## Georg Friedrich Bernhard Riemann

# Georg Friedrich Bernhard Riemann

**1826-1866**

**German Mathematician**

Georg Friedrich Bernhard Riemann was a brilliant German mathematician who recognized the application of his work in non-Euclidean geometry to physics, including the shape of space itself. This mathematical advance made an immense contribution to modern theoretical physics, including laying the foundation for Albert Einstein's (1879-1955) theory of general relativity.

Riemann was born on September 17, 1826, in Breselenz, Germany, the second child of a Lutheran pastor. His family life was happy, and he progressed from his father's early tutelage to the local gymnasium, or secondary school, where his mathematical abilities quickly outstripped those of his teachers. The director of the gymnasium made mathematical texts available to him and allowed him to study them on his own. Otherwise, his education in the standard classical curriculum progressed normally, and he went on to the universities of Gottingen and Berlin, returning to Gottingen for his graduate work. There, after obtaining his father's permission
to transfer from the department of theology, he studied both mathematics and physics, including *Naturphilosophie*, the quest to derive universal principles from natural phenomena.

Riemann obtained his doctorate at Göttingen in 1851, with a thesis entitled "Foundations for a General Theory of Functions of a Complex Variable." Complex numbers are those expressed in terms of a+bi where i is the square root of -1, an imaginary number. Despite the imaginary nature of i, complex variables have many applications in the physical sciences. Riemann's work led to the idea of a multi-layered surface, later called a *Riemann surface*, allowing a multi-valued function of a complex variable to be interpreted as a single-valued function. The thesis won coveted praise from the eminent mathematician Carl Friedrich Gauss (1777-1855).

Riemann's next major body of work was prepared to meet the requirements for admission to the university as a lecturer. This effort led him to an independent formulation of a non-Euclidean geometry. He was apparently unaware that Janos Bolyai (1802-1860) of Hungary and the Russian mathematician Nikolai Lobachevsky (1792-1856) had already developed geometries without Euclid's parallel postulate that states that one and only one line can be drawn parallel to a given line through a point not on the line.

Riemann's geometry differed in that it postulated that there are no lines parallel to another line through a point not on it. He referred to the physical example of two ships traveling on the Earth's curved surface along meridians of longitude and meeting at a pole. It was the idea of space itself having a curved shape, such that it must be described using non-Euclidean geometry, which Einstein drew upon in developing his model of space-time. Riemann's work, which was destined to become a classic in the mathematical field, sufficed to secure his appointment as a *Privatdozent*, an unpaid lecturer dependent upon student fees.

Riemann was soon appointed professor, but suffered from deaths in his family, his own poor health, and the effects of overwork. Still, he was appointed to the Berlin Academy of Sciences in 1859, and continued to publish influential papers. Many important mathematical methods, theorems, and concepts still bear his name. In 1862, he married Elise Koch, a friend of his sister's, and they had one daughter. He died before his 40th birthday, on July 20, 1866, while attempting to recuperate from pleurisy and tuberculosis in Selasca, Italy.

**SHERRI CHASIN CALVO**

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**Georg Friedrich Bernhard Riemann**