, and did not expect to be called to speak on it; but the senior examiner was Carl Friedrich Gauss (1777–1855), and geometry had been a life-long interest of his. Gauss was to announce himself very pleased with what he heard.

“The essay opens with a remark about a darkness that lies at the foundations of geometry, and which, in Riemann’s opinion, is not illuminated by the usual axiomatic presentation. This darkness obscures the connections between what is assumed, which is the notion of space and of constructions in space. It has persisted from Euclid to A.M. Legendre, to name only the most famous of recent authorities, perhaps because the idea of multiply extended magnitudes has not been discussed. Once this is done, said Riemann, it will be seen that even among three-dimensional extended magnitudes there is no unique choice, and so the nature of space itself becomes empirical.

“This is a remarkably bold opening, promising nothing less than the overthrow of Euclidean geometry as the source of all geometric ideas. It is a challenge not just to the scope and reach of mathematics in 1854, but to whatever philosophy orthodoxy might have prevailed in Göttingen at the time; as we shall see, Riemann had a deep immersion in German philosophical thought of the day.

“A multiply-extended magnitude—or manifold (‘*Mannigfaltigkeit*’), as he also called it—is not an unintuitive concept. A one-fold extended magnitude, or one-dimensional manifold, is a curve. Points on a curve require one measurement or coordinate to determine their position. If a curve moves along a line it sweeps out a surface or two-dimensional manifold, in which points require two coordinates to specify their position. Riemann said that examples of multiply-extended magnitudes are the positions of perceived objects and colours. There is no need to stop with three-dimensional manifolds, and indeed Riemann contemplated extended magnitudes of arbitrary multiplicity. So, roughly speaking and without looking for complications, a multiply-extended magnitude is something that is captured or measured by using a certain number of coordinates.

“Once a position of a point in an *n*-dimensional manifold is given by stating its *n* coordinates, the question arises of determining its distance from any other point in the manifold. Evidently this must be measured along a path that lies entirely in the manifold. Riemann noted that Gauss had shown how to do this in general for surfaces in space, and that his methods readily generalised to any number of dimensions. However, the formula that Gauss had used, while natural and correct for the problem he studied, was in general too simple, and it would be necessary to consider more complicated formulae … [Riemann] indicated that one way of doing this would be to write down expressions in the coordinates and their differentials and then invoke the calculus. This is why what he outlined forms a significant shift in our ideas about differential geometry. But the underlying idea is very simple. One has a geometry whenever one has a space of points (a manifold) and a way of measuring distance between points, which would be the case if one always knew the distance between infinitesimally close points. So what one wants is a manifold and an infinitesimal ruler.

“It was evident to Riemann that one might write down a vast number of different formulae for distance even given the same set of points. Presumably each formula would lead to a different geometry on that space of points, so the question was how to proceed given such a bewildering array of alternatives. There was a natural special case to consider. In the plane and on the sphere one may measure lengths by a rigid infinitesimal ruler, which may be put in any position. If the ruler is imagined to be an edge of an infinitesimal rigid body, however, then it is not true that it may be put anywhere on the surface of a pear-shaped surface, for example, because an infinitesimally small ruler adapted to fit the tightly curved dome-shaped region at the top of the pear will not fit the fatter, but still dome-shaped, region at the base, and still less is it adapted to the saddle-shaped region in the middle.

“Surfaces where just one ruler will do, a ruler which one can imagine being one side of an infinitesimally small two-dimensional solid body, have particularly simple formulae for distance, and Riemann proposed to single them out. Among surfaces these include the plane and the cylinder, which cannot be distinguished when only small patches are looked at (which is why printing can be done from a cylindrical drum to a flat piece of paper) and spheres of all radii. These are among the surfaces with constant curvature, to use a term introduced by Gauss in a celebrated memoir [*Disquisitiones generales circa superficies curvas*, 1827] …

“[Riemann] wished to discuss surfaces without reference to any ambient space. His inspiration here was Gauss’s discovery of curvature, which Gauss showed was something that could be determined from quantities measured in the surface alone. Indeed, Riemann’s whole insight into geometry may be summarised by saying that it is about the intrinsic properties of *n*-dimensional manifolds, and that the study of how a manifold inherits properties from a larger ambient space should be reformulated accordingly …

“At this stage in his lecture and in the paper, Riemann had outlined a programme according to which any geometry is to be thought of as a space (an *n*-fold extended magnitude) and distances in the space are to be measured by an infinitesimal ruler … He now proposed to show how this could be applied to the study of space.

“Riemann first observed that, on the assumption that the infinitesimal measuring rod may be put anywhere, and so space has everywhere constant curvature, then the sum of the angles in a triangle is known once it is known in a single triangle … Riemann next distinguished between the metrical aspects of an *n*-dimensional manifold and what he obscurely called the relations of extent. These may be understood as being global, topological, or geometric in nature. The cylinder and the plane are alike metrically, but differ in their relations of extent. The point Riemann observed is that there is a distinction between a space being infinite and being unbounded. The sphere, for example, is not infinite, but it is unbounded—it has no boundary, it does not come to a stop. All the evidence is that space is unbounded, but that does not mean, said Riemann—the first time anyone had thought to say this—that it is infinite. All finite universes considered in astronomy and cosmology before had been bounded, usually by what was called the sphere of the fixed stars.

“Riemann next observed that if actual space is of constant curvature, it must be of very nearly zero curvature. This is because astronomical observations have set bounds on the curvature. But if it is not of constant curvature then it is difficult to say anything about it at all. Finally, he observed that the empirical understanding of the metrical properties of space may break down in very small regions, and if this were to open the way to simpler explanations of natural phenomena we should suppose that they do. Indeed, the universe may not form a continuous (i.e. infinitely divisible) *n*-fold extended magnitude but may be a discrete manifold instead. He then concluded this remarkable paper with the hope that the study of space in such generality may prevent progress from being impeded by too narrow views, and would open the way to the study of phenomena otherwise difficult, if not impossible, to explain …

“Riemann’s vision of geometry … was to lead slowly, and not always directly, to the dominant view of the 20th century, which interprets gravity in geometric terms. The complicated story that leads from Riemann to Einstein, Weyl and the general theory of relativity has still not been fully traced by historians of mathematics, but many of the main features are in place. A number of mathematicians wrote papers solving various of the formidable technicalities of dealing with *n*-dimensional geometries of variable curvature. In 1916, inspired by Einstein and Grossmann’s new theory of general relativity, another Italian mathematician, Tullio Levi-Civita, introduced a profound idea which made it possible to discuss when a vector moves on a curve in space without changing its direction. This was the first time a significant geometric idea had been added to Riemann’s theory of manifolds, and it speedily found its way into the general relativity. But this gap from 1867 to 1916, nearly 50 years, should give us pause” (Gray).

Three vols. bound in one, 4to (259 x 207 mm), pp. [ii], 20 [Ueber die Hypothesen]; [ii], 46 [Über die Darstellbarkeit]; [ii], 50 [Ueber die Fläche] (some light dust soiling to the first and final page, otherwise very clean and crisp throughout). Contemporary half-calf over marbled boards (rubbed). The separate offprints were issued with a simple paper strip glued to the spine (no printed wrappers exist); the spine strips are all present except for part of that on the first paper which has been carefully removed by the binder.

Item #5132

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Price:
$115,000.00
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