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For the general, see Diophantus (general).

Diophantus of Alexandria (Greek: Διόφαντος ὁ Ἀλεξανδρεύς b. between 200 and 214, d. between 284 and 298 AD) was a Hellenized Babylonian/Egyptian (or Jewish) mathematician. He is the author of a series of classical mathematical books called Arithmetica and worked with equations which we now call Diophantine equations; the method to solve those problems is now called Diophantine analysis. The study of Diophantine equations is one of the central areas of number theory. The findings and works of Diophantus have influenced mathematics greatly and caused many other questions to arise. The most famous of these is Fermat's Last Theorem. Diophantus also made advances in mathematical notation and was the first Hellenistic mathematician who frankly recognized fractions as numbers.

Biography

Little is known about the life of Diophantus. He lived in Alexandria, Egypt, probably from between 200 and 214 to 284 or 298 AD.

Almost everything we know about Diophantus comes from a single 5th century Greek anthology, which is a collection of number games and strategy puzzles. Here is one of the puzzles:

“This tomb hold Diophantus. Ah, what a marvel! And the tomb tells scientifically the measure of his life. God vouchsafed that he should be a boy for the sixth part of his life; when a twelfth was added, his cheeks acquired a beard; He kindled for him the light of marriage after a seventh, and in the fifth year after his marriage He granted him a son. Alas! late-begotten and miserable child, when he had reached the measure of half his father's life, the chill grave took him. After consoling his grief by this science of numbers for four years, he reached the end of his life.”

This puzzle reveals that Diophantus lived to be about 84 years old. We cannot be sure if this puzzle is accurate or not.

Origins

Some scholars consider Diophantus to have been "most likely a Hellenized Babylonian."[1] While some scholars consider Diophantus to have been a Greek,[2] most others consider him to have been a non-Greek,[3] possibly either a Hellenized Babylonian,[4] Chaldean, Jew,[5] or Egyptian.[6][7]

His ethnic appearance most likely resembled those of the Fayum mummy portraits, many of which belonged to Hellenized Egyptians with Greek names, much like Diophantus. Under Greco-Roman rule, Egypt hosted several Greek settlements, mostly concentrated in Alexandria and Fayum, but also in a few other cities, where Greek settlers lived alongside some seven to ten million native Egyptians.[8] By the Roman period, much of the "Greek" population of Egypt was made-up of either Hellenized Egyptians or people of mixed Egyptian-Greek origins.[9] The aforementioned Faiyum portraits reflect the complex synthesis of the predominant Egyptian culture and that of the elite Greek minority.[10] According to Walker, the early Ptolemaic Greek colonists married with locals and adopted Egyptian religious beliefs, and by Roman times, their descendants were viewed as Egyptians by the Roman rulers, despite their own self-perception of being Greek.[11] The dental morphology[12] of the Roman-period Faiyum mummies was also compared with that of earlier Egyptian populations, and was found to be "much more closely akin" to that of ancient Egyptians than to Greeks or other European populations.[13]

Arithmetica

See also: Arithmetica

The Arithmetica is the major work of Diophantus and the most prominent work on algebra in Hellenistic and Egyptian 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 Arab books discovered in 1968 are also by DiophantusTemplate:Fix/category[citation needed]. Some Diophantine problems from Arithmetica have been found in Arabic sources.

History

Soon after Diophantus's death, the Dark Ages began, spreading a shadow on math and science, and causing knowledge of Diophantus and the Arithmetica to be lost in Europe for about 1500 years. Possibly the only reason that some of his work has survived is that many Arab scholars studied his works and preserved this knowledge for later generations. 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 . . . .”

The first Latin translation of Arithmetica was by Bombelli who translated much of the work in 1570 but it was never published. Bombelli did however borrow many of Diophantus's problems for his own book Algebra. The editio princeps of Arithmetica was published in 1575 by Xylander. The most famous Latin translation of Arithmetica was by Bachet in 1621 which was the first translation of Arithmetica available to the public.

Margin writing by Fermat and Planudes

The 1621 edition of Arithmetica by Bombelli 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 $a^n+b^n=c^n$ has no solutions in non-zero integers $a,b,c$ . 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 his famous "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 mathematician Maximus Planudes had written "Thy soul, Diophantus, be with Satan because of the difficulty of your theorems" next to the same problem.

Other works

Diophantus did not just write Arithmetica, but very few of his other works 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. Many scholars and researchers believe that The Porisms may have actually been a section included inside Arithmetica or indeed may have been the rest of Arithmetica. Template:Fix/category[citation needed]

Although the Porisms is lost we do know three lemmas contained in The Porisms since Diophantus refers to them in the Arithmetica. One such lemma is 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 numbers $a,b$ then there exist numbers $c,d$ such that $a^3-b^3=c^3+d^3$ .

On polygonal numbers and geometric elements

Diophantus is also known to have written on polygonal numbers. Fragments of one of Diophantus' books on polygonal numbers, a topic of great interest to Pythagoras and his followers, has survived. An extant work called Preliminaries to the Geometric Elements, which has been attributed to Hero of Alexandria, has been studied recently and it is suggested that the attribution to Hero is incorrect, and that the work is actually by Diophantus [14].

Influence

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 seventeenth and eighteenth centuries. Diophantus and his works have 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. As far as we know Diophantus did not affect the lands of the Orient much and how much he affected India is a matter of debate.

The father of algebra?

Diophantus is sometimes called “the father of algebra" because he contributed greatly to number theory, mathematical notation, and because Arithmetica contains early use of syncopated notation.[15] However, it seems that many of the methods for solving linear and quadratic equations used by Diophantus go back to Babylonian mathematics. For this reason, mathematical historian Kurt Vogel writes: “Diophantus was not, as he has often been called, the father of algebra. Nevertheless, his remarkable, if unsystematic, collection of indeterminate problems is a singular achievement that was not fully appreciated and further developed until much later.” According to some historians of mathematics, like Florian Cajori, Diophantus got the first knowledge of algebra from India,[16][17] although other historians disagree.[18] Most historians agree that it is the Islamic mathematician Al-Khwarizmi who deserves the title of "father of algebra" the most for his Al-jabr.[19]

According to mathematics historian Odile Kouteynikoff:

According to the fact that Al-Khwarizmi founded Algebra during the 9th century, it is not surprising that, when being translated into Arabic in the late 9th century by Lebanese Ibn Luqa whose native language was Greek, Diophante’s Arithmetics seemed to be considered as a treatise about Algebra since algebraic vocabulary and way of thinking were most widely shared. Only few people understood that it was actually an arithmetic treatise: Al-Khazin (900–971) did, and therefore he is one of those who laid the foundations for the integer Diophantine analysis.[20]

Diophantine analysis

Today Diophantine analysis is the area of study where integral (whole number) solutions are sought for equations, and Diophantine equations are polynomial equations with integral coefficients to which only integral 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: $ax^2+bx=c\ ,\ ax^2=bx+c\ ,\ ax^2+c=bx$ . 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 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

Diophantus made important advances in mathematical notation. He was the first person 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:

“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.”Template:Fix/category[citation needed]

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 $\frac{12+6n}{n^2-3}$ , 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.

References

1. D. M. Burton (1991, 1995). History of Mathematics, Dubuque, IA (Wm.C. Brown Publishers).
2. Research Machines plc. (2004). The Hutchinson dictionary of scientific biography. Abingdon, Oxon: Helicon Publishing. pp. 312. "Diophantus (lived c.AD 270-280) Greek mathematician who, in solving linear mathematical problems, developed an early form of algebra."
3. H. Hankel (1874, 2nd ed. 1965), Zur Geschichte der Mathematik im Altertum und Mittelalter, Leipzig:
"Here, in the midst of this sad and barren landscape of the Greek accomplishments in arithmetic, suddenly springs up a man with youthful energy: Diophantus. Where does he come from, where does he go to? Who were his predecessors, who his successors? We do not know. It is all one big riddle. He lived in Alexandria. If a conjecture were permitted, I would say he was not Greek; ... if his writings were not in Greek, no-one would ever think that they were an outgrowth of Greek culture..."
4. D. M. Burton (1991, 1995). History of Mathematics, Dubuque, IA (Wm.C. Brown Publishers).
"Diophantos was most likely a Hellenized Babylonian."
5. Oswald Spengler (1923), Der Untergang des Abendlandes, 2 Bande:
"Were Plotin and Diophantus maybe of Jewish or Chaldaic origins?"
6. George Sarton (1936). "The Unity and Diversity of the Mediterranean World", Osiris 2, p. 406-463 [429].
7. 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 privaleged 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."
8. Adams, Winthrope L in Bugh, Glenn Richard. ed. "The Hellenistic Kingdoms". The Cambridge Companion to the Hellenistic World. Cambridge: Cambridge University Press. 2006, p. 39
9. Bagnall, R.S. in Susan Walker, ed. Ancient Faces : Mummy Portraits in Roman Egypt (Metropolitan Museum of Art Publications). New York: Routledge, 2000, pp. 28-29
10. Bagnall, op cit.
11. Walker, Susan, op cit., p. 24
12. Dentition helps archaeologists to assess biological and ethnic population traits and relationships
13. Irish JD (2006). "Who were the ancient Egyptians? Dental affinities among Neolithic through postdynastic peoples.". Am J Phys Anthropol 129 (4): 529-43
14. Knorr, Wilbur: Arithmêtike stoicheiôsis: On Diophantus and Hero of Alexandria, in: Historia Matematica, New York, 1993, Vol.20, No.2, 180-192
15. Carl B. Boyer, A History of Mathematics, Second Edition (Wiley, 1991), page 228
16. Florian Cajori, A History of Elementary Mathematics, 1898
17. Saradakanta Ganguli, Notes on Indian Mathematics. A Criticism of George Rusby Kaye's Interpretation, Isis, Vol. 12, No. 1 (Feb., 1929), pp. 132-145
18. "Heeffer, Albrecht, The Reception of Ancient Indian Mathematics by Western Historians, Ghent University, Belgium."
19. (Boyer 1991, "The Arabic Hegemony" p. 228) "Diophantus sometimes is called "the father of algebra," but this title more appropriately belongs to al-Khwarizmi. It is true that in two respects the work of al-Khwarizmi represented a retrogression from that of Diophantus. First, it is on a far more elementary level than that found in the Diophantine problems and, second, the algebra of al-Khwarizmi is thoroughly rhetorical, with none of the syncopation found in the Greek Arithmetica or in Brahmagupta's work. Even numbers were written out in words rather than symbols! It is quite unlikely that al-Khwarizmi knew of the work of Diophantus, but he must have been familiar with at least the astronomical and computational portions of Brahmagupta; yet neither al-Khwarizmi nor other Arabic scholars made use of syncopation or of negative numbers."
20. History of Mathematics from Medieval Islam to Renaissance Europe: Guillaume Gosselin, an algebraist in Renaissance France

Bibliography

• A. Allard, "Les scolies aux arithmétiques de Diophante d'Alexandrie dans le Matritensis Bibo. Nat. 4678 et les Vaticani gr. 191 et 304," Byzantion 53. Brussels, 1983: 682-710.
• P. Ver Eecke, Diophante d’Alexandrie: Les Six Livres Arithmétiques et le Livre des Nombres Polygones, Bruges: Desclée, De Brouwer, 1926.
• T. L. Heath, Diophantos of Alexandria: A Study in the History of Greek Algebra, Cambridge: Cambridge University Press, 1885, 1910.
• D. C. Robinson and Luke Hodgkin. History of Mathematics, King's College London, 2003.
• P. L. Tannery, Diophanti Alexandrini Opera omnia: cum Graecis commentariis, Lipsiae: In aedibus B.G. Teubneri, 1893-1895.
• Jacques Sesiano, 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.