## Disquisitiones generales circa superficies curvas.

Göttingen: Dieterich, 1828.

First edition, the very rare separately-paginated offprint from *Commentationes Societatis Regiae Scientiarum Göttingensis *(Vol. VI, 1828, pp. 99-146), of this “masterpiece of the mathematical literature” (Zeidler, p. 16). This is an exceptionally fine copy, uncut in its original plain wrappers. “The single most important work in the history of differential geometry is Gauss' paper of 1827 [sic], *Disquisitiones generales circa superficies curvas*” (Spivak, p. 74). “The crowning contribution of the period, and his last great breakthrough in a major new direction of mathematical research, was *Disquisitiones generales circa superficies curvas* (1828), which … was the seed of more than a century of work on differential geometry” (DSB). “A decisive influence on the entire course of development of differential geometry was exerted by the publication of a remarkable paper of Gauss, ‘Disquisitiones generales circa superficies curvas’ (Göttingen, 1828), written in Latin, as was the custom in the seventeenth and eighteenth centuries. It was this paper, carefully polished and containing a wealth of new ideas, that gave this area of geometry more or less its present form and opened a large circle of new and important problems whose development provided work for geometers for many decades” (Kolmogorov & Yushkevitch, p. 7). Gauss’s *Disquisitiones* was, in particular, the basis for Riemann’s famous 1854 Habilitationsschrift ‘Uber die Hypothesen welche die Geometrie zu Grunde liegen’ (see below). “The starting point for Gauss’ work on differential geometry is found on the one hand in his lasting interest since his early years (at least since 1794) in the foundations of geometry (his interest there being focused almost exclusively in ‘finding the truth’ about the independence of Euclid’s axiom of parallels), and on the other hand in his occupation with surveying (which was required professionally of him as Director of the Göttingen Astronomical Observatory)” (Dombrowski, p. 126). ABPC/RBH list only four copies sold in the last 40 years (Gedeon, Honeyman, Norman, and Stanitz).

“The surface theory of Gauss was strongly influenced by Gauss’ work as a surveyor. Under great physical pains, Gauss worked from 1821 to 1825 as a land surveyor in the kingdom of Hannover … In 1822 he submitted his prize memoir “General solution of the problem of mapping parts of a given surface onto another surface in such a way that image and pre-image become similar in their smallest parts” to the Royal Society of Sciences in Copenhagen … When writing his prize memoir, Gauss had apparently already worked on a more general surface theory, because he added the following Latin saying to his title page: *Ab his via sterniture ad maiora* (From here the path to something more important is prepared). The development of the general surface theory, however, was difficult, though the basic ideas were known to Gauss since 1816. On February 19, 1826, he wrote to Olbers: ‘I hardly know any period in my life, where I earned so little real gain for truly exhausting work, as during this winter. I found many, many beautiful things, but my work on other things has been unsuccessful for months.’ Finally on October 8, 1827 Gauss presented the general surface theory. The title of the paper was “Disquisitiones generales circa superficies curvas” (Investigations about curved surfaces). The most important result of this masterpiece in the mathematical literature is the *Theorema egregium*” (Zeidler, p. 15).

“Gauss made the parametric representation of a surface and the corresponding expression for its element of length [the ‘metric’] into the foundations of the *Disquisitiones*. He was the first to formulate clearly and explicitly the concept of intrinsic geometry of a surface, and he proved that the curvature could be measured by a quantity (the Gaussian curvature) that belongs to intrinsic geometry, i.e., does not vary when the surface is deformed. He further developed the theory of geodesic lines [shortest paths on a surface], which also belong to intrinsic geometry …

“Another useful innovation due to Gauss was the use of spherical mapping in geometry, which was usually applied in astronomy. Every oriented line is assigned the point on the unit sphere having radius vector parallel to the line. Thus a region of the surface is mapped to a region on the unit sphere using the normal. Relying on this mapping Gauss introduced the concept of a measure on the curvature *K *(the Gaussian curvature of the surface at the given point) as the ratio of the areas of the corresponding infinitesimal regions on the sphere and on the surface” (Kolmogorov & Yushkevitch, pp. 7-9).

Gauss established the new and unexpected fact that the curvature *K* could be expressed entirely in terms of the metric of the surface, and therefore belongs to the intrinsic geometry of the surface. This, “as Gauss pointed out, leads to his ‘great theorem’ (*Theorema Egregium*): *If a curved surface is developed on any other surface, the measure of the curvature at each point remains invariant* (*ibid*., p. 10). “Gauss’ *Theorema egregium* had an enormous impact on the development of modern differential geometry and modern physics culminating in the principle ‘force equals curvature’. This principle is basic for both Einstein’s theory of general relativity on gravitation and the Standard Model in elementary particle physics” (Zeidler, p. 16).

“The concept of a geodesic, i.e., a shortest line, also belongs to the intrinsic geometry, since geodesic lines remain geodesics under deformation. For that reason Gauss, in studying problems of intrinsic geometry, found the equations of the geodesic lines in curvilinear coordinates and studied their behavior further. He introduced the notion of a geodesic circle, i.e., the geometric locus of the endpoints of geodesic radii of constant length emanating from a single point, and he showed that it was orthogonal to its radii” (Kolmogorov & Yushkevitch, p. 11).

A second major result contained in the present paper, which perhaps had even greater ramifications in mathematics than the *Theorema egregium*, is a version of what is now known as the Gauss-Bonnet theorem. “The remarkable formula found by Gauss for the sum of the angles of a geodesic triangle amounts to the statement that the excess over 180° of the sum of the angles of such a triangle in the case of a surface of positive curvature, or the deficiency in the case of a surface of negative curvature, equals the area of the spherical image of the triangle, called by Gauss the *total curvature* (curvature integra) of the triangle. This formula has a direct connection with Gauss’ reflections and computations on non-Euclidean geometry, which he kept secret during his lifetime” (*ibid*., p. 12). This result was generalized by Pierre-Ossian Bonnet in 1848 and by many later mathematicians, and served as the prototype of results linking the local geometry of a space (e.g. its curvature) with its ‘topology’ (its overall shape), a theme which pervades much of 20^{th} century mathematics. Remarkably, Gauss describes at the end of the present work some measurements he has made to verify the Gauss-Bonnet theorem. He tells us that he has measured the angles of a triangle, the greatest side of which was more than 15 miles, and found that the sum of the angles of the triangle is greater than two right-angles by almost 15 seconds of arc, in agreement with the theorem as it applies to the surface of the Earth.

The discovery that some important geometrical properties of a surface are intrinsic suggested that a surface should be treated as a space with its own geometry; this is the idea that was taken up and generalized to higher dimensions by Riemann in his Habilitationsschrift. “In his approach to differential geometry, Riemann used ideas from Carl Friedrich Gauss’s theory of surfaces, but liberated them from the restriction of being embedded in (three-dimensional) Euclidean space. He started from a determination of the length of a line element as a positive-definite quadratic differential form to derive further notions depending on metrics, in particular that of the geodesic line. Moreover, he introduced the sectional curvature of an infinitely small surface element, derived from the Gaussian curvature of the associated finite surface inside the manifold, which is generated by all geodesic lines starting in the surface element” (*Companion Encyclopedia*, p. 928).

Gauss wrote a draft version of this paper in 1825 under the title *Neue allgemeine Untersuchungen über die krummen Flächen*, but this was not published until 1901, in vol. VIII of the *Werke*. The *Disquisitiones* itself was delivered in the form of a lecture to the Göttingen Royal Scientific Society on 8 October 1827.

Norman 880. I. Grattan-Guinness (ed.), *Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences*, 1994; A. N. Kolmogorov & A. P. Yushkevitch, *Mathematics of the 19th century: Geometry. Analytic Function Theory*, 1996; M. Spivak, *A comprehensive introduction to differential geometry*, vol. 2, 1979 .E. Zeidler, *Quantum Field theory III. Gauge Theory,* 2013, pp. 15-19. For a detailed account of the content and historical antecedents of the work, see P. Dombrowski, ‘150 years after Gauss’ ‘Disquisitiones generales circa superficies curvas’,’ *Astérisque* 62 (1979), pp. 97-153.

4to (264 x 223 mm), pp. 50, uncut (some light foxing). Contemporary (probably original) plain wrappers (spine a bit worn). An exceptional copy in original state.

Item #5288

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Price:
$12,500.00
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