  By retracing research on coexistent magnitudes (grandeurs coexistantes) by
Cauchy (1841), Peano in "Applicazioni geometriche del calcolo infinitesimale"
(1887) defines the "density" (strict derivative) of a "mass" (a distributive
set function) with respect to a "volume" (a positive distributive set
function), proves its continuity (whenever the strict derivative exists) and
shows the validity of the mass-density paradigm: "mass" is recovered from
"density" by integration with respect to "volume". It is remarkable that
Peano's strict derivative provides a consistent mathematical ground to the
concept of "infinitesimal ratio" between two magnitudes, successfully used
since Kepler. In this way the classical (i.e., pre-Lebesgue) measure theory
reaches a complete and definitive form in Peano's Applicazioni geometriche.
  A primary aim of the present paper is a detailed exposition of Peano's work
of 1887 leading to the concept of strict derivative of distributive set
functions and their use. Moreover, we compare Peano's work and Lebesgue's "La
mesure des grandeurs" (1935): in this memoir Lebesgue, motivated by coexistent
magnitudes of Cauchy, introduces a uniform-derivative of certain additive set
functions, a concept that coincides with Peano's strict derivative. Intriguing
questions are whether Lebesgue was aware of the contributions of Peano and
which role is played by the notions of strict derivative or of
uniform-derivative in today mathematical practice.
