Before the Second World War, differential analyzers were the most important machines built for giving numerical solutions for differential equations, both in United States and some European countries. Arithmetical devices were not considered to handle calculations which involve the processes of the Calculus. During this war, Vannevar Bush (1890-1974), who designed and built the first differential analyzer in 1927, planned a huge electronical model, the Rockefeller differential aanalyser project, together with his colleague Samuel H Caldwell (1904-1960). Despite enormous investments mobilized for this project, it was made obsolete by the electronic digital computer ENIAC at the end of the war. This mutation from analog to digital computer has already been the subject of numerous studies, especially from philosophical and sociological perspectives. As he was involved in the construction of these two kinds of engines, Douglas R. Hartree (1897-1958) appears as an appropriate case study to examine this mutation from a mathematical perspective. Beyer professor of applied mathematics (1928-37) at Manchester University, Hartree obtained the plans of the differential analyzer from Bush, and made one model built in Manchester in 1935, and another one for Cambridge in 1938. As he was an expert in numerical calculations even before the building of these machines, he rapidly extended their use to a great deal of general problems, far beyond the self-consistent field problem for which he initially designed them. And at the end of the war, Hartree was invited to USA, to help managing the ENIAC for computations. Plummer professor at Cambridge University from 1946, Hartree worked on the EDSAC and published a synthetic paper on Calculating engines, where he reviewed analog and digital machines. His Numerical Analysis (1952) reported on their computation methods. So the entire career of Hartree, together with his publications and his practice of both differential analyzers and digital machines, made him a particular relevant example to investigate the continuities and ruptures occurred between the two types of machines, with the following issues for both : - what kind of – differential and partial differential – equations were solved ? - how these equations had to be written in order to be solved ? - how control theory, first initiated from the differential Analyzer, could be mathematically transferred in computer ? - what kind of problems were solved ?