 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 of 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 Euclid