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Diophantus

For the general, see Diophantus (general). For the sophist, see Diophantus the Arab. For the intersex soldier, see Diophantus of Abae.

Diophantus of Alexandria[1] (born c. AD 200 – c. 214; died c. AD 284 – c. 298) was a Greek mathematician, who was the author of two main works: On Polygonal Numbers, which survives incomplete, and the Arithmetica in thirteen books, most of it extant, made up of arithmetical problems that are solved through algebraic equations.[2]

His Arithmetica influenced the development of algebra by Arabs, and his equations influenced modern work in both abstract algebra and computer science.[3] The first five books of his work are purely algebraic.[3] Furthermore, recent studies of Diophantus's work have revealed that the method of solution taught in his Arithmetica matches later medieval Arabic algebra in its concepts and overall procedure.[4]

Diophantus was among the earliest mathematicians who recognized positive rational numbers as numbers, by allowing fractions for coefficients and solutions. He coined the term παρισότης (parisotēs) to refer to an approximate equality.[5] 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.

Although not the earliest, the Arithmetica has the best-known use of algebraic notation to solve arithmetical problems coming from Greek antiquity,[6][2] and some of its problems served as inspiration for later mathematicians working in analysis and number theory.[7] In modern use, Diophantine equations are algebraic equations with integer coefficients for which integer solutions are sought. Diophantine geometry and Diophantine approximations are two other subareas of number theory that are named after him.

Biography

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Diophantus was born into a Greek family and is known to have lived in Alexandria, Egypt, during the Roman era, between AD 200 and 214 to 284 or 298.[6][8][9][a] 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 = ⁠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 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.

Arithmetica

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Title page of the Latin translation of Diophantus' Arithmetica by Bachet (1621).
See also: Arithmetica

Arithmetica is the major work of Diophantus and the most prominent work on premodern 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.[14] 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.[15]

History

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Like many other Greek mathematical treatises, Diophantus was forgotten in Western Europe during the 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.[16] In addition, some portion of the Arithmetica probably survived in the Arab tradition (see above). 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 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 Xylander. 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. A later 1895 Latin translation by Paul Tannery was said to be an improvement by Thomas L. Heath, who used it in the 1910 second edition of his English translation.

Margin-writing by Fermat and Chortasmenos

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Problem II.8 in the Arithmetica (edition of 1670), annotated with Fermat's comment which became Fermat's Last Theorem.

The 1621 edition of Arithmetica by Bachet gained fame after Pierre de Fermat wrote his famous "Last Theorem" in the margins of his copy:

If an integer n is greater than 2, then an + bn = cn has no solutions in non-zero integers a, b, and c. I have a truly marvelous proof of this proposition which this margin is too narrow to contain.

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.[16]

Other works

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

The Porisms

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Diophantus himself refers to a work which consists of a collection of lemmas called The Porisms (or Porismata), but this book is entirely lost.[17]

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

a3 − b3 = c3 + d3.

Polygonal numbers and geometric elements

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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.[18]

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.[19]

Influence

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Diophantus' work has had a large influence in history. 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. Diophantus and his works also influenced Arab mathematics and were of great fame among Arab mathematicians. Diophantus' work created a foundation for work on algebra and in fact much of advanced mathematics is based on algebra.[20] How much he affected India is a matter of debate.

Diophantus has been considered "the father of algebra" because of his contributions to number theory, mathematical notations and the earliest known use of syncopated notation in his book series Arithmetica.[2] However this is usually debated, because Al-Khwarizmi was also given the title as "the father of algebra", nevertheless both mathematicians were responsible for paving the way for algebra today.

Diophantine analysis

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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 equations. Diophantus looked at 3 different types of quadratic equations: ax2 + bx = c, ax2 = bx + c, and ax2 + 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 coefficients by considering the given numbers a, b, 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 square root solutions "useless", "meaningless", and even "absurd". To give one specific example, he calls the equation 4 = 4x + 20 'absurd' because it would lead to a negative value for 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 quadratic equations.

Mathematical notation

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See also: Arithmetica § Syncopated algebra, and Syncopated algebra

Diophantus made important advances in mathematical notation, becoming the first person known to use algebraic notation and symbolism. Before him everyone wrote out equations completely. 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. Mathematical historian Kurt Vogel states:[21]

The symbolism that Diophantus introduced for the first time, and undoubtedly devised himself, provided a short and readily comprehensible means of expressing an equation... Since an abbreviation is also employed for the word 'equals', Diophantus took a fundamental step from verbal algebra towards symbolic algebra.

Although Diophantus made important advances in symbolism, he still lacked the necessary notation to express more general methods. This caused his work to be more concerned with particular problems rather than general situations. Some of the limitations of Diophantus' notation are that he only had notation for one unknown and, when problems involved more than a single unknown, Diophantus was reduced to expressing "first unknown", "second unknown", etc. in words. He also lacked a symbol for a general number n. Where we would write ⁠12 + 6n/n2 − 3⁠, Diophantus has to resort to constructions like: "... a sixfold number increased by twelve, which is divided by the difference by which the square of the number exceeds three". Algebra still had a long way to go before very general problems could be written down and solved succinctly.

See also

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Notes

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  1. ^ There have been several fringe theories regarding Diophantus' origins. In modern times, a few authors have described him as possibly being an Arab, a Jew, a Hellenized Egyptian,[10] or a Hellenized Babylonian.[11] Some have even claimed that Diophantus was a convert to Christianity. All of these claims are seen as baseless and speculative.[12][13] These misconceptions about his origin stem due to confusions (e.g. with Diophantus the Arab), conflation of different historical eras, transpositions of mathematical problems into ethnic categories and racialist reasons.[13]

References

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  1. ^ Ancient Greek: Διόφαντος ὁ Ἀλεξανδρεύς, romanized: Diophantos ho Alexandreus
  2. ^ a b c Carl B. Boyer, A History of Mathematics, Second Edition (Wiley, 1991), page 228
  3. ^ a b Hettle, Cyrus (2015). "The Symbolic and Mathematical Influence of Diophantus's Arithmetica". Journal of Humanistic Mathematics. 5 (1): 139–166. doi:10.5642/jhummath.201501.08.
  4. ^ Christianidis, Jean; Megremi, Athanasia (2019). "Tracing the early history of algebra: Testimonies on Diophantus in the Greek-speaking world (4th–7th century CE)". Historia Mathematica. 47: 16–38. doi:10.1016/j.hm.2019.02.002.
  5. ^ 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, arXiv:1210.7750, Bibcode:2012arXiv1210.7750K, doi:10.1162/POSC_a_00101, S2CID 57569974
  6. ^ a b Research Machines plc. (2004). The Hutchinson dictionary of scientific biography. Abingdon, Oxon: Helicon Publishing. p. 312. Diophantus (lived c. A.D. 270-280) Greek mathematician who, in solving linear mathematical problems, developed an early form of algebra.
  7. '^ D. Mary, R. Flamary, C. Theys and C. Aime (2016). Mathematical Tools for Instrumentation & Signal Processing in Astronomy Volume 78-79, 2016. EAS Publications Series. pp. 73–98. Diophantus of Alexandria, a greek mathematician, known as the father of algebra. He studied polynomial equations with integer coefficients and integer solutions, called diophantine equations.{{cite book}}: CS1 maint: multiple names: authors list (link)
  8. ^ Boyer, Carl B. (1991). "Revival and Decline of Greek Mathematics". A History of Mathematics (Second ed.). John Wiley & Sons, Inc. p. 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.
  9. ^ Cooke, Roger (1997). "The Nature of Mathematics". The History of Mathematics: A Brief Course. Wiley-Interscience. p. 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.
  10. ^ 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."
  11. ^ D. M. Burton (1991, 1995). History of Mathematics, Dubuque, IA (Wm.C. Brown Publishers).

    "Diophantos was most likely a Hellenized Babylonian."
  12. ^ Ad Meskens, Travelling Mathematics: The Fate of Diophantos' Arithmetic (Springer, 2010), p. 48: "Since 1500, more than a thousand years after his death, various authors have speculated about the life of Diophantos, identifying him as an Arab, a Jew, a converted Greek or Hellenized Babylonian. None of these characterizations stands up to critical scrutiny though". n. 28: "There may be some confusion here with Diophantus the Arab, Libanius' teacher, who lived during the reign of Julian the Apostate".
  13. ^ a b For an analysis and a refutation of these claims, see: Schappacher, Norbert (2005). Diophantus of Alexandria: a Text and its History. Research Institute Mathématique Avancée.
  14. ^ 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. p. 502.
  15. ^ 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.
  16. ^ a b Herrin, Judith (2013-03-18). Margins and Metropolis: Authority across the Byzantine Empire. Princeton University Press. p. 322. ISBN 978-1400845224.
  17. ^ G. J. Toomer; Reviel Netz. "Diophantus". In Simon Hornblower; Anthony Spawforth; Esther Eidinow (eds.). Oxford Classical Dictionary (4th ed.).
  18. ^ "Diophantus biography". www-history.mcs.st-and.ac.uk. Retrieved 10 April 2018.
  19. ^ Knorr, Wilbur: Arithmêtike stoicheiôsis: On Diophantus and Hero of Alexandria, in: Historia Matematica, New York, 1993, Vol.20, No.2, 180-192
  20. ^ Sesiano, Jacques. "Diophantus - Biography & Facts". Britannica. Retrieved August 23, 2022.
  21. ^ Kurt Vogel, "Diophantus of Alexandria." in Complete Dictionary of Scientific Biography, Encyclopedia.com, 2008.

Sources

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  • 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.

Further reading

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  • 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. Vol. 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. Vol. 4. New York: Scribner.
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