Proofs, Arbitrary Exemplifications, and Inductive Generalizations in Euler’s Mathematical Practice

In this chapter, I will discuss the role of examples in Euler’s mathematical practice. In modern mathematics, examples have the function of clarifying findings, procedures, and theorems; instead, in Euler’s writings, they could also have two other different functions. Firstly, he used examples as a proof for a general statement, namely, he considered a proposition as proved if verified for some examples. Euler felt the need of general and rigorous proofs even pointed out that proofs by example could give rise to errors; however, he never rejected proofs by example entirely and limited himself to advising caution in their use. Secondly, Euler used examples as verification and persuasion tools. In certain cases, he checked the validity of a proof by appropriate examples. In these cases, examples did not substitute the general proof of a statement; rather their aim was to convince the reader of the correctness of certain procedures of the calculus and of the internal coherence of the various branches of the analysis of infinities.