Lipschitz, Rudolf Otto Sigismund (1832-1903)

German mathematician who developed a hypercomplex system of number theory, which became known as Lipschitz algebra. His work in basic analysis provided a condition for the continuity of a function, now known as the Lipschitz condition, subsequently of great importance in proofs of existence and uniqueness, as well as in approximation theory and constructive function theory. Lipschitz was born in Königsberg (now Kaliningrad) and studied there and at Berlin. From 1864 he was professor at the University of Bonn. Lipschitz did extensive work in number theory, Fourier series, the theory of Bessel functions, differential equations, the calculus of variations, co-gradient differentiation, geometry, and mechanics. In investigating the sums of arbitrarily many squares, he derived computational rules for certain symbolic expressions from real transformations. The investigations he began in 1869 into forms of n differentials led to his most valuable contribution to mathematics: the Cauchy-Lipschitz method of approximation of differentials. Lipschitz's book Grundlagen der Analysis 1877-80 was a synthetic presentation of the foundations of mathematics and their applications. The work provided a comprehensive survey of what was then known of the theory of rational integers, differential equations, and function theory.