Editing Charles Babbage (section)

===British Lagrangian School===
The Analytical Society had initially been no more than an undergraduate provocation. During this period it had some more substantial achievements. In 1816, Babbage, Herschel and Peacock published a translation from French of the lectures of [[Sylvestre Lacroix]], which was then the state-of-the-art calculus textbook.<ref>{{cite book|title= Constructing a Bridge: An Exploration of Engineering Culture, Design, and Research in Nineteenth-century France and America|url=https://books.google.com/books?id=6P66Ofj6DiYC&pg=PA110|year=1997|publisher=MIT Press|isbn= 978-0-262-11217-8| page=110}}</ref>

Reference to [[Joseph-Louis Lagrange|Lagrange]] in calculus terms marks out the application of what are now called [[formal power series]]. British mathematicians had used them from about 1730 to 1760. As re-introduced, they were not simply applied as notations in [[differential calculus]]. They opened up the fields of [[functional equation]]s (including the [[difference equation]]s fundamental to the difference engine) and operator ([[D-module]]) methods for [[differential equation]]s.  The analogy of difference and differential equations was notationally changing Δ to D, as a "finite" difference becomes "infinitesimal". These symbolic directions became popular, as [[operational calculus]], and pushed to the point of diminishing returns. The [[(ε, δ)-definition of limit|Cauchy concept of limit]] was kept at bay.<ref name="Guicciardini2003">{{cite book|first=Niccolò | last= Guicciardini|title=The Development of Newtonian Calculus in Britain, 1700–1800|url=https://books.google.com/books?id=nIKVQCeI1FUC&pg=PA138|year=2003|publisher=Cambridge University Press|isbn=978-0-521-52484-1|page=138|author-link=Niccolò Guicciardini}}</ref> Woodhouse had already founded this second "British Lagrangian School" with its treatment of [[Taylor series]] as formal.<ref name="GabbayWoods2008">{{cite book|author1=Dov M. Gabbay|author2=John Woods|title=British Logic in the Nineteenth Century|url=https://books.google.com/books?id=UULl07dutBwC&pg=PA403|year=2008|publisher=Elsevier|isbn= 978-0-08-055701-4|pages=403–404}}</ref>

In this context [[function composition]] is complicated to express, because the [[chain rule]] is not simply applied to second and higher derivatives. This matter was known to Woodhouse by 1803, who took from [[Louis François Antoine Arbogast]] what is now called [[Faà di Bruno's formula]]. In essence it was known to [[Abraham De Moivre]] (1697). Herschel found the method impressive, Babbage knew of it, and it was later noted by [[Ada Lovelace]] as compatible with the analytical engine.<ref>Craik 2005, pp. 122–3.</ref> In the period to 1820 Babbage worked intensively on functional equations in general, and resisted both conventional [[finite difference]]s and Arbogast's approach (in which Δ and D were related by the simple additive case of the [[exponential function|exponential map]]). But via Herschel he was influenced by Arbogast's ideas in the matter of [[iterated function|iteration]], i.e. composing a function with itself, possibly many times.<ref name="GabbayWoods2008"/> Writing in a major paper on functional equations in the ''[[Philosophical Transactions]]'' (1815/6), Babbage said his starting point was work of [[Gaspard Monge]].<ref>{{cite book|first1=Jeremy J. | last1= Gray| first2=Karen | last2= Hunger Parshall|title=Episodes in the History of Modern Algebra (1800–1950)| url=https://books.google.com/books?id=HQijZjoTKnIC&pg=PA19|year=2011|publisher=American Mathematical Society |isbn= 978-0-8218-7257-4|page=19}}</ref>