The making of Peacock's \textit{Treatise on algebra}: A case of creative indecision (Q1302119)

In this article the author looks at the historical background of the first edition of Peacock's treatise of 1830 in view of the intense development of algebra(s) in Britain at that time. He records Peacock's involvement in the English translation (1816) of Lacroix's textbook on the calculus, which was effected with Charles Babbage and John Herschel, his friends in the Analytical Society. The main focus is Peacock's unsuccessful attempt in the treatise to reconcile the gap between algebra based upon its common form as generalized arithmetic, and the then newish algebras of differential operators and functional equations, also inspired by Lagrange (`Analytical' meant more or less `Lagrangian'), which Babbage and Herschel pursued in the 1810s and 1820s with some enthusiasm. The treatment of the philosophy of algebra in the second edition of the treatise (1842-1845) is deemed by the author to be more successful (pp. 173-174). Two senior figures might have gained attention: the Cambridge mathematician Robert Woodhouse and the Frenchman Lazare Carnot. They have been noted to some extent in the pertinent historical literature, some items of which are severely criticized on other grounds in various footnotes. The author laments the lack of sources with which to chart Peacock's progress between the translation and the treatise (p. 141), but he does not consider two encyclopaedia articles of that period (`Arithmetic', in Encyclopaedia metropolitana, Vol. 1 (1829), 269-523; and `Partial differences', Edinburgh encyclopaedia, Vol. 16 (1830), 321-332). He records no significant differences between the philosophy of algebra in the treatise and in a long report an `analysis' which Peacock published in 1834 at the behest of the British Association For The Advancement of Science (pp. 141, 167); but the range of topics has changed somewhat, with more advanced topics such as Fourier analysis considered.