Concerning the History of the Calculus
Chapter X of Robinson's monograph
Non-standard Analysis, North-Holland Publishing Co., Amsterdam, 1966. Revised edition by Princeton University Press, Princeton, 1996, begins:
The history of a subject is usually written in the light of later developments. For over half a century now, accounts of the history of the Differential and Integral Calculus have been based on the belief that even though the idea of a number system containing infinitely small and infinitely large elements might be consistent, it is useless for the development of Mathematical Analysis. In consequence, there is in the writings of this period an noticable contrast between the severity with which the ideas of Leibniz and his successors are treated and the leniency accorded to the lapses of the early proponents of the doctrine of limits. We do not propose here to subject any of these works to a detailed criticism. However, it will serve as a starting point for our discussion to try to give a fair summary of the contemporary impression of the history of the Calculus...
I recomend that you read Robinson's Chapter X. I have often wondered if mathematicians in the time of Weierstrass said things like, 'Karl's epsilon-delta stuff isn't very interesting. All he does is re-prove old formulas of Euler.'
I have a non-standard interest in the history of infinitesimal calculus. It really is not historical. Some of the old derivations like Bernoulli's derivation of Leibniz' formula for the radius of curvature seem to me to have a compelling clarity. Robinson's theory of infiitesimals offers me an opportunity to see what is needed to complete these arguments with a contemporary standard of rigor.
Working on such problems has led me to believe that the best theory to use to underly calculus when we present it to beginners is one based on the kind of derivatives described in Section 2 and not the pointwise approach that is the current custom in the U.S. I believe we want a theory that supports intuitive reasoning like the examples above and pointwise defined derivatives do not.
Created by Mathematica (September 22, 2004)