Consideration of Various Points of Analysis. Offprint from Philosophical Transactions.

London: printed by W. Bulmer [for the Royal Society], 1814.

Very rare separately-paginated offprint of Herschel’s account of the ‘calculus of operations,’ which formed a major part of the programme of reform promoted by the group of Cambridge mathematicians (which included Babbage and Herschel) who called themselves the ‘Analytical Society.’ Koppelman (p. 156) has argued that the reason that the most important contributions of British mathematicians in the first half of the nineteenth century were to abstract algebra “was a direct response of the English to a specific aspect of the work of Continental analysts which became accessible to them. The subject came to be called, by the English, the calculus of operations.” The most important such contributions were made by Duncan Gregory and George Boole. “This technique – the calculus of operations, as it was known – was also important to Boole’s early mathematical research on the calculus of finite differences; and Boole’s knowledge of Gregory’s defence of the technique underlies his algebraic analysis of logic in [Mathematical analysis of logic (1847)]” (Ewald, From Kant to Hilbert, p. 322). OCLC lists one copy only (Northwestern).

Provenance: Marcus Nathan Adler (bookplate). Adler (1803-1890) was the Orthodox Chief Rabbi of the British Empire from 1845 until his death, probably the most prominent nineteenth-century rabbi in the English-speaking world.

“The calculus of operations involved separating symbols of operation – such as differentiation – from symbols of quantity. Using general properties of algebra, the separated symbols of operation were then simplified in order to find a solution to an analytical problem or theorem…

“[In 1812] a group of Cambridge reformers, who called themselves the Analytical Society, established their own journal for the diffusion of what they saw as a radical new approach to mathematics. This approach was an algebraically based conception of the calculus developed by Lagrange that emphasized series expansions instead of limits or infinitesimals. With this new conception, Analytical Society members saw themselves as leaving behind the fluxional, Newtonian notation that had been tenaciously adhered to throughout the eighteenth century in Britain. At the same time, the Society felt that it could connect with a “century of foreign improvement” in the calculus…

“Facing the graduation of almost all its members and with few new recruits, the Analytical Society disintegrated in 1814. After the Society’s demise, Babbage, Herschel and Peacock continued to write reform-minded publications. For example, in 1816, the three published an English translation of Silvestre Lacroix’s Calcul différential et intégral as a way to bring Lacroix’s formulation of the differential calculus to Cambridge and also as a platform from which to promote their own allegiance to the Lagrangian development of the calculus. In this as well as in a series of articles by Herschel from 1814 to 1822, the calculus of operations was ever present. For example, in his 1814 Philosophical Transactions article, “Consideration of Various Points of Analysis,” Herschel professed his belief in the fruitfulness of the method of separating the symbols of operation from those of quantity. He explained that “[t]his method I have, perhaps, extended and carried somewhat farther than has hitherto been customary; but, I trust, without losing sight of its grand and ultimate object, the union of extreme generality with conciseness of expression [p. 12]” (Despaux, pp. 52-4).

S.E. Despeaux, “Very Full of Symbols,” in Episodes in the History of Modern Algebra (1800-1950), (2007), ed. by J.J. Gray & K.H. Parshall; E. Koppelman, ‘The calculus of operations and the rise of abstract algebra,’ Archive for the History of Exact Sciences, Vol. 8 (1971), pp. 155-242.

4to (272 x 213 mm), pp. [ii], [1] 2-29 (the last mis-paginated '33' and corrected in ink manuscript). Later nineteenth-century cloth-backed boards. Inner hinges starting, internally fine and clean.

Item #4283

Price: $4,500.00