He resigned his university post in 1899 so that he could pursue his independent studies, making a living from high-school mathematics teaching.

Vailata's main historical interests concerned mechanics, logic, and geometry, and he was an important contributor to a number of areas, including the study of post-Aristotelian Greek mechanics, of Galileo's predecessors, of the notion and rôle of definition in the work of Plato and Euclid, of mathematical influences on logic and epistemology, and of the non-Euclidean geometry of Gerolamo Saccheri.

Vailati saw the two as differing in approach rather than subject matter, and believed that there should be co-operation between philosophers and scientists in the pursuit of historical studies.

Of a certain interest is the participation of Vailati in the scientific activity of the chair of "Calculus infinitesimal" held by Giuseppe Peano.

The period is full of Prolusions and there are the emergence of disciplines such as anthropology and sociology, which have produced a new scientific concept in the nineteenth century.

[3] The Turin Prolusions (1896-1898) open the horizon to new discoveries and to the formulations of scientific theories and hypotheses concerning the relationship between science and its history.