Editing Analytic geometry (section)

===Western Europe===
{{Descartes}}{{See also|René Descartes#Analytic geometry}}
Analytic geometry was independently invented by [[René Descartes]] and [[Pierre de Fermat]],<ref>{{cite book|first=John|last=Stillwell|author-link=John Stillwell|title=Mathematics and its History |edition=Second |publisher=Springer Science + Business Media Inc.|year=2004|chapter=Analytic Geometry|pages=105|isbn=0-387-95336-1|quote=the two founders of analytic geometry, Fermat and Descartes, were both strongly influenced by these developments.}}</ref><ref>{{harvnb|Boyer|2004|page=74}}</ref> although Descartes is sometimes given sole credit.<ref>{{cite book |first=Roger |last=Cooke |author-link=Roger Cooke (mathematician) |title=The History of Mathematics: A Brief Course |publisher=Wiley-Interscience |year=1997 |chapter=The Calculus |pages=[https://archive.org/details/historyofmathema0000cook/page/326 326] |isbn=0-471-18082-3 |quote=The person who is popularly credited with being the discoverer of analytic geometry was the philosopher René Descartes (1596–1650), one of the most influential thinkers of the modern era. |chapter-url=https://archive.org/details/historyofmathema0000cook/page/326 }}</ref><ref>{{harvnb|Boyer|2004|page=82}}</ref> ''Cartesian geometry'', the alternative term used for analytic geometry, is named after Descartes.

Descartes made significant progress with the methods in an essay titled ''[[La Géométrie]] (Geometry)'', one of the three accompanying essays (appendices) published in 1637 together with his ''Discourse on the Method for Rightly Directing One's Reason and Searching for Truth in the Sciences'', commonly referred to as ''[[Discourse on Method]]''.
<!-- This is a mistranslation of the meaning of that passage, Descartes did not use a pair of axes in his descriptions. Perhaps this can be used elsewhere.<ref>On pages 319-322 of the ''Livre second:  La Geometrie'' (Book 2:  Geometry) of his ''Discourse on Method'', Descartes imposes a pair of perpendicular axes on a plot of a curve and shows how to measure the distances from an arbitrary point C on the curve to the axes.  From p. 321:  ''" … je tire de ce point C la ligne CB parallele a GA, & pourceque CB & BA sont deux quantités indeterminées & inconnuës, je les nomme l'une y & l'autre x."'' ( … I draw from this point C the line CB parallel to GA, and because CB and BA are two undetermined and unknown quantities, I call the one y and the other x.)  Descartes then shows that the curve is described by an equation for a hyperbola, thereby illustrating how analytic geometry can be used to prove that a given curve is an instance of a general class.  See:  René Descartes, ''Discours de la Méthode'' … (Leiden, (Netherlands): Jan Maire, 1637), [http://gallica.bnf.fr/ark:/12148/btv1b86069594/f405.image pp.&nbsp;319–322.]</ref> -->
''La Geometrie'', written in his native [[French language|French]] tongue, and its philosophical principles, provided a foundation for [[calculus]] in Europe. Initially the work was not well received, due, in part, to the many gaps in arguments and complicated equations. Only after the translation into [[Latin]] and the addition of commentary by [[Frans van Schooten|van Schooten]] in 1649 (and further work thereafter) did Descartes's masterpiece receive due recognition.<ref name="Katz 1998 loc=pg. 442">{{harvnb|Katz|1998|loc=pg. 442}}</ref>

Pierre de Fermat also pioneered the development of analytic geometry. Although not published in his lifetime, a manuscript form of ''Ad locos planos et solidos isagoge'' (Introduction to Plane and Solid Loci) was circulating in Paris in 1637, just prior to the publication of Descartes' ''Discourse''.<ref>{{harvnb|Katz|1998|loc=pg. 436}}</ref><ref>Pierre de Fermat, ''Varia Opera Mathematica d. Petri de Fermat, Senatoris Tolosani'' (Toulouse, France:  Jean Pech, 1679), "Ad locos planos et solidos isagoge," [http://gallica.bnf.fr/ark:/12148/bpt6k6213144d/f147.image.langEN pp.&nbsp;91–103.] {{Webarchive|url=https://web.archive.org/web/20150804051846/http://gallica.bnf.fr/ark:/12148/bpt6k6213144d/f147.image.langEN |date=2015-08-04 }}</ref><ref>[http://gallica.bnf.fr/ark:/12148/bpt6k56523g/f73.image.langEN "Eloge de Monsieur de Fermat"] {{Webarchive|url=https://web.archive.org/web/20150804051849/http://gallica.bnf.fr/ark:/12148/bpt6k56523g/f73.image.langEN |date=2015-08-04 }} (Eulogy of Mr. de Fermat), ''Le Journal des Scavans'', 9 February 1665, pp.&nbsp;69–72.  From p. 70:  ''"Une introduction aux lieux, plans & solides; qui est un traité analytique concernant la solution des problemes plans & solides, qui avoit esté veu devant que M. des Cartes eut rien publié sur ce sujet."''  (An introduction to loci, plane and solid; which is an analytical treatise concerning the solution of plane and solid problems, which was seen before Mr. des Cartes had published anything on this subject.)</ref> Clearly written and well received, the ''Introduction'' also laid the groundwork for analytical geometry. The key difference between Fermat's and Descartes' treatments is a matter of viewpoint: Fermat always started with an algebraic equation and then described the geometric curve that satisfied it, whereas Descartes started with geometric curves and produced their equations as one of several properties of the curves.<ref name="Katz 1998 loc=pg. 442"/> As a consequence of this approach, Descartes had to deal with more complicated equations and he had to develop the methods to work with polynomial equations of higher degree. It was [[Leonhard Euler]] who first applied the coordinate method in a systematic study of space curves and surfaces.