ean 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 major achievements in this field (Hilbert’s theory of proofs, the intuitionist logic created by L. Brouwer and his adherents).
At the end of the 19th century and the beginning of the 20th, all branches of mathematics, beginning with the oldest—number theory—underwent extraordinary development that surpassed previous periods not only in the number of works but also in the perfection and power of the methods and the conclusiveness of the results. E. Kummer, L. Kronecker, Dedekind, E. I. Zolotarev, and Hilbert laid the foundations of the modern algebraic theory of numbers. In 1873, C. Hermite proved that e is a transcendental number and in 1882, the German mathematician F. Lindeman proved that π is transcendental. J. Hadamard (1896) and C. J. de la Vallée Poussin (1896) completed Chebyshev’s investigation of the law of diminishing density distribution of the primes in the sequence of natural numbers. H. Minkowski introduced geometric methods into number-theoretic investigations. In Russia number theory after Chebyshev was developed brilliantly by A. N. Korkin, G. K. Voronoi, and A. A. Markov, as well as by E. I. Zolotarev.
The focus of algebraic investigations shifted to new domains, such as the theory of groups, fields, and rings. Many of these branches of algebra found wide application in natural science: in particular, group theory found application in crystallography and later in the problems of quantum physics.
At the boundary between algebra and geometry, S. Lie created (beginning in 1873) the theory of continuous groups, whose methods later penetrated into newer domains of mathematics and natural science.
Elementary and projective geometry attracted the attention of mathematicians chiefly from the standpoint of the study of their logical and axiomatic foundations. However, the greatest scientific efforts were concentrated on differential and algebraic geometry. The differential geometry of Euclidean three-dimensional space was systematically developed by E. Beltrami and J. G. Darboux, among others. Later, the