The 19th century was a watershed era for mathematics. It witnessed the birth of non-Euclidean geometry, the rigorous foundation of analysis, the rise of group theory, the transformation of algebra, and the professionalization of mathematics as a discipline. Few figures are as central to narrating this explosion of ideas as Felix Klein (1849–1925) —a mathematician who not only contributed to many of these fields but also became a towering historian and pedagogue.
By mid-century, Bernhard Riemann, a shy genius from Hanover, shattered the mirror entirely. In his 1854 habilitation lecture (attended by an aging Gauss), Riemann argued that geometry is not about absolute truth, but about measurement. Space could be curved, flat, or wrinkled; its rules depended on a local "metric." The universe, Riemann suggested, might be finite yet unbounded—a mind-bending possibility that would later find its home in Einstein’s relativity. development of mathematics in the 19th century klein pdf
Klein's work on mathematical physics was influenced by the ideas of Maxwell and other physicists. He worked on problems related to electromagnetism and optics, and his contributions to the field helped to establish mathematics as a fundamental tool for understanding physical phenomena. The Development of Mathematics in the 19th Century:
Felix Klein’s Lectures on the Development of Mathematics in the 19th Century There is an English translation published by MIT