The calculus of functions is undoubtedly Babbage's major mathematical invention. The subject, which we will see was taken up by Babbage almost from its origin and developed by a series of ingenious generalisations, has possibilities that have been little explored even in modern mathematics. The calculus of functions is developed on lines analogous to differential equations and difference equations, but is more general than either, and can be considered as including these areas as special cases.

The major ideas are presented in two papers published in the Philosophical Transactions of 1815 and 1816. There are references in other papers which are much shorter, but the two earliest ones are by far the most significant. Both are very long papers and between them occupy 111 pages. Babbage did not develop the subject in his later years, but as he says of the calculus of functions in Passages from the Life of a Philosopher:

The volume referred to may be found in the collection of Babbage relics contained in a cowhide trunk at the Museum of the History of Science, Oxford.