German mathematician Jacobi was born in Potsdam, Prussia in 1804. He made important advances in linear algebra and the theory of functions. Jacobi wrote the classic treatise on elliptic functions  Fundamenta Nova Theoriae Functionum Ellipticarum (1829). He also studied Jacobi theta functions  which are named in his honor. Gauss had already worked out many of the properties of Elliptic Functions,  but never published them. Jacobi was the first, however, to apply elliptic functions  to number theory.  Using this method, he proved Fermat's polygonal number theorem. 

Jacobi also proved that a one-to-one  function of one variable with three distinct periods was impossible and reduced the general quintic equation. The reduction was later carried out even further by Jerrard. Jacobi also put the determinant  in its modern form. Jacobi did important work in celestial mechanics. In 1836, he found the Jacobian integral, but in a sidereal (fixed) coordinate system. Finally, he did much to develop Hamilton-Jacobi theory. 

Additional information:

http://scienceworld.wolfram.com/biography/Jacobi.html

Bell, E. T. "The Great Algorist: Jacobi." Ch. 18 in Men of Mathematics: The Lives and Achievements of the Great Mathematicians from Zeno to Poincaré. New York: Simon and Schuster, pp. 327-339, 1986.

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