theory, and in the geometric interpretation of these solutions he opened a field worked by various writers, notably Casorati and Cayley. To the latter is due (1872) the theory of singular solutions of differential equations of the first order as accepted circa 1900. Reduction to quadratures The primitive attempt in dealing with differential equations had in view a reduction to quadratures. As it had been the hope of eighteenth-century algebraists to find a method for solving the general equation of the nth degree, so it was the hope of analysts to find a general method for integrating any differential equation. Gauss (1799) showed, however, that complex differential equations require complex numbers. Hence, analysts began to substitute the study of functions, thus opening a new and fertile field. Cauchy was the first to appreciate the importance of this view. Thereafter, the real question was no longer whether a solution is possible by means of known functions or their integrals, but whether a given differential equation suffices for the definition of a function of the independent variable or variables, and, if so, what are the characteristic properties. Fuchsian theory Two memoirs by Fuchs inspired a novel approach, subsequently elaborated by Thomé and Frobenius. Collet was a prominent contributor beginning in 1869. His method for integrating a non-linear system was communicated to Bertrand in 1868. Clebsch (1873) attacked the theory along lines parallel to those in his theory of Abelian integrals. As the latter can be classified according to the properties of the fundamental curve that remains unchanged under a rational transformation, Clebsch proposed to classify the transcendent functions defined by differential equations according to the invariant properties of the corresponding surfaces f = 0 under rational one-to-one transformations. Lie's theory From 1870, Sophus Lie's work put the theory of differential equations on a better foundation. He showed that the integration theories of the older mathematicians can, using Lie groups, be referred to a common source, and that ordinary differential equations that admit the same infinitesimal transformations present comparable integration difficulties. He also emphasized the subject of transformations of contact. Lie's group theory of differential equations has been certified, namely: (1) that it unifies the many ad hoc methods known for solving differential equations, and (2) that it provides powerful new ways to find solutions. The theory has applications to both ordinary and partial differential equations. A general solution approach uses the symmetry