Gauss

The distinguishing features of creation g. they are deep organic connection in its studies between teoretich. and by applied mathematics, the extraordinary latitude of problems. Works g. had clout on the development of the highest algebra, theory of the numbers, the differential geometry, theory of attraction, classical theory of electricity and magnetism, geodesy, entire branches of theoretical astronomy. In many the fields of mathematics the transactions g. contributed to an increase in the requirements for logich. of the clearness of proofs; however, g. itself remained apart from the works on a strict substantiation of matematich. of analysis, which were carried out in its time o. Cauchy.

First large SOCh. G. on the theory of the numbers and the highest algebra - arithmetical studies (1801) - in many respects predetermined further development of these disciplines. G. gives here the thorough theory of the quadratic residue, the first proof of quadratic principle of reversibility - one of the central theorems of the theory of the numbers. G. gives also the new detailed account of arifmetich. of the theory of quadratic forms, to that by constructed Zh. By Lagrange, in particular the thorough development of the theory of the composition of the classes of such forms. At the end of the book is presented the theory of the equ. of cyclotomy (i.e. equ. Xⁿ-1 = 0), which in many respects was Galois's prototype theory. Besides the general methods of solving of these equ., g. established the connection between them and by the construction of regular polygons. It, for the first time after others- Greek scientists, made it means, step forward in this question, namely: G. found all those values p, for which correct n- the square can be built by compasses and rule; in particular, after solving equ. xn-1 = by 0, it gave the construction of correct 17- square with the aid of the compasses and rules. G. gave to this discovery very great significance and it bequeathed to engrave the correct 17- square, inscribed into the circle, on its sepulchral monument, that also was completed.

Astronomich. of work G. (1800-20) in essence they are connected with a solution of the problem of the determination of the orbits of minor planets and a study of their disturbances. G. as astronomer obtained wide reputation after the development of the method of enumerating elliptich. of the orbits of planets for three observations, successfully applied by it to the first open minor planets Ceres (1801) and Pallas (1802). The results of studies on the orbit computation g. it published in SOCh. Theory of planetary motion (1809). In 1794-95 opened and into 1821-23 developed osn of matematich. the method of processing nonequivalent observant data (the least squares method). In connection with astronomich. by the calculations, based on the expansion of the integrals of the corresponding differential equ. in infinite series, g. studied a study of a question about the convergence of infinite it was series [ in the work, dedicated to the study of hyper-geometric series (1812) ].

Works g. on geodesy (1820-30) are connected with the commission to conduct geodezich. survey and to compose the detailed map of Hannover kingdom; G. organized the measurement of the arc of the meridian Gettin- gene - Al'tona, as a result of theoretical resolving of problem it created the bases of highest geodesy (studies about the objects of highest geodesy, 1842-47). For the optical signaling g. invented special instrument - heliotrope. The study of the form of the earth's surface required deep general geo-metalRICh of method for investigating the surfaces. Advanced by g. in this region ideas obtained expression in SOCh. General searches about curved surfaces (1827). The leading thought this works consists in the fact that during the study of surface as infinitely thin flexible film osn the value has not an equ. of surface in the Cartesian coordinates, but the differential quadratic form, through which the square of element of length is expressed and the invariants of which are entire sobstv. the properties of surface -, first of all, its curvature at each point. By other words, g. proposed to examine those properties of surface (Vol. n. internal), which do not depend on the bendings of surface, which do not change the lengths of lines on it. Created thus internal geometry of surfaces served as model for creating and- measured Riemann geometry.

The studies g. on the theoretical physicist (1830-40) appear in it means, to measure by the result of close contact and joint scientific work s v. By weber. Together with the weber g. Germany's first electromagnetic telegraph created absolute system of electromagnetic units and designed into 1833. In 1835 it established magnetic observatory with the the Goettingen of astronomich. of observatory. In 1838 it published labor "general theory of terrestrial magnetism". Small SOCh. "about the forces, which act inversely proportional to the square of distance" (1834-40) it contains principles of the theory of potential. To theoretical physicist they adjoin also development (1829) by g. of the Gauss principle of least constraint (see Gauss principle) and work on the theory of capillarity (1830). To the number of physical studies g. they relate and its dioptric studies (1840), in which it placed principles of the theory of the construction of image in the systems of lenses.

Very mn. of the study g. remained unpublished also in the form of descriptions, unfinished works, correspondence with the friends they enter into its scientific heritage. Up to 2-1 world war it was thoroughly developed by the Goettingen scientific society, which published 12 tt of compositions g. most interesting in this heritage they are diary g. materials according to the non-Euclidean geometry and the theory of elliptich. of functions. Diary contains 146 records, which relate to the period of 30 March 1796, when 19-year g. noted the discovery the construction of correct 17- square, on 9 July 1814. These records give the distinct picture of creation g. in first half of its scientific activity; they are very brief, written in the Lat. language and izglagayut usually the essence of the open theorems. The materials, which relate to the non-Euclidean geometry, reveal that g. arrived at the thought about the possibility of construction together with the Euclidean geometry and the geometry of non-Euclidean into 1818, but the fear that these ideas not they will be understood, and, apparently, the insufficient consciousness of their scientific importance were the reason for the fact that g. them did not develop further and did not publish. Moreover, it categorically forbade to publish by their fact, whom it devoted into its views when out of any relation to these attempts g. non-Euclidean geometry it was built and published by n. I. By Lobachevsky, g. related to Lobachevsky's publications with the considerable attention, he was the initiator of the election of his corresponding member. Goettingen scientific society, but its estimation of great discovery Lobachevsky it did not actually give. Archives g. contain also abundant materials according to the theory of elliptich. of functions and their unique theory; however, the merit of independent development and publication of the theory of elliptich. of functions belongs k. Jacobi and N. To Abel.

Works: Werke, Bd 1 -, Gott, 1908 -; in Russ. per. - General studies about the curved surfaces, in the collection: On the bases of geometry, 2 publ., Kazakhstan, 1895; Theoretical astronomy, (lectures, chitannye in Gettingene into 1820 - 26 yr., recorded By kupferom), in the book: Krylov a. n., coll. are working, Vol. 6, M. - l., 1936; Letters P. S. Laplace, K. f. gauss, F. V. Bessel et al. to the academician f. to i. Schubert, in the collection: Scientific inheritance, Vol. 1, M. - l., 1948. s. 801 - 22.

Lit.: Klein f., lectures about the development of mathematics in 19 century, translated from the German, h. 1, M. - L., 1937; Karl Friedrich Gauss. Coll. st., M., 1956.