rodynamics, the theory of magnetism, and thermodynamics. Very important branches of the mechanics of continuous media were developed extensively; only the hydrodynamics of an incompressible ideal fluid had been created as early as the 18th century by D. Bernoulli, Euler, D’Alembert, and Lagrange. The mathematical requirements of technology also grew rapidly. At the start of the 19th century these included problems of thermodynamics of steam engines, technical mechanics, and ballistics. The theory of partial differential equations and especially the potential theory were developed more intensively as the fundamental apparatus for the new branches of mechanics and mathematical physics. Most of the leading analysts of the first half of the century—Gauss, J. Fourier, S. Poisson, Cauchy, P. Dirichlet, G. Green, and M. V. Ostrogradskii—worked in this direction. Ostrogradskii laid the foundations of the calculus of variations for functions of several variables. Vector analysis arose as a result of investigations of the equations of mathematical physics by G. Stokes and other British mathematicians.
Despite the mechanistic conviction of the possibility of describing all natural phenomena with differential equations that prevailed in natural science at the beginning of the 19th century, probability theory was further developed under the pressure of practical requirements. Laplace and Poisson created a powerful new analytic apparatus for this purpose. P. L. Chebyshev gave a rigorous justification of the elements of probability theory and proved his celebrated theorem (1867) combining all previously known forms of the law of large numbers in one general formulation.
As was indicated above, in addition to working on the new problems posed by natural science and technology, mathematicians focused attention on problems of the rigorous substantiation of analysis (Cauchy, 1821, 1823) from the very start of the 19th century. Lobachevskii (1834) and, later, Dirichlet (1837) clearly formulated the definition of function as a wholly arbitrary correspondence. In 1799, Gauss published the first proof of the fundamental theorem of algebra, cautiously formulating the theorem, however, in purely real terms (the decomposability of a real polynomial into real factors of the first and second degrees). Only much later (1831) did Gauss clearly set forth the theory of complex numbers.
The theory of functions of a complex variable arose on the basis of a clear understanding of the nature of complex numbers. Gauss possessed great knowledge in this field but published almost nothing. The general foundations of the theory were laid by Cauchy, and the theory of elliptic functions was developed by Abel and G. Jacobi. Already at this early stage, in contrast to the purely algorithmic approach of the 18th century, attention was focused on clarifying the distinctiveness of the behavior of functions in the complex domain and the fundamental geometric principles that predominate here (beginning with the dependence discovered by Cauchy of the radius of convergence of a Taylor’s series on the location of singular points). This somewhat “qualitative” and geometric character of the theory of functions of a complex variable was intensified still further in the mid-19th century by B. Riemann. Here it turns out that the natural geometric carrier of a many-valued analytic function is not the complex plane but the Riemann surface that corresponds to the given function. K. Weierstrass achieved the same generality as Riemann but on the basis of analysis alone. However, the geometric ideas of Riemann subsequently dominated the entire approach to the theory of functions of a complex variable.
During the period of preoccupation with the theory of functions of a complex variable, Chebyshev was the most prominent investigator of the specific problems of the theory of functions in the real domain. The most explicit expression of this trend was the theory of best approximations, which was formulated (beginning in 1854) by Chebyshev on the basis of the requirements of the theory o