Editing Tangent (section)

==History==

[[Euclid]] makes several references to the tangent ({{lang|grc|ἐφαπτομένη}} ''ephaptoménē'') to a circle in book III of the ''[[Euclid's Elements|Elements]]''  (c. 300 BC).<ref>{{cite web|last1=Euclid|title=Euclid's Elements|url=http://aleph0.clarku.edu/~djoyce/elements/bookIII/bookIII.html|access-date=1 June 2015}}</ref> In [[Apollonius of Perga|Apollonius]]' work ''Conics'' (c. 225 BC) he defines a tangent as being ''a line such that no other straight line could fall between it and the curve''.<ref name="Shenk">{{cite web|last1=Shenk|first1=Al|title=e-CALCULUS Section 2.8|url=http://math.ucsd.edu/~ashenk/Section2_8.pdf|pages=2.8|access-date=1 June 2015}}</ref>

[[Archimedes]] (c.  287 – c.  212 BC) found the tangent to an [[Archimedean spiral]] by considering the path of a point moving along the curve.<ref name="Shenk"/>

In the 1630s [[Fermat]] developed the technique of [[adequality]] to calculate tangents and other problems in analysis and used this to calculate tangents to the parabola. The technique of adequality is similar to taking the difference between <math>f(x+h)</math> and <math>f(x)</math> and dividing by a power of <math>h</math>. Independently [[Descartes]] used his [[method of normals]] based on the observation that the radius of a circle is always normal to the circle itself.<ref>{{cite book|last=Katz|first=Victor J.|year=2008|title=A History of Mathematics|edition=3rd|publisher=Addison Wesley|isbn=978-0321387004|page=510}}</ref>

These methods led to the development of [[differential calculus]] in the 17th century. Many people contributed. [[Gilles de Roberval|Roberval]] discovered a general method of drawing tangents, by considering a curve as described by a moving point whose motion is the resultant of several simpler motions.<ref>{{cite journal|last=Wolfson|first=Paul R.|year=2001|title=The Crooked Made Straight: Roberval and Newton on Tangents| journal=The American Mathematical Monthly | volume=108 | number=3 | pages=206–216 | doi=10.2307/2695381|jstor=2695381}}</ref>
[[René-François de Sluse]] and [[Johannes Hudde]] found algebraic algorithms for finding tangents.<ref>{{cite book|last=Katz|first=Victor J.|year=2008|title=A History of Mathematics|edition=3rd|publisher=Addison Wesley|isbn=978-0321387004|pages=512–514}}</ref> Further developments included those of [[John Wallis]] and [[Isaac Barrow]], leading to the theory of [[Isaac Newton]] and [[Gottfried Leibniz]].
 
An 1828 definition of a tangent was "a right line which touches a curve, but which when produced, does not cut it".<ref>Noah Webster, ''American Dictionary of the English Language'' (New York: S. Converse, 1828), vol. 2, p. 733, [https://archive.org/stream/americandictiona02websrich#page/n733/mode/2up]</ref> This old definition prevents [[inflection point]]s from having any tangent. It has been dismissed and the modern definitions are equivalent to those of [[Gottfried Wilhelm Leibniz|Leibniz]], who defined the tangent line as the line through a pair of [[infinitesimal|infinitely close]] points on the curve; in modern terminology, this is expressed as: the tangent to a curve at a point {{mvar|P}} on the curve is the [[limit (mathematics)|limit]] of the line passing through two points of the curve when these two points tends to {{mvar|P}}.