ABSTRACT
This chapter discusses inspiration from Leonhard Euler’s amazing achievements, and describes a way that Euler’s techniques and results might fruitfully be incorporated into a modern calculus or even precalculus course. It examines modern axioms for infinite and infinitesimal numbers that can be used to give proofs very similar to Euler’s original arguments. The chapter also describes a simple notion of convergence for infinite sums, which are defined without reference to limits, and which appears in one of Euler’s early papers. Euler’s original derivation of the series for the exponential function seems to rely on the fact that one can neglect infinitely many infinitesimals in a summation. Euler stated explicitly at the beginning of the Introductio that by a “function of a variable quantity” he meant a function given by an “analytic expression” and gave examples to show what he meant.
