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 differential geometry of various groups of transformations broader than Euclidean motion developed rapidly, especially the differential geometry of h-dimensional spaces. Stimulated by the emergence of the general theory of relativity, geometric investigations in this direction were initiated by T. Levi-Civita, E. Cartan, and H. Weyl.
Because of the development of more general viewpoints of set theory and the theory of functions of a real variable, the theory of analytic functions at the end of the 19th century lost its previous position as the core of all mathematical analysis. However, it continued to be developed intensively both in accordance with its intrinsic requirements and because of its newly discovered connections with other branches of analysis and natural science. Especially significant in this last direction was the elucidation of the role of conformal mappings in the solution of boundary value problems for partial differential equations (for example, Dirichlet’s problems for the Laplace equation), in the study of planar flows of an ideal fluid, and in problems of the theory of elasticity.
Klein and H. Poincare created the theory of automorphic functions in which Lobachevskii’s geometry found remarkable applications. C. E. Picard, Poincare, Hadamard, and E. Borel worked out in depth the theory of entire functions, which in particular made it possible to derive the theorem of the density distribution of primes that was mentioned above. Poincare, Hilbert, and others developed the geometric theory of functions and the theory of Riemann surfaces. Conformal mappings found application in aeromechanics (N. E. Zhukovskii, S. A. Chaplygin).
As a result of systematic development of mathematical analysis, a new branch of mathematics—the theory of functions of a real variable—arose on the basis of the rigorous arithmetic theory of irrational numbers and set theory. Whereas previously only functions arising “naturally” from various special problems were studied systematically, interest in the complete elucidatio