Editing History of calculus (section)

=== 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>