Editing Algebraic geometry (section)

===Before the 16th century===
Some of the roots of algebraic geometry date back to the work of the [[Hellenistic Greece|Hellenistic Greeks]] from the 5th century BC. The [[Delian problem]], for instance, was to construct a length ''x'' so that the cube of side ''x'' contained the same volume as the rectangular box ''a''<sup>2</sup>''b'' for given sides ''a'' and ''b''. [[Menaechmus]] ({{circa|350 BC}}) considered the problem geometrically by intersecting the pair of plane conics ''ay''&nbsp;=&nbsp;''x''<sup>2</sup> and ''xy''&nbsp;=&nbsp;''ab''.<ref name="Dieudonné">{{Cite Q|Q55886951|author1=Dieudonné, Jean|author-link1=Jean Dieudonné}}</ref> In the 3rd century BC, [[Archimedes]] and [[Apollonius of Perga|Apollonius]] systematically studied additional problems on [[conic sections]] using coordinates.<ref name="Dieudonné"/>{{sfn|Kline|1972|p=108, 90}} [[Apollonius of Perga|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 [[René Descartes|Descartes]] by some 1800 years.<ref>{{Cite journal |last=Molland |first=A. G |date=1976-02-01 |title=Shifting the foundations: Descartes's transformation of ancient geometry |journal=Historia Mathematica |language=en |volume=3 |issue=1 |pages=21–49 |doi=10.1016/0315-0860(76)90004-5 |issn=0315-0860|doi-access=free }}</ref> 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 coordinates using geometric methods like using parabolas and curves.<ref>{{Cite web |title=Apollonius - Biography |url=https://mathshistory.st-andrews.ac.uk/Biographies/Apollonius/ |access-date=2022-11-11 |website=Maths History |language=en}}</ref><ref>{{Cite journal |last=M. |first=G. B. |date=August 1896 |title=Apollonius of Perga: Treatise on Conic Sections |url=https://www.nature.com/articles/054314a0 |journal=Nature |language=en |volume=54 |issue=1397 |pages=314–315 |doi=10.1038/054314a0 |bibcode=1896Natur..54..314G |s2cid=4059946 |issn=1476-4687}}</ref><ref>{{Cite journal |last=Unguru |first=Sabetai |date=June 1976 |title=A Very Early Acquaintance with Apollonius of Perga's Treatise on Conic Sections in the Latin West |url=https://cris.tau.ac.il/en/publications/a-very-early-acquaintance-with-apollonius-of-pergas-treatise-on-c |journal=Centaurus |language=English |volume=20 |issue=2 |pages=112–128 |doi=10.1111/j.1600-0498.1976.tb00924.x |bibcode=1976Cent...20..112U |issn=0008-8994}}</ref> Medieval  mathematicians, including [[Omar Khayyam]], [[Leonardo of Pisa]], [[Gersonides]] and [[Nicole Oresme]] in the [[Middle Ages|Medieval Period]],{{sfn|Kline|1972|p=193}} solved certain cubic and quadratic equations by purely algebraic means and then interpreted the results geometrically. The [[Persian people|Persian]] mathematician [[Omar Khayyám]] (born 1048 AD) believed that there was a relationship between [[arithmetic]], [[algebra]] and [[geometry]].{{sfn|Kline|1972|p=193–195}}<ref>{{cite web |url=http://www-history.mcs.st-andrews.ac.uk/Biographies/Khayyam.html |title=Omar Khayyam |first1=J. J. |last1=O'Connor |first2=E. F. |last2=Robertson |publisher=School of Mathematics and Statistics, University of St Andrews |archive-url=https://web.archive.org/web/20171112123436/http://www-history.mcs.st-andrews.ac.uk/Biographies/Khayyam.html |archive-date=November 12, 2017 |quote=Khayyam himself seems to have been the first to conceive a general theory of cubic equations.}}</ref><ref>{{cite book| last=Rashed |first=Roshdi |date=1994 |url=https://archive.org/stream/RoshdiRashedauth.TheDevelopmentOfArabicMathematicsBetweenArithmeticAndAlgebraSpringerNetherlands1994/Roshdi%20Rashed%20%28auth.%29-The%20Development%20of%20Arabic%20Mathematics_%20Between%20Arithmetic%20and%20Algebra-Springer%20Netherlands%20%281994%29#page/n111/mode/1up |title=The Development Of Arabic Mathematics Between Arithmetic And Algebra |publisher=Springer |pages=102–103}}</ref> This was criticized by Jeffrey Oaks, who claims that the study of curves by means of equations originated with Descartes in the seventeenth century.<ref>{{Cite journal|last=Oaks|first=Jeffrey|title=Excavating the errors in the "Mathematics" chapter of 1001 Inventions|journal=Pp. 151-171 in: Sonja Brentjes, Taner Edis, Lutz Richter-Bernburd Edd., 1001 Distortions: How (Not) to Narrate History of Science, Medicine, and Technology in Non-Western Cultures |date=January 2016 |url=https://www.academia.edu/39325220|url-status=live|archive-url=https://web.archive.org/web/20210227182759/https://www.academia.edu/39325220/Excavating_the_errors_in_the_Mathematics_chapter_of_1001_Inventions |archive-date=2021-02-27 }}</ref>