In 1845 Davidov submitted a thesis The theory of equilibrium of bodies immersed in a liquid for his Master's degree. It is worth noting that the Russian Master's degree was of a similar standard to the current PhD in Britain or the United States. The thesis continued work on the equilibrium of floating bodies which had been started by Euler, Poisson and Dupin. Grigorian write in  that in this thesis:-
[Davidov] was the first to give a general analytic method for determining the position of equilibrium of a floating body, applied his method to the determination of positions of equilibrium of bodies, explained the analytic theory by geometric constructions, and investigated the stability of equilibrium of floating bodies.He began teaching at Moscow University in 1850, and he obtained a doctorate in 1851 for another thesis on fluids, this time on The theory of capillary phenomena. Again we should note that the level of the Russian doctorate was close to the German habilitation. He was promoted to Professor after being awarded his doctorate and worked at Moscow university for the whole of his life.
Before leaving the topic of Davidov's two theses on fluids we note that the quality of these can be judged by the fact that they were both awarded the Demidov Prize by the St Petersburg Academy of Sciences. One of Davidov's colleagues at the Moscow University was Nikolai Dmetrievich Brashman (who had himself been awarded the Demidov Prize fifteen years earlier). He had been appointed to the university in 1834 and was a senior figure there in the area of applied mathematics, in particular mechanics. He greatly influenced Davidov's career and his mathematical research.
Together Davidov and Brashman were joint founders of the Moscow Mathematical Society in 1866, the year of Brashman's death. Davidov the became the first president of the Society and held this role for nearly twenty years until his death in 1885. He also did fine work for the Faculty of Physics and Mathematics at Moscow University being its head for twelve years.
As well as his work on the equilibrium of a floating body, Davidov also worked on partial differential equations, elliptic functions and the application of probability to statistics. He wrote :-
... a number of excellent texts for secondary schools. Of these, the geometry and algebra textbooks enjoyed special success and were republished many times. Through the next half century the geometry text underwent thirty-nine editions and the algebra text twenty-four.
Article by: J J O'Connor and E F Robertson