Диофант — разлика између измена

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Верзија на датум 3. септембар 2022. у 23:34

Насловна страница оригиналног издања из 1621. латинског превода Клода Гаспара Башеа де Мезиријака из Диофантове *Аритметике*

Диофант из Александрије (стгрч. Διόφαντος ὁ Ἀλεξανδρεύς; живео око 250. не), грчки математичар; открио Диофантове једначине.

Мада је био истакнути математичар свог времена, врло мало се зна о Диофантовом животу. Његов рад је сачуван у шест поглавља Аритметике која су доспела до нас (седам поглавља је изгубљено), која је била вероватно најстарији систематски трактат о алгебри. Понекад се Диофант назива оцем алгебре.^[1] Диофант се првенствено интересовао за теорију бројева и решавање једначина и много допринео напретку алгебре употребом симбола за величине, математичке операције и односе, пре тога су ове величине бивале описиване речима. Можда је најпознатији по свом открићу Диофантових једначина, неодређених једначина с рационалним коефицијентима за које се тражи рационално решење.^[2]

Постоји задатак који као решење даје број Диофантових година, а то је уписано на његовом гробу:

„

Шестину века свог као дете се играо,

и још половину шестине с брадом момачком је дочекао.

Усрећи се женом кад превали још седмину,

с којом после лета пет обрадова се сину.

Вољени син поживи пола очевог века само,

и би стргнут оцу судбом која га узе рано.

Два пута два лета оплакиваше родитељ тугу и очај

кад и он угледа тегобном животу своме крај.

”

и решење даје број Диофантових година, наводно 84.

Биографија

Little is known about the life of Diophantus. He lived in Alexandria, Egypt, during the Roman era, probably from between AD 200 and 214 to 284 or 298. Diophantus has variously been described by historians as either Greek,^[3]^[4]^[5] or possibly Hellenized Egyptian,^[6] or Hellenized Babylonian,^[7] The last two of these identifications may stem from confusion with the 4th-century rhetorician Diophantus the Arab.^[8] Much of our knowledge of the life of Diophantus is derived from a 5th-century Greek anthology of number games and puzzles created by Metrodorus. One of the problems (sometimes called his epitaph) states:

'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.'

This puzzle implies that Diophantus' age $x$ can be expressed as

x = .mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num,.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0 0.1em}.mw-parser-output .sfrac .den{border-top:1px solid}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}x/6 + x/12 + x/7 + 5 + x/2 + 4

which gives $x$ a value of 84 years. However, the accuracy of the information cannot be independently confirmed.

In popular culture, this puzzle was the Puzzle No.142 in Professor Layton and Pandora's Box as one of the hardest solving puzzles in the game, which needed to be unlocked by solving other puzzles first.

Аритметика

Аритметика is the major work of Diophantus and the most prominent work on algebra in Greek mathematics. It is a collection of problems giving numerical solutions of both determinate and indeterminate equations. Of the original thirteen books of which Arithmetica consisted only six have survived, though there are some who believe that four Arabic books discovered in 1968 are also by Diophantus.^[9] Some Diophantine problems from Arithmetica have been found in Arabic sources.

It should be mentioned here that Diophantus never used general methods in his solutions. Hermann Hankel, renowned German mathematician made the following remark regarding Diophantus.

“Our author (Diophantos) not the slightest trace of a general, comprehensive method is discernible; each problem calls for some special method which refuses to work even for the most closely related problems. For this reason it is difficult for the modern scholar to solve the 101st problem even after having studied 100 of Diophantos’s solutions”.^[10]

Други радови

Diophantus wrote several other books besides Arithmetica, but very few of them have survived.

The Porisms

Diophantus himself refers to a work which consists of a collection of lemmas called The Porisms (or Porismata), but this book is entirely lost.^[11]

Although The Porisms is lost, we know three lemmas contained there, since Diophantus refers to them in the Arithmetica. 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 $a$ and $b$ , with $a > b$ , there exist $c and d$ , all positive and rational, such that

a 3 - b 3 = c 3 + d 3

.

Polygonal numbers and geometric elements

Diophantus is also known to have written on polygonal numbers, a topic of great interest to Pythagoras and Pythagoreans. Fragments of a book dealing with polygonal numbers are extant.^[12]

A book called Preliminaries to the Geometric Elements has been traditionally attributed to Hero of Alexandria. It has been studied recently by Wilbur Knorr, who suggested that the attribution to Hero is incorrect, and that the true author is Diophantus.^[13]

Референце

^ „HELLENISTIC MATHEMATICS - DIOPHANTUS”. The Story of Mathematics. Приступљено 16. 1. 2020.
^ Katz, Mikhail G.; Schaps, David; Shnider, Steve (2013), „Almost Equal: The Method of Adequality from Diophantus to Fermat and Beyond”, Perspectives on Science, 21 (3): 283—324, Bibcode:2012arXiv1210.7750K, S2CID 57569974, arXiv:1210.7750 , doi:10.1162/POSC_a_00101
^ Research Machines plc. (2004). The Hutchinson dictionary of scientific biography. Abingdon, Oxon: Helicon Publishing. стр. 312. „Diophantus (lived c. A.D. 270-280) Greek mathematician who, in solving linear mathematical problems, developed an early form of algebra.”
^ Boyer, Carl B. (1991). „Revival and Decline of Greek Mathematics”. A History of Mathematics (Second изд.). John Wiley & Sons, Inc. стр. 178. ISBN 0-471-54397-7. „At the beginning of this period, also known as the Later Alexandrian Age, we find the leading Greek algebraist, Diophantus of Alexandria, and toward its close there appeared the last significant Greek geometer, Pappus of Alexandria.”
^ Cooke, Roger (1997). „The Nature of Mathematics”. The History of Mathematics: A Brief Course. Wiley-Interscience. стр. 7. ISBN 0-471-18082-3. „Some enlargement in the sphere in which symbols were used occurred in the writings of the third-century Greek mathematician Diophantus of Alexandria, but the same defect was present as in the case of Akkadians.”
^ Victor J. Katz (1998). A History of Mathematics: An Introduction, p. 184. Addison Wesley, ISBN 0-321-01618-1.
"But what we really want to know is to what extent the Alexandrian mathematicians of the period from the first to the fifth centuries C.E. were Greek. Certainly, all of them wrote in Greek and were part of the Greek intellectual community of Alexandria. And most modern studies conclude that the Greek community coexisted [...] So should we assume that Ptolemy and Diophantus, Pappus and Hypatia were ethnically Greek, that their ancestors had come from Greece at some point in the past but had remained effectively isolated from the Egyptians? It is, of course, impossible to answer this question definitively. But research in papyri dating from the early centuries of the common era demonstrates that a significant amount of intermarriage took place between the Greek and Egyptian communities [...] And it is known that Greek marriage contracts increasingly came to resemble Egyptian ones. In addition, even from the founding of Alexandria, small numbers of Egyptians were admitted to the privileged classes in the city to fulfill numerous civic roles. Of course, it was essential in such cases for the Egyptians to become "Hellenized," to adopt Greek habits and the Greek language. Given that the Alexandrian mathematicians mentioned here were active several hundred years after the founding of the city, it would seem at least equally possible that they were ethnically Egyptian as that they remained ethnically Greek. In any case, it is unreasonable to portray them with purely European features when no physical descriptions exist."
^ D. M. Burton (1991, 1995). History of Mathematics, Dubuque, IA (Wm.C. Brown Publishers).
"Diophantos was most likely a Hellenized Babylonian."
^ Ad Meskens, Travelling Mathematics: The Fate of Diophantos' Arithmetic (Springer, 2010), p. 48 n28.
^ J. Sesiano (1982). Books IV to VII of Diophantus' Arithmetica in the Arabic Translation Attributed to Qusta ibn Luqa. New York/Heidelberg/Berlin: Springer-Verlag. стр. 502.
^ Hankel H., “Geschichte der mathematic im altertum und mittelalter, Leipzig, 1874. (translated to English by Ulrich Lirecht in Chinese Mathematics in the thirteenth century, Dover publications, New York, 1973.
^ G. J. Toomer; Reviel Netz. „Diophantus”. Ур.: Simon Hornblower; Anthony Spawforth; Esther Eidinow. Oxford Classical Dictionary (4th изд.).
^ „Diophantus biography”. www-history.mcs.st-and.ac.uk. Приступљено 10. 4. 2018.
^ Knorr, Wilbur: Arithmêtike stoicheiôsis: On Diophantus and Hero of Alexandria, in: Historia Matematica, New York, 1993, Vol.20, No.2, 180-192

Литература

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.
Bachet de Méziriac, C.G. Diophanti Alexandrini Arithmeticorum libri sex et De numeris multangulis liber unus. Paris: Lutetiae, 1621.
Bashmakova, Izabella G. Diophantos. Arithmetica and the Book of Polygonal Numbers. Introduction and Commentary Translation by I.N. Veselovsky. Moscow: Nauka [in Russian].
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.
Czwalina, Arthur. Arithmetik des Diophantos von Alexandria. Göttingen, 1952.
Heath, Sir Thomas, Diophantos of Alexandria: A Study in the History of Greek Algebra, Cambridge: Cambridge University Press, 1885, 1910.
Robinson, D. C. and Luke Hodgkin. History of Mathematics, King's College London, 2003.
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.
Sesiano, Jacques. The Arabic text of Books IV to VII of Diophantus’ translation and commentary. Thesis. Providence: Brown University, 1975.
Sesiano, Jacques. Books IV to VII of Diophantus’ Arithmetica in the Arabic translation attributed to Qusṭā ibn Lūqā, Heidelberg: Springer-Verlag, 1982. ISBN 0-387-90690-8, doi:10.1007/978-1-4613-8174-7.
Σταμάτης, Ευάγγελος Σ. Διοφάντου Αριθμητικά. Η άλγεβρα των αρχαίων Ελλήνων. Αρχαίον κείμενον – μετάφρασις – επεξηγήσεις. Αθήναι, Οργανισμός Εκδόσεως Διδακτικών Βιβλίων, 1963.
Tannery, P. L. Diophanti Alexandrini Opera omnia: cum Graecis commentariis, Lipsiae: In aedibus B.G. Teubneri, 1893-1895 (online: vol. 1, vol. 2)
Ver Eecke, P. Diophante d’Alexandrie: Les Six Livres Arithmétiques et le Livre des Nombres Polygones, Bruges: Desclée, De Brouwer, 1921.
Wertheim, G. Die Arithmetik und die Schrift über Polygonalzahlen des Diophantus von Alexandria. Übersetzt und mit Anmerkungen von G. Wertheim. Leipzig, 1890.
Bashmakova, Izabella G. "Diophante et Fermat," Revue d'Histoire des Sciences 19 (1966), pp. 289-306
Bashmakova, Izabella G. Diophantus and Diophantine Equations. Moscow: Nauka 1972 [in Russian]. German translation: Diophant und diophantische Gleichungen. Birkhauser, Basel/ Stuttgart, 1974. English translation: Diophantus and Diophantine Equations. Translated by Abe Shenitzer with the editorial assistance of Hardy Grant and updated by Joseph Silverman. The Dolciani Mathematical Expositions, 20. Mathematical Association of America, Washington, DC. 1997.
Bashmakova, Izabella G. “Arithmetic of Algebraic Curves from Diophantus to Poincaré,” Historia Mathematica 8 (1981), 393–416.
Bashmakova, Izabella G., Slavutin, E.I. History of Diophantine Analysis from Diophantus to Fermat. Moscow: Nauka 1984 [in Russian].
Heath, Sir Thomas (1981). A history of Greek mathematics. 2. Cambridge University Press: Cambridge.
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.
Vogel, Kurt (1970). „Diophantus of Alexandria”. Dictionary of Scientific Biography. 4. New York: Scribner.

Спољашње везе

O'Connor, John J.; Robertson, Edmund F. „Диофант”. MacTutor History of Mathematics archive. University of St Andrews.
O'Connor, John J.; Robertson, Edmund F. „Диофант”. MacTutor History of Mathematics archive. University of St Andrews.
Diophantus's Riddle Diophantus' epitaph, by E. Weisstein
Norbert Schappacher (2005). Diophantus of Alexandria : a Text and its History.
Review of Sesiano's Diophantus Review of J. Sesiano, Books IV to VII of Diophantus' Arithmetica, by Jan P. Hogendijk
Latin translation from 1575 by Wilhelm Xylander

[1] „HELLENISTIC MATHEMATICS - DIOPHANTUS”. The Story of Mathematics. Приступљено 16. 1. 2020.

[2] Katz, Mikhail G.; Schaps, David; Shnider, Steve (2013), „Almost Equal: The Method of Adequality from Diophantus to Fermat and Beyond”, Perspectives on Science, 21 (3): 283—324, Bibcode:2012arXiv1210.7750K, S2CID 57569974, arXiv:1210.7750 , doi:10.1162/POSC_a_00101

[Dictionary_of_Scientific_Biography-3] Research Machines plc. (2004). The Hutchinson dictionary of scientific biography. Abingdon, Oxon: Helicon Publishing. стр. 312. „Diophantus (lived c. A.D. 270-280) Greek mathematician who, in solving linear mathematical problems, developed an early form of algebra.”

[4] Boyer, Carl B. (1991). „Revival and Decline of Greek Mathematics”. A History of Mathematics (Second изд.). John Wiley & Sons, Inc. стр. 178. ISBN 0-471-54397-7. „At the beginning of this period, also known as the Later Alexandrian Age, we find the leading Greek algebraist, Diophantus of Alexandria, and toward its close there appeared the last significant Greek geometer, Pappus of Alexandria.”

[5] Cooke, Roger (1997). „The Nature of Mathematics”. The History of Mathematics: A Brief Course. Wiley-Interscience. стр. 7. ISBN 0-471-18082-3. „Some enlargement in the sphere in which symbols were used occurred in the writings of the third-century Greek mathematician Diophantus of Alexandria, but the same defect was present as in the case of Akkadians.”

[Katz_184-6] Victor J. Katz (1998). A History of Mathematics: An Introduction, p. 184. Addison Wesley, ISBN 0-321-01618-1.
"But what we really want to know is to what extent the Alexandrian mathematicians of the period from the first to the fifth centuries C.E. were Greek. Certainly, all of them wrote in Greek and were part of the Greek intellectual community of Alexandria. And most modern studies conclude that the Greek community coexisted [...] So should we assume that Ptolemy and Diophantus, Pappus and Hypatia were ethnically Greek, that their ancestors had come from Greece at some point in the past but had remained effectively isolated from the Egyptians? It is, of course, impossible to answer this question definitively. But research in papyri dating from the early centuries of the common era demonstrates that a significant amount of intermarriage took place between the Greek and Egyptian communities [...] And it is known that Greek marriage contracts increasingly came to resemble Egyptian ones. In addition, even from the founding of Alexandria, small numbers of Egyptians were admitted to the privileged classes in the city to fulfill numerous civic roles. Of course, it was essential in such cases for the Egyptians to become "Hellenized," to adopt Greek habits and the Greek language. Given that the Alexandrian mathematicians mentioned here were active several hundred years after the founding of the city, it would seem at least equally possible that they were ethnically Egyptian as that they remained ethnically Greek. In any case, it is unreasonable to portray them with purely European features when no physical descriptions exist."

[7] D. M. Burton (1991, 1995). History of Mathematics, Dubuque, IA (Wm.C. Brown Publishers).
"Diophantos was most likely a Hellenized Babylonian."

[AM-8] Ad Meskens, Travelling Mathematics: The Fate of Diophantos' Arithmetic (Springer, 2010), p. 48 n28.

[Books_IV-VII_of_Diophantus'_Arithmetica-9] J. Sesiano (1982). Books IV to VII of Diophantus' Arithmetica in the Arabic Translation Attributed to Qusta ibn Luqa. New York/Heidelberg/Berlin: Springer-Verlag. стр. 502.

[10] Hankel H., “Geschichte der mathematic im altertum und mittelalter, Leipzig, 1874. (translated to English by Ulrich Lirecht in Chinese Mathematics in the thirteenth century, Dover publications, New York, 1973.

[11] G. J. Toomer; Reviel Netz. „Diophantus”. Ур.: Simon Hornblower; Anthony Spawforth; Esther Eidinow. Oxford Classical Dictionary (4th изд.).

[12] „Diophantus biography”. www-history.mcs.st-and.ac.uk. Приступљено 10. 4. 2018.

[13] Knorr, Wilbur: Arithmêtike stoicheiôsis: On Diophantus and Hero of Alexandria, in: Historia Matematica, New York, 1993, Vol.20, No.2, 180-192

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@@ Ред 1: / Ред 1: @@
+{{Short description|Greek mathematician (3rd century AD)}}
-'''Диофант из Александрије''' (живео око 250. не), грчки [[математика|математичар]]; открио [[Diofantske jednačine|Диофантове једначине]].
+[[Датотека:Diophantus-cover.jpg|right|thumb|200px|Насловна страница оригиналног издања из 1621. латинског превода [[Claude Gaspard Bachet de Méziriac|Клода Гаспара Башеа де Мезиријака]] из Диофантове ''Аритметике'']]
+'''Диофант из Александрије''' ({{lang-grc|Διόφαντος ὁ Ἀλεξανδρεύς}}; живео око 250. не), грчки [[математика|математичар]]; открио [[Diofantske jednačine|Диофантове једначине]].
-Мада је био истакнути математичар свог времена, врло мало се зна о Диофантовом животу. Његов рад је сачуван у шест поглавља [[Аритметика|Аритметике]] која су доспела до нас (седам поглавља је изгубљено), која је била вероватно најстарији систематски трактат о алгебри. Понекад се Диофант назива оцем алгебре.<ref>{{cite web |title=HELLENISTIC MATHEMATICS - DIOPHANTUS |url=https://www.storyofmathematics.com/hellenistic_diophantus.html |website=The Story of Mathematics |accessdate=16. 1. 2020}}</ref>
-Диофант се првенствено интересовао за теорију бројева и решавање једначина и много допринео напретку алгебре употребом симбола за величине, математичке операције и односе, пре тога су ове величине бивале описиване речима.
+Мада је био истакнути математичар свог времена, врло мало се зна о Диофантовом животу. Његов рад је сачуван у шест поглавља [[Аритметика|Аритметике]] која су доспела до нас (седам поглавља је изгубљено), која је била вероватно најстарији систематски трактат о алгебри. Понекад се Диофант назива оцем алгебре.<ref>{{cite web |title=HELLENISTIC MATHEMATICS - DIOPHANTUS |url=https://www.storyofmathematics.com/hellenistic_diophantus.html |website=The Story of Mathematics |accessdate=16. 1. 2020}}</ref> Диофант се првенствено интересовао за теорију бројева и решавање једначина и много допринео напретку алгебре употребом симбола за величине, математичке операције и односе, пре тога су ове величине бивале описиване речима. Можда је најпознатији по свом открићу Диофантових једначина, неодређених једначина с рационалним коефицијентима за које се тражи рационално решење.<ref>{{citation |last1=Katz |first1=Mikhail G. |author1-link=Mikhail Katz |last2=Schaps |first2=David |last3=Shnider |first3=Steve |author3-link=Steve Shnider |arxiv=1210.7750 |doi= 10.1162/POSC_a_00101|issue=3 |journal=[[Perspectives on Science]] |pages= 283–324|title=Almost Equal: The Method of [[Adequality]] from Diophantus to Fermat and Beyond |volume=21 |year=2013|bibcode=2012arXiv1210.7750K |s2cid=57569974 }}</ref>
-Можда је најпознатији по свом открићу Диофантових једначина, неодређених једначина с рационалним коефицијентима за које се тражи рационално решење.
 Постоји задатак који као решење даје број Диофантових година, а то је уписано на његовом гробу:
+{{цитат 2|
+: Шестину века свог као дете се играо,
+: и још половину шестине с брадом момачком је дочекао.
+: Усрећи се женом кад превали још седмину,
+: с којом после лета пет обрадова се сину.
+: Вољени син поживи пола очевог века само,
+: и би стргнут оцу судбом која га узе рано.
+: Два пута два лета оплакиваше родитељ тугу и очај
+: кад и он угледа тегобном животу своме крај.}}
+и решење даје број Диофантових година, наводно 84.
-{{цитат 2|Шестину века свог као дете се играо,
+== Биографија ==
-и још половину шестине с брадом момачком је дочекао.
+{{рут}}
+Little is known about the life of Diophantus. He lived in [[Alexandria]], [[Egypt]], during the [[Roman Egypt|Roman era]], probably from between AD 200 and 214 to 284 or 298. Diophantus has variously been described by historians as either [[Greeks|Greek]],<ref name="Dictionary of Scientific Biography">{{cite book | title=The Hutchinson dictionary of scientific biography | author = Research Machines plc. | location = Abingdon, Oxon | publisher = Helicon Publishing | year = 2004 | page = 312 | quote = '''Diophantus (lived ''c.'' A.D. 270-280)''' Greek mathematician who, in solving linear mathematical problems, developed an early form of algebra.}}</ref><ref>{{cite book|first=Carl B.|last=Boyer|author-link=Carl Benjamin Boyer|title=A History of Mathematics|edition=Second|publisher=John Wiley & Sons, Inc.|year=1991|chapter=Revival and Decline of Greek Mathematics|page=[https://archive.org/details/historyofmathema00boye/page/178 178]|isbn=0-471-54397-7|quote=At the beginning of this period, also known as the Later [[Alexandrian Age]], we find the leading Greek algebraist, Diophantus of Alexandria, and toward its close there appeared the last significant Greek geometer, Pappus of Alexandria.|chapter-url=https://archive.org/details/historyofmathema00boye/page/178}}</ref><ref>{{cite book|first=Roger|last=Cooke|author-link=Roger Cooke (mathematician)|title=The History of Mathematics: A Brief Course|publisher=Wiley-Interscience|year=1997|chapter=The Nature of Mathematics|page=[https://archive.org/details/historyofmathema0000cook/page/7 7]|isbn=0-471-18082-3|quote=Some enlargement in the sphere in which symbols were used occurred in the writings of the third-century Greek mathematician Diophantus of Alexandria, but the same defect was present as in the case of Akkadians.|chapter-url=https://archive.org/details/historyofmathema0000cook/page/7}}</ref> or possibly [[Hellenization|Hellenized]] [[Egyptians|Egyptian]],<ref name="Katz 184">Victor J. Katz (1998). ''A History of Mathematics: An Introduction'', p. 184. Addison Wesley, {{ISBN|0-321-01618-1}}.
+{{quote|"But what we really want to know is to what extent the Alexandrian mathematicians of the period from the first to the fifth centuries C.E. were Greek. Certainly, all of them wrote in Greek and were part of the Greek intellectual community of Alexandria. And most modern studies conclude that the Greek community coexisted [...] So should we assume that [[Ptolemy]] and Diophantus, [[Pappus of Alexandria|Pappus]] and [[Hypatia]] were ethnically Greek, that their ancestors had come from Greece at some point in the past but had remained effectively isolated from the Egyptians? It is, of course, impossible to answer this question definitively. But research in papyri dating from the early centuries of the common era demonstrates that a significant amount of intermarriage took place between the Greek and Egyptian communities [...] And it is known that Greek marriage contracts increasingly came to resemble Egyptian ones. In addition, even from the founding of Alexandria, small numbers of Egyptians were admitted to the privileged classes in the city to fulfill numerous civic roles. Of course, it was essential in such cases for the Egyptians to become "Hellenized," to adopt Greek habits and the Greek language. Given that the Alexandrian mathematicians mentioned here were active several hundred years after the founding of the city, it would seem at least equally possible that they were ethnically Egyptian as that they remained ethnically Greek. In any case, it is unreasonable to portray them with purely European features when no physical descriptions exist."}}</ref> or Hellenized [[Babylonia]]n,<ref>D. M. Burton (1991, 1995). ''History of Mathematics'', Dubuque, IA (Wm.C. Brown Publishers).
+<br>{{quote|"Diophantos was most likely a Hellenized Babylonian."}}</ref> The last two of these identifications may stem from confusion with the 4th-century rhetorician [[Diophantus the Arab]].<ref name=AM>Ad Meskens, ''Travelling Mathematics: The Fate of Diophantos' Arithmetic'' (Springer, 2010), p. 48 n28.</ref> Much of our knowledge of the life of Diophantus is derived from a 5th-century [[Greek language|Greek]] anthology of number games and puzzles created by [[Metrodorus (grammarian)|Metrodorus]]. One of the problems (sometimes called his epitaph) states:
+:'Here lies Diophantus,' the wonder behold.
-Усрећи се женом кад превали још седмину,
+: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.'
+This puzzle implies that Diophantus' age {{math|''x''}} can be expressed as
-с којом после лета пет обрадова се сину.
+:{{math|''x'' {{=}} {{sfrac|''x''|6}} + {{sfrac|''x''|12}} + {{sfrac|''x''|7}} + 5 + {{sfrac|''x''|2}} + 4}}
-Вољени син поживи пола очевог века само,
+which gives {{math|''x''}} a value of 84 years. However, the accuracy of the information cannot be independently confirmed.
-и би стргнут оцу судбом која га узе рано.
+In popular culture, this puzzle was the Puzzle No.142 in ''[[Professor Layton and Pandora's Box]]'' as one of the hardest solving puzzles in the game, which needed to be unlocked by solving other puzzles first.
-Два пута два лета оплакиваше родитељ тугу и очај
+== ''Аритметика'' ==
-кад и он угледа тегобном животу своме крај.}}
+{{see also|Аритметика}}
+''Аритметика'' is the major work of Diophantus and the most prominent work on algebra in Greek mathematics. It is a collection of problems giving numerical solutions of both determinate and indeterminate [[equation]]s. Of the original thirteen books of which ''Arithmetica'' consisted only six have survived, though there are some who believe that four Arabic books discovered in 1968 are also by Diophantus.<ref name="Books IV-VII of Diophantus' Arithmetica">{{cite book | title=Books IV to VII of Diophantus' ''Arithmetica'' in the Arabic Translation Attributed to Qusta ibn Luqa | author = J. Sesiano | location = New York/Heidelberg/Berlin | publisher = Springer-Verlag | year = 1982 | page = 502}}</ref> Some Diophantine problems from ''Arithmetica'' have been found in Arabic sources.
-и решење даје број Диофантових година, наводно 84.
+It should be mentioned here that Diophantus never used general methods in his solutions. [[Hermann Hankel]], renowned German mathematician made the following remark regarding Diophantus.
+“Our author (Diophantos) not the slightest trace of a general, comprehensive method is discernible; each problem calls for some special method which refuses to work even for the most closely related problems.  For this reason it is difficult for the modern scholar to solve the 101st problem even after having studied 100 of Diophantos’s solutions”.<ref>Hankel H., “Geschichte der mathematic im altertum und mittelalter, Leipzig, 1874. (translated to English by Ulrich Lirecht in Chinese Mathematics in the thirteenth century, Dover publications, New York, 1973.</ref>
+== Други радови ==
+Diophantus wrote several other books besides ''Arithmetica'', but very few of them have survived.
+===The ''Porisms''===
+Diophantus himself refers to a work which consists of a collection of [[Lemma (mathematics)|lemmas]] called ''The Porisms'' (or ''Porismata''), but this book is entirely lost.<ref>{{cite encyclopedia|title=Oxford Classical Dictionary|entry=Diophantus |url=https://www.oxfordreference.com/view/10.1093/acref/9780199545568.001.0001/acref-9780199545568-e-2229?rskey=YDxfom&result=2|author1=G. J. Toomer|author2=Reviel Netz|edition=4th|editor1=Simon Hornblower|editor2=Anthony Spawforth|editor3=Esther Eidinow}}</ref>
+Although ''The Porisms'' is lost, we know three lemmas contained there, since Diophantus refers to them in the ''Arithmetica''. 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}}}}.
+===Polygonal numbers and geometric elements===
+Diophantus is also known to have written on [[polygonal number]]s, a topic of great interest to [[Pythagoras]] and [[Pythagoreans]]. Fragments of a book dealing with polygonal numbers are extant.<ref>{{cite web|url=http://www-history.mcs.st-and.ac.uk/Biographies/Diophantus.html|title=Diophantus biography|website=www-history.mcs.st-and.ac.uk|access-date=10 April 2018}}</ref>
+A book called ''Preliminaries to the Geometric Elements'' has been traditionally attributed to [[Hero of Alexandria]]. It has been studied recently by [[Wilbur Knorr]], who suggested that the attribution to Hero is incorrect, and that the true author is 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>
 == Референце ==
 {{reflist}}
+== Литература ==
+{{refbegin|30em}}
+* 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.
+* Bachet de Méziriac, C.G. ''Diophanti Alexandrini Arithmeticorum libri sex et De numeris multangulis liber unus''. Paris: Lutetiae, 1621.
+* Bashmakova, Izabella G. ''Diophantos. Arithmetica and the Book of Polygonal Numbers. Introduction and Commentary'' Translation by I.N. Veselovsky. Moscow: Nauka [in Russian].
+* 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.
+* Czwalina, Arthur. ''Arithmetik des Diophantos von Alexandria''. Göttingen, 1952.
+* [[T. L. Heath|Heath, Sir Thomas]], ''Diophantos of Alexandria: A Study in the History of Greek Algebra'', Cambridge: Cambridge University Press, 1885, 1910.
+* Robinson, D. C. and Luke Hodgkin. ''History of Mathematics'', [[King's College London]], 2003.
+* 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.
+* Sesiano, Jacques. ''The Arabic text of Books IV to VII of Diophantus’ translation and commentary''. Thesis. Providence: Brown University, 1975.
+* Sesiano, Jacques. ''Books IV to VII of Diophantus’ Arithmetica in the Arabic translation attributed to Qusṭā ibn Lūqā'',  Heidelberg: Springer-Verlag, 1982. {{isbn|0-387-90690-8}}, {{DOI|10.1007/978-1-4613-8174-7}}.
+* Σταμάτης, Ευάγγελος Σ. ''Διοφάντου Αριθμητικά. Η άλγεβρα των αρχαίων Ελλήνων. Αρχαίον κείμενον – μετάφρασις – επεξηγήσεις''. Αθήναι, Οργανισμός Εκδόσεως Διδακτικών Βιβλίων, 1963.
+* 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])
+* Ver Eecke, P. ''Diophante d’Alexandrie: Les Six Livres Arithmétiques et le Livre des Nombres Polygones'', Bruges: Desclée, De Brouwer, 1921.
+* Wertheim, G. ''Die Arithmetik und die Schrift über Polygonalzahlen des Diophantus von Alexandria''. Übersetzt und mit Anmerkungen von G. Wertheim. Leipzig, 1890.
+* Bashmakova, Izabella G. "Diophante et Fermat," ''Revue d'Histoire des Sciences'' 19 (1966), pp. 289-306
+* Bashmakova, Izabella G. ''[[Diophantus and Diophantine Equations]]''. Moscow: Nauka 1972 [in Russian]. German translation: ''Diophant und diophantische Gleichungen''. Birkhauser, Basel/ Stuttgart, 1974. English translation: ''Diophantus and Diophantine Equations''. Translated by Abe Shenitzer with the editorial assistance of Hardy Grant and updated by Joseph Silverman. The Dolciani Mathematical Expositions, 20. Mathematical Association of America, Washington, DC. 1997.
+* Bashmakova, Izabella G. “Arithmetic of Algebraic Curves from Diophantus to Poincaré,” ''Historia Mathematica'' 8 (1981), 393–416.
+* Bashmakova, Izabella G., Slavutin, E.I. ''History of Diophantine Analysis from Diophantus to Fermat''. Moscow: Nauka 1984 [in Russian].
+* {{cite book|last=Heath|first=Sir Thomas|title=A history of Greek mathematics|url=https://archive.org/details/ahistorygreekma00heatgoog|year=1981|publisher=Cambridge |location=Cambridge University Press|volume=2}}
+* 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.
+* {{cite encyclopedia|last=Vogel|first=Kurt|title=Diophantus of Alexandria|encyclopedia=Dictionary of Scientific Biography|volume=4|publisher=Scribner|location=New York|year=1970}}
+{{refend}}
 == Спољашње везе ==
+{{Commons category|Diophantus}}
-{{портал|Античка Грчка}}
+* {{MacTutor Biography|id=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]]
 {{Грчка математика}}
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