Editing Analytic geometry (section)

==History==

===Ancient Greece===
The [[Ancient Greece|Greek]] mathematician [[Menaechmus]] solved problems and proved theorems by using a method that had a strong resemblance to the use of coordinates and it has sometimes been maintained that he had introduced analytic geometry.<ref>{{cite book |first=Carl B. |last=Boyer |author-link=Carl Benjamin Boyer |title=A History of Mathematics |edition=Second |publisher=John Wiley & Sons, Inc. |year=1991 |isbn=0-471-54397-7 |chapter=The Age of Plato and Aristotle |pages=[https://archive.org/details/historyofmathema00boye/page/94 94–95] |quote=Menaechmus apparently derived these properties of the conic sections and others as well. Since this material has a strong resemblance to the use of coordinates, as illustrated above, it has sometimes been maintained that Menaechmus had analytic geometry. Such a judgment is warranted only in part, for certainly Menaechmus was unaware that any equation in two unknown quantities determines a curve. In fact, the general concept of an equation in unknown quantities was alien to Greek thought. It was shortcomings in algebraic notations that, more than anything else, operated against the Greek achievement of a full-fledged coordinate geometry. |chapter-url=https://archive.org/details/historyofmathema00boye/page/94 }}</ref>

[[Apollonius of Perga]], in ''[[Apollonius of Perga#De Sectione Determinata|On Determinate Section]]'', dealt with problems in a manner that may be called an analytic geometry of one dimension; with the question of finding points on a line that were in a ratio to the others.<ref>{{cite book |first=Carl B. |last=Boyer |author-link=Carl Benjamin Boyer |title=A History of Mathematics |edition=Second |publisher=John Wiley & Sons, Inc. |year=1991 |isbn=0-471-54397-7 |chapter=Apollonius of Perga |pages=[https://archive.org/details/historyofmathema00boye/page/142 142] |quote=The Apollonian treatise ''On Determinate Section'' dealt with what might be called an analytic geometry of one dimension. It considered the following general problem, using the typical Greek algebraic analysis in geometric form: Given four points A, B, C, D on a straight line, determine a fifth point P on it such that the rectangle on AP and CP is in a given ratio to the rectangle on BP and DP. Here, too, the problem reduces easily to the solution of a quadratic; and, as in other cases, Apollonius treated the question exhaustively, including the limits of possibility and the number of solutions. |chapter-url=https://archive.org/details/historyofmathema00boye/page/142 }}</ref> Apollonius in the ''Conics'' further developed a method that is so similar to analytic geometry that his work is sometimes thought to have anticipated the work of [[Descartes]] by some 1800 years. His application of reference lines, a diameter and a tangent is essentially no different from our modern use of a coordinate frame, where the distances measured along the diameter from the point of tangency are the abscissas, and the segments parallel to the tangent and intercepted between the axis and the curve are the ordinates. He further developed relations between the abscissas and the corresponding ordinates that are equivalent to rhetorical equations (expressed in words) of curves. However, although Apollonius came close to developing analytic geometry, he did not manage to do so since he did not take into account negative magnitudes and in every case the coordinate system was superimposed upon a given curve ''a posteriori'' instead of ''a priori''. That is, equations were determined by curves, but curves were not determined by equations. Coordinates, variables, and equations were subsidiary notions applied to a specific geometric situation.<ref>{{cite book |first=Carl B. |last=Boyer |author-link=Carl Benjamin Boyer |title=A History of Mathematics |edition=Second |publisher=John Wiley & Sons, Inc. |year=1991 |isbn=0-471-54397-7 |chapter=Apollonius of Perga |pages=[https://archive.org/details/historyofmathema00boye/page/156 156] |quote=The method of Apollonius in the ''Conics'' in many respects are so similar to the modern approach that his work sometimes is judged to be an analytic geometry anticipating that of Descartes by 1800 years. The application of references lines in general, and of a diameter and a tangent at its extremity in particular, is, of course, not essentially different from the use of a coordinate frame, whether rectangular or, more generally, oblique. Distances measured along the diameter from the point of tangency are the abscissas, and segments parallel to the tangent and intercepted between the axis and the curve are the ordinates. The Apollonian relationship between these abscissas and the corresponding ordinates are nothing more nor less than rhetorical forms of the equations of the curves. However, Greek geometric algebra did not provide for negative magnitudes; moreover, the coordinate system was in every case superimposed ''a posteriori'' upon a given curve in order to study its properties. There appear to be no cases in ancient geometry in which a coordinate frame of reference was laid down ''a priori'' for purposes of graphical representation of an equation or relationship, whether symbolically or rhetorically expressed. Of Greek geometry we may say that equations are determined by curves, but not that curves are determined by equations. Coordinates, variables, and equations were subsidiary notions derived from a specific geometric situation; [...] That Apollonius, the greatest geometer of antiquity, failed to develop analytic geometry, was probably the result of a poverty of curves rather than of thought. General methods are not necessary when problems concern always one of a limited number of particular cases. |chapter-url=https://archive.org/details/historyofmathema00boye/page/156 }}</ref>

===Persia===
The 11th-century Persian mathematician [[Omar Khayyam]] saw a strong relationship between geometry and algebra and was moving in the right direction when he helped close the gap between numerical and [[geometric algebra]]<ref name="Boyer Omar Khayyam positive roots"/> with his geometric solution of the general [[cubic equation]]s,<ref>{{cite journal
| last1=Cooper | first1=Glen M.
| date=2003
| title=Review: Omar Khayyam, the Mathmetician by R. Rashed, B. Vahabzadeh
| journal=[[The Journal of the American Oriental Society]]
| volume=123
| issue=1
| pages=248–249
| doi=10.2307/3217882
| jstor=3217882}}</ref> but the decisive step came later with Descartes.<ref name="Boyer Omar Khayyam positive roots">{{cite book|last=Boyer|author-link=Carl Benjamin Boyer|title=A History of Mathematics|chapter-url=https://archive.org/details/historyofmathema00boye|chapter-url-access=registration|year=1991|chapter=The Arabic Hegemony|pages=[https://archive.org/details/historyofmathema00boye/page/241 241–242]|isbn=9780471543978 |quote=Omar Khayyam (ca. 1050–1123), the "tent-maker," wrote an ''Algebra'' that went beyond that of al-Khwarizmi to include equations of third degree. Like his Arab predecessors, Omar Khayyam provided for quadratic equations both arithmetic and geometric solutions; for general cubic equations, he believed (mistakenly, as the sixteenth century later showed), arithmetic solutions were impossible; hence he gave only geometric solutions. The scheme of using intersecting conics to solve cubics had been used earlier by Menaechmus, Archimedes, and Alhazan, but Omar Khayyam took the praiseworthy step of generalizing the method to cover all third-degree equations (having positive roots). For equations of higher degree than three, Omar Khayyam evidently did not envision similar geometric methods, for space does not contain more than three dimensions, ... One of the most fruitful contributions of Arabic eclecticism was the tendency to close the gap between numerical and geometric algebra. The decisive step in this direction came much later with Descartes, but Omar Khayyam was moving in this direction when he wrote, "Whoever thinks algebra is a trick in obtaining unknowns has thought it in vain. No attention should be paid to the fact that algebra and geometry are different in appearance. Algebras are geometric facts which are proved."}}</ref> Omar Khayyam is credited with identifying the foundations of [[algebraic geometry]], and his book ''Treatise on Demonstrations of Problems of Algebra'' (1070), which laid down the principles of analytic geometry, is part of the body of Persian mathematics that was eventually transmitted to Europe.<ref>[[#refmathmaster|Mathematical Masterpieces: Further Chronicles by the Explorers]], p. 92</ref> Because of his thoroughgoing geometrical approach to algebraic equations, Khayyam can be considered a precursor to Descartes in the invention of analytic geometry.<ref name ="Cooper">Cooper, G. (2003). Journal of the American Oriental Society,123(1), 248-249.</ref>{{rp|248}}

===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.