rily connected with the activities of various academies, and universities played a lesser role. The remoteness of the major mathematicians from university teaching was compensated for by the energy with which, beginning with Euler and Lagrange, they all wrote textbooks and extensive treatises on individual topics.
All the branches of mathematical analysis that were created in the 17th and 18th centuries continued to develop intensively in the 19th and 20th centuries. The range of their applications to the problems advanced by natural science and technology also expanded greatly. However, in addition to this quantitative growth, a number of essentially new features can be observed in the development of mathematics in the late 18th century and in the early 19th century.
Expansion of the subject of mathematics. The enormous amount of factual material that was accumulated in the 17th and 18th centuries necessitated a deeper logical analysis and consolidation of the material from new viewpoints. Among the new developments at the turn of the 19th century were discovery and introduction into use of the geometric interpretation of complex numbers by the Danish surveyor C. Wessel (1799), and the French mathematician J. Argand(1806), N. Abel’s proof (1824) of the unsolvability of a general fifth-degree algebraic equation by radicals, development by A. Cauchy of the foundations of the theory of functions of a complex variable and Cauchy’s works on the rigorous substantiation of infinitesimal analysis, the creation of non-Euclidean geometry by N. I. Lobachevskii (1826, published 1829-30) and J. Bolyai (1832), and the works of K. Gauss (1827) on the intrinsic geometry of surfaces.
The close relation of mathematics to natural science now acquired more complex forms. Major new theories arose not only as a result of the direct requirements of natural science and technology but also from the intrinsic requirements of mathematics itself. Such was the development of the theory of functions of a complex variable, which occupied the central position in all mathematical analysis in the first half of the 19th century. Lobachevskii’s “imaginary geometry” is another noteworthy example of a theory that arose as a result of the development within mathematics itself.
Still another example can be cited of how the reevaluation begun at the end of the 18th century and the first half of the 19th of previously obtained concrete mathematical facts from more general points of view found strong support in the second half of the 19th century and in the 20th in the new requirements of natural science. Group theory originated with Lagrange’s examination (1771) of groups of substitution in connection with the problem of the solvability of algebraic equations of higher degrees by radicals. E. Galois (1830-32, published 1832, 1846), using the theory of substitution groups, gave a conclusive answer to the question of the conditions for the solvability by radicals of algebraic equations of any degree. In the mid-19th century A. Cayley gave a general “abstract” definition of group. S. Lie worked out the theory of continuous groups on the basis of general problems of geometry. Only after this did E. S. Fedorov (1890) and the German scientist A. Schonflies (1891) establish that the structure of crystals conforms to theoretical group principles; still later, group theory became a powerful research tool in quantum physics.
The development of vector calculus and tensor calculus was more directly related to the requirements of mechanics and physics. The extension of vector and tensor concepts to infinite-dimensional quantities took place within the framework of functional analysis and was closely connected to the requirements of modern physics.
Thus, the range of quantitative relations and spatial forms studied by mathematics expanded greatly as a result both of the requirements of mathematics itself and of the new needs of natural science. It included such problems as the relations among the elements of an arbitrary group, vectors, and operators in function spaces and t