Colin Maclaurin in his 1742 textbook “A Treatise of Fluxions” attempted to clarify the foundations of calculus and do away with infinitesimals. He influenced D’Alembert and they both rejected the infinitesimals as real quantities.

In the eighteenth century the calculus methods separated themselves from the applications of mechanics and physics. Lagrange focused on power series expansions and the calculus.. Through this focus the term analytic function became associated with at the existence of a convergent Taylor series representation of a function. Lagrange did not see a simple series expansion as based on the limit process. This approach was improved upon by Cauchy.

In the first half of the nineteenth century Cauchy provide clearer definitions of limit, continuity and of the derivative.