Observations notation employed calculus functions von babbage charles (4 Ergebnisse)
Weitere BilderVerlag: University Press, Cambridge, 1821
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In den WarenkorbFirst edition. 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 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 ind.
Weitere BilderVerlag: J. Smith, Cambridge, 1820
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In den WarenkorbZustand: Very good. First Edition. Presentation copy, inscribed to a mathematician and fellow co-founder of the Astronomical Society, of Babbage's major early-career paper on mathematical notation. Before Babbage invented the first mechanical computer, he had already distinguished himself as one of the most innovative mathematic…ians 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." 11'' x 9''. Early drab paper wrappers. [2], 14; 389-454, [8] pages. Ink inscription, "Dr Gregory / from the author" to title page. Bound with: AN ESSAY TOWARDS THE CALCULUS OF FUNCTIONS 389-423 pages; A.P. Wilson Philip, SOME ADDITIONAL EXPERIMENTS AND OBSERVATIONS ON THE RELATION WHICH SUBSISTS BETWEEN THE NERVOUS AND SANGUIFEROUS SYSTEMS, 424-446 pages; errata 447; INDEX TO THE PHILOSOPHICAL TRANSACTIONS FOR THE YEAR 1815, [8] pages. Small shelf label to front corner of upper wrapper, pencil notation "1820"; some pencil annotations to margins, including mathematical calculations. Spine extensively repaired and reinforced with shallow chipping to fore-edges. Some browning and foxing to first imprint, else clean. Housed in custom quarter goatskin clamshell box. Signed.
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In den WarenkorbLondon, Cambridge University Press, 1822. 4to. In plain white paper-wrappers with title-page of journal volume pasted on to front wrapper. In "Transactions of the Cambridge Philosophical Society", Volume 1. Fine and clean. Pp. (63)-76 First appearance of Babbage paper on the notation employed in the Calculus of Functions."Babbag…e's major Contribution to mathematics was his calculus of functions, which he became interested in as early as 1809 and continued to develop during his years at Cambridge. Babbage presents his major ideas on the subject in the above two papers, published in the "Philosophical Transactions" in 1815 and 1816. "It can be said with some assurance that no mathematician prior to Babbage had treated the calculus of functions in such systematic way.Babbage must be given full credit as the inventor of a distinct and importent branch of mathematics" (Dubbey 1978, 90). Elsewhere Dubby states 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 most fascinating (Dubbey 1989, 18-19)." (Hook a. Norman No. 19).Erwin Tomash B47.
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In den WarenkorbLondon, Cambridge University Press, 1822. 4to. In recent paper wrappers. Extracted from the "Transactions of the Cambridge Philosophical Society", Volume 1, bound with the title-page of the volume. Fine and clean. (2), (63)-76 First appearance of Babbage paper on the notation employed in the Calculus of Functions."Babbage's majo…r Contribution to mathematics was his calculus of functions, which he became interested in as early as 1809 and continued to develop during his years at Cambridge. Babbage presents his major ideas on the subject in the above two papers, published in the "Philosophical Transactions" in 1815 and 1816. "It can be said with some assurance that no mathematician prior to Babbage had treated the calculus of functions in such systematic way.Babbage must be given full credit as the inventor of a distinct and importent branch of mathematics" (Dubbey 1978, 90). Elsewhere Dubby states 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 most fascinating (Dubbey 1989, 18-19)." (Hook a. Norman No. 19).
