Editing Charles Babbage (section)

==After Cambridge==
Considering his reputation, Babbage quickly made progress. He lectured to the [[Royal Institution]] on astronomy in 1815, and was elected a [[Fellow of the Royal Society]] in 1816.<ref name= "Essinger">{{cite book| first=James | last= Essinger |title=Jacquard's Web |year=2007 |publisher=Oxford University Press |isbn= 978-0-19-280578-2 |pages= 59, 98}}</ref> After graduation, on the other hand, he applied for positions unsuccessfully, and had little in the way of a career. In 1816 he was a candidate for a teaching job at [[Haileybury College]]; he had recommendations from [[James Ivory (mathematician)|James Ivory]] and [[John Playfair]], but lost out to [[Henry Walter (antiquary)|Henry Walter]].<ref>{{cite book| first1=Raymond | last1=Flood | authorlink1=Raymond Flood (mathematician) | first2=Adrian | last2= Rice| first3=Robin | last3=Wilson | authorlink3=Robin Wilson (mathematician) |title=Mathematics in Victorian Britain|url= https://books.google.com/books?id=QmzfJUShGYUC&pg=PA145|year= 2011| publisher=Oxford University Press|isbn=978-0-19-960139-4|page=145}}</ref> In 1819, Babbage and Herschel visited Paris and the [[Society of Arcueil]], meeting leading French mathematicians and physicists.<ref>{{cite book| title= George Green: Mathematician and Physicist, 1793–1841: The Background to His Life and Work|url=https://books.google.com/books?id=x2Y2eb9IzwwC&pg=PA255|year=2001|publisher=SIAM|isbn= 978-0-89871-463-0| page=255 note 19}}</ref> That year Babbage applied to be professor at the [[University of Edinburgh]], with the recommendation of [[Pierre Simon Laplace]]; the post went to [[William Wallace (mathematician)|William Wallace]].<ref>{{cite book| first=Roger | last=Hahn|title=Pierre Simon Laplace: 1749–1827; a Determined Scientist|url=https://books.google.com/books?id=ROu0P-pYQekC&pg=PA295|access-date=8 May 2013|year=2005|publisher= Harvard University Press|isbn= 978-0-674-01892-1|pages=295 note 34}}</ref><ref>{{cite ODNB|id=28545|title= Wallace, William|first=Maria|last=Panteki}}</ref><ref>{{cite book|title=The Edinburgh magazine, and literary miscellany, a new series of The Scots magazine|url=https://books.google.com/books?id=4NsEAAAAQAAJ&pg=PA369|year= 1819| page=369}}</ref>

With Herschel, Babbage worked on the [[Electromagnetism|electrodynamics]] of [[Arago's rotations]], publishing in 1825. Their explanations were only transitional, being picked up and broadened by [[Michael Faraday]]. The [[Phenomenon|phenomena]] are now part of the theory of [[eddy current]]s, and Babbage and Herschel missed some of the clues to unification of [[electromagnetic theory]], staying close to [[Ampère's force law]].<ref>{{cite book | first=L. Pearce | last=Williams |title= Michael Faraday |year=1965 |publisher=Da Capo Press |isbn=978-0-306-80299-7 |pages=[https://archive.org/details/michaelfaradaybi0000will_g3v0/page/170 170–172] |url=https://archive.org/details/michaelfaradaybi0000will_g3v0/page/170 }}</ref>

Babbage purchased the actuarial tables of [[George Barrett (actuary)|George Barrett]], who died in 1821 leaving unpublished work, and surveyed the field in 1826 in ''Comparative View of the Various Institutions for the Assurance of Lives''.<ref>{{cite ODNB|id=1523|title= Barrett, George|first=Robert|last=Brown}}</ref> This interest followed a project to set up an insurance company, prompted by [[Francis Baily]] and mooted in 1824, but not carried out.<ref>{{cite book| first1= Bruce | last1=Collier| first2=James | last2= MacLachlan|title=Charles Babbage: And the Engines of Perfection|url=https://books.google.com/books?id=-vzMEwf-bHEC&pg=PA29|year=2000|publisher= Oxford University Press|isbn= 978-0-19-514287-7| pages=29–30}}</ref> Babbage did calculate actuarial tables for that scheme, using [[Equitable Society]] mortality data from 1762 onwards.<ref>{{cite book| first=Anthony | last=Hyman |title=[[Charles Babbage: Pioneer of the Computer]]|year=1985|publisher=[[Princeton University Press]]|isbn=978-0-691-02377-9|page=[https://books.google.com/books?id=YCddaWqWK2cC&pg=PA59 59]}}</ref>

During this whole period, Babbage depended awkwardly on his father's support, given his father's attitude to his early marriage, of 1814: he and Edward Ryan wedded the Whitmore sisters. He made a home in [[Marylebone]] in London and established a large family.<ref>{{cite book| author=James|title=Remarkable Engineers|url=https://books.google.com/books?id=0bwb5bevubwC&pg=PA45|access-date=26 April 2013|publisher= Cambridge University Press|isbn= 978-1-139-48625-5|page=45|date=25 February 2010}}</ref> On his father's death in 1827, Babbage inherited a large estate (value around £100,000, equivalent to £{{Format price|{{Inflation|UK|100000|1827}}}} or ${{Format price|{{To USD|{{Inflation|UK|100000|1827}}|GBR}}}} today), making him independently wealthy.<ref name="ODNB"/> After his wife's death in the same year he spent time travelling. In Italy he met [[Leopold II, Grand Duke of Tuscany]], foreshadowing a later visit to [[Piedmont]].<ref name="Essinger"/> In April 1828 he was in [[Rome]], and relying on Herschel to manage the difference engine project, when he heard that he had become a professor at Cambridge, a position he had three times failed to obtain (in 1820, 1823 and 1826).<ref>{{cite book| first=Kevin C. | last=Knox|title=From Newton to Hawking: A History of Cambridge University's Lucasian Professors of Mathematics|url=https://books.google.com/books?id=QGX_rAeia4kC&pg=PA242|access-date=26 April 2013|date=6 November 2003|publisher=Cambridge University Press|isbn=978-0-521-66310-6|pages= 242; 258–72}}</ref>

===Royal Astronomical Society===
Babbage was instrumental in founding the [[Royal Astronomical Society]] in 1820, initially known as the Astronomical Society of London.<ref>{{Cite web|url=http://www-history.mcs.st-and.ac.uk/Biographies/Babbage.html|title=Babbage biography|website=History.mcs.st-and.ac.uk|access-date=21 December 2017}}</ref> Its original aims were to reduce astronomical calculations to a more standard form, and to circulate data.<ref>{{cite book| first= Martin | last=Campbell-Kelly | authorlink= Martin Campbell-Kelly |title=The History of Mathematical Tables: From Sumer to Spreadsheets| at=[https://books.google.com/books?id=O170gWPZ7M8C&pg=PA8 p. 8]|year= 2003|publisher=Oxford University Press|isbn=978-0-19-850841-0|title-link= The History of Mathematical Tables}}</ref> These directions were closely connected with Babbage's ideas on computation, and in 1824 he won its [[Gold Medal of the Royal Astronomical Society|Gold Medal]], cited "for his invention of an engine for calculating [[Mathematical table|mathematical]] and [[Ephemeris|astronomical tables]]".<ref>{{Cite book|last1=Buxton|first1=Harry Wilmot|url= https://books.google.com/books?id=jrUmAAAAMAAJ|title=Memoir of the Life and Labours of the Late Charles Babbage Esq., F.R.S.|last2=Tomash|first2=Erwin|date=1988|publisher=MIT Press|isbn=978-0-262-02269-9|location= Cambridge|pages=77|language=en}}</ref>

Babbage's motivation to overcome errors in tables by mechanisation had been a commonplace since [[Dionysius Lardner]] wrote about it in 1834 in the ''[[Edinburgh Review]]'' (under Babbage's guidance).<ref>{{cite book|first1=Raymond | last1=Flood | authorlink1=Raymond Flood (mathematician) | first2=Adrian | last2= Rice| first3=Robin | last3=Wilson | authorlink3= Robin Wilson (mathematician) |title=Mathematics in Victorian Britain|url=https://books.google.com/books?id=YruifIx88AQC&pg=PA34|year=2011|publisher=Oxford University Press|isbn= 978-0-19-162794-1|page=34}}</ref><ref>{{cite book| first=Harro | last= Maas|title=William Stanley Jevons and the Making of Modern Economics|url=https://books.google.com/books?id=aGWvA6SiWfMC&pg=PA201|year= 2005|publisher=Cambridge University Press|isbn=978-0-521-82712-6|page= 201}}</ref> The context of these developments is still debated. Babbage's own account of the origin of the difference engine begins with the Astronomical Society's wish to improve ''[[The Nautical Almanac]]''. Babbage and Herschel were asked to oversee a trial project, to recalculate some part of those tables. With the results to hand, discrepancies were found. This was in 1821 or 1822, and was the occasion on which Babbage formulated his idea for mechanical computation.<ref>{{cite book| first=M. Norton | last= Wise|title=The values of precision|url=https://books.google.com/books?id=g8uvNtuPhgUC&pg=PA320|year=1997|publisher=Princeton University Press|isbn=978-0-691-01601-6|page= 320}}</ref> The issue of the ''Nautical Almanac'' is now described as a legacy of a polarisation in British science caused by attitudes to [[Sir Joseph Banks]], who had died in 1820.<ref>{{cite book|first1= Eleanor |last1= Robson| first2=Jacqueline |last2= Stedall|title=The Oxford Handbook of the History of Mathematics|url=https://books.google.com/books?id=HIMiWHVKLDkC&pg=RA1-PR34|access-date=25 April 2013|date=18 December 2008|publisher=Oxford University Press|isbn= 978-0-19-160744-8|page=xxxiv}}</ref>
[[File:Difference engine plate 1853.jpg|thumb|A portion of the [[difference engine]]]]
Babbage studied the requirements to establish a modern [[postal system]], with his friend [[Thomas Frederick Colby]], concluding there should be a uniform rate that was put into effect with the introduction of the [[Uniform Fourpenny Post]] supplanted by the [[Uniform Penny Post]]<ref>{{cite book|author=Anthony Hyman|title=[[Charles Babbage: Pioneer of the Computer]]|year=1985|publisher= [[Princeton University Press]]|isbn= 978-0-691-02377-9|page=[https://books.google.com/books?id=YCddaWqWK2cC&pg=PA115 115]}}</ref> in 1839 and 1840. Colby was another of the founding group of the Society.<ref name="Hyman1985">{{cite book|first=Anthony |last= Hyman|title=[[Charles Babbage: Pioneer of the Computer]]|year=1985|publisher=Princeton University Press|isbn=978-0-691-02377-9|page=[https://books.google.com/books?id=YCddaWqWK2cC&pg=PA45 45]}}</ref> He was also in charge of the [[Survey of Ireland]]. Herschel and Babbage were present at a celebrated operation of that survey, the remeasuring of the [[Lough Foyle]] baseline.<ref>{{cite book|publisher=Royal Institution of Great Britain|title=Proceedings|url=https://books.google.com/books?id=LxNcKnmdDTIC&pg=PA518|year=1858|page=518}}</ref>

===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>