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[[File:Parabola.svg|right|thumb|A [[parabola]], one of the simplest curves, after (straight) lines]]
In [
Intuitively, a curve may be thought of as the trace left by a moving [[point (geometry)|point]]. This is the definition that appeared more than 2000 years ago in [[Euclid's Elements|Euclid's ''Elements'']]: "The [curved] line{{efn|In current mathematical usage, a line is straight. Previously lines could be either curved or straight.}} is […] the first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which […] will leave from its imaginary moving some vestige in length, exempt of any width."<ref>In (rather old) French: "La ligne est la première espece de quantité, laquelle a tant seulement une dimension à sçavoir longitude, sans aucune latitude ni profondité, & n'est autre chose que le flux ou coulement du poinct, lequel […] laissera de son mouvement imaginaire quelque vestige en long, exempt de toute latitude." Pages 7 and 8 of ''Les quinze livres des éléments géométriques d'Euclide Megarien, traduits de Grec en François, & augmentez de plusieurs figures & demonstrations, avec la corrections des erreurs commises és autres traductions'', by Pierre Mardele, Lyon, MDCXLV (1645).</ref>
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==History==
[[File:Newgrange Entrance Stone.jpg|thumb|225px|[[Megalithic art]] from [[Newgrange]] showing an early interest in curves]]
Interest in curves began long before they were the subject of mathematical study. This can be seen in numerous examples of their decorative use in art and on everyday objects dating back to prehistoric
times.<ref name="Lockwood">Lockwood p. ix</ref> Curves, or at least their graphical representations, are simple to create, for example with a stick on the sand on a beach.
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In particular, the length <math> s </math> of the [[graph of a function|graph]] of a continuously differentiable function <math> y = f(x) </math> defined on a closed interval <math> [a,b] </math> is
:<math>
s = \int_{a}^{b} \sqrt{1 + [f'(x)]^{2}} ~ \mathrm{d}{x}
</math>
which can be thought of intuitively as using the [[Pythagorean theorem]] at the infinitesimal scale continuously over the full length of the curve.<ref>{{Cite book|url=https://books.google.com/books?id=OS4AAAAAYAAJ&dq=length+of+a+curve+formula+pythagorean&pg=RA2-PA108|title=The Calculus|last1=Davis|first1=Ellery W.|last2=Brenke|first2=William C.|date=1913|publisher=MacMillan Company|isbn=9781145891982|page=108|language=en}}</ref>
More generally, if <math> X </math> is a [[metric space]] with metric <math> d </math>, then we can define the length of a curve <math> \gamma: [a,b] \to X </math> by
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