Diofant

Diofant iz Aleksandrije (stgrč. Διόφαντος ὁ Ἀλεξανδρεύς; živeo oko 250. ne), grčki matematičar; otkrio Diofantove jednačine.

Mada je bio istaknuti matematičar svog vremena, vrlo malo se zna o Diofantovom životu. Njegov rad je sačuvan u šest poglavlja Aritmetike koja su dospela do nas (sedam poglavlja je izgubljeno), koja je bila verovatno najstariji sistematski traktat o algebri. Ponekad se Diofant naziva ocem algebre.^[1] Diofant se prvenstveno interesovao za teoriju brojeva i rešavanje jednačina i mnogo doprineo napretku algebre upotrebom simbola za veličine, matematičke operacije i odnose, pre toga su ove veličine bivale opisivane rečima. Možda je najpoznatiji po svom otkriću Diofantovih jednačina, neodređenih jednačina s racionalnim koeficijentima za koje se traži racionalno rešenje.^[2] Postoji zadatak koji kao rešenje daje broj Diofantovih godina, a to je upisano na njegovom grobu.

Biografija

Malo se zna o životu Diofanta. Živeo je u Aleksandriji, Egipat, tokom rimske ere, verovatno između 200. i 214. do 284. ili 298. godine. Istoričari su Diofanta na različite načine opisali ili kao Grka,^[3]^[4]^[5] ili možda helenizovanog Egipćana,^[6] ili helenizovanog Vavilonca,^[7] Poslednje dve od ovih identifikacija mogu proizaći iz zabune sa retoričarom iz 4. veka Diofantom Arapom.^[8] Veliki deo našeg znanja o Diofantovom životu potiče iz grčke antologije igara brojeva i zagonetki iz 5. veka koju je stvorio Metrodor. Jedan od problema (koji se ponekad naziva i njegov epitaf) glasi:

„

Šestinu veka svog kao dete se igrao,

i još polovinu šestine s bradom momačkom je dočekao.

Usreći se ženom kad prevali još sedminu,

s kojom posle leta pet obradova se sinu.

Voljeni sin poživi pola očevog veka samo,

i bi strgnut ocu sudbom koja ga uze rano.

Dva puta dva leta oplakivaše roditelj tugu i očaj

kad i on ugleda tegobnom životu svome kraj.

”

Ova zagonetka implicira da se $x$ Diofantovog doba može izraziti kao

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

što daje $x$ vrednost od 84 godine. Međutim, tačnost informacija ne može se nezavisno potvrditi.

U popularnoj kulturi, ova zagonetka je bila Slagalica br. 142 u Profesoru Lejtonu i Pandorinoj kutiji kao jedna od najtežih rešavanja zagonetki u igri, koju je trebalo otključati rešavanjem drugih zagonetki prvo.

Aritmetika

Aritmetika je glavno Diofantovo delo i najistaknutije delo o algebri u grčkoj matematici. To je skup zadataka koji daje numerička rešenja određenih i neodređenih jednačina. Od originalnih trinaest knjiga od kojih se Aritmetika sastojala samo je šest sačuvano, mada postoje neki koji veruju da su četiri arapske knjige otkrivene 1968. takođe Diofantove.^[9] Neki diofantski problemi iz Aritmetike pronađeni su u arapskim izvorima.

Ovde treba napomenuti da Diofant u svojim rešenjima nikada nije koristio opšte metode. Herman Hankel, poznati nemački matematičar, dao je sledeću primedbu u vezi sa Diofantom.

„Naš autor (Diofant) ne daje se ni najmanji trag opšte, sveobuhvatne metode; svaki problem zahteva neki poseban metod koji odbija da radi čak i za najteže povezane probleme. Zbog toga je modernom naučniku teško da reši 101. problem čak i nakon što je proučio 100 Diofantovih rešenja”.^[10]

Drugi radovi

Diofant je napisao još nekoliko knjiga pored Aritmetike, ali je vrlo malo njih preživelo.

Porizmi

Sam Diofant se poziva na delo koje se sastoji od zbirke lema pod nazivom Porizmi (ili Porizmata), ali ova knjiga je u potpunosti izgubljena.^[11]

Iako su Porizmi izgubljeni, poznate su tri leme koje su se tamo nalaze, pošto ih Diofant pominje u Aritmetici. Jedna lema navodi da je razlika kubova dva racionalna broja jednaka zbiru kubova dva druga racionalna broja, tj. za bilo koje $a$ i $b$ , sa $a > b$ , postoje $c$ i $d$ , svi od kojih su pozitivni i racionalni, tako da

a 3 - b 3 = c 3 + d 3

.

Poligonalni brojevi i geometrijski elementi

Poznato je i da je Diofant pisao o poligonalnim brojevima, što je tema od velikog interesa za Pitagoru i Pitagorejce.^[12]^[13] Sačuvani su fragmenti knjige koja se bavi poligonalnim brojevima.^[14]

Knjiga pod nazivom Preliminari geometrijskih elemenata tradicionalno se pripisuje Heronu od Aleksandrije. Nedavno ju je proučavao Vilbur Nor, koji je sugerisao da je pripisivanje Heron netačno i da je pravi autor Diofant.^[15]

Reference

^ „HELLENISTIC MATHEMATICS - DIOPHANTUS”. The Story of Mathematics. Pristupljeno 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. str. 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 izd.). John Wiley & Sons, Inc. str. 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. str. 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. str. 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 (20. 12. 2012). „Diophantus”. Ur.: Simon Hornblower; Anthony Spawforth; Esther Eidinow. Oxford Classical Dictionary (4th izd.). Oxford University Press. ISBN 978-0-19-954556-8. CS1 održavanje: Format datuma (veza)
^ „Archived copy” (PDF). Arhivirano iz originala (PDF) 2011-06-15. g. Pristupljeno 2010-06-13.
^ „Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers” (PDF). Arhivirano iz originala (PDF) 29. 05. 2013. g. Pristupljeno 04. 09. 2022.
^ „Diophantus biography”. www-history.mcs.st-and.ac.uk. Pristupljeno 10. 4. 2018.
^ Knorr, Wilbur: Arithmêtike stoicheiôsis: On Diophantus and Hero of Alexandria. Historia Matematica, New York. 20 (2): 180—192. 1993. Nedostaje ili je prazan parametar |title= (pomoć)

Literatura

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 (1982). Books IV to VII of Diophantus' Arithmetica. Sources in the History of Mathematics and Physical Sciences. 3. ISBN 978-1-4613-8176-1. 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.
Wells, David (4. 9. 1997). The Penguin Dictionary of Curious and Interesting Numbers. Penguin. ISBN 0-14-026149-4. Tekst „pages ” ignorisan (pomoć)CS1 održavanje: Format datuma (veza)
F. Tapson (1999). The Oxford Mathematics Study Dictionary (2nd izd.). Oxford University Press. str. 88–89. ISBN 0-19-914-567-9.

Spoljašnje veze

O'Connor, John J.; Robertson, Edmund F. „Diofant”. MacTutor History of Mathematics archive. University of St Andrews.
O'Connor, John J.; Robertson, Edmund F. „Diofant”. 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
Polygonal numbers at PlanetMath Arhivirano na sajtu Wayback Machine (20. februar 2016)
Weisstein, Eric W. „Polygonal Numbers”. MathWorld.

[1] „HELLENISTIC MATHEMATICS - DIOPHANTUS”. The Story of Mathematics. Pristupljeno 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. str. 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 izd.). John Wiley & Sons, Inc. str. 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. str. 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. str. 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 (20. 12. 2012). „Diophantus”. Ur.: Simon Hornblower; Anthony Spawforth; Esther Eidinow. Oxford Classical Dictionary (4th izd.). Oxford University Press. ISBN 978-0-19-954556-8. CS1 održavanje: Format datuma (veza)

[siam_07-003s-12] „Archived copy” (PDF). Arhivirano iz originala (PDF) 2011-06-15. g. Pristupljeno 2010-06-13.

[13] „Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers” (PDF). Arhivirano iz originala (PDF) 29. 05. 2013. g. Pristupljeno 04. 09. 2022.

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

[15] Knorr, Wilbur: Arithmêtike stoicheiôsis: On Diophantus and Hero of Alexandria. Historia Matematica, New York. 20 (2): 180—192. 1993. Nedostaje ili je prazan parametar |title= (pomoć)

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