Two parts of analysis to which Leonhard Euler contributed in the 1740s and 1750s are the calculus of variations and the theory of infinite series. Certain concepts from these subjects occupy a fundamental place in modern analysis, but do not appear in the work of either Euler or his contemporaries. In the case of variational calculus there is the concept of the invariance of the variational equations; in the case of infinite series there is the concept of summability. However, some modern mathematicians have suggested that early forms of these concepts are implicitly present in Euler’s writings. We examine Euler’s work in calculus of variations and infinite series and reflect on this work in relation to modern theories.

This appendix collects mathematical tools that are needed in the main text. In addition, it gives a brief description of some essential background topics. It is assumed that the reader knows elementary calculus. The topics are grouped in four sections. First, we consider some useful methods of indirect proofs. Second, we introduce basic results for complex numbers and polynomials. The third topic concerns series expansions. Finally, some further calculus is presented, including difference calculus, Stieltjes integrals, and multivariable constrained optimization (covering also the case of inequality constraints).