‘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.
Cambridge: University Press, 1821. 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 simplest improvements. ‘The first analyst who employed the very natural abbreviation of a2 instead of aa little contemplated the existence of fractional negative and imaginary exponents, at the moment when he adopted this apparently insignificant mode of abridging his labor, so great however is the connection that subsists between all branches of pure analysis, that we cannot employ a new symbol or make a new definition, without at once introducing a whole train of consequences, and in defiance of ourselves, the very sign we have created, and on which we have bestowed a meaning, itself almost prescribes the path our future investigations are to follow’ (pp. 63-4). “The symbolism introduced in the first place for convenience, opens up totally new possibilities for the branch of reasoning being considered. This is admirably illustrated by Babbage's example of the use of a2, a3, a4 for aa, aaa, aaaa, ... suggesting the possibility of quantities with negative or fractional indices. One could cite the case of the differential notation in the calculus similarly suggesting possibilities that would be hidden forever in the fluxionary notation. “Babbage now applies this principle to the construction of a suitable notation for the calculus of functions. “He uses the letter f to denote the general functional operation and suggests that a repetition of the operation involving f should be written as f2(x) rather than ffx. Similarly f3(x) will be written for fffx … “From this he produces the obvious, but important, identity (A) that fn+m(x) = fnfm(x), where n and m are whole numbers … The equation (A), which can be proved inductively whenever the indices are positive integers, can now be used to assign meanings to the function raised to a fractional, surd or negative index. All we have to do is to use the equation (A) as a definition as far as these quantities are concerned, and a whole range of new functions are introduced in a way logically continuous with those already known. ‘The index n was now defined by means of the equation (A) and was said to indicate such a modification of the function to which it is attached that the equation shall be verified’ (p. 65). “Putting n = 0 in (A), it follows that fm(x) = f0fm(x). Now put y = fm(x) and the equation becomes y = f°(y). Since y can be taken as any variable, this implies that the symbol f0 is the unitary function, that is the function which leaves the variable unaltered. It is consequently independent of the particular operation f, and so, as Babbage points out, is analogous to the algebraic result x0 = l. “Another result can be obtained by putting n = 1, m = – 1. Then f0(x) = f1f-1(x) so f1f-1(x) = x. ‘f-1(x) must therefore signify such a function of x, that if we perform upon it the operation denoted by f it shall be reduced to x’ (p. 66) … “He quotes as an example of possible ambiguity the function f(x) = xn. Let r1, ..., rn be the roots of vn – 1 = 0, then f-1(x) may be any of the n functions r1x1/n, …, rnx1/n, for if the operation f ismperformed on any of these, the result is x. Thus we have f(f-1(x)) = x for f-1(x) being any of these n functions. “On the other hand, f-1f(x) = rix, where i can be any integer between 1 and n, depending on which function is taken for f-1, and only one of these gives f-1f(x) = x. ‘Consequently, we must define f-1(x) not as any function which has the property ff-1(x) = x, but the one which at the same time makes f-1f(x) = x. Only then can we call f-1 the inverse of f. ‘It was more necessary to make this observation, because several errors have arisen from not attending to it, and because that particular form of f-1which gives f-1f(x) = x possesses peculiar properties: f-1(x) is then the inverse function of fx; and if we have the equation f(x) = y, we may indicate its resolution thus x = f-1(y)’ ( p. 66) … “Babbage now introduces the more complicated idea of the function of two variables, and develops the concept analogically to his work on the functions of a single variable … “This paper provides a splendid example of Babbage's principles on notation. He has shown how a notation may be logically derived from the subject matter, and how its simplicity and conciseness can lead to a whole series of unsuspected results. In addition, we have this remarkable set of problems in which he has shown that the art of inventing notation can itself be reduced to a mathematical procedure. The great economy which this notation is shown to ensure is an admirable illustration of his contention that there should be as few signs of magnitude as possible” (Dubbey, pp. 155-162). Dubbey, The Mathematical Work of Charles Babbage, 1978.
4to (289 x 230 mm), pp. [ii], xxiii, 201, with 7 leaves of plates (loosely inserted cancels for pp. 51/52, 89/90, 99/100, 133/134, 137/138, and additional leaf facing p. 146). Original never-bound sheets (very minor soiling to title page). A virtually mint copy.
Item #6404
Price: $500.00
