f mechanisms.
In algebra, after the aforementioned proof of the unsolvability by radicals of a fifth-degree general equation (P. Ruffini, Abel), Galois demonstrated that the Galois group of an equation is the key to its solvability by radicals. The general abstract study of groups was proposed by Cayley. It should be noted that even in algebra the importance of group theory was universally recognized only after the works of C. Jordan appeared in the 1870’s. The concept of the field of algebraic numbers, which led to the creation of a new science—the algebraic theory of numbers— also originated in the works of Galois and Abel. Work on the old problems of number theory, which were connected with the simplest properties of ordinary integers, also rose to an essentially new level in the 19th century. In 1801, Gauss worked out the theory of representation of numbers by quadratic forms, and in 1848 and 1850, Chebyshev obtained fundamental results on the density distribution of prime numbers in the sequence of natural numbers. In 1837, Dirichlet proved the theorem of the existence of an infinite number of primes in arithmetic progressions.
The differential geometry of surfaces was created by Gauss (1827) and K. M. Peterson (1853). As indicated above, the creation of non-Euclidean geometry by Lobachevskii was of fundamental importance to the development of new views on the subject of geometry. Projective geometry, which is also associated with significant changes in the old views of space, developed in parallel with and for a long time independently of non-Euclidean geometry (J. Poncelet, J. Steiner, K. von Staudt). J. Pliicker constructed a geometry by considering lines as the fundamental elements, and H. Grassmann created the affine and metric geometry of n-dimensional vector space.
Already in the Gaussian intrinsic geometry of surfaces, differential geometry was in essence freed from its supposedly unbreakable bond with Euclidean geometry: it is incidental to this theory that a surface lies in three-dimensional Euclidean space. On this basis, Riemann created (1854 published 1856) the concept of an n -dimensional manifold with a metric geometry defined by a differential quadratic form. This gave rise to the general differential geometry of n-dimensional manifolds. Riemann was also responsible for the first ideas on the topology of n -dimensional manifolds.
Late 19th century and early 20th century. It was only in the early 1870’s that F. Klein obtained a model of Lobachevskii’s non-Euclidean geometry that finally eliminated all doubts regarding its consistency. In 1872, Klein subordinated the study of the various “geometries” of spaces of different dimensions that had been constructed by that time to the study of the invariants of a given group of transformations. At the same time (1872), works on the substantiation of analysis acquired the necessary foundation in the form of the rigorous theory of irrational numbers (R. Dedekind, G. Cantor, and Weierstrass). The fundamental works of Cantor on the general theory of infinite sets were published between 1879 and 1884. Only then could the modern general concepts of the subject of mathematics, the structure of mathematical theory, the role of axiomatics, and so on be formulated. The extensive dissemination of these concepts required several more decades. The general recognition of the modern concept of the structure of geometry is usually associated with the publication in 1899 of D. Hibert’s Foundations of Geometry.
The further investigations into the foundations of mathematics concentrated on surmounting the logical difficulties that arose in general set theory and studying the structure of mathematical theory and the methods of constructive solution of mathematical problems by the procedures of mathematical logic. These investigations grew into a large independent branch of mathematics—mathematical logic. The foundations of mathematical logic were created in the 19th century by G. Boole, P. S. Poretskii, F. Schroder, G. Frege, and G. Peano. The beginning of the 20th century saw 