Editing Cubic equation (section)

==History==

Cubic equations were known to the ancient Babylonians, Greeks, Chinese, Indians, and Egyptians.<ref>{{Citation|last = Høyrup|first = Jens|title = Amphora: Festschrift for Hans Wussing on the Occasion of his 65th Birthday|chapter = The Babylonian Cellar Text BM 85200 + VAT 6599 Retranslation and Analysis|pages = 315–358|publisher = [[Birkhäuser]]|year = 1992|doi = 10.1007/978-3-0348-8599-7_16|isbn = 978-3-0348-8599-7}}</ref><ref name="oxf"/><ref name=wae/> [[Babylonia]]n (20th to 16th centuries BC) cuneiform tablets have been found with tables for calculating cubes and cube roots.<ref>{{cite book|last=Cooke|first=Roger|title=The History of Mathematics|url=https://books.google.com/books?id=CFDaj0WUvM8C&pg=PT63|date=8 November 2012|publisher=John Wiley & Sons|isbn=978-1-118-46029-0|page=63}}</ref><ref name="nen">{{cite book|last= Nemet-Nejat|first=Karen Rhea|title=Daily Life in Ancient Mesopotamia|url=https://archive.org/details/dailylifeinancie00neme|url-access= registration|year=1998|publisher=Greenwood Publishing Group|isbn=978-0-313-29497-6|page=[https://archive.org/details/dailylifeinancie00neme/page/306 306]}}</ref> The Babylonians could have used the tables to solve cubic equations, but no evidence exists to confirm that they did.<ref name=co>{{cite book|last=Cooke|first=Roger|title=Classical Algebra: Its Nature, Origins, and Uses|url=https://books.google.com/books?id=JG-skeT1eWAC&pg=PA64|year=2008|publisher=John Wiley & Sons|isbn=978-0-470-27797-3|page=64}}</ref> The problem of [[doubling the cube]] involves the simplest and oldest studied cubic equation, and one for which the ancient Egyptians did not believe a solution existed.<ref>{{Harvtxt|Guilbeau|1930|p=8}} states that "the Egyptians considered the solution impossible, but the Greeks came nearer to a solution."</ref> In the 5th century BC, [[Hippocrates of Chios|Hippocrates]] reduced this problem to that of finding two mean proportionals between one line and another of twice its length, but could not solve this with a [[compass and straightedge construction]],<ref name=Guilbeau>{{Harvtxt|Guilbeau|1930|pp=8–9}}</ref> a task which is now known to be impossible. Methods for solving cubic equations appear in ''[[The Nine Chapters on the Mathematical Art]]'', a [[Chinese mathematics|Chinese mathematical]] text compiled around the 2nd century BC and commented on by [[Liu Hui]] in the 3rd century.<ref name="oxf">{{cite book|last1=Crossley|first1=John|last2=W.-C. Lun|first2=Anthony|title=The Nine Chapters on the Mathematical Art: Companion and Commentary|url=https://books.google.com/books?id=eiTJHRGTG6YC&pg=PA176|year=1999|publisher=Oxford University Press|isbn=978-0-19-853936-0|page=176}}</ref> 

In the 3rd century AD, the [[Greek mathematics|Greek mathematician]] [[Diophantus]] found integer or rational solutions for some bivariate cubic equations ([[Diophantine equation]]s).<ref name="wae">Van der Waerden, Geometry and Algebra of Ancient Civilizations, chapter 4, Zurich 1983 {{ISBN|0-387-12159-5}}</ref><ref>{{cite book|title=Diophantus of Alexandria: A Study in the History of Greek Algebra|last=Heath|first=Thomas L.|author-link=Thomas Little Heath|pages=[https://archive.org/details/diophantusofalex00heatiala/page/87 87]–91|url=https://archive.org/details/diophantusofalex00heatiala|date=April 30, 2009|isbn=978-1578987542|publisher=Martino Pub}}</ref> Hippocrates, [[Menaechmus]] and [[Archimedes]] are believed to have come close to solving the problem of doubling the cube using intersecting [[conic sections]],<ref name="Guilbeau" /> though historians such as Reviel Netz dispute whether the Greeks were thinking about cubic equations or just problems that can lead to cubic equations. Some others like [[T. L. Heath]], who translated all of Archimedes's works, disagree, putting forward evidence that Archimedes really solved cubic equations using intersections of two [[Conic section|conics]], but also discussed the conditions where the roots are 0, 1 or 2.<ref>{{cite book|title=The works of Archimedes|author=Archimedes|author-link=Archimedes|others=Translation by T. L. Heath|date=October 8, 2007|isbn= 978-1603860512|publisher=Rough Draft Printing}}</ref> 
[[Image:Graph of cubic polynomial.svg|200px|left|thumb|[[Graph of a function|Graph]] of the cubic function ''f''(''x'') = 2''x''<sup>3</sup>&nbsp;&minus;&nbsp;3''x''<sup>2</sup>&nbsp;&minus;&nbsp;3''x''&nbsp;+&nbsp;2 = 
 (''x''&nbsp;+&nbsp;1)&nbsp;(2''x''&nbsp;&minus;&nbsp;1)&nbsp;(''x''&nbsp;&minus;&nbsp;2)]]

In the 7th century, the [[Tang dynasty]] astronomer mathematician [[Wang Xiaotong]] in his mathematical treatise titled [[Jigu Suanjing]] systematically established and solved [[Numerical analysis|numerically]] 25 cubic equations of the form {{math|''x''<sup>3</sup> + ''px''<sup>2</sup> + ''qx'' {{=}} ''N''}}, 23 of them with {{math|''p'', ''q'' ≠ 0}}, and two of them with {{math|''q'' {{=}} 0}}.<ref>{{Citation
 |first= Yoshio
 |last= Mikami
 |author-link= Yoshio Mikami
 |title= The Development of Mathematics in China and Japan
 |chapter= Chapter 8 Wang Hsiao-Tung and Cubic Equations
 |pages= 53&ndash;56
 |publisher= Chelsea Publishing Co.
 |location= New York
 |year= 1974
 |edition= 2nd
 |orig-year= 1913
 |isbn= 978-0-8284-0149-4
 }}</ref>

In the 11th century, the Persian poet-mathematician, [[Omar Khayyam]] (1048–1131), made significant progress in the theory of cubic equations. In an early paper, he discovered that a cubic equation can have more than one solution and stated that it cannot be solved using compass and straightedge constructions. He also found a geometric solution.<ref>A paper of Omar Khayyam, Scripta Math. 26 (1963), pages 323–337</ref>{{efn|In {{MacTutor|id=Khayyam|title=Omar Khayyam}} one may read ''This problem in turn led Khayyam to solve the cubic equation'' {{math|''x''<sup>3</sup> + 200''x'' {{=}} 20''x''<sup>2</sup> + 2000}} ''and he found a positive root of this cubic by considering the intersection of a rectangular hyperbola and a circle. An approximate numerical solution was then found by interpolation in trigonometric tables''. The geometric construction was perfectly suitable for Omar Khayyam, as it occurs for solving a problem of geometric construction. At the end of his article he says only that, for this geometrical problem, if approximations are sufficient, then a simpler solution may be obtained by consulting [[Generating trigonometric tables|trigonometric tables]]. Textually: ''If the seeker is satisfied with an estimate, it is up to him to look into the table of chords of Almagest, or the table of sines and versed sines of Mothmed Observatory.'' This is followed by a short description of this alternate method (seven lines).}} In his later work, the ''[[Treatise on Demonstration of Problems of Algebra]]'', he wrote a complete classification of cubic equations with general geometric solutions found by means of intersecting [[conic section]]s.<ref>J. J. O'Connor and E. F. Robertson (1999), [https://mathshistory.st-andrews.ac.uk/Biographies/Khayyam/ Omar Khayyam], [[MacTutor History of Mathematics archive]], states, "Khayyam himself seems to have been the first to conceive a general theory of cubic equations."<!-- quotation is in MacTutor--></ref><ref>{{Harvtxt|Guilbeau|1930|p=9}} states, "Omar Al Hay of Chorassan, about 1079 AD did most to elevate to a method the solution of the algebraic equations by intersecting conics."</ref> Khayyam made an attempt to come up with an algebraic formula for extracting cubic roots. He wrote: <blockquote>“We have tried to express these roots by algebra but have failed. It may be, however, that men who come after us will succeed.”<ref>{{Cite book |last=Berggren |first=J. L. |url=https://books.google.com/books?id=I-jwDQAAQBAJ&dq=%E2%80%9CWe+have+tried+to+express+these+roots+by+algebra+but+have+failed.+It+may+be%2C+however%2C+that+men+who+come+after+us+will+succeed.%E2%80%9D&pg=PA14 |title=Episodes in the Mathematics of Medieval Islam |date=2017-01-18 |publisher=Springer |isbn=978-1-4939-3780-6 |language=en}}</ref></blockquote>

In the 12th century, the Indian mathematician Bhaskara II attempted the solution of cubic equations without general success. However, he gave one example of a cubic equation: {{math|''x''<sup>3</sup> + 12''x'' {{=}} 6''x''<sup>2</sup> + 35}}.<ref>{{Citation|last1 = Datta|first1 = Bibhutibhushan|author-link = Bibhutibhushan Datta|last2 = Singh|first2 = Avadhesh Narayan|title = History of Hindu Mathematics: A Source Book|volume = 2|page = 76|chapter = Equation of Higher Degree|publisher = Bharattya Kala Prakashan|location = Delhi, India|year = 2004|isbn = 81-86050-86-8|title-link = History of Hindu Mathematics: A Source Book}}</ref> In the 12th century, another [[Mathematics in medieval Islam|Persian]] mathematician, [[Sharaf al-Dīn al-Tūsī]] (1135–1213), wrote the ''Al-Muʿādalāt'' (''Treatise on Equations''), which dealt with eight types of cubic equations with positive solutions and five types of cubic equations which may not have positive solutions. He used what would later be known as the ''[[Horner method#Horner–Ruffini method|Horner–Ruffini method]]'' to [[Numerical analysis|numerically approximate]] the [[root of a function|root]] of a cubic equation. He also used the concepts of [[maxima and minima]] of curves in order to solve cubic equations which may not have positive solutions.<ref>{{MacTutor|id=Al-Tusi_Sharaf|title=Sharaf al-Din al-Muzaffar al-Tusi}}</ref> He understood the importance of the [[discriminant]] of the cubic equation to find algebraic solutions to certain types of cubic equations.<ref>{{Citation |first=J. L. |last=Berggren |year=1990 |title=Innovation and Tradition in Sharaf al-Dīn al-Ṭūsī's Muʿādalāt |journal=Journal of the American Oriental Society |volume=110 |issue=2 |pages=304–309 |doi= 10.2307/604533|jstor = 604533}}</ref>

In his book ''Flos'', Leonardo de Pisa, also known as [[Fibonacci]] (1170–1250), was able to closely approximate the positive solution to the cubic equation {{math|''x''<sup>3</sup> + 2''x''<sup>2</sup> + 10''x'' {{=}} 20}}. Writing in [[Babylonian numerals]] he gave the result as 1,22,7,42,33,4,40 (equivalent to 1&nbsp;+&nbsp;22/60&nbsp;+&nbsp;7/60<sup>2</sup>&nbsp;+&nbsp;42/60<sup>3</sup>&nbsp;+&nbsp;33/60<sup>4</sup>&nbsp;+&nbsp;4/60<sup>5</sup>&nbsp;+&nbsp;40/60<sup>6</sup>), which has a [[relative error]] of about 10<sup>−9</sup>.<ref>{{MacTutor|id=Fibonacci|title=Fibonacci}}</ref>

In the early 16th century, the Italian mathematician [[Scipione del Ferro]] (1465–1526) found a method for solving a class of cubic equations, namely those of the form {{math|''x''<sup>3</sup> + ''mx'' {{=}} ''n''}}. In fact, all cubic equations can be reduced to this form if one allows {{mvar|m}} and {{mvar|n}} to be negative, but [[negative number]]s were not known to him at that time. Del Ferro kept his achievement secret until just before his death, when he told his student Antonio Fior about it.
[[Image:Niccolò Tartaglia.jpg|thumb|160px|Niccolò Fontana Tartaglia]]
In 1535, [[Niccolò Tartaglia]] (1500–1557) received two problems in cubic equations from [[Zuanne da Coi]] and announced that he could solve them. He was soon challenged by Fior, which led to a famous contest between the two. Each contestant had to put up a certain amount of money and to propose a number of problems for his rival to solve. Whoever solved more problems within 30 days would get all the money. Tartaglia received questions in the form {{math|''x''<sup>3</sup> + ''mx'' {{=}} ''n''}}, for which he had worked out a general method. Fior received questions in the form {{math|''x''<sup>3</sup> + ''mx''<sup>2</sup> {{=}} ''n''}}, which proved to be too difficult for him to solve, and Tartaglia won the contest.

Later, Tartaglia was persuaded by [[Gerolamo Cardano]] (1501–1576) to reveal his secret for solving cubic equations. In 1539, Tartaglia did so only on the condition that Cardano would never reveal it and that if he did write a book about cubics, he would give Tartaglia time to publish. Some years later, Cardano learned about del Ferro's prior work and published del Ferro's method in his book ''[[Ars Magna (Gerolamo Cardano)|Ars Magna]]'' in 1545, meaning Cardano gave Tartaglia six years to publish his results (with credit given to Tartaglia for an independent solution). 

Cardano's promise to Tartaglia said that he would not publish Tartaglia's work, and Cardano felt he was publishing del Ferro's, so as to get around the promise. Nevertheless, this led to a challenge to Cardano from Tartaglia, which Cardano denied. The challenge was eventually accepted by Cardano's student [[Lodovico Ferrari]] (1522–1565). Ferrari did better than Tartaglia in the competition, and Tartaglia lost both his prestige and his income.<ref>{{Cite book |last=Katz |first=Victor |title=A History of Mathematics |page=[https://archive.org/details/historyofmathema00katz/page/220 220] |location=Boston |publisher=Addison Wesley |year=2004 |isbn=9780321016188 |url=https://archive.org/details/historyofmathema00katz/page/220 }}</ref>

Cardano noticed that Tartaglia's method sometimes required him to extract the square root of a negative number. He even included a calculation with these [[complex number]]s in ''Ars Magna'', but he did not really understand it. [[Rafael Bombelli]] studied this issue in detail<ref name="Bombelli">{{Citation|last2 = Mazur|first2 = Barry|author2-link = Barry Mazur|last1 = La Nave|first1 = Federica|journal = [[The Mathematical Intelligencer]]|title = Reading Bombelli|volume = 24|issue = 1|pages = 12–21|year = 2002|doi = 10.1007/BF03025306|s2cid = 189888034}}</ref> and is therefore often considered as the discoverer of complex numbers.

[[François Viète]] (1540–1603) independently derived the trigonometric solution for the cubic with three real roots, and [[René Descartes]] (1596–1650) extended the work of Viète.<ref name=Nickalls/>