ries was undertaken by D’Alembert. Euler, Lagrange, and particularly Legendre laid the foundations for the study of elliptic integrals—the first type of nonelementary function to be subjected to an intensive study.
Considerable attention was devoted to differential equations. In particular, Euler devised (1739, published 1743) the first method of solving a linear differential equation of any order with constant coefficients, D’Alembert examined systems of differential equations, and Lagrange and Laplace developed the general theory of linear differential equations of any order. Euler, G. Monge, and Lagrange laid the foundations of the general theory of first-order partial differential equations; the same was done for second-order equations by Euler, Monge, and Laplace. Of particular interest is the introduction into analysis of the expansion of functions into trigonometric series since it led to a controversy between Euler, D. Bernoulli, D’Alembert, Monge and Lagrange which centered on the concept of function. This controversy prepared the groundwork for the fundamental results of the 19th century concerning the relationship between functions defined by means of analytic expressions and functions defined in an arbitrary manner. Finally, a new branch of analysis—the calculus of variations—was created by Euler and Lagrange in the 18th century. On the basis of various achievements of the 17th and 18th centuries, De Moivre, Jakob Bernoulli, and Laplace formulated the basic principles of probability theory.
In geometry Euler brought the system of elementary analytic geometry to its culmination. Euler, A. Clairaut, Monge, and J. Meusnier laid the groundwork for the differential geometry of space curves and surfaces. Lambert developed the theory of perspective, and Monge gave descriptive geometry its final form.
The survey presented above shows that 18th-century mathematics, while proceeding from the ideas of the 17th century, far surpassed previous centuries in the scale of work. This flowering of mathematics was primarily 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 