Rates of change and derivatives

  • Karl Menger


Ever since Lagrange initiated a new epoch in pure analysis by defining the derivative of a function, the logical clarity of applied mathematics has suffered from a confusion of those derivatives with the rate of change of one variable quantity with respect to another. Yet a mere count of the ideas involved in the two concepts clearly demonstrates that the situations studied in pure and in applied mathematics are basically unlike. The derivative associates a function with one function; for instance, the cosine function with the sine function. The rate of change associates a variable quantity with two variable quantities; for instance, the velocity with the distance travelled and the time.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    K. Monger, Algebra of analysis, Notre Dame Math. Lectures 1944.Google Scholar
  2. [2]
    K. Monger, The ideas of variable and function Proc. Nat. Acad. Sci. U. S. A. 39 (1953), p. 956–961.MathSciNetCrossRefGoogle Scholar
  3. [3]
    K. Monger, On variables in mathematics and in natural science Br. J. Phil. Sci. 5 (1954), p. 134–142.MathSciNetCrossRefGoogle Scholar
  4. [4]
    K. Monger, Variables de diverses natures Bull. Sciences Mathématiques 78 (1954), p. 229–234.MathSciNetzbMATHGoogle Scholar
  5. [5]
    K. Monger, Random variables and the general theory of variables, Proc. 3rd Berkeley Symposium Math. Stat. & Prob., vol. II, 1954, p. 215–229.Google Scholar
  6. [6]
    K. Monger, Calculus. A modern approach Boston 1955Google Scholar

Copyright information

© Springer-Verlag Wien 2003

Authors and Affiliations

  • Karl Menger
    • 1
  1. 1.ChicagoUSA

Personalised recommendations