Editing Diophantus

{{Short description|3rd-century Greek mathematician}}
{{Infobox scientist
| honorific_prefix  = 
| name              = Diophan
| native_name       = Διόφαντος 
| native_name_lang  = grc
| image             = Διόφαντος - Diophantos - ДИОФАНТ.jpg
| birth_name        = <!-- if different from "name" -->
| birth_date        = {{circa}} 3rd century CE
| birth_place       = [[Alexandria]]
| death_date        = {{circa}} 3rd century CE
| known_for         = [[Algebra]]
| notable_students  = [[Anatolius of Alexandria]] (disputed)
}}
{{for-multi|the general|Diophantus (general)|the sophist|Diophantus the Arab|the intersex soldier|Diophantus of Abae}}
'''Diophantus of Alexandria''' ({{langx|grc|Διόφαντος|Diophantos}}) ({{IPAc-en|d|aɪ|oʊ|ˈ|f|æ|n|t|ə|s}}; {{fl|250 CE}}) was a [[Greek mathematics|Greek mathematician]] who was the author of the ''[[Arithmetica]]'' in thirteen books, ten of which are still extant, made up of arithmetical problems that are solved through [[algebraic equation]]s. 

Although [[Joseph-Louis Lagrange]] called Diophantus "the inventor of [[algebra]]" he did not invent it; however, his exposition became the standard within the Neoplatonic schools of [[Late antiquity]], and its translation into Arabic in the 9th century AD and had influence in the development of later algebra: Diophantus' method of solution matches medieval Arabic algebra in its concepts and overall procedure. 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.

In modern use, [[Diophantine equation|Diophantine equations]] are algebraic equations with [[integer]] coefficients for which integer solutions are sought. [[Diophantine geometry]] and [[Diophantine approximation]]s are two other subareas of [[number theory]] that are named after him. Some problems from the ''Arithmetica'' have inspired modern work in both [[abstract algebra]] and [[number theory]].<ref>{{Cite journal |last=Hettle |first=Cyrus |date=2015 |title=The Symbolic and Mathematical Influence of Diophantus's Arithmetica |url=http://scholarship.claremont.edu/jhm/vol5/iss1/8/ |journal=Journal of Humanistic Mathematics |volume=5 |issue=1 |pages=139–166 |doi=10.5642/jhummath.201501.08|doi-access=free }}</ref> 

==Biography==
The exact details of Diophantus life are obscure. Although he probably flourished in the third century CE, he may have lived anywhere between 170 BCE, roughly contemporaneous with [[Hypsicles]], the latest author he quotes from, and 350 CE, when [[Theon of Alexandria]] quotes from him.{{sfn|Christianidis|Oaks|2023|pp=4-6}} [[Paul Tannery]] suggested that a reference to an "Anatolius" as a student of Diophantus in the works of [[Michael Psellos]] may refer to the early Christian bishop [[Anatolius of Alexandria]], who may possibly the same Anatolius mentioned by [[Eunapius]] as a teacher of the pagan [[Neopythagorean]] philosopher [[Iamblichus]], either of which would place him in the 3rd century CE.{{sfn|Christianidis|Oaks|2023|pp=4-6}}

The only definitive piece of information about his life is derived from a set of mathematical puzzles attributed to the 5th or 6th century CE grammarian [[Metrodorus (grammarian)|Metrodorus]] preserved in book 14 of the [[Greek Anthology]]. One of the problems (sometimes called Diophantus' epitaph) states:<blockquote>Here lies Diophantus, the wonder behold. Through art algebraic, the stone tells how old: 'God gave him his boyhood one-sixth of his life, One twelfth more as youth while whiskers grew rife; And then yet one-seventh ere marriage begun; In five years there came a bouncing new son. Alas, the dear child of master and sage After attaining half the measure of his father's life chill fate took him. After consoling his fate by the science of numbers for four years, he ended his life.'</blockquote>This puzzle implies that Diophantus' age {{math|''x''}} can be expressed as

:{{math|''x'' {{=}} {{sfrac|''x''|6}} + {{sfrac|''x''|12}} + {{sfrac|''x''|7}} + 5 + {{sfrac|''x''|2}} + 4}}

which gives {{math|''x''}} a value of 84 years. However, the accuracy of the information cannot be confirmed.{{sfn|Christianidis|Oaks|2023|pp=3-4}}

==''Arithmetica'' ==
[[File:Diophantus-cover.png|thumb|Title page of the Latin translation of Diophantus' ''Arithmetica'' by [[Claude Gaspard Bachet de Méziriac|Bachet]] (1621).|328x328px]]
''Arithmetica'' is the major work of Diophantus and the most prominent work on premodern [[algebra]] in Greek mathematics. It is a collection of 290 [[algebra]]ic problems giving numerical solutions of determinate [[equations]] (those with a unique solution) and [[indeterminate equation]]s. ''Arithmetica'' was originally written in thirteen books, but only six of them survive in Greek,<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> while another four books survive in Arabic, which were discovered in 1968.{{sfn|Sesiano|2012}} The books in Arabic correspond to books 4 to 7 of the original treatise, while the Greek books correspond to books 1 to 3 and 8 to 10.{{sfn|Sesiano|2012}}

''Arithmetica'' is the earliest extant work present that solve arithmetic problems by algebra. Diophantus however did not invent the method of algebra, which existed before him.{{sfn|Christianidis|Oaks|2023|p=80}} Algebra was practiced and diffused orally by practitioners, with Diophantus picking up technique to solve problems in arithmetic.{{sfn|Christianidis|Oaks|2013|pp=158-160}}

Equations in the book are presently called [[Diophantine equation]]s. The method for solving these equations is known as [[Diophantine analysis]].  Most of the ''Arithmetica'' problems lead to [[quadratic equation]]s.

===Notation===
Diophantus introduced an algebraic symbolism that used an abridged notation for frequently occurring operations, and an abbreviation for the unknown and for the powers of the unknown. 

Similar to medieval Arabic algebra, Diophantus uses three stages to solution of a problem by algebra:{{sfn|Christianidis|Oaks|2023|pp=53–66}}
# An unknown is named and an equation is set up
# An equation is simplified to a standard form (''al-jabr'' and ''al-muqābala'' in Arabic)
# Simplified equation is solved

Diophantus does not give classification of equations in six types like [[Al-Khwarizmi]] in extant parts of ''Arithmetica''. He does says that he would give solution to three terms equations later, so this part of work is possibly just lost.{{sfn|Christianidis|Oaks|2013|pp=158-160}}

The main difference between Diophantine notation and modern algebraic notation is that the former lacked special symbols for operations, relations, and exponentials.{{sfn|Cooke|1997|pp=167-168}}{{sfn|Derbyshire|2006|pp=35-36}} So for example, what would be written in modern notation as
<math display="block">x^3 - 2x^2 + 10x -1 = 5,</math>
which can be rewritten as
<math display=block>\left({x^3}1 + {x}10\right) - \left({x^2}2 + {x^0}1\right) = {x^0}5,</math>
would be written in Diophantus's notation as

:<math>\Kappa^{\upsilon} \overline{\alpha} \; \zeta \overline{\iota} \;\, \pitchfork \;\, \Delta^{\upsilon} \overline{\beta} \; \Mu \overline{\alpha} \,\;</math>{{lang|grc|ἴ}}<math>\sigma\;\, \Mu \overline{\varepsilon}</math>

{| class="wikitable"
|-
! Symbol
! What it represents
|-
| &nbsp;<math>\overline{\alpha}</math>
| 1 ([[Alpha]] is the 1st letter of the [[Greek alphabet]])
|-
| &nbsp;<math>\overline{\beta}</math>
| 2 ([[Beta]] is the 2nd letter of the Greek alphabet)
|-
| &nbsp;<math>\overline{\varepsilon}</math>
| 5 ([[Epsilon]] is the 5th letter of the Greek alphabet)
|-
| &nbsp;<math>\overline{\iota}</math>
| 10 ([[Iota]] is the 9th letter of the [[History of the Greek alphabet|{{em|modern}} Greek alphabet]] but it was the 10th letter of an [[Archaic Greek alphabets|ancient archaic Greek alphabet]] that had the letter [[digamma]] (uppercase: Ϝ, lowercase: ϝ) in the 6th position between [[epsilon]] ε and [[zeta]] ζ.)
|-
| {{lang|grc|ἴσ}}
| "equals" (short for {{lang|grc|[[wiktionary:ἴσος|ἴσος]]}})
|-
| <math>\pitchfork</math>
| represents the subtraction of everything that follows <math>\pitchfork</math> up to {{lang|grc|ἴσ}}
|-
| <math>\Mu</math>
| the zeroth power (that is, a constant term)
|-
| <math>\zeta</math>
| the unknown quantity (because a number <math>x</math> raised to the first power is just <math>x,</math> this may be thought of as "the first power")
|-
| <math>\Delta^{\upsilon}</math>
| the second power, from Greek {{lang|grc|δύναμις}}, meaning strength or power
|-
| <math>\Kappa^{\upsilon}</math>
| the third power, from Greek {{lang|grc|κύβος}}, meaning a cube
|-
| <math>\Delta^{\upsilon}\Delta</math>
| the fourth power
|-
| <math>\Delta\Kappa^{\upsilon}</math>
| the fifth power
|-
| <math>\Kappa^{\upsilon}\Kappa</math>
| the sixth power
|-
|}

Unlike in modern notation, the coefficients come after the variables and addition is represented by the juxtaposition of terms. A literal symbol-for-symbol translation of Diophantus's equation into a modern equation would be the following:{{sfn|Derbyshire|2006|pp=35-36}}
<math display=block>{x^3}1 {x}10 - {x^2}2 {x^0}1 = {x^0}5</math>
where to clarify, if the modern parentheses and plus are used then the above equation can be rewritten as:{{sfn|Derbyshire|2006|pp=35-36}}
<math display=block>\left({x^3}1 + {x}10\right) - \left({x^2}2 + {x^0}1\right) = {x^0}5</math>

===Contents===
In Book 3, Diophantus solves problems of finding values which make two linear expressions simultaneously into squares or cubes. In book 4, he finds rational powers between given numbers. He also noticed that numbers of the form <math>4n + 3</math> cannot be the sum of two squares. Diophantus also appears to know that [[Lagrange's four-square theorem|every number can be written as the sum of four squares]]. If he did know this result (in the sense of having proved it as opposed to merely conjectured it), his doing so would be truly remarkable:  even Fermat, who stated the result, failed to provide a proof of it and it was not settled until [[Joseph-Louis Lagrange]] proved it using results due to [[Leonhard Euler]].

==Other works==
Another work by Diophantus, ''On Polygonal Numbers'' is transmitted in an incomplete form in four Byzantine manuscripts along with the ''Arithmetica''.{{sfn|Christianidis|Oaks|2023|p=11}} Two other lost works by Diophantus are known: ''Porisms'' and ''On Parts''.{{sfn|Christianidis|Oaks|2023|p=15}} 

Recently, [[Wilbur Knorr]] has suggested that another book, ''Preliminaries to the Geometric Elements'', traditionally attributed to [[Hero of Alexandria]], may actually be by Diophantus.<ref>Knorr, Wilbur: Arithmêtike stoicheiôsis: On Diophantus and Hero of Alexandria, in: Historia Matematica, New York, 1993, Vol.20, No.2, 180-192</ref>

===On polygonal numbers===
This work on [[polygonal number]]s, a topic that was of great interest to the [[Pythagoreans]] consists of a preface and five propositions in its extant form.{{sfn|Christianidis|Oaks|2023|p=10-11}} The treatise breaks off in the middle of a proposition about how many ways a number can be a polygonal number.{{sfn|Christianidis|Oaks|2023|p=11}}

===The ''Porisms''===
The ''Porisms'' was a collection of [[Lemma (mathematics)|lemmas]] along with accompanying proofs. Although ''The Porisms'' is lost, we know three lemmas contained there, since Diophantus quotes them in the ''Arithmetica'' and refers the reader to the ''Porisms'' for the proof.{{sfn|Christianidis|Oaks|2023|p=15}} 

One lemma states that the difference of the cubes of two rational numbers is equal to the sum of the cubes of two other rational numbers, i.e. given any {{math|''a''}} and {{math|''b''}}, with {{math|''a'' > ''b''}}, there exist {{math|''c'' and ''d''}}, all positive and rational, such that

:{{math|''a''{{sup|3}} − ''b''{{sup|3}} {{=}} ''c''{{sup|3}} + ''d''{{sup|3}}}}.

===''On Parts''===
This work, on [[fractions]], is known by a single reference, a [[Neoplatonic]] [[scholia|scholium]] to [[Iamblichus]]' treatise on [[Nicomachus]]' ''[[Introduction to Arithmetic]]''.{{sfn|Christianidis|Oaks|2023|p=15}} Next to a line where Iamblichus writes "Some of the Pythagoreans said that the [[Unit (number)|unit]] is the borderline between number and parts" the scholiast writes "So Diophantus writes in ''On Parts'', for parts involve progress in diminution carried to infinity."{{sfn|Christianidis|Oaks|2023|p=15}}

==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>

==Notes==
{{Reflist|60em}}

==Editions and translations==
* Bachet de Méziriac, C.G. ''Diophanti Alexandrini Arithmeticorum libri sex et De numeris multangulis liber unus''. Paris: Lutetiae, 1621.
* Diophantus Alexandrinus, Pierre de Fermat, Claude Gaspard Bachet de Meziriac, ''Diophanti Alexandrini Arithmeticorum libri 6, et De numeris multangulis liber unus''. Cum comm. C(laude) G(aspar) Bacheti et observationibus P(ierre) de Fermat. Acc. doctrinae analyticae inventum novum, coll. ex variis eiu. Tolosae 1670, {{doi|10.3931/e-rara-9423}}.
* Tannery, P. L. ''Diophanti Alexandrini Opera omnia: cum Graecis commentariis'', Lipsiae: In aedibus B.G. Teubneri, 1893-1895 (online: [https://archive.org/details/diophantialexan03plangoog vol. 1], [https://archive.org/details/diophantialexan00plangoog vol. 2])
* Sesiano, Jacques. ''The Arabic text of Books IV to VII of Diophantus’ translation and commentary''. Thesis. Providence: Brown University, 1975.
* {{cite book |last1=Sesiano |first1=Jacques |title=Books IV to VII of Diophantus’ Arithmetica: in the Arabic Translation Attributed to Qustā ibn Lūqā |date=6 December 2012 |publisher=Springer Science & Business Media |isbn=978-1-4613-8174-7 |url=https://books.google.com/books?id=G93TBwAAQBAJ |access-date=3 May 2025 |language=en}}
* {{cite book |last1=Christianidis |first1=Jean |last2=Oaks |first2=Jeffrey A. |title=The Arithmetica of Diophantus: a complete translation and commentary |date=2023 |publisher=Routledge |location=Abingdon, Oxon New York, NY |isbn=1138046353}}

==References==
* {{Cite journal |last1=Christianidis |first1=Jean |last2=Oaks |first2=Jeffrey  |year=2013 |title=Practicing algebra in late antiquity: The problem-solving of Diophantus of Alexandria |journal=Historia Mathematica |volume=40 |issue=2 |pages=158–160 |doi=10.1016/j.hm.2012.09.001|doi-access=free }}
* {{Cite journal |last=Christianidis |first=Jean |last2=Megremi |first2=Athanasia |date=2019 |title=Tracing the early history of algebra: Testimonies on Diophantus in the Greek-speaking world (4th–7th century CE) |url=https://linkinghub.elsevier.com/retrieve/pii/S0315086018301411 |journal=Historia Mathematica |language=en |volume=47 |pages=16–38 |doi=10.1016/j.hm.2019.02.002|doi-access=free}}
* {{cite book|last=Cooke|first=Roger|title=The History of Mathematics: A Brief Course|publisher=Wiley-Interscience|year=1997|isbn=0-471-18082-3}}
* {{cite book|last=Derbyshire|first=John|author-link=John Derbyshire|title=Unknown Quantity: A Real And Imaginary History of Algebra|publisher=Joseph Henry Press|year=2006|isbn=0-309-09657-X}}

==Further reading==
* Allard, A. "Les scolies aux arithmétiques de Diophante d'Alexandrie dans le Matritensis Bibl.Nat.4678 et les Vatican Gr.191 et 304" ''Byzantion'' 53. Brussels, 1983: 682–710.
* Christianidis, J. "Maxime Planude sur le sens du terme diophantien "plasmatikon"", ''Historia Scientiarum'', 6 (1996)37-41.
* Christianidis, J. "Une interpretation byzantine de Diophante", ''Historia Mathematica'', 25 (1998) 22–28.
* {{cite book |last1=Katz |first1=Victor J. |last2=Parshall |first2=Karen Hunger |title=Taming the Unknown: A History of Algebra from Antiquity to the Early Twentieth Century |date=2014 |publisher=Princeton University Press |isbn=978-0-691-14905-9}}
* Rashed, Roshdi, Houzel, Christian. ''Les Arithmétiques de Diophante : Lecture historique et mathématique'', Berlin, New York : Walter de Gruyter, 2013.
* Rashed, Roshdi, ''Histoire de l’analyse diophantienne classique : D’Abū Kāmil à Fermat'', Berlin, New York : Walter de Gruyter.
* Rashed, Roshdi. ''L’Art de l’Algèbre de Diophante''. éd. arabe. Le Caire : Bibliothèque Nationale, 1975.
* Rashed, Roshdi. ''Diophante. Les Arithmétiques''. Volume III: Book IV; Volume IV: Books V–VII, app., index. Collection des Universités de France. Paris (Société d’Édition "Les Belles Lettres"), 1984.

==External links==
{{commonscat}}
{{wikiquote}}
{{EB1911 poster|Diophantus}}
* {{MacTutor Biography|id=Diophantus}}
* [http://mathworld.wolfram.com/DiophantussRiddle.html Diophantus's Riddle] Diophantus' epitaph, by E. Weisstein
* Norbert Schappacher (2005). [http://irma.math.unistra.fr/~schappa/NSch/Publications_files/1998cBis_Dioph.pdf Diophantus of Alexandria : a Text and its History].
* [http://www.jphogendijk.nl/reviews/sesiano.html Review of Sesiano's Diophantus] Review of J. Sesiano, Books IV to VII of Diophantus' Arithmetica, by Jan P. Hogendijk
* [http://echo.mpiwg-berlin.mpg.de/ECHOdocuViewfull?mode=imagepath&url=/mpiwg/online/permanent/library/W770Y3H9/pageimg&viewMode=images Latin translation from 1575] by [[Wilhelm Xylander]]
* [https://wilbourhall.org/index.html#apollonius Scans of Tannery's edition of Diophantus] at wilbourhall.org

{{Ancient Greek mathematics}}
{{Authority control}}

[[Category:3rd-century births]]
[[Category:3rd-century deaths]]
[[Category:3rd-century Greek writers]]
[[Category:3rd-century Egyptian people]]
[[Category:Roman-era Alexandrians]]
[[Category:Ancient Greek mathematicians|Diophantus of Alexandria]]
[[Category:Egyptian mathematicians]]
[[Category:Number theorists|Diophantus of Alexandria]]
[[Category:3rd-century writers]]
[[Category:3rd-century mathematicians]]