Gauss, Karl Friedrich


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Gauss, Karl Friedrich

 

Born Apr. 30, 1777, in Braunschweig; died Feb. 23, 1855, in Göttingen, a German mathematician who also made a fundamental contribution to astronomy and geodesy. He was born into the family of a plumber. From 1795 through 1798 he studied at the University of Göttingen. In 1799 he received an assistant professorship at Braunschweig and in 1807 a professorship of mathematics and astronomy at the University of Göttingen, which carried with it the position of director of the Göttingen Astronomical Observatory. Gauss held this post until his death.

The distinguishing features of Gauss’ work are the deep organic relationship in his studies between theoretical and applied mathematics and the extraordinary breadth of the problems he covered. Gauss’ works exerted a great influence on the development of higher algebra, the theory of numbers, differential geometry, the theory of gravitation, the classical theory of electricity and magnetism, geodesy, and entire branches of theoretical astronomy. In many fields of mathematics Gauss’ works helped increase the demands on the logical rigor of proofs, but Gauss himself did not engage in work on the strict substantiation of mathematical analysis that was conducted in his time by A. Cauchy.

Gauss’ first major work on the theory of numbers and higher algebra—Disquisitiones arithmeticae—largely predetermined the subsequent development of these disciplines. Here Gauss provided a detailed theory of quadratic residues and the first proof of the quadratic law of reciprocity—one of the central theorems of number theory. Gauss also gave a detailed new presentation of the arithmetic theory of quadratic forms, which had previously been constructed by J. Lagrange, and in particular gave careful treatment to the theory of the composition of the classes of these forms. The theory of cyclotomic equations (that is, equations of the type xn — 1 = 0), which was largely the prototype for Galoís’ theory, is presented at the end of the book.

In addition to the general methods of solving these equations, Gauss established a relation between them and the construction of regular polygons. He was the first since the ancient Greek scholars to take a significant stride forward in this problem: he found all values of n for which a regular n-gon can be constructed with compass and ruler; in particular, by solving the equation x17 — 1 = 0, he gave a construction of a regular 17-sided polygon by means of compass and ruler. Gauss ascribed very great importance to this discovery and ordered that a regular 17-sided polygon be inscribed in a circle on his tombstone, which was in fact done.

Gauss’ astronomical work (1800-20) was mainly associated with the solution of the problem of determining the orbits of the asteroids and with the study of their perturbations. As an astronomer, Gauss became well known after developing a method of computing the elliptical orbits of planets from three observations, a method he successfully applied to the first asteroids to be discovered—Ceres (1801) and Pallas (1802). Gauss published the results of his investigations on orbital computation in his work Theory of the Motion of Heavenly Bodies. In 1794-95 he discovered, and in 1821-23 worked out, a basic mathematical method of handling non-equivalent observational data (the method of least squares). In connection with astronomical computations based on the expansion of the integrals of corresponding differential equations into infinite series, Gauss dealt with the problem of the convergence of infinite series (in a work devoted to the study of hypergeometric series, 1812).

Gauss’ works on geodesy (1820-30) were associated with the task of making a geodetic survey and compiling a detailed map of the kingdom of Hanover; he undertook to measure the arc of the Göttingen-Altona meridian, and as a result of his theoretical handling of the problem created the foundations for higher geodesy (Research on Objects of Higher Geodesy, 1842—47). He invented a special device—the heliotrope—for optical signaling. The study of the shape of the earth’s surface required a deeper general geometric method for studying surfaces. The ideas advanced by Gauss in this field found expression in the work General Research on Curved Surfaces (1827). The guiding idea of this work was that in the study of a surface as an infinitely fine, flexible film, the differential quadratic form, through which the square of the length component is expressed and whose invariants are all inherent properties of the surface (primarily its curvature at each point), is of primary importance, rather than the equation for the surface in Cartesian coordinates. In other words, Gauss proposed that those properties of a surface (so-called internal properties) be considered that do not depend on buckling of the surface that does not change the length of lines inscribed on it. The internal surface geometry thus created served as a model for devising n-dimensional Riemannian geometry.

Gauss’ research in theoretical physics (1830-40) is in large measure the result of close correspondence and joint scientific work with W. Weber. Together with Weber, Gauss devised an absolute system of electromagnetic units and in 1833 designed the first electromagnetic telegraph in Germany. In 1835, Gauss founded a magnetic observatory at the Gçttingen Astronomical Observatory. In 1838 he published his work General Theory of Terrestrial Magnetism. His short work On Forces Acting in Inverse Proportion to the Square of Distance (1834-40) contains the foundations of the theory of potential. Gauss’ development (1829) of the principle of least constraint and his work on the theory of capillarity (1830) also border on theoretical physics. Among Gauss’ research in physics are his Dioptric Research (1840), in which he laid the foundations of the theory of image construction in lens systems.

A great deal of Gauss’ research has remained unpublished and, in the form of essays, unfinished works, and correspondence with friends, is part of his scientific heritage. Until World War II it was carefully handled by the Royal Society of Sciences in Gçttingen, which published 12 volumes of his works. Gauss’ diary and his materials on non-Euclidean geometry and the theory of elliptic functions are the most interesting items in this heritage. The diary contains 146 notes dating to the period between Mar. 30, 1796, when the 19-year-old Gauss commented on his discovery of the construction of a regular 17-sided polygon, to July 9, 1814. These notes give a clear picture of his creativity in the first half of his scientific endeavor; they are very brief, written in Latin, and usually present the essence of the discovered theorems. The materials relating to non-Euclidean geometry show that in 1818 Gauss concluded that the construction of a non-Euclidean geometry, in addition to Euclidean geometry, was possible, but the apprehension that these ideas would not be understood and apparently insufficient recognition of their scientific importance were the reasons why he did not further develop or publish them. Moreover, he categorically prohibited those whom he initiated into his views to publish them. When, entirely independently of Gauss’ attempts, a non-Euclidean geometry was constructed and published by N. I. Lobachevskii, Gauss paid great attention to Lobachevskii’s publications and initiated his election as corresponding member of the Göttingen Royal Society of Sciences but essentially made no assessment of Lobachevskii’s great discovery. Gauss’ archives also contain copious materials on the theory of elliptic functions and a unique theory of them; however, credit for the independent development and publication of the theory of elliptic functions belongs to K. Jacobi and N. Abel.

WORKS

Werke, vols. 1—. Göttingen, 1908—.
In Russian translation:
“Obshchie issledovaniia o krivykh poverkhnostiakh.” In the collection Ob osnovaniiakh geometrii, 2nd ed. Kazan, 1895.
“Teoreticheskaia astronomiia” (lectures delivered in Göttingen in 1820-26 and recorded by Kupfer). In A. N. Krylov, Sobr. trudov, vol. 6, Moscow-Leningrad, 1936.
“Pis’ma P. S. Laplasa, K. F. Gaussa, F. V. Besselia i dr. k akademiku F. I. Shubertu.” In the collection Nauchnoe nasledstvo, vol. 1. Moscow-Leningrad, 1948. Pages 801-22.

REFERENCES

Klein, F. Lektsii o razvitii matematiki v 19 stoletii, part 1. Moscow-Leningrad, 1937. (Translated from German.)
Karl Fridrikh Gauss: Sb. st. Moscow, 1956.