Diophantus of Alexandria

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Diophantos of Alexandria ( ancient Greek Διόφαντος ὁ Ἀλεξανδρεύς Dióphantos ho Alexandreús , German also Diophant ) was an ancient Greek mathematician . He is considered the most important algebraist of antiquity, he is even considered one of the essential founders of algebra and number theory. He came from Alexandria in Egypt.

Life

It is not known exactly when Diophantus lived. The information fluctuates between 100 BC and 350 AD. However, since Diophant quoted Hypsicles of Alexandria (in the treatise on polygonal numbers that probably came from him ), he must after 150 BC. Lived before AD 364 when Theon of Alexandria mentioned Diophant's work (the first such mention in literature). Theon's daughter Hypatia wrote a (lost) commentary on part of the books of Arithmetic by Diophantine. It is also believed that he lived around 250 AD, as he dedicated his Arithmetica to a Dionysius, who could be Dionysius the Great , who became Bishop of Alexandria in 248 AD. There are also some other indications that speak for the time around 250.

Almost nothing is known about the real life of the Diophant. There is a riddle in the form of an alleged grave inscription by Diophantus in a later Greek anthology, according to which he was 84 years old, married at 33 and had a son who died at 42, but the truth of the matter is very uncertain. Only his works are known, and these, too, have been interpreted very differently over time (in the formulation of Norbert Schappacher : The Arithmetica are almost as elusive as their author ).

The inscription on his tombstone was based on a poem handed down in the Anthologia Palatina:

Here the tomb covers Diophantus - a miracle to look at:
Through the art of the asleep, the stone teaches you his age.
To remain a boy a sixth of the life a god gave him;
Adding the twelfth, he let his cheek sprout;
Put the torch of the wedding on for him after the seventh,
And five years later he gave him a son.
Sore! unhappy child, so loved! The father half had it
Reached old age when Hades, the gruesome, took on.
Four more years of soothing the pain by telling the numbers
Finally he himself reached the goal of being.

The son was 30 years and 8 months old.

That makes
x = 84 years

Another interpretation that starts instead of on the right-hand side comes to 65 years 4 months; it corresponds to the German translation, but not the text of the Greek original.

It deals with equations with a unique solution as well as with multiple solutions, with Diophant only allowing positive rational numbers as solutions. Whether he also counted with negative numbers was controversial or for a long time it was the prevailing opinion that this would not be the case (Isabella Basmakova and Klaus Barner provided examples of this). The first book deals with linear and quadratic equations with specific solutions and was essentially known even before Diophant; the new is found in the following books (rational solutions of algebraic equations in several variables), with which he became the founder of arithmetic algebraic geometry and number theory has been. Today such algebraic equations for which integer solutions are sought are called Diophantine equations . The theory of the Diophantine approximation is also named after him . Diophantine's arithmetic contains no references to other mathematical literature or authors. His work was an important influence on the development of algebra among the Arabs in the Middle Ages and was accessible in Europe from the 15th century and had a significant influence on Francois Viète and Pierre de Fermat .

Because of the originality of his work and the special role it plays in Greek mathematics (which deals primarily with geometry or uses a geometric language), it has been suggested that Diophantus is a compilation of other authors similar to Euclid's elements. This was represented, for example, by Paul Tannery (1879) and Thomas Little Heath in the second edition of his Diophant book in 1910 after he still took the view in the first edition in 1885 that he was dealing with an original work by a single author (what also before that, for example, Georg Heinrich Ferdinand Nesselmann 1842) and in his analysis at that time also found no forerunners among the ancient Greek mathematicians. The occasion was an anonymous manuscript from the 12th century ( Hieronymus Zeuthen , Johan Ludvig Heiberg ) with algebraic tasks in the style of Diophant (which, however, did not come from Diophant's arithmetic), which he dated to the time between Euclid and Diophant, which had meanwhile been found. Today that is considered obsolete.

The moon crater Diophantus is named after him.

Works

Diophantos, Arithmetica in the manuscript Rome written in 1296, Biblioteca Apostolica Vaticana , Vaticanus graecus 191, fol. 388v
Cover of the 1621 published Arithmetica -Issue
In the 15th century six books (originally: scrolls), namely volumes 1 to 3 and 8 to 10 (in Greek) were found again. In 1968 the science historian Fuat Sezgin discovered another 4 books - books 4 to 7 - in Arabic translations. The last three books have disappeared. Editions of the work exist in a Latin translation by Wilhelm Xylander (Basel 1575), in Greek and with an improved Latin translation and commentaries by Bachet de Meziriac (Paris 1621). The philologist Johann Otto Leopold Schulz (1782–1849) (Berlin 1822) created a German translation . Pierre de Fermat later provided his copy of the Latin edition of Bachet de Meziriac with handwritten annotations, such as the comment in which he formulated Fermat's Great Theorem, which is now named after him .
  • De numeris polygonis , excerpt from the tenth volume of the Arithmetika.
This work was translated into German by Friedrich Theodor Poselger (Leipzig 1810).

Text editions and translations

  • Diophante: Les Arithmétiques. Livre IV, Livres V-VI-VII . Texts établi et traduit par Roshdi Rashed . Les Belles Lettres, Paris 1984.
  • Roshdi Rashed , Christian Houzel : Les “Arithmétiques” de Diophante. Lecture historique et mathématique. Berlin: De Gruyter, 2013, ISBN 978-3-11-033593-4 , doi : 10.1515 / 9783110336481 .
  • Diophantus, Qusta Ibn-Luqa al-Balabakki , Jacques Sesiano: Books IV (four) to VII of Diophantus Arithmetica in the Arabic translation attributed to Qusta Ibn-Luqa. Springer, New York – Heidelberg – Berlin 1982. Sources in the history of mathematics and physical sciences, 3. Arithmetica (Eng.) - Text partly Arabic. u. Engl., Zugl .: Providence, Brown Univ., Diss. J. Sesiano, 1975 (reprint).
  • Arthur Czwalina : Arithmetic of Diophantos from Alexandria. From d. Greek transfer u. explained by Arthur Czwalina. Vandenhoeck & Ruprecht, Göttingen 1952 (Appendix 1 to the treatises from the Mathematical Seminar of the University of Hamburg).
  • Diophantus of Alexandria: Arithmetica. The arithmetic and the writing about polygonal numbers (German). (Arithmeticorum libri 6 et de numeris multangulis libri 1, German). Trans. U. with note cf. by G (ustav) Wertheim (1890). Teubner, Leipzig 1890, urn : nbn: de: bvb: 12-bsb00082428-8
  • Pierre de Fermat, Maximilian Miller: Comments on Diophant. Observationes (German). From d. Lat. trans. u. with note ed. by Max Miller. Akad. Verlagsges., Leipzig 1932 (Ostwald's Classic of Exact Sciences No. 234).
  • Thomas Heath Diophantus of Alexandria , Cambridge Univ. Press 1910, with English translation of the arithmetic into English and supplements to Fermat and Euler .
  • Paul Tannery : Diophantus Alexandrinus, Opera Omnia , 2 volumes, Teubner 1893, 1895, Reprint Bibliotheca Teubneriana 1974 (with explanations also on the biography)

literature

Overview representations

Investigations

Web links

Remarks

  1. ↑ Schappacher, Diophantus of Alexandria, a text and its history, p. 7 (English extended version of his Euler lecture Potsdam 1998), pdf
  2. ↑ Egmont Colerus : From multiplication tables to integral. Paul Zsolnay Verlag; Berlin, Vienna, Leipzig, 1941, pp. 157 and 158
  3. ↑ Schappacher, Diophantus of Alexandria, A text and its history (English version of his essay Wer war Diophant?, Mathem. Semesterberichte 1998), p. 5, online
  4. ^ Heath, Diophantos of Alexandria. A study in the history of greek algebra, Cambridge UP 1885, 2nd edition 1910
  5. ↑ See Klaus Barner, Negative Größe bei Diophant I, NTM 15, 2007, p. 23
  6. ^ Through Regiomontanus in Venice 1463, Christa Binder in Dauben, Scriba Writing the history of mathematics , Birkhäuser 2002, p. 214
  7. ↑ Jan P. Hogendijk, Review of J. Sesiano, Books IV to VII of Diophantus' Arithmetica , in: Historia Mathematica 12 (1985), pp. 82-85, online , accessed July 2, 2018
personal data
SURNAME Diophantus of Alexandria
ALTERNATIVE NAMES Διόφαντος ὁ Ἀλεξανδρεύς (ancient Greek)
BRIEF DESCRIPTION ancient Greek mathematician
DATE OF BIRTH between 100 BC Chr. And 350
DATE OF DEATH between 100 BC Chr. And 350