Editing History of calculus (section)

====Greece====
{{See also|Ancient Greek mathematics}}
[[File:Parabolic segment and inscribed triangle.svg|thumb|upright=.7|Archimedes used the [[method of exhaustion]] to calculate the area under a parabola in his work ''[[Quadrature of the Parabola]]''.]]
From the age of Greek mathematics, [[Eudoxus of Cnidus|Eudoxus]] (c. 408–355&nbsp;BC) used the [[method of exhaustion]], which foreshadows the concept of the limit, to calculate areas and volumes, while [[Archimedes]] (c. 287–212&nbsp;BC) [[The Method of Mechanical Theorems|developed this idea further]], inventing [[heuristics]] which resemble the methods of integral calculus.<ref>Archimedes, ''Method'', in ''The Works of Archimedes'' {{isbn|978-0-521-66160-7}}</ref> Greek mathematicians are also credited with a significant use of [[infinitesimal]]s. [[Democritus]] is the first person recorded to consider seriously the division of objects into an infinite number of cross-sections, but his inability to rationalize discrete cross-sections with a cone's smooth slope prevented him from accepting the idea. At approximately the same time, [[Zeno of Elea]] discredited infinitesimals further by his articulation of the [[Zeno's paradoxes|paradoxes]] which they seemingly create.

Archimedes developed this method further, while also inventing heuristic methods which resemble modern day concepts somewhat in his ''[[The Quadrature of the Parabola]]'', ''[[Archimedes use of infinitesimals|The Method]]'', and ''[[On the Sphere and Cylinder]]''.<ref>MathPages — [http://mathpages.com/home/kmath343.htm Archimedes on Spheres & Cylinders] {{Webarchive|url=https://web.archive.org/web/20100103045422/http://mathpages.com/home/kmath343.htm |date=2010-01-03 }}</ref> It should not be thought that infinitesimals were put on a rigorous footing during this time, however. Only when it was supplemented by a proper geometric proof would Greek mathematicians accept a proposition as true. It was not until the 17th century that the method was formalized by [[Bonaventura Cavalieri|Cavalieri]] as the [[method of Indivisibles]] and eventually incorporated by [[Isaac Newton|Newton]] into a general framework of [[integral calculus]]. Archimedes was the first to find the tangent to a curve other than a circle, in a method akin to differential calculus. While studying the spiral, he separated a point's motion into two components, one radial motion component and one circular motion component, and then continued to add the two component motions together, thereby finding the tangent to the curve.<ref>{{cite book |first=Carl B. |last=Boyer |author-link=Carl Benjamin Boyer |title=A History of Mathematics |edition=2nd |publisher=Wiley |year=1991 |isbn=978-0-471-54397-8 |chapter=Archimedes of Syracuse |pages=[https://archive.org/details/historyofmathema00boye/page/127 127] |quote=Greek mathematics sometimes has been described as essentially static, with little regard for the notion of variability; but Archimedes, in his study of the spiral, seems to have found the tangent to a curve through kinematic considerations akin to differential calculus. Thinking of a point on the spiral 1=''r'' = ''aθ'' as subjected to a double motion — a uniform radial motion away from the origin of coordinates and a circular motion about the origin — he seems to have found (through the parallelogram of velocities) the direction of motion (hence of the tangent to the curve) by noting the resultant of the two component motions. This appears to be the first instance in which a tangent was found to a curve other than a circle.<br/>Archimedes' study of the spiral, a curve that he ascribed to his friend [[Conon of Samos|Conon of Alexandria]], was part of the Greek search for the solution of the three famous problems. |chapter-url=https://archive.org/details/historyofmathema00boye/page/127 }}</ref>