Close encounters with the Stirling numbers of the second kind

This is a historical introduction to the theory of Stirling numbers of the second kind S(n,k) from the point of view of analysis. We tell the story of their birth in the book of James Stirling (1730) and show how they mature in the works of Johann Grunert (1843). We demonstrate their usefulness in several differentiation formulas. The reader can also see the connection to Bernoulli numbers, to Euler polynomials and to power sums.