Editing History of calculus (section)

==Modern precursors==
=== Integrals ===
[[Johannes Kepler]]'s work ''Stereometrica Doliorum'' published in 1615 formed the basis of integral calculus.<ref>{{cite web |title=Johannes Kepler: His Life, His Laws and Times |date=24 September 2016 |url=https://www.nasa.gov/kepler/education/johannes |accessdate=2021-06-10 |publisher=NASA |archive-date=2021-06-24 |archive-url=https://web.archive.org/web/20210624003856/https://www.nasa.gov/kepler/education/johannes/ |url-status=dead }}</ref> Kepler developed a method to calculate the area of an ellipse by adding up the lengths of many radii drawn from a focus of the ellipse.<ref name=EB1911>{{cite EB1911 |wstitle=Infinitesimal Calculus/History |display=Infinitesimal Calculus § History |volume=14 |page=537}}</ref>

A significant work was a treatise inspired by Kepler's methods<ref name=EB1911/> published in 1635 by [[Bonaventura Cavalieri]] on his [[method of indivisibles]]. He argued that volumes and areas should be computed as the sums of the volumes and areas of infinitesimally thin cross-sections. He discovered [[Cavalieri's quadrature formula]] which gave the area under the curves ''x''<sup>''n''</sup> of higher degree. This had previously been computed in a similar way for the parabola by Archimedes in ''[[The Method of Mechanical Theorems|The Method]]'', but this treatise is believed to have been lost in the 13th century, and was only rediscovered in the early 20th century, and so would have been unknown to Cavalieri. Cavalieri's work was not well respected since his methods could lead to erroneous results, and the infinitesimal quantities he introduced were disreputable at first.

[[Evangelista Torricelli|Torricelli]] extended Cavalieri's work to other curves such as the [[cycloid]], and then the formula was generalized to fractional and negative powers by Wallis in 1656. In a 1659 treatise, Fermat is credited with an ingenious trick for evaluating the integral of any power function directly.<ref name=quadrature>{{Cite web | last1 = Paradís | first1 = Jaume | last2 = Pla | first2 = Josep | last3 = Viader | first3 = Pelagrí | title = Fermat's Treatise On Quadrature: A New Reading | url = http://www.econ.upf.edu/docs/papers/downloads/775.pdf | access-date = 2008-02-24 | archive-date = 2007-01-07 | archive-url = https://web.archive.org/web/20070107221624/http://www.econ.upf.edu/docs/papers/downloads/775.pdf | url-status = dead }}</ref> Fermat also obtained a technique for finding the centers of gravity of various plane and solid figures, which influenced further work in quadrature.

=== Derivatives ===
In the 17th century, European mathematicians [[Isaac Barrow]], [[René Descartes]], [[Pierre de Fermat]], [[Blaise Pascal]], [[John Wallis]] and others discussed the idea of a [[derivative]]. In particular, in ''Methodus ad disquirendam maximam et minima'' and in ''De tangentibus linearum curvarum'' distributed in 1636, Fermat introduced the concept of [[adequality]], which represented equality up to an infinitesimal error term.<ref>{{cite book|author-link=André Weil |last=Weil |first=André |title=Number theory: An approach through History from Hammurapi to Legendre |location=Boston |publisher=Birkhauser Boston |year=1984 |isbn=0-8176-4565-9 |page=28}}</ref> This method could be used to determine the maxima, minima, and tangents to various curves and was closely related to differentiation.<ref name=Pellegrino>{{cite web | last = Pellegrino | first = Dana | title=Pierre de Fermat | url=http://www.math.rutgers.edu/~cherlin/History/Papers2000/pellegrino.html| access-date=2008-02-24}}</ref>

[[Isaac Newton]] would later write that his own early ideas about calculus came directly from "Fermat's way of drawing tangents."<ref name=Simmons>{{cite book | last = Simmons | first = George F. | title = Calculus Gems: Brief Lives and Memorable Mathematics | url = https://archive.org/details/calculusgemsbrie0000simm | url-access = registration | publisher = Mathematical Association of America | year = 2007 | page = [https://archive.org/details/calculusgemsbrie0000simm/page/98 98] | isbn = 978-0-88385-561-4}}</ref>

=== 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>

=== Other developments ===
[[File:hyperbola E.svg|thumb|Shaded area of one unit square measure when ''x'' = 2.71828... The discovery of [[Euler's number]] e, and its exploitation with functions e<sup>x</sup> and natural logarithm, completed integration theory for calculus of rational functions.]]
One prerequisite to the establishment of a calculus of functions of a [[real number|real]] variable involved finding an [[antiderivative]] for the [[rational function]] <math>f(x) \ = \ \frac{1}{x} .</math> This problem can be phrased as [[quadrature (mathematics)|quadrature]] of the rectangular hyperbola ''xy'' = 1. In 1647 [[Gregoire de Saint-Vincent]] noted that the required function ''F'' satisfied <math>F(st) = F(s) + F(t) ,</math> so that a [[geometric sequence]] became, under ''F'', an [[arithmetic sequence]]. [[A. A. de Sarasa]] associated this feature with contemporary algorithms called ''logarithms'' that economized arithmetic by rendering multiplications into additions. So ''F'' was first known as the [[hyperbolic logarithm]]. After [[Euler]] exploited e = 2.71828..., and ''F'' was identified as the [[inverse function]] of the [[exponential function]], it became the [[natural logarithm]], satisfying <math>\frac{dF}{dx} \ = \ \frac{1}{x} .</math>

The first proof of [[Rolle's theorem]] was given by [[Michel Rolle]] in 1691 using methods developed by the Dutch mathematician [[Johann van Waveren Hudde]].<ref>{{cite book
|title=A Transition to Advanced Mathematics: A Survey Course
|first1=William
|last1=Johnston
|first2=Alex
|last2=McAllister
|publisher=Oxford University Press US
|year=2009
|isbn=978-0-19-531076-4
|page=333
|url=https://books.google.com/books?id=LV21vHwnkpIC}}, [https://books.google.com/books?id=LV21vHwnkpIC&pg=PA333 Chapter 4, p. 333]
</ref> The mean value theorem in its modern form was stated by [[Bernard Bolzano]] and [[Augustin-Louis Cauchy]] (1789–1857) also after the founding of modern calculus. Important contributions were made by Barrow, [[Christiaan Huygens|Huygens]], and many others.

Barrow has been credited by some authors as having invented calculus, however, Swiss mathematician [[Florian Cajori]] notes that while Barrow did work out a set of "geometric theorems suggesting to us constructions by which we can find lines, areas and volumes whose magnitudes are ordinarily found by the analytical processes of the calculus", he did not create "what by common agreement of mathematicians has been designated by the term differential and integral calculus", and further notes that "Two processes yielding equivalent results are not necessarily the same". Cajori finishes with stating that "The invention belongs rightly belongs to Newton and Leibniz".<ref>{{Cite journal |last=Cajori |first=Florian |author-link=Florian Cajori |date=1919 |title=Who Was the First Inventor of the Calculus? |url=http://www.jstor.org/stable/2974042?origin=crossref |journal=The American Mathematical Monthly |volume=26 |issue=1 |pages=15 |doi=10.2307/2974042}}</ref>