  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 $e^{v}$.
