It is unanimously recognized that the lively Russian scene was unable to carve out a significant place in the XIX century. The historian Demidov identified Nikolai Bugaev and Pafnutij Čebyšëv (1821-1894) as the most influential figures in the circuits of Moscow and Saint Petersburg, respectively, which followed opposite trends. Bugaev disliked the traditional approach, which focused on continuous functions (§ 2.2), and he ‘preached the idea of building a new mathematics, whose center should be the theory of discontinuous functions’ [165, p. 74]; Bugaev’s approach, which he himself called ‘arithmology’, was based on the application of piecewise smooth functions and a combination of mathematics and philosophy.