Cambidge: printed by J. Smith . . . and sold by Deighton & Sons [etc.], 1813.
First edition, extremely rare, of the only volume of the Memoirs of the Analytical Society, written entirely by Babbage and Herschel while students at Cambridge University. The aim of the society was to promote the Leibnizian approach to calculus as opposed to Newtonian fluxions or, as Babbage punningly described it, to promote “The Principle of pure D-ism in opposition to the Dot-age of the University” (Babbage 1864, p. 29). “The brief period of the society’s existence is traditionally seen as a crucial moment in the adoption of the differential calculus within English mathematics” (ODNB).
❧Origins of Cyberspace 17 (lacking last two leaves of the Babbage paper). OCLC lists two copies in the US (Brown and NYPL) and one in UK.
The preface to the volume, which sets out the manifesto of the Society, draws attention to what was to be a lasting preoccupation for Babbage, “the importance of adopting a clear and comprehensive notation” (p. xvi), which found its most important expression in the development of his symbolic notation for the action of the Difference and Analytical Engines.
As well as the 22-page preface, discussed in detail below, this volume contains Babbage’s first published work, ‘On continued products,’ (pp. 1-31), and two papers by Herschel: ‘On Trigonometrical Series, Particularly Those Whose Terms Are Multiplied by the Tangents, Cotangents, Secants, &c. of Quantities in Arithmetic Progression, Together with Some Singular Transformations; with Notes Relating to a Variety of Subjects Connected with the Preceding Memoir,’ pp. 33–64; “On equations of differences and their application to the determination of functions from given conditions”, pp. 65-114.
Cambridge in the late eighteenth century was intellectually stagnant, as a result of its attachment to Newtonianism and prejudice against French ideas. British mathematics of the time was a tool for understanding Isaac Newton’s Principia—which was viewed as the cornerstone of human knowledge of both God and the natural world. Modern French algebraic analysis was viewed by many as a symbol of the recent political upheavals in France; it was the horrifying result of human intellect set free from all social restraints. The abstract nature of a pure algebraic analysis seemingly allowed the mind to wander into fantasy through the meaningless manipulation of symbols. It was further connected with a mechanical and industrial view of the mind. Pure mathematics was thus not seen as an appropriate part of a Cambridge education. Instead all textbooks were dependent on geometric figures and were applicable to specific physical foundations, rather than algebraic abstractions and mathematical generalization.
It was in this context that the Analytical Society was founded by Babbage (1791-1871) and Herschel (1792-1871), at the time both undergraduate students; other early participants included George Peacock and William Whewell. Its purpose was to import the most powerful techniques of continental mathematical analysis and replace the Newtonian fluxional ‘dot’ notation with the ds of Leibnizian differentials. They hoped thereby to connect with a “century of foreign improvement” in the calculus (p. xv). Its inspiration derived from predominantly French mathematicians and men of science such as Pierre-Simon Laplace, Joseph Louis Lagrange, and Sylvestre Lacroix. The latter’s Sur le calcul différentiel et integral (1802), in particular, was deemed by Babbage to be “so perfect that any comment was unnecessary” (Babbage 1864, p. 28). The society met for the first time in May 1812 and held monthly meetings during the Cambridge terms. Although there is no evidence of formal activity by the society after the end of 1813, a few years later Part I of Lacroix’s work, on differential calculus, was translated by Babbage; Part II, on integral calculus, was translated jointly by Peacock and Herschel. These, together with an “Appendix” written by Herschel, and notes by Herschel and Peacock, were published in 1816 as An elementary treatise on the differential and integral calculus.
“The Analytical Society’s philosophical claims were set out in the lengthy preface to the Memoirs of the Analytical Society published in December 1813. ‘Our business’, they wrote, ‘is exclusively with the pure Analytics’ (p. ii, note). It is not a coincidence that in the preface the history of the differential calculus is played out in synchronization with its mathematical principles. Notation mirrored this development, since with the advancement of science and complex calculations the need for a clear and comprehensive notation was shown to be necessary. Thus as science progressed so did notation, and, more important, the process of discovery was accelerated. The ultimate extension of notation came with ‘defining the result of every operation that can be performed on quantity, by the general term of function, and expressing this generalization by a characteristic letter’ (p. xvi). The true operation of the mind was based on the principle that linked all discoveries. The tools for this were abstract operations, an algebraic machine that chiseled out such discoveries. The Cambridge establishment was furious when George Peacock, a former—albeit inactive—member of the Analytical Society, introduced the d notation into the exam papers of 1817” (ODNB).
Whewell also introduced continental analysis into the Cambridge mathematical tripos through his popular An elementary treatise on mechanics (1819) but, to Babbage’s great disappointment, fully clothed within the traditional dress of Newtonian mechanics. The Analytical Society had failed to realize its ambitions at Cambridge and looked south to London as a more fertile place to launch their project. In the end Herschel and Babbage joined a number of other like-minded people and established the Astronomical Society of London in 1820.
Babbage and Herschel remained lifelong friends. Herschel supported Babbage through personal encouragement, service on the Royal Society committees specially convened to advise on the utility of Babbage’s Engines, and through his advocacy for continued government investment in their construction. In 1827, shattered by the death of his wife, father and second son in the space of a year, Babbage sought solace with the Herschels in Slough in the rural outskirts of London. When Babbage left shortly after for an extended tour of the Continent, it was to Herschel that he entrusted the construction of the Engine. Herschel attempted to restrain his friend’s excesses, especially Babbage’s public diatribes against the Royal Society and his sarcastic attacks on prominent figures. When shown a pre-publication draft of Babbage’s vehement attack on the scientific establishment, Reflections on the Decline of Science in England, and on some of its Causes (1830), Herschel wrote that he deserved ‘a good slap in the face’ and that he should burn the manuscript. Babbage took no heed, published and was damned. The contrast between the two friends, and their relative success in their lifetime, is telling. Herschel died five months before Babbage. In a letter to his widow Babbage described Herschel as ‘one of the earliest and most valued friends of my life’.
Babbage, Passages from the Life of a Philosopher, 1864; A. W. Van Sinderen, ‘The printed papers of Charles Babbage,’ Annals of the History of Computing, Vol. 2, 1980, pp. 169-850, no. 1.
Large 4to (310 x 240 mm), pp. [iv], xxii, [ii], 114. Uncut in contemporary cloth, spine lettered in gilt with original printed front wrapper pasted onto upper cover and original blank rear wrapper bound in at end, ex-Royal Society of Edinburgh copy with some small ink stamps and letter of de-accession loosely inserted.