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Reference group
Analytical Society (act. 1812–1813) was founded by a small group of disillusioned Cambridge undergraduates who sought to challenge what they considered the dilapidated intellectual climate of Cambridge University. The aim of the society was to import the most powerful techniques of mathematical analysis and replace the Newtonian fluxional notation with the ds of Leibnizian differentials. Its inspiration derived from predominantly French mathematicians and men of science such as Pierre-Simon Laplace, Joseph Louis Lagrange, and Sylvestre Lacroix. 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.

The Analytical Society met for the first time in May 1812 and held monthly meetings during the Cambridge terms. There is no evidence of formal activity by the society after the end of 1813. The two main instigators of the society were Charles Babbage and John Herschel; other notable members included Edward Bromhead, George Peacock, and William Whewell. All these figures, in varying degrees except Bromhead, went on to play important roles in establishing the Cambridge Philosophical Society (1819), the Astronomical Society of London (1820), and the British Association for the Advancement of Science (1831). At the time Babbage, Herschel, and Bromhead veered towards what would have been considered politically radical views, while Peacock was a moderate whig and Whewell by far the most conservative.

Although the precise membership of the society cannot be established, various sources identify other Cambridge undergraduates involved either as members or on the fringes: Charles ffrench Bromhead (1795–1855), Alexander Charles Louis D'Arblay (1794/5–1837), Richard Gwatkin (1791–1870), John Philips Higman (1793–1855), Joseph William Jordan (b. 1789), Frederick Maule (1790–1813), William Hodge Mill, Thomas Robinson, Edward Ryan, Michael Slegg (1790/91–1873), John William Whittaker (1791–1854), and Henry Wilkinson (1791/2–1838).

Cambridge in the late eighteenth century has traditionally been viewed as intellectually stagnant, as a result of its attachment to Newtonianism and prejudice against French ideas. The mathematics taught there developed within an environment designed to produce clergymen for the Church of England; indeed British geometry was as much a national institution as the Church of England. Mathematical study was promoted as a means of reinforcing Christian faith rather than to discover new facts; it was to train the mind to appreciate the evidence of the record and the teachings of scripture. To adopt French abstract mathematics would lead to atheism: one writer raged that ‘the writings of some eminent French mathematicians abound in infidel principles, but we tremble for the fate of the young’ (Eclectic Review, May 1813).

Modern French algebraic analysis was viewed by many as a symbol of the unnatural events in France; it was the horrifying result of human intellect set free from all social restraints. The result was a product of human fantasy and folly, cut loose from established religion, tradition, and reality. 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.

A strict Anglican education defined the mathematical tripos in which gentlemanly geometry was a means to train the mind to conform, and not go beyond what was known from prescription, experience, and what could be seen representing real things. It was the most important tool for understanding Isaac Newton's Principia—which was viewed as the cornerstone of human knowledge of both God and the natural world. The task of the Anglican universities was to inculcate on the student those ideas that would teach him to take his place as a leader and not a critic of society and its established institutions. As a former senior wrangler and fellow of Trinity College, Daniel Mitford Peacock, warned in 1819: ‘There is in truth no one point on which the University should be more upon its guard, than against the introduction of mere algebraic or analytical speculations into its public examinations … Academical education should be strictly confined to subjects of real utility’, unlike ‘the lubrications of French analysts’ which ‘have no immediate bearing upon philosophy’ (Peacock, 85).

It was within this context that Babbage, as he recalled in his memoirs, came up with the idea of forming the Analytical Society. He
drew up the sketch of a society to be instituted for translating the small work of Lacroix on the Differential and Integral Calculus. It proposed that the Society should have periodical meetings for the propagation of D's and consigned to perdition all who supported the heresy of dots. It maintained that the work of Lacroix was so perfect that any comment was unnecessary. (Babbage, 20)
In addition to challenging the existing notion of what constituted a liberal education, the society wanted to make mathematics a recognized profession and an engine for generating new facts. The Analytical Society sought to overturn the entrenched approach to mathematics and introduce the foreign notation in an attempt to keep up with developments on the continent in the physical sciences.

It was the Analytical Society's stress on systematization and economizing mental labour that had led them to Lagrange's algebraic calculus. With its emphasis on abbreviation, symmetry, and unity, Lagrangian calculus sat well within their agenda. Symbolical algebra provided a powerful tool for by-passing the restrictions of arithmetic. Within its enclosed language the symbols did not depend on an external referent for their meaning, but simply on their initial relationship within the system. Analysis was a tool for discovering knowledge while geometry was derived from observation. The analytical method could be applied to anything. Through analysis, wrote the analyticals, an ‘arbitrary symbol can neither convey, nor excite any idea foreign to its original definition’. With each improvement came a new symbol which in turn shortened the ‘ancient paths’. Further, a whole train of consequences developed that prescribed the path for future investigations (Babbage and Herschel, i).

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’ (Babbage and Herschel, ii n). 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’ (Babbage and Herschel, 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 chiselled 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.

By now the one-time analytical sympathizer Whewell had become suspicious of French analysis. His first pedagogical text on mechanics in 1819 was carefully constructed to repel such extremism:
It appears as if their [French mathematicians'] attachment to the forms and processes of pure analysis, which they have cultivated with such signal success, had given them a disrelish for the more physical and inductive part of reasoning, and made them completely indifferent as to the manner in which they arrive at that part of the subject where the machinery of analysis begins to work. Hence those principles which mechanics must borrow of experiment are often made to depend on abstract reasonings and artificial definitions: or introduced as self evident, with some slight notice as to their agreement with matter of fact. (Whewell, vi)
Herschel and Babbage expected to find the external world in the structure of analysis, while Whewell expected first to find the external structure and then see if analysis could be applied. Herschel could barely restrain his disappointment, claiming that the work would have ‘been much more widely circulated, had you conformed a little more to the taste of the age and a little less to that of the University’ (Herschel to Whewell, 1 Dec 1819, Whewell papers, Trinity College, Cambridge).

Whewell's treatise introduced continental analysis into the Cambridge mathematical tripos but 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.

William J. Ashworth


C. Babbage, Passages from the life of a philosopher (1864) · D. M. Peacock, A comparative view of the principles of the fluxional and differential calculus (1819) · C. Babbage and J. Herschel, Memoirs of the Analytical Society (1813) · W. Whewell, An elementary treatise on mechanics (1819) · H. Becher, ‘Radicals, whigs and conservatives: the middle and lower classes in the analytical revolution at Cambridge in the age of aristocracy’, British Journal for the History of Science, 28 (1995), 405–6 · P. Enros, ‘The Analytical Society (1812–1813): precursor of the renewal of Cambridge mathematics’, Historica Mathematica, 10 (1983), 24–47 · M. Fisch, ‘The problematic history of nineteenth century British algebra’, British Journal for the History of Science, 27 (1994), 247–76 · W. J. Ashworth, ‘Memory, efficiency and symbolic analysis: Charles Babbage, John Herschel and the industrial mind’, Isis, 87 (1996), 629–53 · Venn, Alum. Cant.