'Observations on the notation employed in the calculus of functions,' pp. 63-76 in: Transactions of the Cambridge Philosophical Society, Vol. I, Part I. Read May 1, 1820

BABBAGE'S 'CALCULUS OF NOTATION'. First edition, journal issue in never-bound sheets, of Babbage's first and most important paper on mathematical notation, which was essential for his major mathematical invention, the calculus of functions. Babbage's preoccupation with systems of notation found its most important expression in the development of his symbolic notation for the action of the Difference and Analytical Engines, the invention of which has earned him the title 'father of the computer'. "Throughout his career he emphasised the vital importance of a good working symbolism. At Cambridge he crusaded successfully for the reformation of notation in the differential calculus. Later he published his views at greater length, not only devising a set of rules that all mathematical notations were to follow but even constructing a kind of notational calculus. We can see continuity from the mathematical to the less mathematical part of his life, when he later devised, as an essential feature of his drawings for the engines, a workable convention which he described as the 'mechanical notation'" (Dubbey, p. 9). In the calculus of functions, Babbage "took a branch of mathematics barely considered by his predecessors and transformed it into a systematic calculus, the analysis containing some very original stratagems and devices" (ibid., p. 8). "Babbage believed that his new scheme would serve as a generalized calculus to include all problems capable of analytical formulation, and it is possible to see here a hint of the inspiration for his concept of the Analytical Engine. While the work on the engines and his other scientific, social and political activities caused him virtually to abandon mathematical research at the age of thirty, the calculus of functions was the area he often yearned to continue. In fact the calculus of functions was not taken up by other workers, and it is the aspect of Babbage's mathematical work that modern mathematicians find the most fascinating" (Dubbey, in The Works of Charles Babbage, Vol. 1 (2014), p. 21). Many years later, in his Passages from the Life of a Philosopher, Babbage referred to the calculus of functions as his 'earliest step' and 'one to which I would willingly recur if other demands on my time permitted.' RBH lists one copy of the offprint, no copy of the journal issue. "Babbage was very much involved in notational reform in his early career. Later, he wrote three long and interesting papers on the subject of notation, the first of which was 'Observations on the notation employed in the calculus of functions.' "This is quite a remarkable paper, for the writer not only demonstrates conclusively the excellence of his notation introduced for his own invented subject, the calculus of functions, but even performs a series of calculations to prove the conciseness of the symbolism. One might almost describe the paper as another branch of mathematics invented by Babbage, the 'calculus of notations'. "Before considering any of the calculations, I will quote Babbage's introduction at length: 'Amongst the various causes which combine in enabling us by the use of analytical reasoning to connect through a long succession of intermediate steps the data of a question with its solution, no one exerts a more powerful influence than the brevity and compactness which is so peculiar to the language employed. The progress of improvement in leading us from the simpler up to the most complex relations has gradually produced new codes of shortening the ancient paths, and the symbols which have thus been invented in many instances from a partial view, or for very limited purposes, have themselves given rise to questions far beyond the expectations of their authors, and which have materially contributed to the progress of the science. Few indeed have been so fortunate as at once to perceive all the bearings and foresee all the consequences which result either necessarily, or analogically even from some of the simple.…