mathematics: In the Nineteenth Century

The modern period of mathematics dates from the beginning of the 19th cent., and its dominant figure is C. F. Gauss. In the area of geometry Gauss made fundamental contributions to differential geometry, did much to found what was first called analysis situs but is now called topology, and anticipated (although he did not publish his results) the great breakthrough of non-Euclidean geometry. This breakthrough was made by N. I. Lobachevsky (1826) and independently by János Bolyai (1832), the son of a close friend of Gauss, whom each proceeded by establishing the independence of Euclid's fifth (parallel) postulate and showing that a different, self-consistent geometry could be derived by substituting another postulate in its place. Still another non-Euclidean geometry was invented by Bernhard Riemann (1854), whose work also laid the foundations for the modern tensor calculus description of space, so important in the general theory of relativity.

In the area of arithmetic, number theory, and algebra, Gauss again led the way. He established the modern theory of numbers, gave the first clear exposition of complex numbers, and investigated the functions of complex variables. The concept of number was further extended by W. R. Hamilton, whose theory of quaternions (1843) provided the first example of a noncommutative algebra (i.e., one in which ab ≠ ba). This work was generalized the following year by H. G. Grassmann, who showed that several different consistent algebras may be derived by choosing different sets of axioms governing the operations on the elements of the algebra.

These developments continued with the group theory of M. S. Lie in the late 19th cent. and reached full expression in the wide scope of modern abstract algebra. Number theory received significant contributions in the latter half of the 19th cent. through the work of Georg Cantor, J. W. R. Dedekind, and K. W. Weierstrass. Still another influence of Gauss was his insistence on rigorous proof in all areas of mathematics. In analysis this close examination of the foundations of the calculus resulted in A. L. Cauchy's theory of limits (1821), which in turn yielded new and clearer definitions of continuity, the derivative, and the definite integral. A further important step toward rigor was taken by Weierstrass, who raised new questions about these concepts and showed that ultimately the foundations of analysis rest on the properties of the real number system.

Sections in this article:

• Introduction
• In the Twentieth Century
• In the Nineteenth Century
• Western Developments from the Twelfth to Eighteenth Centuries
• Chinese and Middle Eastern Advances
• Greek Contributions
• Development of Mathematics
• Applied Mathematics
• Geometry
• Analysis
• Algebra
• Foundations
• Bibliography

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