If you can find a copy, George Berkeley's The Analyst was a cogent criticism of the foundations of calculus in his day. (He didn't have answers, but he identified real problems.) A variety of books were written by mathematicians in response. Uniformly these were much lower quality, and consisted of defenses of the foundations of mathematics, rather than acknowledgements that there were real issues. His criticisms were not taken seriously until that mathematical framework completely fell apart when Fourier series constructed "obviously impossible" things.
Today we are used to a very general notion of a "function". But historically a square wave simply wasn't a function. (Because if it was, how did you know how it interacted with infinitesmals?) And getting one out of the sum of a bunch of well behaved sins, with some simple integration, was a huge shock.