Editing Diophantus (section)

==Influence==
Diophantus' work has had a large influence in history. Although [[Joseph-Louis Lagrange]] called Diophantus "the inventor of [[algebra]]", he did not invent it, however his work ''Arithmetica''{{sfn|Christianidis|Oaks|2023|p=80}} created a foundation for work on algebra and in fact much of advanced mathematics is based on algebra.<ref>{{cite web |url=https://www.britannica.com/biography/Diophantus |title= Diophantus - Biography & Facts|last=Sesiano |first=Jacques|website=Britannica |access-date= August 23, 2022}}</ref> Diophantus and his works influenced [[mathematics in the medieval Islamic world]], and editions of ''Arithmetica'' exerted a profound influence on the development of algebra in Europe in the late sixteenth and through the 17th and 18th centuries.{{sfn|Christianidis|Megremi|2019}} 

===Later antiquity===
After its publication, Diophantus' work continued to be read in the Greek-speaking Mediterranean from the 4th through the 7th centuries.{{sfn|Christianidis|Megremi|2019}}  The earliest known reference to Diophantus, in the 4th century, is the ''Commentary on the [[Almagest]]'' [[Theon of Alexandria]], which quotes from the introduction to the ''Arithmetica''.{{sfn|Christianidis|Megremi|2019|pp=18-20}} According to the [[Suda]], [[Hypatia]], who was Theon's daughter and frequent collaborator, wrote a now lost commentary on Diophantus' ''Arithmetica'', which suggests that this work may have been closely studied by [[Neoplatonism|Neoplatonic]] mathematicians in Alexandria during [[Late antiquity]].{{sfn|Christianidis|Megremi|2019|pp=18-20}} References to Diophantus also survive in a number of Neoplatonic [[scholia]] to the works of [[Iamblichus]].{{sfn|Christianidis|Megremi|2019|pp=22}} A 6th century Neoplatonic commentary on [[Porphyry of Tyre|Porphyry]]'s ''[[Isagoge]]'' by [[Pseudo-Elias]] also mentions Diophantus; after outlining the [[quadrivium]] of [[arithmetic]], [[geometry]], [[music]], and [[astronomy]]  and four other disciplines adjacent to them ("logistic", "geodesy", "music in matter" and "spherics"), it mentions that [[Nicomachus]] (author of the ''Introduction to Arithmetic'') occupies the first place in arithmetic but Diophantus occupies the first place in "logistic", showing that, despite the title of ''Arithmetica'', the more algebraic work of Diophantus was already seen as distinct from arithmetic prior to the medieval era.{{sfn|Christianidis|Megremi|2019|pp=23}} 

===Medieval era===
Like many other Greek mathematical treatises, Diophantus was forgotten in Western Europe during the [[Dark Ages (historiography)|Dark Ages]], since the study of ancient Greek, and literacy in general, had greatly declined.  The portion of the Greek ''Arithmetica'' that survived, however, was, like all ancient Greek texts transmitted to the early modern world, copied by, and thus known to, medieval Byzantine scholars. Scholia on Diophantus by the Byzantine Greek scholar [[John Chortasmenos]] (1370–1437) are preserved together with a comprehensive commentary written by the earlier Greek scholar [[Maximos Planudes]] (1260 – 1305), who produced an edition of Diophantus within the library of the [[Chora Monastery]] in Byzantine [[Constantinople]].{{sfn|Herrin|2013|p=322}}

''Arithmetica'' became known to [[Mathematics in medieval Islam|mathematicians in the Islamic world]] in the ninth century, when [[Qusta ibn Luqa]] translated it into Arabic.<ref>{{cite book |year= 1998 |editor-last= Magill |editor-first= Frank N. |title= Dictionary of World Biography |url= https://books.google.com/books?id=_CMl8ziTbKYC&pg=PA362 |volume= 1 |publisher= Salem Press |page= 362 |isbn= 9781135457396 }}</ref>

In 1463 German mathematician [[Regiomontanus]] wrote:"No one has yet translated from the Greek into Latin the thirteen books of Diophantus, in which the very flower of the whole of arithmetic lies hidden." ''Arithmetica'' was first translated from Greek into [[Latin]] by [[Rafael Bombelli|Bombelli]] in 1570, but the translation was never published. However, Bombelli borrowed many of the problems for his own book ''Algebra''. The ''[[editio princeps]]'' of ''Arithmetica'' was published in 1575 by [[Guilielmus Xylander|Xylander]].

===Fermat===
[[File:Diophantus-II-8-Fermat.jpg|right|thumb|200px|[[Diophantus_II.VIII|Problem II.8]] in the ''Arithmetica'' (edition of 1670), annotated with Fermat's comment which became [[Fermat's Last Theorem]].]]
The Latin translation of ''Arithmetica'' by [[Bachet]] in 1621 became the first Latin edition that was widely available. [[Pierre de Fermat]] owned a copy, studied it and made notes in the margins. The 1621 edition of ''Arithmetica'' by [[Bachet]] gained fame after [[Pierre de Fermat]] wrote his famous "[[Fermat's Last Theorem|Last Theorem]]" in the margins of his copy: <blockquote>If an integer {{math|''n''}} is greater than 2, then {{math|''a''{{sup|''n''}} + ''b''{{sup|''n''}} {{=}} ''c''{{sup|''n''}}}} has no solutions in non-zero integers {{math|''a''}}, {{math|''b''}}, and {{math|''c''}}. I have a truly marvelous proof of this proposition which this margin is too narrow to contain.</blockquote>Fermat's proof was never found, and the problem of finding a proof for the theorem went unsolved for centuries. A proof was finally found in 1994 by [[Andrew Wiles]] after working on it for seven years.  It is believed that Fermat did not actually have the proof he claimed to have. Although the original copy in which Fermat wrote this is lost today, Fermat's son edited the next edition of Diophantus, published in 1670. Even though the text is otherwise inferior to the 1621 edition, Fermat's annotations—including the "Last Theorem"—were printed in this version.

Fermat was not the first mathematician so moved to write in his own marginal notes to Diophantus; the Byzantine scholar [[John Chortasmenos]] (1370–1437) had written "Thy soul, Diophantus, be with Satan because of the difficulty of your other theorems and particularly of the present theorem" next to the same problem.<ref>{{Cite book|url=https://books.google.com/books?id=-zrGDDwQLo8C&pg=PA322|title=Margins and Metropolis: Authority across the Byzantine Empire|last=Herrin|first=Judith|date=2013-03-18|publisher=Princeton University Press|isbn=978-1400845224|page=322|language=en}}</ref>

Diophantus was among the first to recognise positive [[rational number]]s as numbers, by allowing fractions for coefficients and solutions. He coined the term παρισότης (''parisotēs'') to refer to an approximate equality.<ref>{{citation |last1=Katz |first1=Mikhail G. |title=Almost Equal: The Method of [[Adequality]] from Diophantus to Fermat and Beyond |journal=[[Perspectives on Science]] |volume=21 |issue=3 |pages=283–324 |year=2013 |arxiv=1210.7750 |bibcode=2012arXiv1210.7750K |doi=10.1162/POSC_a_00101 |s2cid=57569974 |last2=Schaps |first2=David |last3=Shnider |first3=Steve |author1-link=Mikhail Katz |author3-link=Steve Shnider}}</ref> This term was rendered as ''adaequalitas'' in Latin, and became the technique of [[adequality]] developed by [[Pierre de Fermat]] to find maxima for functions and tangent lines to curves.

===Diophantine analysis===
{{see also|Diophantine equation}}
Today, Diophantine analysis is the area of study where integer (whole-number) solutions are sought for equations, and Diophantine equations are polynomial equations with integer coefficients to which only integer solutions are sought. It is usually rather difficult to tell whether a given Diophantine equation is solvable. Most of the problems in ''Arithmetica'' lead to [[quadratic equation]]s. Diophantus looked at 3 different types of quadratic equations: {{math|''ax''{{sup|2}} + ''bx'' {{=}} ''c''}}, {{math|''ax''{{sup|2}} {{=}} ''bx'' + ''c''}}, and {{math|''ax''{{sup|2}} + ''c'' {{=}} ''bx''}}. The reason why there were three cases to Diophantus, while today we have only one case, is that he did not have any notion for zero and he avoided [[negative numbers|negative]] coefficients by considering the given numbers {{math|''a''}}, {{math|''b''}}, {{math|''c''}} to all be positive in each of the three cases above.  Diophantus was always satisfied with a rational solution and did not require a whole number which means he accepted fractions as solutions to his problems. Diophantus considered negative or [[irrational number|irrational]] square root solutions "useless", "meaningless", and even "absurd". To give one specific example, he calls the equation {{math|4 {{=}} 4''x'' + 20}} 'absurd' because it would lead to a negative value for {{math|''x''}}. One solution was all he looked for in a quadratic equation. There is no evidence that suggests Diophantus even realized that there could be two solutions to a quadratic equation. He also considered [[simultaneous equations|simultaneous]] quadratic equations.

=== Rediscovery of books IV-VII ===
In 1968, [[Fuat Sezgin]] found four previously unknown books of ''Arithmetica'' at the shrine of Imam Rezā in the holy Islamic city of [[Mashhad]] in northeastern Iran.<ref>{{cite web|title= Review of J. Sesiano, Books IV to VII of Diophantus' Arithmetica |last= Hogendijk |first= Jan P. |author-link= Jan Hogendijk |year= 1985 |url= http://www.jphogendijk.nl/reviews/sesiano.html |access-date= 2014-07-06 }}</ref> The four books are thought to have been translated from Greek to Arabic by [[Qusta ibn Luqa]] (820–912).<ref>{{cite book |year= 1998 |editor-last= Magill |editor-first= Frank N. |title= Dictionary of World Biography |url= https://books.google.com/books?id=_CMl8ziTbKYC&pg=PA362 |volume= 1 |publisher= Salem Press |page= 362 |isbn= 9781135457396 }}</ref> Norbert Schappacher has written:
<blockquote>
[The four missing books] resurfaced around 1971 in the [[Central Library of Astan Quds Razavi|Astan Quds Library]] in Meshed (Iran) in a copy from 1198.  It was not catalogued under the name of Diophantus (but under that of [[Qusta ibn Luqa]]) because the librarian was apparently not able to read the main line of the cover page where Diophantus’s name appears in geometric [[Kufic|Kufi calligraphy]].<ref>{{cite web|title= Diophantus of Alexandria : a Text and its History |last= Schappacher |first= Norbert |url= http://www-irma.u-strasbg.fr/~schappa/NSch/Publications_files/1998cBis_Dioph.pdf
 |date= April 2005 |page= 18 |access-date= 2015-10-09}}</ref>
</blockquote>