Throughout the 19th century mathematics became increasingly abstract. Carl Friedrich Gauss (1777–1855) epitomizes成为…的典范 this trend. He did revolutionary work on functions of complex variables复变函数, in geometry, and on the convergence of series极数的收敛性, leaving aside his many contributions to science. He also gave the first satisfactory proofs of the fundamental theorem of algebra. 代数基本定理
This century saw the development of the two forms of non-Euclideanw非欧几里得 geometry, where the parallel postulate假设 of Euclidean geometry no longer holds. The Russian mathematician Nikolai Ivanovich Lobachevsky and his rival竞争对手, the Hungarian mathematician János Bolyai, independently defined and studied hyperbolic双曲线的 geometry, where uniqueness唯一性 of parallels no longer holds. In this geometry the sum of angles in a triangle add up to less than 180°. Elliptic椭圆 geometry was developed later in the 19th century by the German mathematician Bernhard Riemann; here no parallel can be found and the angles in a triangle add up to more than 180°. Riemann also developed Riemannian geometry, which unifies统一 and vastly generalizes广泛概况 the three types of geometry, and he defined the concept of a manifold流形, which generalizes the ideas of curves曲线 and surfaces曲面.