Toscead betweox fadungum "Rīmagiefung"

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Þā staðolas stæfrīmtācninge cunnon wesan gefunden on þǣm gamolum [[Babylonisc rīmlār|Babyloniscum]] ,<ref>Struik, Dirk J. (1987). ''A Concise History of Mathematics''. New York: Dover Publications.</ref> þā forðodon forðode rīmmetende endebyrdnesse þurh þā þe hīe cūðdon dōn rīmweorcunga in rīmendebyrdnessiscre wīsan. Þā Babyloniscan forðodon endebyrdnessa tō weorcienne arāflunga cnottena þe sindgewunelīce arāfled tōdæg be þǣre nytte [[līnlicra efenweorþfindunga]], [[fēowerscētra efenweorþfindunga]], and [[unrīm efenweorþfindung|unrīmra līnlicra efenweorþfindunga]]. Tō ungelīcnesse, þæt mǣste dǣl [[Egyptisc rīmlār|Egyptiscena]] þisre tīde, and [[Crēcisc rīmlār|Crēciscena]] and [[Cīnisc rīmlār|Cīniscena]] rīmlārmanna in þǣm [[1 þūsendgēare fōre Crīste]], gewunelīce arāflede swelca efenweorþfindunga þurh eorþmetlica endebyrdnesse, swelce þās amearcod in þǣm ''[[Rhind Rīmlāriscan Papyrus]]'', [[Euclides Gesceaft|Euclides ''Gesceaft'']], and ''[[Þā Nigone Hēafodwearda on þǣm Rīmcræfte]]''. Þæt eorþmetlice weorc þāra Crēciscena, ēowod in þǣre bēc ''Gesceaft'', macode þæt grundweorc for þǣre gewuneliclǣcunge endebyrdnessa begeondan þā arāflunge sumra cnottena on maniga gewunelica endebyrdnessa secgunge and arāflunge efenweorþfindunga, þēah þe þis ne scolde wesan gecūþ oþ þā [[rīmcræft in midealdum Alladōme|midealdan Alladōmiscan rīmcræftmenn]].
 
Þā Crēciscan rīmcræftmenn [[Hero Alexandrie]] and [[Diophantus]] <ref>[http://library.thinkquest.org/25672/diiophan.htm Diophantus, Father of Algebra]</ref> and Indisce rīmcræftmenn, swelce [[Brahmagupta]], hēoldon þā þēawas [[Ægypte]]s and [[Babylōn]]ie, þēah þe Diophantus ''[[Arithmetica]]'' and Brahmaguptan ''[[Brahmasphutasiddhanta]]'' standaþ on hīeran efenette.<ref>[http://www.algebra.com/algebra/about/history/ History of Algebra]</ref> Tō bȳsne, sēo forme fulle rīmcræftisce arāfling (befōne nāwihtrīm and underran arāflunga) [[fēowerscētra efenweorþfindunga]] wæs amearcod fram Brahmaguptan in his bēc ''Brahmasphutasiddhanta''. Þǣræfter, Arabisce and Muslime rīmcræftmenn forðodon stæfrīmtācningisca endebyrdnessa on swīðe hīeran strengðe manigfealdnesse. Þēah Diophantus and þā Babylōniscan brucon oftost ānlica endebyrdnessa tō arāflenne efenweorþlǣcinga, Al-Khwarizmi wæs se forma þe arāflede efenweorþlǣcinga þā hwīle þe hē brēac brāda endebyrdnessa. Hē arāflede þā līnlican unfæstlican efenweorþlǣcinga, fēowerscēta efenweorþlǣcinga, unfæstlica efenweorþlǣcinga þǣre ōðerre endebyrdnesse and efenweorþlǣcinga mid manigfealdum unfæstrīmum.
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The [[Hellenistic civilization|Hellenistic]] mathematicians [[Hero of Alexandria]] and [[Diophantus]] <ref>[http://library.thinkquest.org/25672/diiophan.htm Diophantus, Father of Algebra]</ref> as well as [[Indian mathematics|Indian mathematicians]] such as [[Brahmagupta]] continued the traditions of Egypt and Babylon, though Diophantus' ''[[Arithmetica]]'' and Brahmagupta's ''[[Brahmasphutasiddhanta]]'' are on a higher level.<ref>[http://www.algebra.com/algebra/about/history/ History of Algebra]</ref> For example, the first complete arithmetic solution (including zero and negative solutions) to [[quadratic equation]]s was described by Brahmagupta in his book ''Brahmasphutasiddhanta''. Later, Arabic and Muslim mathematicians developed algebraic methods to a much higher degree of sophistication. Although Diophantus and the Babylonians used mostly special ad hoc methods to solve equations, Al-Khwarizmi was the first to solve equations using general methods. He solved the linear indeterminate equations, quadratic equations, second order indeterminate equations and equations with multiple variable.
[[File:Gerolamo Cardano.jpg|thumb|200px|In 1545, the Italian mathematician [[Girolamo Cardano]] published ''[[Ars Magna (Gerolamo Cardano)|Ars magna]]'' -''The great art'', a 40-chapter masterpiece in which he gave for the first time a method for solving the general [[quartic equation]].]]
The [[Greeks|Greek]] mathematician [[Diophantus]] has traditionally been known as the "father of algebra" but in more recent times there is much debate over whether al-Khwarizmi, who founded the discipline of ''al-jabr'', deserves that title instead.<ref>Carl B. Boyer, ''A History of Mathematics, Second Edition'' (Wiley, 1991), pages 178, 181</ref> Those who support Diophantus point to the fact that the algebra found in ''Al-Jabr'' is slightly more elementary than the algebra found in ''Arithmetica'' and that ''Arithmetica'' is syncopated while ''Al-Jabr'' is fully rhetorical.<ref>Carl B. Boyer, ''A History of Mathematics, Second Edition'' (Wiley, 1991), page 228</ref> Those who support Al-Khwarizmi point to the fact that he introduced the methods of "[[Reduction (mathematics)|reduction]]" and "balancing" (the transposition of subtracted terms to the other side of an equation, that is, the cancellation of [[like terms]] on opposite sides of the equation) which the term ''al-jabr'' originally referred to,<ref name=Boyer-229>{{Harv|Boyer|1991|loc="The Arabic Hegemony" p. 229}} "It is not certain just what the terms ''al-jabr'' and ''muqabalah'' mean, but the usual interpretation is similar to that implied in the translation above. The word ''al-jabr'' presumably meant something like "restoration" or "completion" and seems to refer to the transposition of subtracted terms to the other side of an equation; the word ''muqabalah'' is said to refer to "reduction" or "balancing" - that is, the cancellation of like terms on opposite sides of the equation."</ref> and that he gave an exhaustive explanation of solving quadratic equations,<ref>{{Harv|Boyer|1991|loc="The Arabic Hegemony" p. 230}} "The six cases of equations given above exhaust all possibilities for linear and quadratic equations having positive root. So systematic and exhaustive was al-Khwarizmi's exposition that his readers must have had little difficulty in mastering the solutions."</ref> supported by geometric proofs, while treating algebra as an independent discipline in its own right.<ref>Gandz and Saloman (1936), ''The sources of al-Khwarizmi's algebra'', Osiris i, p. 263–277: "In a sense, Khwarizmi is more entitled to be called "the father of algebra" than Diophantus because Khwarizmi is the first to teach algebra in an elementary form and for its own sake, Diophantus is primarily concerned with the theory of numbers".</ref> His algebra was also no longer concerned "with a series of [[problem]]s to be resolved, but an [[Expository writing|exposition]] which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study." He also studied an equation for its own sake and "in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems."<ref name=Rashed-Armstrong>{{Citation | last1=Rashed | first1=R. | last2=Armstrong | first2=Angela | year=1994 | title=The Development of Arabic Mathematics | publisher=[[Springer Science+Business Media|Springer]] | isbn=0792325656 | oclc=29181926 | pages=11–2}}</ref>