Diophantus

Diophantus, fl. A.D. 250, Greek algebraist. He pioneered in solving a type of indeterminate algebraic equation where one seeks integer values for the unknowns; work in this field is known as Diophantine analysis. Only 6 of the 13 books with which he is credited are extant. The standard edition in Greek was edited by Paul Tannery (2 vol., 1893-95).

Diophantus of Alexandria (Greek: Διόφαντος ὁ Ἀλεξανδρεύς b. between 200 and 214, d. between 284 and 298 AD), sometimes called "the father of algebra", a title some claim should be shared by a Persian mathematician al-Khwārizmī, born some 500 years later. Diophantus was an Alexandrian mathematician and the author of a series of books called Arithmetica that deal with solving algebraic equations, many of which are now lost. Pierre de Fermat studied Arithmetica and made a fateful note in the margin of his copy of the book that a certain equation similar to the Pythagorean equation considered by Diophantus has no solutions and he found "a truly marvelous proof of this proposition", the celebrated Fermat's Last Theorem. This led to tremendous advances in number theory, and the study of diophantine equations ("diophantine geometry") and of diophantine approximations remain important areas of mathematical research. Diophantus was the first Greek mathematician who recognized fractions as numbers; thus he allowed positive rational numbers for the coefficients and solutions. In modern use, diophantine equations are usually algebraic equations with integer coefficients, for which integer solutions are sought. Diophantus also made advances in mathematical notation.

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, although Klaus Barner states that he may have been active as early as 150 BC, thus the life span of about 400 years. While most scholars consider Diophantus to have been a Greek, others speculate him to have been a non-Greek, possibly either a Hellenized Babylonian, an Egyptian, a Jew, or a Chaldean. Much of our knowledge the life of Diophantus is derived from a 5th century Greek anthology of number games and strategy puzzles. One of the problems 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 lived to be about 84 years old. However, the accuracy of the information cannot be independently confirmed. 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

The Arithmetica 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 Arab books discovered in 1968 are also by Diophantus. 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”

History

During the Dark Ages, Diophantus was forgotten and like many other mathematical treatises from the classical period, Arithmetica survived through the Arab tradition. 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 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 best known Latin translation of Arithmetica was made by Bachet in 1621 and became the first Latin edition that was widely available. Pierre de Fermat owned a copy, studied it, and made notes in the margins.

Margin writing by Fermat and Planudes

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 $a^n\; +\; b^n\; =\; c^n$ 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 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 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. Some scholars think that The porisms may have actually been a section of Arithmetica that is now lost.

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

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.

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 .

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 often called “the father of algebra" because he contributed greatly to number theory, mathematical notation, and because Arithmetica contains the earliest known use of syncopated notation . 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.”

Diophantine analysis

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: $ax^2\; +\; bx\; =\; c$, $ax^2\; =\; bx\; +\; c$, and $ax^2\; +\; c\; =\; bx$. 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.”

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 $(12\; +\; 6n)/(n^2\; -3)$, 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.

See also

• Muhammad ibn Mūsā al-Khwārizmī

References

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.

External links

• Diophantus's Riddle Diophantus' epitaph, by E. Weisstein
• Norbert Schappacher (2005). Diophantus of Alexandria : a Text and its History
• Tannery's edition of the Works of Diophantus, now in the public domain (Classical Greek)