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 the various forms of spaces of arbitrary dimension. With this broad understanding of the terms “quantitative relations” and “spatial forms,” the definition of mathematics given at the beginning of the article is also applicable to the new, modern stage of the development of mathematics.
An essential innovation in 19th-century mathematical development was that problems related to the necessary expansion of the range of quantitative relations and spatial forms to be studied became the objects of conscious and active interest on the part of mathematicians. Whereas previously the introduction, for example, of negative and complex numbers and the precise formulation of the rules governing operations involving such numbers required much work and effort, mathematics now required the development of techniques for the planned creation of new geometric systems and new “algebras” with noncommutative or even nonassociative multiplication needed in one or another connection. For example, the creation of a new “algebra” with new rules of operation in order to analyze and synthesize some new type of relay-contact circuits is now a matter of everyday scientific technical practice and no longer generates particular surprise. It would be difficult to overrate the importance of the reorganization of the entire frame of mathematical thinking that had to take place during the 19th century for this change in attitude to set in. From this conceptual viewpoint, the discovery of Lobachevskii’s non-Euclidean geometry was the most significant discovery of the early 19th century. It was precisely on the basis of this geometry that belief in the inviolability of the axioms that had been elucidated by a millennium of mathematical development was overcome and the possibility of creating essentially new mathematical theories by selectively setting aside previously imposed restrictions that had no intrinsic logical necessity was understood; it was also perceived that with time such an abstract theory may find increasingly broad and quite specific