Leibnitz differential

Leibnitz represents the differential of an independent variable as an infinitesimally small constant, say $dx=1-0.9999\dots$ . And he represents the number axis to be divided into infinitesimally small intervals of the same length $dx$. In the lecture one shows haw to realize these ideas with modern level of rigor, even without help of Nonstandard Calculus with its ultrafilters. The development of this approach leads to concepts of differential forms and integrals, which in a natural way includes the generalized functions.