Göttingen: Dieterich, 1811, 1813, 1816, 1820, 1828.
First editions, journal issues, of thirteen important papers by Gauss, including works on the fundamental theorem of algebra, number theory, hypergeometric functions, approximation theory, differential geometry, gravitation, and celestial mechanics. “Gauss ranks, together with Archimedes and Newton, as one of the greatest geniuses in the history of mathematics” (Printing & the Mind of Man, p. 155). “In his first years at Göttingen, Gauss experienced a second upsurge of ideas and publications in various fields of mathematics. Among the latter were several notable papers inspired by his work on the tiny planet Pallas, perturbed by Jupiter [including Disquisitio de elementis ellipticis palladis]; Disquisitlones generates circa seriem infinitam, an early rigorous treatment of series and the introduction of the hypergeometric functions, ancestors of the “special functions” of physics; Methodus nova integralium valores per approximationem invenlendi, an important contribution to approximate integration; … and Determinatio attractionis quam in punctum quodvis positionis datae exerceret planeta si eius massa per totam orbitam ratione temporis quo singulae partes describuntur uniformiter esset dispertita, which showed that the perturbation caused by a planet is the same as that of an equal mass distributed along its orbit in proportion to the time spent on an arc. At the same time Gauss continued thinking about unsolved mathematical problems … By 1817 Gauss was ready to move toward geodesy … the period of preoccupation with geodesy was in fact one of the most scientifically creative of Gauss’s long career. Already in 1813 geodesic problems had inspired his Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum methodus nova tractata, a significant early work on potential theory … Surveying problems also motivated Gauss to develop his ideas on least squares and more general problems of what is now called mathematical statistics. The result was the definitive exposition of his mature ideas in the Theoria combinationis obseruationum erroribus minimis obnoxiae (with supplement) [only the supplement present in the offered volumes] … However, the crowning contribution of the period, and his last breakthrough in a major new direction of mathematical research, was Disquisitiones generates circa superficies curvas, which grew out of his geodesic meditations of three decades and was the seed of more than a century of work on differential geometry” (DSB).
Gauss’s first great work was his doctoral thesis, Demonstratio Nova Theorematis Omnem Functionem Algebraicam Rationalem Integram unius Variabilis in Factores Reales Primi vel Secundi Gradus resolvi posse (1799), in which he gave the first rigorous proof of the fundamental theorem of algebra. This theorem, which states that every algebraic equation in one unknown has a root, was stated by Albert Girard, Descartes, Newton, and Maclaurin. Attempts at a proof were made by d’Alembert, Euler, and Lagrange, but Gauss was the first to furnish a rigorous demonstration. However, Gauss’s proof of 1799, which used geometric notions, was itself not completely rigorous (its gaps were not completely filled until 1920), and Gauss himself was not satisfied with it. In 1816 he published two further proofs, Demonstratio nova altera theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse and Theorematis de resolubilitate functionum algebraicarum integrarum in factores reales demonstratio tertia. The first of these is almost strictly algebraic, but it is not really complete as it presupposes that every real equation of odd degree has a real root and that every quadratic equation with complex coefficients has two complex roots. The third proof is easier but uses calculus.
Two of the papers in these volumes deal with analysis (i.e., calculus). Disquisitlones generales circa seriem “has several claims to fame: it considers x as a complex variable; and it contains the earliest rigorous argument for the convergence of a power series and a study of the behaviour of the function at a point on the boundary of the circle of convergence, as well as a thorough examination of continued fraction expansions for certain quotients of hypergeometric functions” (Gray, Linear Differential Equations and Group Theory from Riemann to Poincaré (2010), p. 5).
In Methodus nova integralium valores per approximationem inveniendi, Gauss introduced new quadrature (i.e., integration) formulas with a degree of accuracy considerably improved as compared with the Newton-Cotes formula. The latter formula gives an approximate value for the integral of a function over an interval in terms of the values of the function at the intermediate points obtained by dividing the interval into n equal parts. Gauss asked what could be accomplished if these intermediate points were available as parameters to be chosen? He answered this question completely, leading to the so-called Gauss quadrature formula. He showed that if these division points are taken to be the roots of a Legendre polynomial of degree n, the quadrature formula is exact for the integration of polynomials of degree up to 2n – 1. In fact, Gauss did not use Legendre polynomials in his proof, he used his hypergeometric function instead (which contain the Legendre polynomials as a special case). “Gauss’s method is not only elegant but it differs from the more modern approaches in that he arrives at his result from first principles, and not, as is now done, by showing how the result follows from properties of Legendre’s polynomials” (Goldstine, A History of Numerical Analysis (1977), p. 225).“
Three of the papers deal with gravitation and celestial mechanics. “Gauss used the method of least squares to determine the orbit of Ceres in 1801 just after the discovery of the planet by Piazzi. When a second minor planet, Pallas, was later discovered by Olbers, Gauss again set about determining its precise orbit, and he explained his approach in a memoir dated 1810 [Disquisitio de elementis ellipticis palladis]. Starting from six observations made when the planet was in opposition — and so close to the Earth — Gauss obtained 12 equations between 6 unknowns (the mean anomaly, the mean diurnal movement, the longitude of the perihelion, the eccentricity, the longitude of the node, the inclination). After obtaining an approximate solution, he determined 12 linear equations that must satisfy corrections to be made to the 6 unknowns. Rejecting the tenth because it was too imprecise on account of the observation, he used 11 equations from which he derived 6 normal equations and finally 6 corrections. In order to make it easier to deal with the algebraic solution of the system formed by the normal equations, Gauss made certain remarks at the end of the Theory of the Movement of Celestial Bodies (1809) which he developed in his Memoir on the minor planet Pallas (1810), and then in the second part of the Theory of the Combination of Observations which appeared in 1823 [the penultimate paper offered here]. These remarks remind us of what is now called, in linear algebra, Gaussian elimination, or the Gauss pivot method, and the transformation of a quadratic into quadratic form, that is into a sum of squares, which produces a diagonal matrix” (Chabert, A History of Algorithms (1999), p. 291). “In 1810, in his Disquisitio de Elementis ellipticis Palladis, Gauss sets out to determine details (‘elliptical elements’) about the orbit of Pallas, the second-largest asteroid of the solar system. He obtains a system of linear equations in six unknowns, where not all equations can be satisfied simultaneously. Hence he needs to determine values for the unknowns that will minimize the total squared error. Instead of merely solving the problem at hand, Gauss digresses and introduces a method for dealing with such systems of linear equations in general” (Althoen & McLaughlin, ‘Gauss-Jordan reduction: a brief history,’ The American Mathematical Monthly 94 (1987), p. 134).
In Determinatio attractionis, Gauss’s astronomical work led to different mathematical ideas. In this paper he showed that the gravitational attraction due to a planet is the same as that of an equal mass distributed along its orbit in proportion to the time spent on an arc, and that the attraction caused by such a ‘Gaussian ring’ could be expressed in terms of elliptic integrals. He related the evaluation of these integrals to the ‘arithmetic-geometric mean’. This was the only work he published on elliptic integrals (although much more remained in manuscript and was published after his death).
Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum methodus nova tractate, which is arguably the first work in potential theory, contains the first indication of what was later called the ‘divergence theorem’, which relates integrals over a volume in three-dimensional space with integrals over its bounding surface. “The contributions Gauss’s paper makes are twofold: first, the idea of the surface of a body and its general local infinitesimal area, and secondly, the process of lines entering and exiting from the surface. These two concepts remained long after the rest of the paper had been forgotten” (Harman, Wranglers and Physicists (1985), pp. 114-5). The divergence theorem was proved in complete generality by George Green in 1828.
The number-theoretic papers in these volumes all relate to the law of quadratic reciprocity, which Gauss privately referred to as the Aureum Theorema (Golden Theorem). This was discovered by Euler in 1783 but the first proof was given by Gauss in his Disquisitiones arithmeticae (1801). In Theorematis fundamentalis in doctrina de residuis quadraticis demonstrationes et ampliationes novae Gauss provided his fifth and sixth proofs which are particularly important since, as Gauss explains in the introduction to this work, the techniques they employ (‘Gauss’s lemma’ and ‘Gauss sums,’ respectively) can be applied to the study of cubic and biquadratic reciprocity. Gauss sums were studied in his Summatio quarumdam serierum singularium.
“Gauss’ fifth proof of the quadratic reciprocity is Gauss’ most elementary proof of the quadratic reciprocity law… for which Gauss has given altogether eight proofs, six of which were published, and two more were found in his papers after his death. The fifth proof was published in 1818, together with the sixth proof, under the title Theorematis Fundamentalis in Doctrina de Residuis Quadraticis Demonstrationes et Ampliationes Novae. But Gauss found it probably already between 1807 and 1808… [it is] based on Gauss’ lemma” (Laudal & Piene, The legacy of Niels Henrik Abel (2004), p. 276).
“The sixth and last of Gauss’ published proofs of the law of quadratic reciprocity was published in 1818… He mentions in the introduction to this paper that for years he had searched for a method that would generalize to the cubic and biquadratic case and that finally his untiring efforts were crowned with success… The purpose of publishing this sixth proof, he states, was to bring to a close that part of the higher arithmetic dealing with quadratic residues and to say, in a sense, farewell…” (Ireland & Rosen, A classical introduction to modern number theory (1998), p. 76).
The attempt to generalize quadratic reciprocity for powers higher than the second was one of the main goals that led nineteenth century mathematicians, including Gauss, Dedekind, Dirichlet, Eisenstein, Hilbert, Jacobi and Kummer to the study of general algebraic number fields; notably, Kummer invented ‘ideal numbers’ in order to state and prove higher reciprocity laws. The ninth in the list of 23 unsolved problems which David Hilbert proposed to the International Congress of Mathematicians in 1900 asked for the “Proof of the most general reciprocity law for an arbitrary number field.” The first steps were taken by Gauss himself in Theoria residuorum biquadraticorum [of which the first part appears in this collection – a second part appeared in Vol. VII]. In this paper Gauss stated (but did not prove) the law of biquadratic [i.e., quartic] reciprocity.
The last paper in the collection is the most important, Disquisitiones generales circa superficies curvas. The most important result of this masterpiece in the mathematical literature is the Theorema egregium” (Zeidler, Quantum Field theory III. Gauge Theory (2013), p. 15). “A decisive influence on the entire course of development of differential geometry was exerted by the publication of [this] paper of Gauss… 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, Mathematics of the 19th century: Geometry. Analytic Function Theory (1996), p. 7).
The Theorema egregium shows that a certain measure of how curved a surface is (now called its Gaussian curvature) is ‘intrinsic,’ i.e. can be determined by measurements made only on the surface and without reference to the ambient three-dimensional space. “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). Gauss also shows that certain other features of a surface, such as its geodesic lines (the shortest paths), are intrinsic.
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. Gauss showed that the total Gaussian curvature of the interior of a triangle on a surface with geodesic sides is the amount by which the sum of the angles of the triangle exceeds its Euclidean value π. 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 20th 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).
Many, and possibly all, of these papers were also issued in offprint form, although most of them are extremely rare. While in many cases offprints were published later than the journal, in this case the offprints probably take precedence. For example, Methodus nova integralium valores per approximationem inveniendi appeared as an offprint in 1815 but in the journal in 1816, and Determinatio attractionis appeared as an offprint in 1818 but in the journal in 1820.
4to (238 x 188 mm), pp. xx, 564 with 17 folding plates (Vol. I); xxiv, 565 with 11 folding plates (Vol. II); pp. xxiv, 564 with 16 folding plates (Vol. III); xxiv, 512 with 18 folding plates (Vol. IV); xxiv, 579 with 24 folding plates (Vol. VI). In vols. I & II, each article is separately-paginated; in vols. III-VI, each section is separately-paginated. Vol. I in contemporary full calf, upper compartment of spine missing, Vol. II, III, IV, and VI in uniform contemporary half-calf. All volumes with old ex-library rubber stamps.