We reconstruct essential features of Lagrange's theory of analytical functions by exhibiting its structure and basic assumptions, as well as its main shortcomings. We explain Lagrange's notions of function and algebraic quantity, and concentrate on power-series expansions, on the algorithm for derivative functions, and the remainder theorem --- especially the role this theorem has in solving geometric and mechanical problems. We thus aim to provide a better understanding of Enlightenment mathematics and to show that the foundations of mathematics did not, for Lagrange, concern the solidity of its ultimate bases, but rather purity of method --- the generality and internal organization of the discipline.