Euler's Series for Sine and Cosine. An Interpretation in Nonstandard Analysis

Abstract

In chapter VIII of Introductio in analysin infinitorum, Euler derives a series for sine, cosine, and the formula $e^{iv}=\cos v+i\sin v$ His arguments employ infinitesimal and infinitely large numbers and some strange equalities. We interpret these seemingly inconsistent objects within the field of hyperreal numbers. We show that any non-Archimedean field provides a framework for such an interpretation. Yet, there is one implicit lemma underlying Euler's proof, which requires specific techniques of non-standard analysis. Analyzing chapter III of Institutiones calculi differentialis reveals Euler's appeal to the rules of an ordered field which includes infinitesimals -- the same ones he applies deriving series for $\sin v$, $\cos v$, and $e^v$.