"the calculus of functions [...] is the aspect of Babbage's mathematical work that modern mathematicians find the most fascinating" — John Michael Dubbey

Before Babbage invented the first mechanical computer, he had already distinguished himself as one of the most innovative mathematicians of his lifetime. "Observations" was "quite a remarkable paper," showing by unimpeachable mathematical calculations the reasons for his chosen system of notation for the calculus of functions. The notation of a mathematical theory is fundamental to its progress: without a usable system, the true potential of a theory remains by default unrealized. In other words, a great mathematical innovation cannot be great until it can be properly communicated through notation. In this paper, Babbage has put forth a compact and muscular system of notation for his biggest purely mathematical innovation, the calculus of functions.

This 1820 paper is bound in early wrappers with a few further Royal Society papers, including another foundational work in Babbage's early mathematical career, his 1815 "Essay Towards the Calculus of Functions [part I]." The calculus of functions was "undoubtedly Babbage's major mathematical invention" (Dubbey 51). While studying at Cambridge, Babbage realized that the patriotic dispute between Newton and Leibniz as contemporaneous inventors of calculus had led English education into staleness: Europeans continued to build on Leibniz's system of notations for calculus, while Cambridge mathematicians required rote memorization of Newton's work. Many of Babbage's earliest published mathematical treatises on the calculus of functions, including those here, not only brought the English approach up to date, but sought to innovate further in the field — essentially, the first substantive mathematical innovations by an Englishman since Newton. Babbage's calculus of functions, an area of mathematics he essentially invented, "has possibilities that have been little explored even in modern mathematics" (Dubbey 51).

The 1820 "Observations" was published the same year that Babbage, along with 13 others, founded the Astronomical Society of London (soon to be renamed the Royal Astronomical Society). Among its founding members was Babbage's friend Olinthus Gilbert Gregory, a professor of mathematics who was almost certainly the recipient of this presentation copy. Gregory and Babbage were part of the committee charged with developing the by-laws of the society the same year. It was, in fact, Babbage's activites with other Astronomical Society members that led to his first proposal to develop a difference engine for accomplishing large masses of calculations without error. When Babbage successfully constructed his first engine in 1823, the Astronomical Society presented him with a gold medal, and "this may have given him the encouragement he needed to go ahead with the task of constructing a much enlarged difference engine" (Dubbey 182). When Babbage sought government funding for what would be the first mechanical computer, Gregory wrote Babbage in full support of the idea: "The application of machinery to the purposes of computation, in the way you have so happily struck out [...] cannot fail" (quoted in Dubbey 183).

Presentation copies of Babbage papers are quite scarce, let alone examples from a formative time in his career — at the peak of his innovations for pure mathematics, and on the cusp of the world-changing mechanical ones that would earn him the nickname "father of the computer."

Read more: Dubbey, Mathematical Work of Charles Babbage; Collier and MacLachlan, Charles Babbage and the Engines of Perfection.