Editing History of calculus (section)

=== Fundamental theorem of calculus ===
The formal study of calculus brought together Cavalieri's infinitesimals with the [[calculus of finite differences]] developed in Europe at around the same time, and Fermat's adequality. The combination was achieved by [[John Wallis]], [[Isaac Barrow]], and [[James Gregory (astronomer and mathematician)|James Gregory]], the latter two proving predecessors to the second [[fundamental theorem of calculus]] around 1670.<ref>{{Cite journal|last=Hollingdale|first=Stuart|date=1991|title=Review of Before Newton: The Life and Times of Isaac Barrow|url=https://www.jstor.org/stable/531707|journal=[[Notes and Records of the Royal Society of London]]|volume=45|issue=2|pages=277–279|doi=10.1098/rsnr.1991.0027|issn=0035-9149|jstor=531707|s2cid=165043307|quote=The most interesting to us are Lectures X-XII, in which Barrow comes close to providing a geometrical demonstration of the fundamental theorem of the calculus... He did not realize, however, the full significance of his results, and his rejection of algebra means that his work must remain a piece of mid-17th century geometrical analysis of mainly historic interest.}}</ref><ref>{{Cite journal|last=Bressoud|first=David M.|author-link=David Bressoud|date=2011|title=Historical Reflections on Teaching the Fundamental Theorem of Integral Calculus|url=https://www.tandfonline.com/doi/full/10.4169/amer.math.monthly.118.02.099|journal=[[The American Mathematical Monthly]]|volume=118|issue=2|pages=99|doi=10.4169/amer.math.monthly.118.02.099|s2cid=21473035}}</ref>

[[James Gregory (astronomer and mathematician)|James Gregory]], influenced by Fermat's contributions both to tangency and to quadrature, was then able to prove a restricted version of the second fundamental theorem of calculus, that integrals can be computed using any of a function's antiderivatives.<ref name= sherlock >
See, e.g., Marlow Anderson, Victor J. Katz, Robin J. Wilson, ''Sherlock Holmes in Babylon and Other Tales of Mathematical History'', Mathematical Association of America, 2004, [https://books.google.com/books?id=BKRE5AjRM3AC&pg=PA114 p.&nbsp;114].
</ref><ref name=geometriae>{{cite book| last=Gregory | first=James | title=Geometriae Pars Universalis | url=https://archive.org/details/gregory_universalis | publisher= Patavii: typis heredum Pauli Frambotti | year=1668 | location=[[Museo Galileo]] }}</ref>

The first full proof of the fundamental theorem of calculus was given by [[Isaac Barrow]].<ref name= barrowGeomLect >{{cite book |title=The geometrical lectures of Isaac Barrow, translated, with notes and proofs, and a discussion on the advance made therein on the work of his predecessors in the infinitesimal calculus |publisher=Open Court |location=Chicago |year=1916 |url=https://archive.org/details/geometricallectu00barruoft }} Translator: J. M. Child (1916)</ref>{{rp|p.61 when arc ME ~ arc NH at point of tangency F fig.26}}<ref name= revChildsTranslat >[https://www.ams.org/journals/bull/1918-24-09/S0002-9904-1918-03122-4/S0002-9904-1918-03122-4.pdf Review of J.M. Child's translation (1916) The geometrical lectures of Isaac Barrow] reviewer: Arnold Dresden (Jun 1918) p.454 Barrow has the fundamental theorem of calculus</ref>