This text concludes the series of historical studies of the mathematics of the 19th century. Previous volumes have studied logic, algebra, number theory, probability, geometry, and analytic functions. The third volume encompasses: an essay on the development of Chebyshev's theory of approximation of functions, later called "constructive function theory"; a systematic analysis of the history of the theory of ordinary differential equations from Cauchy and certain of his predecessors up to Poincare and A.M. Lyapunov; an essay of the development of the calculus of variations; and a study of the history of finite differences, in whose development Soviet mathematicians have played a prominent role. This book should be a valuable source for understanding the development of these areas. It provides the general reader, as well as historians of science, not only with the descriptive history but also with a good perception of the outer and inner motivations and the results and effects of the presented concepts and methods.